FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA Apostolos Kantzas, PhD P. Eng. Jonathan Bryan, PhD, P. Eng. Saeed Taheri, PhD
[Type the document subtitle]
1.
CHAPTER 1........................................................................................................................................... 11 INTRODUCTION ....................................................................................................................................... 11
2.
CHAPTER 2........................................................................................................................................... 15 THE POROUS MEDIUM............................................................................................................................ 15 HOMOGENEITY ................................................................................................................................... 16 ANISOTROPY ....................................................................................................................................... 18 POROSITY ............................................................................................................................................ 18 PORE SIZE DISTRIBUTION .................................................................................................................... 28 SPECIFIC SURFACE AREA ..................................................................................................................... 32 COMPRESSIBILITY OF POROUS ROCKS ................................................................................................ 33 PERMEABILITY ..................................................................................................................................... 37 SATURATION ....................................................................................................................................... 50 FORMATION RESISTIVITY FACTOR ...................................................................................................... 53 MULTI-PHASE SATURATED ROCK PROPERTIES ................................................................................... 60 RELATIVE PERMEABILITY ................................................................................................................... 100
3.
CHAPTER 3......................................................................................................................................... 171 MOLECULAR DIFFUSION ....................................................................................................................... 171 Introduction ...................................................................................................................................... 171 Fick’s Law of Binary Diffusion ........................................................................................................... 171 Diffusion Coefficient ......................................................................................................................... 174
4.
CHAPTER 4......................................................................................................................................... 204 Immiscible Displacement ...................................................................................................................... 204 Introduction ...................................................................................................................................... 204 Buckley-Leverett Theory, .................................................................................................................. 204 Water Injection Oil Recovery Calculations........................................................................................ 215 Vertical and Volumetric Sweep Efficiencies...................................................................................... 230
5.
CHAPTER 5......................................................................................................................................... 242 Miscible Displacement .......................................................................................................................... 242 Introduction ...................................................................................................................................... 242 FLUID PHASE BEHAVIOR.................................................................................................................... 245 First Contact Miscibility Process ....................................................................................................... 255
1
Multiple Contact Miscibility Processes ............................................................................................. 257 Determination of Miscibility Condition............................................................................................. 267 Fluid properties in miscible displacement ........................................................................................ 277 The Equation of Continuity ............................................................................................................... 305 The Equation of Continuity in Porous Media .................................................................................... 306
2
Figure 1-1: World supply of primary energy by fuel type. .......................................................................... 11 Figure 1-2: Simplified Illustrations of Vertical and Horizontal Wells .......................................................... 12 Figure 1-3: Elementary Trap in Sectional View ........................................................................................... 14 Figure 2-1: Dependence of Permeability on Sample Volume ..................................................................... 16 Figure 2-2 –A Probability Density Function can be used to find the homogeneity or heterogeneity type of a porous medium ........................................................................................................................................ 17 Figure 2-3 – Microscopic Cross Section Image of a Porous Medium .......................................................... 18 Figure 2-4 – Dead-end pore ........................................................................................................................ 20 Figure 2-5 - Storage and connecting pore model for shale or any other type of rock with interconnected pore systems ............................................................................................................................................... 20 Figure 2-6 - The Various Cross-Sections of Connecting Pores .................................................................... 21 Figure 2-7 – a) Three-dimension distribution of connecting; b) Complete storage-connecting pore system. ........................................................................................................................................................ 21 Figure 2-8 – Typical ordered porous medium structures ........................................................................... 23 Figure 2-9 – Effect of sorting and grain size distribution on porosity......................................................... 24 Figure 2-10 – Porosimeter Based on Boyle - Mariotte’s Law ..................................................................... 27 Figure 2-11 – Colored Thin Section Microscopic Image.............................................................................. 27 Figure 2-12 - Schematic Shape of pore and Pore Throat ............................................................................ 29 Figure 2-13 – Sieve Analysis Tools .............................................................................................................. 29 Figure 2-14 - Schematic representation of pores. ...................................................................................... 31 Figure 2-15 – Tomographic image. ............................................................................................................. 32 Figure 2-16 – Specific Surface Area............................................................................................................. 32 Figure 2-17 – Porosity Reduction as an Effect of Compaction Increment by Depth .................................. 34 Figure 2-18 – a) Experimental Equipment for Measuring Pore Volume Compaction and Compressibility 36 Figure 2-19 – a) Formation Compaction Component of Total Rock Compressibility ................................. 36 Figure 2-20 – Flow through a Pipe .............................................................................................................. 38 Figure 2-21 – Metallic cast of pore space in a consolidated sand .............................................................. 39 Figure 2-22 – Schematic Drawing of Darcy Experiment of Flow of Water through Sand ........................... 39 Figure 2-23 – Linear Flow through Layered Bed ......................................................................................... 42 Figure 2-24 – Linear Flow through Series Beds........................................................................................... 44 Figure 2-25 – Plot of Experimental Results for Calculation of Permeability ............................................... 46 Figure 2-26 – Permeability of Core Sample to Three Different Gases and Different Mean Pressure ........ 48 Figure 2-27 - Effect of Permeability on the Magnitude of the Klinkenberg Effect ..................................... 48 Figure 2-28 – ASTM Extraction Apparatus .................................................................................................. 53 Figure 2-29 - The Influence of Pore Structure on the Electrical Conductivity ............................................ 55 Figure 2-30 – Formation Resistivity Factor vs. Porosity .............................................................................. 56 Figure 2-31 - Apparent Formation Factor vs. Water Resistivity for Clayey and Clean Sands ..................... 57 Figure 2-32 - Water-Saturated Rock Conductivity as a Function of Water Conductivity ........................... 58 Figure 2-33- Illustration of surface tension (Surface molecules pulled toward liquid causes tension in surface). ...................................................................................................................................................... 60 Figure 2-34- Simplified Models for Interfacial Tension Determination. ..................................................... 62 Figure 2-35- Pressure relations in capillary tubes....................................................................................... 62 3
Figure 2-36- Illustration of Wettability ....................................................................................................... 63 Figure 2-37- Equilibrium of Forces at a Liquid-Gas-Solid Interface. ........................................................... 64 Figure 2-38- Rock-Fluid-Fluid Interactions Effect on the Contact Angle..................................................... 66 Figure 2-39- liquid drop spreading on a solid surface ................................................................................ 67 Figure 2-40- Amott Wettability Test ........................................................................................................... 68 Figure 2-41- Amott Index Ternary Diagram ................................................................................................ 69 Figure 2-41- Amott Index Calculation ......................................................................................................... 69 Figure 2-42- USBM Index Calculation ......................................................................................................... 70 Figure 2-43- Pressure Relation in capillary Tube ........................................................................................ 72 Figure 2-44- Dependency of Water Column to (a). Capillary Radius, (b). Wettability................................ 73 Figure 2-45- Principle radii for wetting fluid and spherical grain ............................................................... 74 Figure 2-46- Wetting and non-Wetting fluid distribution about inter grain contact of sphere. ................ 75 Figure 2-47- Non-Wetting fluid entering the capillary tube. ...................................................................... 76 Figure 2-48- Non-Wetting fluid entering the non-uniform capillary tube. ................................................. 76 Figure 2-49- Non-Wetting Fluid enter to a bubble and exit it. ................................................................... 77 Figure 2-50- Capillary Pressure versus wetting phase saturation .............................................................. 78 Figure 2-51- Variation of Capillary Pressure with Permeability .................................................................. 79 Figure 2-52- Flow into a Constriction (cone). ............................................................................................. 79 Figure 2-53- Flow out of a Constriction ...................................................................................................... 80 Figure 2-54- Flow in a Capillary Tube. ......................................................................................................... 81 Figure 2-55- porous diaphragm capillary pressure device.......................................................................... 83 Figure 2-56- Centrifugal apparatus ............................................................................................................. 84 Figure 2-57- Mercury Injection Method ..................................................................................................... 85 Figure 2-58- Pore size distribution from mercury injection test................................................................. 86 Figure 2-59- Dynamic Measurement of Capillary pressure. ....................................................................... 87 Figure 2-60- Capillary Pressure Curve. ........................................................................................................ 88 Figure 2-61- Contact Angle Hysteresis during the Displacement ............................................................... 89 Figure 2-62- Dynamic Contact Angle Behavior. .......................................................................................... 89 Figure 2.63: Static values of advancing and receding contact angles at rough surfaces versus values at smooth surfaces (where E refers to smooth surface measurements........................................................ 90 Figure 2-64- Non-Wetting fluid Enter to a capillary tube with square cross section. ................................ 91 Figure 2-65- Side view after snap-off .......................................................................................................... 91 Figure 2-66- Trapping in a porous media. ................................................................................................... 92 Figure 2-67- Typical non-wetting phase trapping characteristics of some reservoir rocks. ....................... 92 Figure 2-68- Pore Doublet Model. .............................................................................................................. 93 Figure 2-69- Imbibition and Drainage mechanisms in a pore doublet model ............................................ 94 Figure 2-70- Pore doublet model for illustration for displacement and trapping of oil. ............................ 95 Figure 2-71- Trapping of a droplet in a capillary tube. ............................................................................... 97 Figure 2-72- J-function correlation of capillary pressure data in Edwards Jourdanton field...................... 99 Figure 2-73- Typical relative permeability curve. ..................................................................................... 103 Figure 2-74- Typical Gas-Oil Relative Permeability Curve......................................................................... 105 Figure 2-75- Relative Permeability Curve, (a) Drainage, (b) Imbibition .................................................... 105 4
Figure 2-76- Hafford Relative Permeability Apparatus ............................................................................. 107 Figure 2-77- Fluid Saturation during Steady-State Test ............................................................................ 108 Figure 2-78- Unsteady state apparatus. ................................................................................................... 109 Figure 2-79- (a) Unsteady State Water Flood Procedure, (b) Typical Relative Permeability Curve ......... 110 Figure 2-80- (a) Average Water saturation vs. Water Injection, (b) Injectivity Ratio ............................... 111 Figure 2-81: Amott Ternary Wettability Diagram ..................................................................................... 119 Figure 2-82: Comparison between capillary pressure and relative permeability curves ......................... 122 Figure 2-83 Relative permeability curves of Berea Sandstone. Strongly Water-wet conditions (Sankar 1979) ......................................................................................................................................................... 125 Figure 2-84 Relative permeability of water wet and oil wet systems. ..................................................... 134 Figure 2-85 Comparison between experimental and predicted values (after 3). .................................... 140 Figure 2-86 Compare with results of Corey et al. (after 11) ..................................................................... 145 Figure 2-87 Compare with results of Dalton et al. (after 11).................................................................... 145 Figure 2-88 Stone’s Method 2 predictions and Corey et al.’s experimental data (after 12) ................... 146 Figure 2-89 Stone’s prediction of Sor and Holmgren-Morse’s data (after 12) ........................................ 146 Figure 2-90 Effect of capillary number on relative permeability (after 9) ................................................ 153 Figure 2-91 Low IFT systems (after 21) ..................................................................................................... 155 Figure 2-92 High IFT systems (after 21) .................................................................................................... 155 Figure 2-93 Effects of viscous force on Swir (after 22). .............................................................................. 156 Figure 2-94 Effects of flow rate on relative permeability (after 22) ......................................................... 158 Figure 2-95 Effect of viscosity on relative permeability (after 26) ........................................................... 160 Figure 3-1 - Simple diffusion experiment.................................................................................................. 172 Figure 3-2 – Diffusion across a thin film ................................................................................................... 173 Figure 3-3 – Prediction overall diffusion from intrinsic diffusion ............................................................. 177 Figure 3-4 – Diffusion process in a control volume with a concentration dependent diffusion coefficient .................................................................................................................................................................. 178 Figure 3-5 – In a porous medium fluid generally flowing at about 45o with respect to average direction of flow ....................................................................................................................................................... 180 Figure 3-6 – Pressure decay test cell. ....................................................................................................... 182 Figure 3-7 - Refraction of light at the interface between two media. ...................................................... 183 Figure 3-8 - Sample of light refraction results a) initial time b) after diffusion occurred ......................... 184 Figure 3-9 - (a). Hydrogen nuclei behave as a tiny bar magnets aligned with the spin axes of the nuclei. (b). Spinning protons with random nuclear magnetic axes in the absence of an external magnetic field. .................................................................................................................................................................. 185 Figure 3-10 – Line up nuclear spins in an external magnetic field............................................................ 186 Figure 3-11 – Polarization/Relaxation curve. ............................................................................................ 187 Figure 3-12 – the Tipping process. ............................................................................................................ 187 Figure 3-13 – Net magnetization return to equilibrium by turning off the B1, (the arrow represent net magnetization) .......................................................................................................................................... 188 Figure 3-14 – de-phasing (loss of phase coherence) during T2. ............................................................... 189 Figure 3-15 – Spin-echo sequence. ........................................................................................................... 190 Figure 3-16 – CPMG pulse sequence. ....................................................................................................... 191 5
Figure 3-17 - The amplitudes of the decaying spin echoes yield an exponentially decaying curve with time constant T2. ....................................................................................................................................... 191 Figure 3-18 - The echo train (echo amplitude as a function of time) is mapped to a T 2 distribution (porosity as a function of T2). .................................................................................................................... 192 Figure 3-19 – Typical NMR spectrum for pure bitumen, pure solvent, and a mixture of them. .............. 194 Figure 3-20 – two samples of NMR calibration for bitumen-solvent mixture,. ........................................ 195 Figure 3-21 – Diffusion coefficient as a function time, NMR experiment result. ..................................... 197 Figure 3-22 - Schematic view of CAT scanning using x-ray. ...................................................................... 199 Figure 3-23 – Calibration curves for the CAT scanner, (a) Liquid calibration curve, (b)Liquid-solid calibration curve. ...................................................................................................................................... 199 Figure 3-24 – Image sample of diffusion process ..................................................................................... 200 Figure 3-25 – Medium Domain ................................................................................................................. 201 Figure 3-26 - Sample of diffusion in sand saturated with oil. ................................................................... 202 Figure 3-27 - Average diffusion coefficients for pentane, hexane and octane in heavy oil. .................... 202 Figure 3-28 - Comparison of the diffusion coefficients of pentane in heavy oil in absence/presence sand. .................................................................................................................................................................. 203 Figure 4-1 – Semilog plot of relative permeability ratio versus saturation .............................................. 205 Figure 4-2 – Fractional flow curve ............................................................................................................ 206 Figure 4-3 – Horizontal bed containing oil and water. ............................................................................. 207 Figure 4-4 – Cubic reservoir under active water drive.............................................................................. 208 Figure 4-5 – Water fractional flow ant its derivative ................................................................................ 209 Figure 4-6 – Fluid Distribution at 60, 120, 240 days ................................................................................. 210 Figure 4-7 - Water saturation distribution as a function of distance, prior to breakthrough .................. 211 Figure 4-8 - Tangent to the fractional flow curve from Sw = Swc ............................................................... 212 Figure 4-9 – xD vs. tD for a linear waterflooding. ....................................................................................... 213 Figure 4-10 – Saturation Profile at tD = 0.28 ............................................................................................. 214 Figure 4-11 – Saturation History at xD = 1, producing face of the medium .............................................. 214 Figure 4-12 - Saturation distribution after 240 days................................................................................. 215 Figure 4-13 - Application of the Welge graphical technique to determine: (a) The front saturation, (b) Oil recovery after breakthrough .................................................................................................................... 219 Figure 4-14 - Water saturation distribution as a function of distance between injection and production wells for (a) ideal or piston-like displacement and (b) non-ideal displacement ...................................... 219 Figure 4-15 - (a) Microscopic displacement (b) Residual oil remaining after a water flood .................... 221 Figure 4-16 – (a) Relative Permeability Curves, (b) Fractional Flow Curve............................................... 223 Figure 4-17 – Graphical determination of front saturation and water fractional flow. ........................... 223 Figure 4-18 – ............................................................................................................................................. 224 Figure 4-19 – dimensionless pore volume oil recovery vs. dimensionless pore volume water injection 225 Figure 4-20 - Fractional flow plots for different oil-water viscosity ratios ............................................... 227 Figure 4-21 - Water Saturation Distributions in Systems for Different Oil/Water Viscosity Ratios ......... 228 Figure 4-22 - Typical injection/production well configurations and associated flooding patterns .......... 229 Figure 4-23 - Schematic representation of the two components of the volumetric sweep: (a) areal sweep; (b) vertical sweep in stratified formation. .................................................................................... 231 6
Figure 4-24 – a) Bottom coning at oil-water or gas-oil contact, b) Edge coning at oil-water or gas-oil contact ...................................................................................................................................................... 237 Figure 4-25 - Stable Cone. ......................................................................................................................... 237 Figure 4-26 – Flow rate versus time.......................................................................................................... 239 Figure 5-1 – Miscible Displacement, a) secondary recovery, b)Tertiary recovery. .................................. 243 Figure 5-2 - A typical phase diagram for a pure component. ................................................................... 245 Figure 5-3 – Typical P-T diagram for a multicomponent system. ............................................................. 246 Figure 5-4 – (a). Phase diagram of ethane-normal heptane, (b) Critical loci for binary mixtures. ........... 246 Figure 5-5 – Typical P-X diagram for the Methane-normal Butane system. ............................................ 248 Figure 5-6 – Pressure-composition diagram for mixture of C1 with a liquid mixture of C1-nC4-C10. ..... 249 Figure 5-7 – ternary phase diagram for a system consisting of components A, B, and C which are miscible in all proportions. ...................................................................................................................................... 251 Figure 5-8 – All mixture of M1 and M2 would be along line . ...................................................... 252 Figure 5-9 – Example 5-2 a) Ternary diagram, b) Chemicals to be mixed. ............................................... 252 Figure 5-10 - Error! Reference source not found...................................................................................... 253 Figure 5-11 – Ternary Phase Diagram. ...................................................................................................... 254 Figure 5-12 – Pressure effect on the miscibility, P1
Figure 5-35 – Estimated breakthrough recovery as a function of viscosity ratio [32]. ............................. 290 Figure 5-36 – Impact of viscous instability on secondary CO2 flood oil recovery efficiency .................... 291 Figure 5-37– Gravity segregation in displacement processes. A) Gravity override ρo<ρs. b) Gravity Underride ρs<ρo ....................................................................................................................................... 292 Figure 5-38 – Flow regimes for miscible displacement in a vertical cross section. .................................. 292 Figure 5-39 – Flow regimes in a two-dimensional, uniform linear system (Schematic) [29]. .................. 293 Figure 5-40 – volumetric sweep efficiency at breakthrough as a function of the viscous/gravity force ratio. .......................................................................................................................................................... 294 Figure 5-41 - Model for determining stability criterion in a dipping reservoir ......................................... 295 Figure 5-42 - Schematic of the experimental apparatus. ......................................................................... 298 Figure 5-43 - Plot of mole fraction of solvent in the effluent as a function of the pore volumes of solvent injected. .................................................................................................................................................... 299 Figure 5-44 - Cumulative recovery for rate reduction and pressure pulsing as a function of pore volumes of solvent injected .................................................................................................................................... 300 Figure 5-45 - Mole fraction of solvent in the effluent as a function of the pore volumes of solvent injected. Rate reduction and pressure pulsing case. ................................................................................ 301 Figure 5-46 – profile of the solvent effluent concentration produced from a capillary tube in an equal viscosity and density. ................................................................................................................................ 301 Figure 5-47 – Concentration profile for injection of a slug of solvent to displace oil. .............................. 302 Figure 5-48 – Mixing of solvent and oil by longitudinal and transverse dispersion. ................................ 303 Figure 5-49 – dispersion porous medium being viewed as a series of mixing tank. ................................ 304 Figure 5-50 - Stagnant volume models ..................................................................................................... 304 Figure 5-51 - The effects of capacitance. .................................................................................................. 305 Figure 5-52 – Dispersion caused by variation of flow paths in a porous medium.................................... 305 Figure 5-53 Development of the mixing zone as a function of time during laminar flow in a capillary (after Nunge and Gill, 1970)...................................................................................................................... 309 Figure 5-54 Summary of the regions of applicability of various analytical solutions for dispersion in capillary tubes with step change in inlet concentration as a function of dimensionless time and Peclet number (from Dullien, 1992). ................................................................................................................... 311 Figure 5-55 Dependence of dispersion coefficient on Peclet number in different flow regimes. The scales on the axes depend on porous medium and other factors. The curve shown approximates longitudinal dispersion in unconsolidated random packs (after Perkins and Johnston, 1963). ................................... 313 Figure 5-56 Dependence of Peclet number on Reynolds number for an aqueous system (from Perkins and Johnston, 1963).................................................................................................................................. 315 Figure 5-57 Range of dispersion coefficients for various sandpacks. The lower curves are for coarse sand packed to a porosity of about 34%. The upper curves are for finer sand or looser packings (porosity > 34%). Fine sand 200-270 mesh, medium sand 40-200 mesh, coarse sand 20-30 mesh (after Blackwell, 1962). ........................................................................................................................................................ 315 Figure 5-58 Concentration as a function of transformed distance for different values of dispersion coefficient or time, calculated from eq. (5-72), infinite system. .............................................................. 319 Figure 5-59 Typical probability plot for determination of longitudinal dispersion coefficient. ............... 320
8
Figure 5-60 Effluent concentration profiles for a range of Peclet numbers, calculated from eq. (5-78), infinite system........................................................................................................................................... 321 Figure 5-61 Effluent concentration profiles for a range of Peclet numbers, finite system with Danckwerts’ boundary conditions (Brenner, 1962)....................................................................................................... 323 Figure 5-62 Effect of boundary conditions on solutions to the convection-dispersion equation at different Peclet numbers. ......................................................................................................................... 324 Figure 5-63 The effects of capacitance. ................................................................................................... 326 Figure 5-64 Typical shape of a Langmuir adsorption isotherm. ............................................................... 330 Figure 5-65 Effect of adsorption model parameters on adsorbate effluent concentrations. .................. 332
9
PREFACE The purpose of this manuscript is to provide the reader with the basic principles of flow in porous media and their association to hydrocarbon production from underground formations. The intended audience is undergraduate and graduate students in petroleum engineering and associated disciplines, as well as practicing engineers and geoscientists in the oil and gas industry. The material draws from the experiences of the authors in the Western Canadian Sedimentary Basin and, wherever possible, draws from unconventional resources.
10
1 .
C H A P T E R
1
INTRODUCTION Beginning with the industrial revolution of the early nineteenth century, man has turned more and more to the use of mineral fuels to supply the energy to operate his machines. The first commercial well drilled solely for oil was completed in the United States in 1859. Following the success of this well petroleum production and processing rapidly grew into a major industry in United States. Today, in satisfying the world’s energy needs, fossil fuels playing the prominent role. Although their share in the energy mix is expected to fall, it remains over 80% throughout the period to 2030. The leading role in the energy mix will continue to be played by oil, with its share remaining above 30%, albeit falling over time (Figure 1-1).
Figure 1-1: World supply of primary energy by fuel type 1.
Today petroleum is used not only as a fuel but as a raw material for many industrial materials such as paint, plastic, rubber, lubricants, and so forth.
What is Petroleum? Petroleum is a mixture of naturally occurring hydrocarbons which may exist in the solid, liquid, or gaseous state, depending upon the pressure and temperature to which it is subjected. Virtually all petroleum is produced from the earth in either liquid or gaseous form, and commonly these materials are referred to as either crude oil or natural gas, depend upon the state of hydrocarbon mixture.
1
http://www.opec.org/opec_web/en/
11
Since the vast majority of oil and gas bearing formations are several hundred meters to several thousand meters beneath the earth’s surface, the oil well offers the only tool for accessing the reservoir from the surface. The well-bore region and the collected core (if any) offer a snapshot of the reservoir properties, in a fashion similar to a line drawn in a three dimensional volume. In other words, results from core tests do not describe the reservoir accurately but they do help describe the physics. Up until the late 1980s there was practically no variability in the manner in which a well was drilled. However, horizontal drilling (a Soviet invention of the 1920s) revolutionized the drilling industry as it allowed for one well to access a formation at several horizontal locations (Figure 1-2).
Figure 1-2: Simplified Illustrations of Vertical and Horizontal Wells
Origin of Oil Many theories of the origin of petroleum have been advanced. The theories of the origin of petroleum can be classified as either organic or as inorganic. The inorganic theory attempts to explain the formation of petroleum by assuming chemical reaction between water, carbon dioxide and various inorganic substances such as carbonates, in the earth. The organic theories assume that petroleum evolved from the decomposition of vegetable and animal organisms that lived during previous geological ages. Organic theories are commonly acceptable. Source beds as organic rich formations are the necessity of petroleum generation. Petroleum migration occurs after formation from source beds toward the reservoir or storage beds. Reservoir rocks have void spaces and are permeable to fluids, in other words they have interconnected void spaces.
Lithology Reservoir rocks are categorized as either sandstone or carbonate. Sandstones are formed from grains that have undergone sedimentation, compaction and cementation. The major characteristics of sandstone reservoirs are as follows:
Composed of silica grains (mainly quartz and some feldspar),
Consolidated or unconsolidated formations,
May contain shale,
12
May contain minerals (such as iron oxide and iron sulfides),
May include clays (Note: Clays have a negative effect on the reservoir quality).
Carbonates are formed from the remnants of hard-shelled organisms that existed in coral reef environments. The major characteristics of carbonates are:
Limestone (CaCO3) and/or dolomite (CaMg(CO3)2),
May contain shale,
Minerals, such as pyro-bitumen and anhydrite
Pore space is comprised of areas of dissolution (vugs), fractures and inter-crystalline spaces.
About 60% of the conventional oil reservoir rocks are sandstones and about 39% of them are carbonates.
Trap External forces such as buoyancy which force the petroleum to migrate from source rock to reservoir rock could push oil to reach the surface. So presence of a barrier over the reservoir formation is vital in accumulation of oil in the reservoir rocks. This barrier is known as “trap” in petroleum engineering. Traps associated with oil fields are complex. Different reservoir according to type of their trap can be classified as follows (Figure 1-3):
Convex Trap reservoirs which are surrounded by edge water and the trap is due to convexity alone, Permeability trap reservoir that the barrier is due to the loss of permeability in reservoir rock, Pinch out trap reservoir, which the periphery partly defined by edge water and partly by the margin due to the pinch out of reservoir bed, Fault trap reservoir that has a fault boundary.
13
Convex Trap
Pinchout Trap
Permeability Trap
Fault Trap
Figure 1-3: Elementary Trap in Sectional View
14
2 .
C H A P T E R
2
THE POROUS MEDIUM Porous materials are encountered literally everywhere in everyday life, in technology and in nature. With the exception of metals, some dense rocks, and some plastics, virtually all solid and semi-solid materials are “porous” to varying degrees. A material or structure must have these two properties in order to qualify as a porous medium: 1. It must contain spaces, so-called voids or pores, free of solids, imbedded in the solid or semisolid matrix. The pores usually contain some fluid, such as air, water, oil or a mixture of different fluids. 2. It must be permeable to a variety of fluids, i.e., fluids should be able to penetrate through one face of a sample of material and emerge on the other side. There are many examples where porous media play important roles in technology and, conversely, many different technologies that depend on porous media. Among the most important technologies that depend on the properties of porous media are: 1. Hydrology, which relates to water movement in earth and sand structures, such as water flow to wells from water-bearing formations. 2. Petroleum engineering which is mainly concerned with petroleum and natural gas exploration and production. The petroleum engineer is concerned with the quantities of fluid content within the rocks, transmissibility of fluids through the rocks, and other related properties. These properties depend on the rock and frequently upon the distribution of character of the fluid occurring within the rock. Knowledge of the physical properties of the rock and the existing interaction between the hydrocarbon system (gas, oil and water) and the formation is essential in understanding and evaluating the performance of a given reservoir. Rock properties are determined by performing laboratory analyses on cores from the reservoir to be evaluated. The cores are removed from the reservoir environment through the well during the drilling operations. There are primarily two main categories of core analysis tests that are performed on core samples regarding physical properties of reservoir rocks. These are:
Routine core analysis tests
Porosity Permeability Saturation
Special core analysis tests
Capillary pressure Relative permeability 15
Wettability Surface and Interfacial Tension Electrical Conductivity Pore size Distribution
These properties constitute a set of fundamental parameters by which the rock can be quantitatively described. They are essential for reservoir engineering calculations as they directly affect both the quantity and the distribution of hydrocarbons and, when combined with fluid properties, control the flow of the existing phases (i.e., gas, oil, and water) within the reservoir.
HOMOGENEITY Homogeneous usually means describing a material or system that has the same properties at every point in space; in other words, uniform without irregularities. It also describes a substance or an object whose properties do not vary with position. To apply this term on a porous medium we should define a “macroscopic system”. A few sand grains cemented together constitute a microscopic rather than macroscopic porous medium. The properties of a microscopic sample are not expected to be representative of the macroscopic porous medium from which it was removed. Let us suppose that a macroscopic pore structure parameter such as porosity or permeability is determined in a series of samples of increasing size, taking from a large porous sample, and the result is plotted against the sample’s size. It could be seen that initially the calculated properties change with sample size and there are some fluctuations in the results with increasing sample size. With increasing sample size the amplitude of these fluctuations decreases and gradually diminishes until finally a smooth line is obtained after a certain sample size (Figure 2-1). By definition “the sample said to be macroscopically representative whenever the macroscopic measured property (such as porosity and permeability) is not fluctuating any more when including more material around the initial sampling point, but its variation can be represented by a smooth line”2. When the properties of a porous medium do not change with changes in the macroscopic representative sample the medium is said to be macroscopically homogeneous.
Figure 2-1: Dependence of Permeability on Sample Volume 2
F.A.L. Dullien (1979)
16
Assume we run a test to determine the homogeneity of the porous medium. In this test along an arbitrary direction we choose different macroscopic samples and we measure their macroscopic properties. By increasing the number of samples we can plot a probability density function (PDF) for the measured properties. The PDF plot could have different shapes according to the homogeneity or heterogeneity of the medium. The simplest case is when the PDF is a delta function (vertical line) that shows the property is constant regardless to the sampling position (Figure 2-2.a). Thus we can confirm the homogeneity of the medium. While proceeding in an arbitrary direction in the medium, we may find that the measured macroscopic property first remains constant but it suddenly changes to a different but constant value. In this case the medium is said to be “macroscopically heterogeneous in the discontinuous sense” (Figure 2-2.b). Processing in any chosen direction in the medium, we may find that the property continuously changes according to the position of the samples. In this case the medium is said to be “macroscopically heterogeneous in continuous sense” (Figure 2-2.c). Finally it may happen that the function that representing the variation of parameter with position is piecewise continuous. In this case the medium is said to be “macroscopically heterogeneous in both continuous and discontinuous sense” (Figure 2-2.d).
F (P)
F (P)
(a)
(c)
P (As a special macroscopic property)
(b)
F (P)
P (As a special macroscopic property)
(d)
P (As a special macroscopic property)
F (P)
P (As a special macroscopic property)
Figure 2-2 –A Probability Density Function can be used to find the homogeneity or heterogeneity type of a porous medium3 a) Macroscopically Homogeneous System; b) Macroscopically Heterogeneous in Discontinuous Sense c) Macroscopically Heterogeneous in Continuous Sense; d) Macroscopically Heterogeneous In both Continuous and Discontinuous Sense
3
Greenkorn and Kessler (1970)
17
ANISOTROPY Anisotropy means that some properties of the porous medium do not have the same value in different directions. In an anisotropic porous medium, the permeability, formation resistivity factor, and breakthrough capillary pressure depend on the direction. In the most general case these properties are function of both location in the medium and orientation. Thus the probability density function for each property can be described with five independent variables (x, y, z) for location and ( , ) for orientation. If the probability density distribution is independent of the angular coordinates, the medium is isotropic, otherwise it is anisotropic. In the reservoir formation generally anisotropy can be caused by periodic layering. So generally we have different properties such as permeability at z direction compare to the x and y direction.
POROSITY The rock texture consists of mineral grains of various shapes and sizes and its pore structure is extremely complex. The most important factors of the pore structure are how much space there is between these grains and what their shapes are. That is because the spaces between these grains serve to either mainly transport fluids forming connecting pores, or to store the fluids forming storage pores. From the reservoir engineering standpoint, porosity is one of the most important rock properties, a measure of space available for storage of hydrocarbons. Quantitatively, porosity is the ratio of the pore volume to the total volume (bulk volume). This important rock property is determined mathematically by the following generalized relationship (Figure 2-3):
(2-1)
Where φ = porosity
Grain Pore
Figure 2-3 – Microscopic Cross Section Image of a Porous Medium
As the sediments were deposited and the rocks were being formed during geological times, some void spaces that developed became isolated from the other void spaces by excessive cementation. Thus, 18
many of the void spaces are interconnected while some of the pore spaces are completely isolated. This leads to two distinct types of porosity, namely:
Absolute porosity Effective porosity
Absolute porosity The absolute porosity is defined as the ratio of the total pore space in the rock to that of the bulk volume. A rock may have considerable absolute porosity and yet have no conductivity to fluid for lack of pore interconnection. The absolute porosity is generally expressed mathematically by the following relationship: (2-2)
Where φa = absolute porosity.
Effective porosity From the standpoint of flow through a porous medium only interconnected pores are of interest, hence the concept of effective porosity defined as the percentage of interconnected pore space with respect to the bulk volume, or
(2-3)
Where e = effective porosity. The effective porosity is used in all reservoir engineering calculations because it represents the interconnected pore space that contains the recoverable hydrocarbon fluids. Transportation of fluids is controlled mainly by connected pores. For intergranular materials, poorly to moderately well cemented, the total porosity is approximately equal to effective porosity. For more cemented materials and some carbonates, significant difference in total porosity and effective porosity values may occur. Another type of pores that seem to belong to the class of interconnected pores but contribute very little to the flow, are dead-end pores or stagnant pockets (Figure 2-4). These pores have just a constricted opening to the flow path so the fluid in them is practically stagnant. In certain mechanisms of flow such as diffusion and dispersion it is important to pay attention to the effects of dead-end pores.
19
Dead-end pore
Main flow channel Figure 2-4 – Dead-end pore
A pore-structure concept for rocks (Figure 2-5) was presented (Katsube and Collett, 1973) in the 1970s, which consisted of total connecting porosity ( C) for connecting pores that contribute mainly to fluid migration and of storage porosity ( S) for storage pores that contribute mainly to fluid storage, where their sum is effective porosity ( e): S+
C=
(2-4)
e
The storage pore shapes can be characterized by vugular or intergranular, as shown in Figure 2-5.
Figure 2-5 - Storage and connecting pore model for shale or any other type of rock with interconnected pore
systems4
There are two major types of connecting pores: sheet-like, circular and/or tubular pores. Many more types of connecting pores can be considered, but these are most representative for describing the extreme differences between their types. For example, the cross-section of connecting pores can have many shapes (Figure 2-6). A representative cross-section of a sheet-like pore is shown in Figure 2-6.a. Cross-sections of circular and/or tubular pores can take many shapes (Figure 2-6.b). The total connecting porosity ( C) value of a rock includes connecting pores in all three directions (Figure 2-7.a). However, when considering fluid or electrical current flow through the rock, only pores in two directions are
4
T.J. Katsube (2010)
20
considered for sheet-like pores (Figure 2-7.a), and only in one direction for circular and/or tubular pores (Figure 2-7.b).
Figure 2-6 - The Various Cross-Sections of Connecting Pores a) Tortuous Sheet-Like Pore; b) Various Shapes of Tubular Pores
The fluid flow in a rock is controlled mainly by connecting pores. The term “connecting pores” implies all pores except for isolated pores. However, some of these connecting- or storage- pore systems can be dead-end. The connecting pores that contribute to electrical current- or fluid flow through the rock have to be interconnected from one end of the rock to the other, and not include dead-end pores. These are distinguished as end-to-end connecting pores. The actual porosity of the end-to-end connecting pores should be smaller than the value of C, since it does not include the porosity of the dead-end pores. The storage-connecting pore system and its porosity descriptions are shown in Figure 2-7.b for a rock section that includes vugular storage and isolated pores. The end-to-end connecting porosity of the pore system in Figure 5 is represented by CF.
Figure 2-7 – a) Three-dimension distribution of connecting; b) Complete storage-connecting pore system.5
Where: 5
Bowers and Katsube (2002)
21
Total Porosity
,
Storage flow porosity
Effective Porosity
,
Storage blind (Dead-end) porosity
Storage porosity
,
Connecting flow porosity
Blind porosity
,
Connecting storage porosity
Connecting porosity
,
Connecting blind porosity
Porosity may be classified according to the mode of origin as “original” and “induced”. The original porosity is that developed in the process of deposition that forms the rock, while induced or secondary porosity added at a later stage by some geologic and chemical process. The inter-granular porosity of sandstones and the inter-crystalline and oolitic porosity of some limestones typify original porosity. Induced porosity is typified by fracture development as found in shales and limestones and by the vugs or solution cavities commonly found in limestones. Rocks having original porosity are more uniform in their characteristics than those rocks in which a large part of the porosity is included. Materials having induced porosity such as carbonate rocks have complex pore configuration. In fact two or more systems of pore openings may occur in such rocks. The basic rock material is usually finely crystalline and is referred to as the matrix. The matrix contains uniformly small pore openings which comprise one system of pores. One or more systems of larger openings usually occur in carbonate rocks as a result of leaching or fracturing of the primary rock material. Fractures and vugs are highly variable in size and distribution. Therefore even more than for intergranular materials, laboratory measurements are required for quantitative evaluation of porosity. For direct quantitative measurement of porosity, reliance must be placed on formation samples obtained by coring. Many porous media are made of discrete large and small grains or particles that are loose (unconsolidated porous media). Consolidated sedimentary rocks are derived from initially unconsolidated grains that have gone significant cementation at areas of grain contact. Early investigations of the porosity were conducted to a large extend by investigation in the fields of ground water geology, chemical engineering, and ceramics. Therefore much interest was centered on the investigation of the porosity of unconsolidated materials. The porosity of unconsolidated materials depends on:
Grain shape Grain packing Grain sorting Grain size distribution Compaction
The porosity of consolidated materials depends mainly on the degree of cementation and consolidation but also on the above mentioned parameters.
22
Grain shape and packing Consider simple models, such as a regular packing of uniform sphere or rods. Graton and Fraser (1935) analyzed the porosity of variable packing arrangements of uniform spheres. The least compact arrangement of uniform spheres is that of cubical packing with a porosity of 47.6%. The most compact packing of uniform spheres is the rhombohedra or close-packed, where the porosity is 26.0%. In these and other cases of sphere of equal size, the porosity is independent of the radius of spheres. Cross view of the unit cell of two of the mentioned packing are shown in Figure 2-8.
Cubic packing of uniform spheres
Rhombohedra or close-packed of uniform spheres
Figure 2-8 – Typical ordered porous medium structures
Often porous materials with spherical grains have lower porosity than materials composed of nonspherical grain.
Example 2-1 Calculate the cubic packing of uniform spheres porosity (Figure 2-8).
Solution The unit cell is a cube with sides equal to 2r where r is the radius of sphere. Therefore
Since there are 8 (1/8) spheres in the unit cell
The porosity is therefore is
The interesting point is that the radii cancel in the formula and the porosity of packing uniform spheres is a function of packing only.
Grain size distribution and grain sorting Naturally occurring materials are composed of a variety of particle sizes. The particle size distribution may appreciably affect the resulting porosity, as small particles may occupy pores formed between large particles, thus reducing the porosity (Figure 2-9-a). On the other hand sometimes porosity increases during a phenomenon called bridging (Figure 2-9-b). 23
a)
Porosity reduction (well sorted)
b)
Porosity increase (bridging)
Figure 2-9 – Effect of sorting and grain size distribution on porosity
In naturally occurring materials porosity increases by decreasing the grain sizes. An increase in range of particle size tends to decrease porosity.
Cementation and compaction During the cementation process in consolidated rocks as the pore space is filled with cementing material, significant reduction in porosity may take place. Because compaction forces vary with depth, porosity will also vary with depth especially in clays and shales. Krumbein and Sloss (1951) indicate a reduction in sandstone porosity from 52 to 41% and in shale from 60 to 6% as depth increases from 0 to 2000m. Most of the pore reduction is due to the inelastic, hence irreversible, effects of intergranular movement. Reservoir rocks may generally show large variations in porosity vertically but do not show great variations in porosity parallel to the bedding planes.
LABORATORY POROSITY MEASUREMENT A great many methods have been developed for determining porosity, mainly of consolidated rocks having intergranular porosity (encountered in oil reservoir). Most of the methods developed have been designed for small samples. From the definition of porosity it is obvious that common to all methods is the need to determine two of three volumes: total or bulk volume of the sample, its pore volume, and/or the volume of its solid matrix. The various methods based on such volume determination, called “direct methods”, differ from each other in the way these volumes are determined. Other methods are available, called “indirect methods” based on the measurement of some properties of the void space. Examples of such properties are the electrical conductivity of electrically conducting fluid filling the void space of the sample, or the absorption of radioactive particles by a fluid filling the void space of the sample. The porosity of the larger portion of rock is determined statistically from the results obtained on numerous small samples.
Bulk volume The simplest direct method for determining bulk volume of a consolidated sample with a well design geometric shape is to measure its dimensions. The method is applicable to cylindrical core with smoothed flat surfaces. The usual procedure is to determine the volume of fluid displaced by the sample. This method is particularly desirable for irregular shaped samples. The fluid volume that the 24
sample displaces can be determined volumetrically or gravimetrically. In both methods the displaced fluid should be prevented from penetrating the pore space of the sample. There are 3 strategies to do that:
Coating the rock sample with paraffin By saturating the rock with the fluid into which it is to be immersed By using mercury, which according to its surface tension and wettability characteristics does not tend to enter to small pores of most intergranular samples?
Gravimetric determination of the bulk volume can be accomplished by measuring the loss in the weight of sample when it is immersed in the fluid or observing the change in the weight of pycnometer when is filled with mercury and when is filled with mercury and core sample.
Example 2-2 Bulk Volume Calculation of coated sample immersed in water. A = weight of dry sample in air = 30.0 g B = weight of the sample after coating with paraffin = 31.8 g; Paraffin density = 0.9 g/cm3 C = weight of paraffin = B – A = 1.8 g D = volume of paraffin = 1.8 / 0.9 = 2cm3 E = weight after immersing of the coated sample in the water = 20 g, water density = 1 g/cm3 Volume of water displaced = (B – E) / water density = 11.8 cm3 Bulk volume of rock = volume of water displaced – volume of paraffin = 9.8 cm3
Example 2-3 Calculate the volume of a dry sample immersed in mercury pycnometer. A = weight of dry sample in air = 30 g B = weight of pycnometer filled with mercury at 20oC = 360 g, mercury density = 13.546 g/cm3 C = weight of pycnometer filled with mercury and sample at 20oC = 245.9 g D = weight of sample + weight of mercury filled pycnometer = A + B = 390 g E = Weight of mercury displaced = D – C = 144.1 g Bulk volume of rock = E/mercury density = 144.1 / 13.546 = 10.6 cm3.
Grain volume The grain volume can be determined from the dry sample weight and the grain density. For many purposes, result with sufficient accuracy can be obtained by using the density of quartz (2.65 g/cm3) as the grain density.
25
A method of determining the gain volume is crushing the sample after determining the bulk volume, thus removing all pores including the non-interconnecting ones. The volume of solids is then determined by fluid displacement in a pycnometer.
Pore volume There are methods to measure the pore volume of the rock sample directly with no need to determine the grain volume. Actually, all these methods measure effective porosity. The methods are based on the extraction of a fluid from the sample or intrusion of a fluid into the pore space of the rock sample. Mercury injection method: Both the bulk and pore volume are determined in this method. The tested sample is placed in a chamber filled to a certain level with mercury, with a known volume of air at a known pressure (e.g. atmospheric pressure) above it. The volume of mercury displaced by the sample gives the bulk volume. When the pressure of mercury is increased by a volumetric pump, the mercury penetrates the pore space of the sample. Total effective pore volume could be determined by gradually increasing the pressure. In general the method is not suitable for low permeability samples as very high pressure are required. Gas expansion method: This method is based on the Boyle-Marriote Gas Law. It may be the most widely used method for determine porosity. The test usually carried out at the constant temperature. Basically two chambers with known volumes are connected by a valve (Figure 2-10). The tested sample is placed in the chamber of volume V1. The pressure in this chamber is P1. The second chamber (volume V2), initially at pressure P0, is connected to the first one by opening the valve between them, thus permitting the gas to expand isothermally. If the final pressure is P2, from Boyle-Marriote law we have:
,
(2-5)
Where Vs = grain volume. Usually, for simplification, helium as an approximately ideal gas, at low pressure, is used and according the ideal gas assumption Z=1;
WE NEED TO ADD REFERENCES IN EACH SECTION OR AT THE END
26
P1
P2
V1
V2
Gas inlet Test Sample
Valve Figure 2-10 – Porosimeter Based on Boyle - Mariotte’s Law
Imbibition method: Reservoir rocks have the ability of imbibe water spontaneously. This property is used to determine effective porosity of the rock. In this method the weight of a dry sample is measured and then the sample is immersed under vacuum in water or any other fluid that rock has the tendency to imbibe. After enough time, up to several days, the saturated sample is weighted. Utilizing the density of the liquid we can find the imbibed fluid volume and subsequently the effective porosity of the sample. Optical methods: The porosity of a sample is equal to the “areal porosity” provided that pore structure is random. The areal porosity is determined on polished thin sections of a sample. It is often necessary to impregnate the pore with some material such as wax, plastic, color, or wood’s metal in order to make pores more visible and/or distinguishing interconnected pore from the isolated pores (Figure 1-1). When impregnating the sample with a resin only the interconnected pores will be invaded. Whenever there are very small pores present along with large ones, it is very difficult to make sure that all the small pores have been accounted for by the measurement. This is one of the reasons why the porosity measurements by the optical methods may differ significantly from the results obtained by other methods.
Figure 2-11 – Colored Thin Section Microscopic Image
27
Statistical methods: A pin is dropped many times in a random manner on an enlarged photomicrograph of a section of consolidated porous material, the porosity of which is to be determined. It can be shown that the probability of a random point falling within the pore space of this section is equal to the porosity. Therefore, as the number of tosses increases, the number of times the pin’s point falls in the pore space to the total number of tosses approaches the value of the porosity. X-ray tomography methods: ADD TEXT Magnetic resonance methods: ADD TEXT
WELL LOGGING Well logging, also known as borehole logging is the practice of making a detailed record (a well log) of the geologic formations penetrated by a borehole. The log may be based either on visual inspection of samples brought to the surface (geological logs) or on physical measurements made by instruments lowered into the wellbore (geophysical logs). Well logging is done during all phases of a well's development; drilling, completing, producing and abandoning. Logging measurements are quite sophisticated. The prime target is the measurement of various geophysical properties of the subsurface rock formations. Of particular interest is porosity. Logging tools provide measurements that allow for the mathematical interpretation of porosity. There are different types of well logging that used to estimate the porosity of the formation around the well, such as: CNL (compensated neutron) logs: also called neutron logs, determine porosity by assuming that the reservoir pore spaces are filled with either water or oil and then measuring the amount of hydrogen atoms (neutrons) in the pores. These logs underestimate the porosity of rocks that contain gas. FDC (formation density compensated) logs: also called density logs, is a porosity log that measures electron density of a formation and determine porosity by evaluating the density of the rocks. Because these logs overestimate the porosity of rocks that contain gas they result in "crossover" of the log curves when paired with Neutron logs. NMR (nuclear magnetic resonance) logs: may be the well logs of the future. These logs measure the magnetic response of fluids present in the pore spaces of the reservoir rocks. In so doing, these logs measure porosity and permeability, as well as the types of fluids present in the pore spaces.
PORE SIZE DISTRIBUTION There is not a unique definition of “pore diameter” or “pore size”. Every method of pore size determination defines a pore size in terms of a pore model which is best suited to the quantity measured in the particular experiment. There is the same situation for the definition of void space. For simplicity it is usually restricted to the pore space enclosed between “solid” surfaces. Given that the 28
pore space consists of an irregular network of pores, there are terms to distinguish between pore spaces that are relatively narrow and the interconnected relatively larger spaces. The narrow constrictions that interconnect relatively larger spaces are called pore throats or pore necks, with pore throats being the more common term. While the relatively larger pore spaces are called pore bodies, node pores or bulge pores, with pore bodies being the most commonly used term (Figure 2-12).
Pore Throat
Pore body Figure 2-12 - Schematic Shape of pore and Pore Throat
In the vast majority of porous media, the pore sizes are distributed over a wide spectrum of values, called “pore size distribution”. On the other word pore size distribution is a probability density function giving the distribution of pore volume by a characteristic pore size. If the pores were separated objects then each pore could be assigned a size according to some consistent definition, and the pore size distribution would become analogous to the particle size distribution obtained, for example, by sieve analysis. A sieve analysis (or gradation test) is a practice or procedure used to assess the particle size distribution (also called gradation) of a granular material. A sieve analysis is performed on a sample of aggregate in a laboratory. A typical sieve analysis involves a nested column of sieves with wire mesh cloth (screen) (Figure 2-13). A representative weighed sample is poured into the top sieve which has the largest screen openings. Each lower sieve in the column has smaller openings than the one above. At the base is a round pan, called the receiver. The column is typically placed in a mechanical shaker. The shaker shakes the column, usually for some fixed amount of time. After the shaking is complete the material on each sieve is weighed. The weight of the sample of each sieve is then divided by the total weight to give a percentage retained on each sieve. The size of the average particles on each sieve then being analysis to get the cut-point or specific size range captured on screen.
A mechanical shaker used for sieve analysis
Sieves used for sieve test
Figure 2-13 – Sieve Analysis Tools
29
The pore in the interconnected pore space, however are not separated objects, and the volume assigned to a particular pore size depends on both the experimental method and the pore structure model used.
Methods of measurement The most popular methods of determining pore size distribution are mercury intrusion porosimetery sorption isotherm and image analysis. The first of these is used mostly but not exclusively to determine the size of relatively larger pores whereas sorption isotherms are best suited in the case of smaller pores. The use of imaging to analyze section of a sample has some advantages over the other two methods, but the most complete information on pore size distribution may be obtained if all three methods are used jointly. Mercury porosimetry: Mercury porosimetry characterizes a material’s porosity by applying various levels of pressure to a sample immersed in mercury. The pressure required to intrude mercury into the sample’s pores is inversely proportional to the size of the pores, so at the same times it finds pore size distribution. Theory and key assumption: a key assumption in mercury porosimetry is the pore shape. Essentially all instruments assume a cylindrical pore geometry using a modified Young-Laplace equation:
(
)
(2-6)
It relate the pressure difference across the curved mercury interface (r1 and r2 describe the curvature of that interface) to the corresponding pore size using the surface tension of mercury ( ) and the contact angle between solid and mercury. The real pore shape is however quite different and cylinder pore shape assumption may lead to major differences between reality and analysis. As indicated in equation (2-5), we need to know surface tension and contact angle for the given sample and then measure the pressure and the intruded volume in order to obtain the pore volume-pore size relation. Mercury is completely non wetting phase for each sample and in general, the surface tension of mercury is not of any great concern with respect to errors in the determination of pore size distribution. A value 0.485 N/m at 25oC is commonly accepted by most researchers. The contact angle is a parameter which clearly affects the analysis results and numerous paper have demonstrated the wide range of contact angle between mercury and various different or even very similar solid surface. However in most practical situations and out of convenience users often apply a fixed value irrespective of the specific sample material, e.g. 130o or 140o. Mercury porosimetry also has limitations. One of the most important limitations is the fact that mercury porosimetry does not actually measure the internal pore size, but it rather determines the largest connection (throat or pore channel) from the sample surface towards that pore. Thus, mercury porosimetry results will always show smaller pore sizes compare to the image analysis method results. For obvious reasons it can also not be used to analyze closed pores, since the mercury has no way of entering that pore. The smallest pore size, which can be filled with mercury, is limited by the maximum
30
pressure, which can be achieved by the instrument, e.g. 3.5 nm diameter at 400 MPa assuming a contact angle of 140°. Isolated Pore
Dead end pores
Cross linked pores
Throat pores
Figure 2-14 - Schematic representation of pores.
Sorption Method: Sorption (Adsorption – Desorption) measurements involve the measurement of the surface area and the small size pores of a given medium. This method is based on the commencement of condensation with increasing capillary pressure (Note: condensation starts in the smallest pores and proceeds to increasingly larger pores). There are many models available for the adsorption of gases onto solids where the volume adsorbed is a function of pressure with constant temperature. Basic equation and method: the fundamental equation to find the pore size distribution from capillary condensation isotherms is follow: ∫
(2-7)
Where VS is the volume of adsorbate at saturation vapor pressure (equal to the total pore volume), V the volume of adsorbate at intermediate vapor pressure P”, L(R)dR the total length of pores whose total length fall between R and R + dR, R pore radius, and (t) the multilayer thickness that is built up at pressure P”. This equation states the fact that the volume of adsorbed at pressure P” is equal to the volume of pores that has not yet been filled. Micro-tomography: A modern tool for the determination of pore properties is x-ray micro-tomography. With this technique as small sample (less than 1cm in diameter) is exposed to x-rays and a three dimensional reconstruction of the sample is created typically at a resolution of ~1 micron (Figure 2-15). From the three dimensional object information about the pore space, connectivity, total porosity and mineral content can be obtained. The limitation of the technique is related to the resolution of the created images. There are very powerful micro-tomography systems associated with synchrotron facilities and bench top devices. Other image analysis methods
31
Figure 2-15 – Tomographic image.
SPECIFIC SURFACE AREA The specific surface of a porous material is defined as the interstitial surface area of the voids and pores either per unit mass (S) or per unit bulk volume (SV) of the porous material (Figure 2-16). The specific surface based on the solid volume is denoted by SO.
Figure 2-16 – Specific Surface Area
For example the specific surface of a porous material made of identical spheres of radius R in a cubic packing is: It thus becomes obvious that the fine materials will exhibit much greater specific surface than will coarse materials. Some fine porous materials contain an enormous specific area. For example the specific area of sandstone may be in the order of
. Carman (1938) gives the range of
for specific surface of the sand. The specific area of a porous material is affected by porosity, by mode of packing, by the grain size and by the shape of the grains. For example disc shaped particles will exhibit a much larger specific area than will spherical ones. Specific surface plays an important role in a variety of different application of porous media. It is a measure of the adsorption capacity of various industrial adsorbent; it plays an important role in determining the effectiveness of catalysts and filters. In petroleum and rheology study it is related to the fluid conductivity or permeability of porous media.
32
Obviously the specific surface of natural porous media can be determined only by indirect or statistical methods such as: Statistical method: A needle of length “L” is drop at random a great many times on an enlarged photomicrograph of a section of porous material. A count is kept the number of times ( ) the pin’s end point falls within the void space and the number of times ( ) the pin intersects the perimeter of pores. The specific surface is then found from: (2-8) This method is considered one of the best methods. Many other matrix properties can be derived with it. Adsorption Method: These are based on the adsorption of a gas or a vapor by solid surface. The solid’s surface area is determined from the quantity of gas adsorbed on it, assuming the gas covers the entire surface of the solid with a uniform monomolecular film. Fluid Flow: This method suggests a relation between permeability of a medium and its specific area. Using this relationship one can obtain the specific area by conducting the experiments leading to the determination of the permeability of the medium.
COMPRESSIBILITY OF POROUS ROCKS Under natural conditions, a porous medium volume at some depth in a ground water aquifer or in an oil reservoir is subjected to an internal stress or hydrostatic pressure of the fluid saturation the medium, which is a hydrostatic pressure that has the same values at different direction, and to an external stress exerted by the formation in which the particular volume is surrounded and may have different value at different directions. The external stress of the formation can lead to the compaction of the porous medium that is a function of the formation depth. Krumbein and Sluss (1951) showed that porosity of the sedimentary rocks is a function of the degree of compaction of the rock (Figure 2-17).
33
60
Shales Sandstones
Porosity (%)
50 40 30 20 10 0 0
1000
2000
3000
4000
5000
6000
Depth of burial, ft Figure 2-17 – Porosity Reduction as an Effect of Compaction Increment by Depth
Compaction effect on the porosity that leads to porosity reduction is principally due to the packing rearrangement after compaction. The porosity of shales is greatly reduced by compaction largely because “bridging’ is eliminated by the greater forces. Addition to the effect of compaction on the grain arrangement, rocks also are compressible. Three kinds of compressibility must be distinguished in rocks:
Rock matrix compressibility Rock bulk compressibility Pore compressibility
Rock matrix compressibility is the fractional change in volume of the solid rock materials (grains) with a unit change in pressure. Rock bulk compressibility is the change in volume of the bulk volume of the rock with a unit change in pressure. Pore compressibility is the fractional change in the pore volume of the rock with a unit change in pressure. The depletion of fluids from the pore space of a reservoir rock results in a change in the internal pressure in the rock while the external pressure in constant, thus results a change in the net pressure. This change in the net stress could leads to a change in grain, pore and bulk volume of the rock. Pore volume change is an interesting subject to the reservoir engineer. Bulk volume change is an important subject in the areas that surface subsidence could cause appreciable property damage. Volume change under the pressure effect can be expressed as compressibility coefficient. The coefficient of solid matrix compressibility, pore compressibility and bulk compressibility are defined for of a saturated porous medium as the fractional change in the volume with a unit change in the pressure: (2-9) (2-10) (2-11)
34
The value of (in some literature mentioned as as rock compressibility) can be determined by saturating the rock with a fluid, immersing the rock in a pressure vessel containing the saturating fluid, then imposing a hydrostatic pressure on the fluid and observing the change in the volume (or ) of the rock sample. The compressibility of solid matrix ( or ) is considered for most rock to be independent of the imposed pressure. But reservoir rocks are under other conditioning of loading than this experiment. A rock buried at depth is subjected to an overburden load due to the overlying sediments which is in general greater than the internal hydrostatic pressure of the formation fluids. (Figure 2-18.a) shows an experimental apparatus that simulate this condition for a sample rock. A core sample is enclosed in a copper jacket which is then immersed in a pressure vessel and connected to a Jurguson sight glass gauge. The hydraulic pressure system is arranged so that a saturated core can be subjected to variable internal (or pore) pressure and external (or overburden) pressure. The resulting internal volume change is indicated by the position of the mercury slug level in the sight glass. Typical curve are obtained shown if (Figure 2-18.b). The ordinate is the reduction in pore space resulting from a change in overburden pressure. The slop of the curve is the compressibility of the form (
)(
)
(2-12)
It may be noted that the slop of the curves can be considered constant over most of the pressure range above 1000 psi. Hall (1953) ran some similar tests. He designate the compressibility term (2-12) as formation compaction component as total rock compressibility and develop a correlation of this function with porosity (Figure 2-19.a). Also he investigated ( ) (
)
at constant overburden
pressure. This he designated as effective rock compressibility and correlated with porosity (Figure 2-19.b). In Figure 2-19.a and b, it may be noted that compressibility decreases as the porosity increases. The value of can be determined by measuring the change in the bulk volume of a jacketed sample by varying the external hydrostatic pressure while maintaining a constant internal pressure. For sandstones and shale it can be shown that: (2-13) (2-14)
35
(a)
(b)
Figure 2-18 – a) Experimental Equipment for Measuring Pore Volume Compaction and Compressibility b) Rock compressibility test result6
(a)
(b)
Figure 2-19 – a) Formation Compaction Component of Total Rock Compressibility b) Effective reservoir rock compressibility7
6 7
Carpenter and Spencer (1940) Hall (1953)
36
eq. (2-14) provided that Cr is much less than CB. Therefore (2-15) This equation states that total change in volume is equal to the change in the pore volume. Geertsma (1957) stated that in reservoir only the vertical component of overburden pressure is constant and the stress components in horizontal plane are characterized by boundary condition that there is no bulk deformation in those directions. For these boundary conditions he developed the following approximation for sandstone: (2-16) Thus, the effective pore compressibility for reservoir rocks under the depletion of pore pressure is only one-half of that determined by present methods in the laboratory. In summary it can be stated that the pore volume compressibilities of consolidated sandstones are of the order of reciprocal psi.
PERMEABILITY Permeability is a property of the porous medium that measures the capacity and ability of the formation to transmit fluids. The rock permeability, k, is a very important rock property because it controls the directional movement and the flow rate of the reservoir fluids in the formation. This rock characterization was first defined mathematically by Henry D’ Arcy in 1856. By analogy with electrical conductors, permeability represents reciprocal of residence which porous medium offers to fluid flow. Poiseuille’s equation for viscous flow in a cylindrical tube is a well-known equation (2-17) Where:
= fluid velocity, cm/sec = tube diameter, cm = pressure loss over length L, = fluid viscosity, centipoise = length over which pressure loss is measured, cm
A more convenient form of Poiseuille’s equation is (2-18) If assume that the rock is consist of a lot of tube in different group with different radius, total flow rate from this system by using the equation (2-18)
37
= number of tubes of radius
∑
= number of groups of tubes of different radii
Example 2-4 Derive Poiseuille’s equation for viscous flow in a horizontal cylindrical tube.
Solution Consider a horizontal flow in a circular pipe. Assume a disc shape element of the fluid in the middle of the cylinder that is concentric with the tube and with radius equal to rw and length equal to . The forces on the disc are due to the pressure on the upstream and downstream face of the disc and shear force over the rim of the element Figure 2-20. Flow
P
(P+P)
r
rw
Figure 2-20 – Flow through a Pipe
According to the steady state condition: (2-19) After simplification (2-20) (2-21) (2-22) According to the shear stress definition and using equation (2-21) and then rearranged and integration on both side (2-23) Maximum velocity at r=0 so (2-24) For viscous flow in the cylindrical tube ( )
̅ (2-25)
From (2-21), (2-24) and (2-25) 38
̅
(2-26)
d = 2rw , so after rearrangement : Poiseuille’s equation
A cast of the flow channel in a rock formation is shown in (Figure 2-21). It is seen that the flow channels are of varying sizes and shapes and are randomly connected. So it is not correct to use the Poiseuille’s equation for flow in the porous media.
Figure 2-21 – Metallic cast of pore space in a consolidated sand8
In 1856, D’ Arcy (commonly known as Darcy) developed a fluid flow equation that has since become one of the standard mathematical tools of the petroleum engineer. He investigated the flow of water through sand filter for water purification. His experimental apparatus showed schematically in (Figure 2-22)
Figure 2-22 – Schematic Drawing of Darcy Experiment of Flow of Water through Sand9 8
Amyx (1960)
39
Darcy interpreted his observation so as to yield result essentially as following equation: (2-27) Here Q represents the volume rate of flow of water downward through the cylindrical sand pack of cross sectional area A and height h and K is a proportionality constant. The sand pack was assumed to be fully saturated with water. Later investigator found that Darcy’s law could be extended to other fluid as well as water and that the constant of proportionality K could be written as . The final form of the Darcy equation for horizontal linear flow of an incompressible fluid is established through a core sample of length L and a cross-section of area A is (2-28) Where: K = Proportionality constant or permeability, Darcy’s = Viscosity of flowing fluid, cp = Pressure drop per unit length, = Volumetric flow rate, This is a linear law, similar to Newton’s law of viscosity, Ohm’s law of electricity, Fourier ‘s law of heat conduction, and Fick’s law of diffusion. Substituting the relationship = Q/A, in equation (2-28) results in (2-29) The velocity, , in Equation 2-29 is not the actual velocity of the flowing fluid but is the apparent velocity determined by dividing the flow rate by the cross-sectional area across which fluid is flowing. “Darcy” is a practical unit of permeability (in honor of Henry Darcy). A porous material has permeability equal to 1 Darcy if a pressure difference of 1 atm will produce a flow rate of 1 viscosity through a cube having side 1 cm in length. Thus (
of a fluid with 1 cP
)
(2-30) (
)
One Darcy is a relatively high permeability as the permeabilities of most reservoir rocks are less than one Darcy. In order to avoid the use of fractions in describing permeabilities, the term millidarcy is used. As the term indicates, one millidarcy, i.e., 1 md, is equal to one-thousandth of one Darcy. The negative sign in equation (2-29) is necessary as the pressure increases in one direction while the length increases in the opposite direction.
Permeability correlations Radial Version of Darcy’s Law Permeability and conductivity 9
Hubbert (1954)
40
Example 2-5 Find the Darcy’s equation for isothermal flow of ideal gas.
Solution Multiply both side of equation (2-28) by density so we have a mass flow equation:
But because of isothermal flow, constant temperature:
which “b” is as a “base condition”. For ideal gas at . Therefore (2-31)
Separating variable and integrating (2-32) Define ̅ as
and ̅ as flow rate at
̅ . Then: ̅̅̅̅
. Substituting in equation (2-32)
̅
(2-33)
There for flow rate of ideal gas could be found from the Darcy’s law for the incompressible fluid when the flow rate defined at the algebraic mean pressure.
Permeability of combination layers The foregoing flow equations were all derived on the basis one continuous value of permeability between the inflow and out flow face. It is rare to encounter a homogeneous reservoir in actual practice. Most formations have space variations of permeability. They are more variable than porosity and more difficult to measure. Yet an adequate knowledge of permeability distribution is critical to the prediction of reservoir depletion by any recovery process. In many cases, the reservoir contains distinct layers, blocks, or concentric rings of varying permeabilities. Also, because smaller-scale heterogeneities always exist, core permeabilities must be averaged to represent the flow characteristics of the entire reservoir or individual reservoir layers (units). The proper way of averaging the permeability data depends on how permeabilities were distributed as the rock was deposited. There are three simple permeability-averaging techniques that are commonly used to determine an appropriate average permeability to represent an equivalent homogeneous system. These are:
Weighted-average permeability 41
Harmonic-average permeability Geometric-average permeability
Weighted-Average Permeability: This averaging method is used to determine the average permeability of layered-parallel beds with different permeabilities. Consider the case where the flow system is comprised of three parallel layers that are separated from one another by thin impermeable barriers, i.e., no cross flow, as shown in Figure 2-23. All the layers have the same width w with a cross-sectional area of A.
Figure 2-23 – Linear Flow through Layered Bed
The flow from each layer can be calculated by applying Darcy’s equation in a linear form as expressed by equation (2-28), to give: Layer 1
Layer 2
Layer 3
The total flow rate from the entire system is expressed as
The total flow rate Qt is equal to the sum of the flow rates through each layer or:
Substitution of value for each term:
42
So:
General expression for average permeability of a parallel layered formation with n layers is: ∑
(2-34)
The presented general formula for a parallel layered formation which each layer has its own width (‘w’ in Figure 2-23) is as follows: ∑
: cross-sectional area of layer j : width of layer j
Harmonic-Average Permeability Permeability variations can occur laterally in a reservoir as well as in the vicinity of a well bore. Consider Figure 2-24 which shows an illustration of fluid flow through a combination of series beds with different permeabilities. For a steady-state flow, the flow rate is constant and the total pressure drop is equal to the sum of the pressure drops across each bed, or
Substituting for the pressure drop by applying Darcy’s equation, i.e., Equation (2-28), gives:
After simplification ⁄
⁄
⁄
General expression for average permeability of a series layered formation with n layer is: ∑ ∑
⁄
(2-35)
43
Figure 2-24 – Linear Flow through Series Beds
For radial systems with series layers the following general equation can be produced for average permeability: ⁄ ∑
[
( ⁄
) ]
(2-36)
Fractured Systems In many sand and carbonate reservoirs the formation frequently contains solution channels and natural or artificial fractures. These channels and fractures do not change the permeability of the matrix but do change the effective permeability of the system. In order to determine the contribution made by a fracture or channel to the total conductivity of the medium, it is necessary to express their conductivity in terms of Darcy’s law. Channels: Recalling Poiseuille’s equation for fluid conductivity of capillary tubes: (2-18) The total area available to flow is
So that the equation reduces to
From Darcy’s law: (2-28) After equating two last equations we have
44
(2-37) If r is in centimeter, then K in darcy is given by:
Fracture: For flow through slots of fine clearance and unit width, Buckingham reported that (2-38) By analogues to Darcy’s law: (2-39) Where h is in cm, the permeability of a fracture in darcy is given by
Measurement of Permeability The permeability of a porous medium can be determined from the samples extracted from the formation or by in place testing such as well logging and well testing. Measurement of permeability in the case of isotropic media is usually performed on linear, mostly cylindrical shaped, “core” samples. Cores are cylinders with approximately 3.81cm (1.5 inch) diameter and 5 cm (2 inch) length. Sometimes the permeability tests run on a whole core samples about 30-50 cm long. The experiment can be arranged so as to have horizontal or vertical flow through the sample. Permeability is reduced by overburden pressure, and this factor should be considered in estimating permeability of the reservoir rock in deep wells because permeability is an anisotropic property of the porous rock, that is, it is directional. Routine core analysis is generally concerned with plug samples drilled parallel to bedding planes and, hence, parallel to the direction of flow in the reservoir. These yield horizontal permeabilities (Kh). The measured permeabilities on plugs that are drilled perpendicular to the bedding planes are referred to as vertical permeability (Kv). Both liquids and gases have been used to measure permeability. However liquids sometimes change the pore structure and therefore the permeability. For example injection of water to a sample with some amount of clay leads to decreasing permeability due to swelling of the clays. There are several factors that could lead to some of error in determining reservoir permeability. Some of these factors are:
Core sample may not be representative of the reservoir rock because of reservoir heterogeneity. Core recovery may be incomplete. Permeability of the core may be altered when it is cut, or when it is cleaned and dried in preparation for analysis. This problem is likely to occur when the rock contains reactive clays.
As core samples usually contain water and oil, it is necessary to prepare the core samples for the test. Cores are dried in an oven or extracted by a Soxhlet extractor and then they are subsequently dried. The residual fluids are thus removed and the core samples become 100% saturated with air. In principle 45
measurement at a steady single flow rate permits the “routine” calculation of the permeability from Darsy’s law. The core is inserted in a core holder. A pressure applied on the surface of the core as confining pressure. An appropriate pressure gradient is adjusted across the core sample and the rate of flow of air through the plug is observed. The permeability could be found from either equation (2-32) or (2-33). However there is considerable experimental error in this experiment so the requirement that permeability be determined for conditions of viscous flow is best satisfied by obtaining data at several flow rates and plotting the flow rate versus pressure drop, as shown in Figure 2-25. A straight line is fitted to the data points. According to the Darcy’s law, the slope of this line is ⁄ , and this line must pass through the origin. But at ultralow flow rates, the flow rate is not proportional to pressure drop. Darcy’s law should not be extrapolated to the origin. Deviation from the straight line at high flow rates is an indication of turbulent flow (Figure 2-25). This deviation shows that the pressure drop in turbulent flow is higher than viscous flow. By increasing the pressure drop we can reach to a maximum flow rate capacity of the medium, after that flow rate will not increase by increasing the pressure drop. 4 Turbulent Flow
𝑄̅ ⁄𝐴
3 2 1
Viscous Flow
0 0
0.5
1
𝑃
1.5
𝑃 ⁄𝐿
Figure 2-25 – Plot of Experimental Results for Calculation of Permeability
The same experiment can be done with water or other liquids. In this case the core sample should be fully saturated with the testing liquid. When liquid is used as the testing fluid, care must be taken that it does not react with the solids in the core sample. The permeability of a core sample measured by flowing air is always greater than the permeability obtained when a liquid is the flowing fluid. Klinkenberg (1941) postulated, on the basis of his laboratory experiments, that liquids had a zero velocity at the sand grain surface, while gases exhibited some finite velocity at the sand grain surface. This resulted in a higher flow rate for the gas at a given pressure differential. Correction can be applied for the change in permeability because the reduction in confining pressure on the sample.
Example 2-6 The following data obtained during a routine permeability test at 70oF. Find the permeability.
Flow rate = 2000 cc of air at 1 atm in 400 sec Downstream pressure = 1atm Viscosity of air at test temperature = 0.02 cP Core cross sectional area = 3 cm2 Core length = 5 cm 46
Upstream Pressure = 1.75 atm
Solution ̅ ̅̅ ̅
̅
̅
̅ ̅
Measurements of permeability on large core samples generally yield better indication of permeability of limestone than do with small core sample. Rocks which contain fractures in situ separate along natural plane of weakness when cored. Therefore the conductivity of such fractures will not be included in the laboratory data.
Effect of reactive liquid on permeability While water used as testing liquid in permeability determination, in samples with clay material water act as a reactive liquid in connection with permeability determination. Reactive liquids alter the internal geometry of the porous medium which causes permeability change. The effect of clay swelling in presence of water when water used as testing fluid in permeability test is the most known effect of a reactive testing fluid. The degree of swelling is a function of water salinity. While the fresh water may cause swelling of the cementation material in the core it is a reversible process. Highly saline water can pass through the core and return the permeability to its original value.
The Klinkenberg Effect Klinkenberg (1941) reported a variation in the permeability test results with the pressure when gas is used as testing fluid. Klinkenberg found that for a given porous medium as the mean pressure increased the calculated permeability decreased. Mean pressure is defined as follow: (2-40) This variation caused by “gas slippage” phenomenon. The phenomenon of gas slippage occurs when the diameter of the capillary opening approaches the mean free path of the gas. As mentioned before in flowing of the gas through the porous media the velocity at the solid wall cannot, in general, be considered zero, but a so called “slip” or “drift” velocity at the wall must be taken into account. This effect becomes significant when the mean free path of the gas molecules is of comparable magnitude as the pore size. When the mean free path is such smaller than the pore size, the slip velocity becomes negligibly small. As in liquids the mean free path of molecules is of the order of the molecular diameter, so the no-slip condition always applied in liquid flow. The mean free path of a gas is a function of molecular size and the kinetic energy of the gas. Therefore the “Klinkenberg Effect” is a function of the gas that is used as testing fluid and the conditions of the test 47
like as pressure and temperature. Figure 2-26 is a plot of the permeability of the porous medium as determined at various mean pressures using hydrogen, nitrogen and carbon monoxide as the testing fluids.
Permeability
Carbon dioxide Nitrogen Hydrogen
Liquid or absolute permeability 0
0.2
0.4
0.6
0.8
1
1.2
⁄ Figure 2-26 – Permeability of Core Sample to Three Different Gases and Different Mean Pressure
Note that for each gas a straight line is observed for the observed permeability as a function of ⁄ . The data obtained with lowest molecular weight gas yields the straight line with greater slop, indicative of a greater slippage effect. All the line when extrapolated to infinite mean pressure, ⁄ , intercept the permeability axes at a common point. This point is the equivalent liquid permeability, . It is established that the permeability of a porous medium to a single phase liquid is equal to the equivalent liquid permeability. The magnitude of the Klinkenberg effect varies with the core permeability as well as type of the gas used in the experiment as shown in Figure 2-27.
Permeability
Low Permeability Intermediate Permeability High Permeability
-0.1
6E-16
0.1
0.2
⁄
0.3
0.4
0.5
0.6
Figure 2-27 - Effect of Permeability on the Magnitude of the Klinkenberg Effect
The resulting straight-line relationship can be expressed as: ⁄
(2-41)
48
(2-42) Where:
b = Klinkenberg factor
Kg = measured gas permeability
Pm = mean pressure KL = equivalent liquid permeability, i.e., absolute permeability, k m = slope of the line b = constant for a given gas in a given medium
Klinkenberg suggested that the “Klinkenberg factor” is a function of:
Type of the gas used in measuring the permeability Pore throat size distribution
Since permeability is, in effect a measure of size opening in porous medium, it is found that b is a function of permeability. Jones (1972) studied the gas slip phenomena for a group of cores for which porosity, liquid permeability KL (absolute permeability), and air permeability were determined. He correlated the parameter b with the liquid permeability by the following expression: (2-43) The usual measurement of permeability is made with air at mean pressure just above atmospheric pressure. To obtain accurate measurement it is required to do approximately 12 flow tests. Permeability should be determined for 4 flow rates, each at three different mean pressures. This procedure permits the obtaining three value of permeability at three mean pressure values, from which permeability to liquid can be graphically determined. In the absence of such data, Equations (2-36) and (2-34) can be combined and arranged to give: (2-44) Equation (2-44) can be used to calculate the absolute permeability when only one gas permeability measurement (kg) of a core sample is made at pm. This nonlinear equation can be solved iteratively by using the Newton- Raphson iterative methods. The proposed solution method can be conveniently written as: ́
(2-45)
Where:
Ki initial guess of the absolute permeability, md Ki+1 new permeability value to be used for the next iteration i iteration level Equation 2-44 as evaluated by using the assumed value of Ki. ́ first-derivative of Equation (2-44) as evaluated at Ki 49
The first derivative of Equation (2-44) with respect to Ki is: ́ Example (2-7) The permeability of a core plug is measured by air. Only one measurement is made at a mean pressure of 2.152 psi. The air permeability is 46.6 md. Estimate the absolute permeability of the core sample. Compare the result with the actual absolute permeability of 23.66 md. Solution Substitute the given values of pm and kg into Equations (2-44) and (2-45) ́ Assume ki = 30 and apply the Newton-Raphson method to find the required solution as shown below. ́ i Ki 1 30.000 25.12 3.45 22.719 2 22.719 -0.466 3.29 22.861 3 22.861 0.414 3.29 22.848 After three iterations, the Newton-Raphson method converges to an absolute value for the permeability of 22.848 md.
Effect of overburden pressure All the confining forces release and rock matrix expands when the core is removed from the formation. Fluid flow path in the rock changes by expansion of the rock’s matrix. Compaction of the core due to the overburden pressure may cause as much as 60% reduction in the permeability of various formations. So there is a need to an empirical correlation to correct the surface permeability for overburden pressure. It is noted that some formations are more compressible than others, so we need more data to develop this correlation. Forscheimer Equation
SATURATION In most oil formation it is believed that the formation was fully saturated with water prior to the oil migration and trapping in the formation. The less dense hydrocarbons are considered to migrate to positions of hydrostatic and dynamic equilibrium by displacing the initial water. The oil will not displace all the water originally occupied these pores. Thus reservoir rocks normally contain both hydrocarbon and water (frequently referred to as connate water or interstitial water). Saturation is defined as that fraction, or percent, of the pore volume occupied by a particular fluid (oil, gas, or water). This property is expressed mathematically by the following relationship:
50
All saturations are based on pore volume not gross volume of the reservoir. The saturation of each individual phase ranges between zero to 100 percent. By definition, the sum of the saturations is 100%, therefore
Connate (interstitial) water saturation Swc is important primarily because it reduces the amount of space available between oil and gas. It is generally not uniformly distributed throughout the reservoir but varies with permeability, lithology, and height above the free water table. Another particular phase saturation of interest is called the critical saturation and it is associated with each reservoir fluid. The definition and the significance of the critical saturation for each phase is described below. Critical oil saturation, Soc For the oil phase to flow, the saturation of the oil must exceed a certain value which is termed critical oil saturation. At this particular saturation, the oil remains in the pores and, for all practical purposes, will not flow. Residual oil saturation, Sor During the displacing process of the crude oil system from the porous media by water or gas injection (or encroachment) there will be some remaining oil left that is quantitatively characterized by a saturation value that is larger than the critical oil saturation. This saturation value is called the residual oil saturation, Sor. The term residual saturation is usually associated with the non-wetting phase when it is being displaced by a wetting phase. Movable oil saturation, Som Movable oil saturation Som is another saturation of interest and is defined as the fraction of pore volume occupied by movable oil as expressed by the following equation: Som 1 Swc Soc Critical gas saturation, Sgc As the reservoir pressure declines below the bubble-point pressure, gas evolves from the oil phase and consequently the saturation of the gas increases as the reservoir pressure declines. The gas phase remains immobile until its saturation exceeds certain saturation, called critical gas saturation, above which gas begins to move. Critical water saturation, Swc The critical water saturation, connate water saturation, and irreducible water saturation are extensively used interchangeably to define the maximum water saturation at which the water phase will remain immobile.
Determination Fluid Saturation from Rock Sample The methods that are used to measure values of original rock saturation can be classified to two classes: Evaporation of the fluids in the rock and Leaching out the fluids in the rock by extraction with a solvent.
51
Retort method: This method takes a sample and heats as to vaporize water and oil, which is condensed and collected in a small receiving vessel. This method has some disadvantage. First to vaporize all the in situ oil the core sample should reached to a high temperature around 1100oF. At this high temperature the water of crystallization within the rock is driven off, causing the water recovery values to be greater than just interstitial water. The second error is that the oil at this high temperature range tends to cock and crack. This change in the molecule type causes decreasing in the liquid volume and coats the internal walls of the core sample. Before using of the retort test calibration curves should be used to correct the errors resulted from the cocking and cracking at different temperature. ASTM method: This method is based on the extracting with a solvent during a distillation process. The core is placed and a vapor of toluene, gasoline, or naphtha rises through the core and is condensed to reflux back over the core. This process leaches out oil and water in the core. The water and extracting fluid are condensed and collect in a graduated receiving tube (Figure 2-28). The water settles to the bottom of graduated tube because of its higher density. The process continues until no more water is collected in the graduated vessel. After the process the water saturation can be determined directly. The oil saturation is an indirect determination. By knowing the weight of core sample before the test, the weight of the dried sample after the test, and the weight of extracted water we can determine the oil saturation: (2-46)
Condenser
Graduated tube Thimble and Core Solvent Electronic heater
Figure 2-28 – ASTM Extraction Apparatus
52
Centrifugal method: The water and oil are extracted from the sample core with solvent as the ASTM method. The difference is that the extraction force is applied by a centrifugal force. The solvent removes all the water and oil in the sample under the centrifugal force and the extracted fluid collected in a container to determine the oil and water saturation in the same way of ASTM method. The use of centrifuge provides a very rapid method.
FORMATION RESISTIVITY FACTOR Porous media consist of mineral, rock fragments and void space. The solids with exception of certain clay minerals (such as shaly sands where clay shales produce electrical conductivity) are nonconductive. Generally the electrical property of a rock depends on void space geometry and the fluids that occupy the void space. The fluid of interest for petroleum engineers are oil, gas and water. Oil and gas are nonconductor and water is conductive when contains dissolved salts. Water has electrolyte conductivity, because electricity conducted by movement of ions. The resistivity term, as the reciprocal of conductivity, is used to define the ability of a material to conduct current: (2-47) Where:
r = resistance, ohm R = resistivity of, ohm-cm A = Cross section area, cm2 L = Length, cm
“Formation Resistivity Factor” is the most fundamental concept in considering electrical properties of rock: (2-48) Where:
= Resistivity of fully water saturated rock = Saturating water resistivity
It is only defines for porous matrices of negligible electrical conductivity. It is evident that F is always greater than unity in the absence of electrical conductive layers. The formation resistivity factor measure the influence of pore structure on the resistance of sample. In the absence of a conductive mineral layer the electrical current can flow only through the fluid in the rock interconnected pores. This implies that F is related to the porosity of the rock. The influence of pore structure on the electrical conductivity may be divided into two contributions: the reduction of the cross section which is available for conduction and the orientation and length of conduction path (Figure 2-29). For isotropic disordered media, the ratio of the cross section available for conduction to the bulk cross section is equal to the bulk porosity, i.e. F is inversely related to porosity: 53
Figure 2-29.:
⁄ ⁄ ⁄ Figure 28.III:
( )
⁄ ⁄
⁄
Where is defined as tortuosity.
54
a = Effective cross section, A = Cross sectional area of model L = Length of element or model, Le = Effective length of capillary
Figure 2-29 - The Influence of Pore Structure on the Electrical Conductivity
Measurements showed that F varies more than just in inverse proportion to porosity. The first relationship between F and was suggested by Archie (1942): (2-49) Where m is “cementation exponent” and its value is usually between 1.3 to 2.5 for various types of rocks. For clean and uniform size sands: a = 1 and m = 2. More general form of the Archie’s law is: (2-50) Where a is an intercept. Generally a logarithmic plot of F versus is used to find a and m value for a special sample (Figure 2-30). According to the presented formula it can be shown that the following variables have effects on the resistivity of natural porous media:
Temperature Water salinity Porosity Pore geometry Formation stress Rock composition Degree of cementation Type of pore system inter-crystalline Sorting and Packing (in particulate system)
The last six factors have effect through the influence on the conduction path. Confinement or overburden pressure may cause a significant increase in resistivity by blocking of some conduction paths 55
Formation Resistivity Factor
and reduction in the cross sections which are available for flow. This usually occurs in rocks with low porosity or that are not well cemented. Older data was collected using measurements in unconfined core samples. So to have the same condition as in the reservoir resistivity measurements and formation factors determination under confining pressures are recommended for improved analysis. 1000
𝐹=𝑎×
100
−𝑚
10
1 0.1
Porosity (Fraction)
1
Figure 2-30 – Formation Resistivity Factor vs. Porosity
Several generalized relationships have been reported to relate F and . The widely used ones are: Table 1 - Generalized Relationships between F and
Equation
⁄
Investigator
Year Comments
Winsauer et al
1952
data from 30 samples – 28 sandstones core plugs, one limestone plug, and one unconsolidated sand sample
Carothers
1968
Data from 793 sandstone reservoirs
Timur et al
1972
Data from 1833 sandstone samples
Porter and Carothers
1970
1575 F-φ data points from 11 wells from offshore CaliforniaPliocene and four wells from offshore of Texas-Louisiana Miocene
Porter and Carothers
1970
720 F-φ data points from 11 wells from offshore CaliforniaPliocene and four wells from offshore of Texas-Louisiana Miocene
Carothers
1968
Data from 188 carbonate samples
Schlumberger and Shell
1979
For compact rocks, low porosity and non-fractured carbonate
Schlumberger and Shell
1979
For compact rocks, low porosity and non-fractured carbonate
Pérez-Rosales
1982
analytical relationship
EFFECT OF CONDUCTIVE SOLIDS The clay minerals present in natural rock act as a conductor and are sometimes referred to as “conductive solids”, but it is actually the water in the clay and the ions in the water that are the conducting materials. Figure 2-31 shows variation of formation factor with water resistivity for clean and clayey sands. The effect of the clay on the resistivity of the rock is dependent upon the amount, type and manner of distribution of the clay in the rock. 56
Apparent Formation Factor
100
10
1
Comparable Clean Sand Stevens Sand, California 0.1 0.1
1
water resistivity
10
100
Figure 2-31 - Apparent Formation Factor vs. Water Resistivity for Clayey and Clean Sands
As shown in Figure 2-31 the formation factor for a clay-free sand is constant. However, the formation factor for a clayey sand increases with decreasing water resistivity and approaches a constant value at a water resistivity of about 0.1 ·m. The apparent formation factor Fa was calculated from the definition of the formation factor and observed values of Roa and Rw (Fa = Roa/Rw). Wyllie proposed that the observed effect of clay minerals was similar to having two electrical circuits in parallel: the conducting clay minerals and the water-filled pores. Thus,
(2-51) Where Roa is the resistivity of shaly sand when 100% saturated with water of resistivity Rw. Rc is the resistivity due to the clay minerals. FRw is the resistivity due to the distributed water, and F is the true formation factor of the rock (the constant value when the rock contains low-resistivity water). Rearrange equation (2-51)
⁄
⁄
(2-52)
From equation (2-52)
Therefore, Fa approaches F as a limit as Rw becomes small. This was observed in Figure 2-31. The data presented in Figure 2-32 shows the relationship expressed in the previous equation. The plots are linear and are of the general form: (2-53)
57
Where C is the slope of the line and b is the y-intercept. Comparing with the equation (2-51), it may be noted that C = 1/F and b = 1/Rc. The line in which b = 0 indicates a clean sand. 1
Suite 1 no. 40 Suite 2 No. 21 Suite 1 no.4 Suite 6 no.2
1/Roa
0.8 0.6 0.4 0.2 0 0
5
10
1/Rw
15
20
25
Figure 2-32 - Water-Saturated Rock Conductivity as a Function of Water Conductivity
RESISTIVITY INDEX, RI Another fundamental electrical property of porous rocks is the resistivity index, RI. It is defined as: (2-54) Where Ro is the resistivity of the sample saturated with electrolyte (salted water) and Rt (true resistivity) is the resistivity of the sample which is saturated with electrolyte and hydrocarbon. As previously mentioned hydrocarbons are non-conductive materials. So always . SI value decreases by increasing the water saturation till unity at fully water saturation. Various empirical correlations have been found that relate the RI to water saturation in the medium. They are of the form (2-55) Where n is the resistivity index exponent (also known as saturation exponent) and c is some function of tortuosity. Archie’s correlation gives n = 2.0 for consolidated sands, while William’s correlation gives n = 2.7. Combination of equations (2-50), (2-51) and (2-55) yields: √
√
√
(2-56)
58
According to equation (2-56) initial water saturation of the formation could be predicted by knowing the value of Rt, and Rw. Usually Rt and porosity evaluated by the log data and Rw calculated according to the formation water salinity.
59
MULTI-PHASE SATURATED ROCK PROPERTIES In the previous sections the physical properties of the rocks are defined in terms of single fluid system. Such a simplified case that rock in fully saturated with just a single phase is seldom found in actual petroleum reservoirs. In petroleum reservoirs, generally there are at least two phases in contact with each other. Simultaneous presentation of more than one phase in the rock requires defining new properties such as wettability, capillary pressure and relative permeability.
SURFACE AND INTERFACIAL TENSION In dealing with multiphase systems, it is necessary to consider the effect of the forces at the interface when two immiscible fluids are in contact. When these two fluids are liquid and gas, the term surface tension is used to describe the forces acting on the interface. Surface tension is a property of the surface of a liquid that allows it to resist an external force and liquid surface acts like a thin elastic sheet. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water. When the interface is between two liquids, the acting forces are called interfacial tension. The cohesive forces among the liquid molecules are responsible for this phenomenon of surface tension. In the bulk of the liquid, each molecule is pulled equally in every direction by neighboring liquid molecules, resulting in a net force of zero. The molecules at the surface do not have the same other molecules on all sides of them and therefore are pulled inwards. This creates some internal pressure and forces liquid surfaces to contract to the minimal area. The unbalanced attraction force between the molecules creates a membrane like surface with a measurable tension, i.e., surface/interfacial tension (Figure 2-33).
Figure 2-33- Illustration of surface tension (Surface molecules pulled toward liquid causes tension in surface).
A certain amount of work is required to move a water molecule from within the body of the liquid through the interface. This work is frequently referred to as the free surface energy of the liquid. Free surface energy may be defined as the work necessary to create a new unit area of the surface. The interfacial/surface tension is the force per unit length required to create new surface which is numerically equal to the surface energy. The surface or interfacial tension has the units of force per unit of length, e.g., dynes/cm, and is usually denoted by the symbol .
60
Simple models for interfacial tension Determination (IFT): Soap Film: Consider a liquid film enclosed by a wire loop (Figure 2-34.a) where one side of the loop is allowed to slide back and forth. A force (f) is applied to counter the contractive force due to the surface tension of the liquid. Working against the surface tension an incremental increase in the force will do work W in pulling the slide out a distance dx: (2-57)
We define the Surface Tension in terms of this force such that: (2-58)
Where, ‘L’ is the length along the slide and the factor of ‘2’ is required to account for the film’s two sides. Then we relate the surface tension to the change in area (A) of the film via: (2-59) Where, ‘dA’ is the increase in area of the surface of the liquid film. Soap Bubble: Consider a soap bubble of radius r, (Figure 2-34.b), where energy is described by: (2-60) Where, is the area of the bubble as a sphere. If the radius is decreased by dr, then the change in surface energy is: (2-61) Since shrinking decreases the surface energy, the tendency to do so must be balanced by a pressure difference across the film, P, such that the work against this pressure difference is equal to the decrease in surface free energy. In other words: (2-62) So: (2-63) Where, P = (Inside pressure) - (Outside pressure). Therefore, the smaller the soap bubble, the larger the air pressure inside compared to outside.
61
Wire Frame L
r
f
Soap Film
dr dx
Movable Wire
(a)
(b)
Figure 2-34- Simplified Models for Interfacial Tension Determination.
Capillary rise Method: This methodology employs a small bore capillary which is inserted into the liquid whose surface tension is to be determined (Figure 2-35). The height to which the liquid rises in the tube is proportional to the surface tension. Assuming the radius of the capillary tube is r, the total upward force Fup, which holds the liquid up, is equal to the force per unit length of surface times the total length of surface, or (2-64) Where
is contact angle.
The upward force is counteracted by the weight of the water, which is equivalent to a downward force of mass times acceleration, or (2-65) Where
h = height to which the liquid is held, cm g = acceleration due to gravity, cm/sec2 3 w = density of water, gm/cm 3 air = density of gas, gm/cm
Density of air is negligible compare to the water density. Equating eq.(2-64) and eq.(2-65) and solving for the surface tension gives: (2-66) And for water-oil system: (2-67) Figure 2-35- Pressure relations in capillary tubes
62
WETTABILITY There exists a surface tension between a fluid and a solid, in the same way that a surface tension exists between two immiscible fluids. Wettability is defined as the tendency of one fluid to spread on or adhere to a solid surface in the presence of other immiscible fluids. This spreading tendency can be expressed more conveniently by measuring the angle of contact at the liquid-solid surface. This angle, which is always measured through the liquid to the solid, is called the contact angle that is known also as wetting angle. The contact angle has achieved significance as a measure of wettability. As shown in Figure 2-36, as the contact angle decreases, the wetting characteristics of the liquid increase. Complete wettability would be evidenced by a zero contact angle, and complete non-wetting would be evidenced by a contact angle of 180°. There have been various definitions of intermediate wettability but, in much of the published literature, contact angles of 60° to 90° will tend to repel the liquid. When one fluid preferentially covers the surface, it is called the wetting fluid, and the other fluid is called the non-wetting fluid. The origin of these surface tensions arises in the different strengths of molecular level interactions taking place between the pairs of fluids. For example a quartz sandstone grain generally develops greater molecular forces between itself and water than between itself and oils. Clean sandstones are therefore commonly water wet.
Figure 2-36- Illustration of Wettability
When two fluids are in contact with a solid surface, the equilibrium configuration of the two fluid phases (for example air and water) depends on the relative values of the surface tension between each pair of the three phases (Figure 2-37). Each surface tension acts upon its respective interface, and define the angle at which the liquid contacts the surface. Equilibrium considerations allow us to calculate the wetting angle from the surface tensions: (2-68)
This equation is known as Young’s equation (1805).
63
Figure 2-37- Equilibrium of Forces at a Liquid-Gas-Solid Interface.
The problems with using the Young’s equation approach include:
The values of cannot be measured independently in an experiment, Surface roughness effects are not considered.
To account for surface roughness it was suggested that (Wenzel, 1936): (2-69) Where,
is the apparent contact angle and fr is a roughness parameter defined as the ratio of the
true area of the solid-liquid interface to the apparent area of the solid-liquid interface. fr > 1 for any real surface. Table 2 shows some contact angles and surface tensions for common fluids in the hydrocarbon industry. Table 2- Contact angles and interfacial tension for common fluid-fluid interfaces
Interface
Contact angel ( )
Air-Water
0
1.000
72
Oil-Water
30
0.866
48
Air-Oil
0
1.000
24
140
-0.756
480
Air-Mercury
Interfacial Tension (dyne/cm)
Spreading Coefficients When two interfaces are present in a pore and are approaching each other, such as a water/oil interface and a water/gas interface, there is a point where the fluids are spontaneously redistributed in the pore. In that, oil spreads between water and gas thus eliminating the water/gas interface and creating an oil/gas interface. This condition where oil spreads over water in the presence of gas is best treated in terms of what is referred to as the spreading coefficient, defined by the equation:
⁄
(2-70)
64
Where, ⁄ is the final spreading coefficient of oil over water, are the interfacial tensions when all three fluids are in thermodynamic equilibrium. The numerical value of the spreading coefficient, as computed from the above equation, can be a positive or a negative value. The spreading parameter for non-equilibrium situation is the energy gained when covering one unit area of the water with a flat oil film of macroscopic thickness. When the spreading coefficient is positive, oil is expected to spread over water, whereas a negative spreading coefficient indicated that oil will not spread over water and the situation where the water is covered by the oil film is not favourable. The saturation of a watergas interface with a monolayer of oil results in an interfacial value, wg’ which is smaller than the interfacial tension of pure water against gas, wg, measured at the same temperature and pressure. The value of the spreading coefficient is an important parameter in some enhanced oil recovery methods. The spreading coefficient also defined for the gas-liquid-solid system. Positive spreading coefficient means that the solid is covered by the liquid film and the system gains some energy because of that. For positive S, in the G-L-S system, the liquid wet the solid completely, in which case a liquid drop will spread with time after its deposition on the surface, with its initially nonzero contact angle moving towards its limiting equilibrium, zero value.
Contact angle Hysteresis According to the definition ‘hysteresis’ occurs when a measured variable depends on the direction of change of an independent variable. Some factors that may contribute to contact angle hysteresis are as follows:
Surface roughness, Solid surface heterogeneity and differential adsorption of wettability aerating compounds, Surface immobility which prevents fluid motion that is necessary for the system to reach a threephase contact point equilibrium. For example, a surfactant desorbing from the solid-liquid interface into the bulk liquid, and the solid film at the oil-water interface, Contamination, Components in different phases equilibrate among the phases. This can occur when surfactants have solubility in oil and water and adsorb at both interfaces. The solid phase contains soluble components and/or when oil contains compounds that adsorb onto solids. The adsorption of oil components is slow and the contact angle changes over hours or months; systems that are initially water-wet may change to oil-wet.
Figure 2-38 shows the significant effect of rock-fluid-fluid interactions on the contact angle measurement.
65
Figure 2-38- Rock-Fluid-Fluid Interactions Effect on the Contact Angle
Laboratory Determination There are several methods for determining wettability of a rock to various fluids. The main ones are: Microscopic observation: This involves the direct observation and measurement of wetting angles on small rock samples. One of the most popular methods for measuring the contact angle is the ‘sessile drop method’, which involves depositing a liquid drop on a smooth solid surface and measuring the angle between the solid surface and the tangent to the drop profile at the drop edge. In most of the case the drop is small enough, thus, the gravity action can be neglected. The deposited drop deforms from its initial spherical shape, flattens to form a small cap of liquid and eventually reaches its equilibrium state. Figure 2-39 shows a schematic diagram of a liquid drop spreading on a solid surface. During the spreading process, the liquid drop will form the so-called ‘dynamic contact angle’ with the solid surface,
(t), and spread out along the horizontal axis. With the increase of the contact radius of
the drop, R(t), the drop will become thinner and its central height h(t), will decrease in order to meet a constant value. The spreading process will continue until the so-called ‘equilibrium contact angle’ is achieved. This contact angle represents the wettability of the solid–liquid–fluid system and can be related to the interfacial tensions of the system by the Young equation.
66
h(t) h(t) R(t) Immediate after drop formation
R(t) After some times
Figure 2-39- liquid drop spreading on a solid surface
The measurements are extremely difficult, and good data relies more on luck than judgment. There are several issues that must be addressed when measuring contact angles for determining reservoir properties:
Surface roughness and history of which fluid first contacted the surface will affect the measured value of the contact angle.
Rock-fluid interaction (e.g. solubility, pH, ions in the aqueous phase, polar groups in crude oil, etc.) will affect the value of the contact angle.
Polished solids (quartz, calcite) may not be representative of solid surfaces in porous media.
Time to reach equilibrium (when the contact angle is independent of time) may vary from seconds to days or years. Consequently, the contact angle measured in the laboratory may not represent the natural wettability of the system under examination.
Amott wettability measurements: This is a macroscopic mean wettability of a rock to given fluids. It involves the measurement of the amount of fluids spontaneously and forcibly imbibed by a rock sample. It has no validity as an absolute measurement, but is industry standard for comparing the wettability of various core plugs. The Amott method (Figure 2-40) involves four basic measurements. Figure 2-42 shows how to use measured Amott test data: 1) 2) 3) 4)
The amount of water or brine spontaneously imbibed = the amount of produced oil in step 1, AB. The amount of water or brine forcibly imbibed = the amount of produced oil in step 2, BC. The amount of oil spontaneously imbibed = the amount of produced water in step 3, CD The amount of oil forcibly imbibed = the amount of produced water in step 4, DA
67
Step 1 (A to B)
Step 2 (B to C )
Step 3 (C to D)
Step 4 (D to A)
Figure 2-40- Amott Wettability Test10
Initially the core sample is saturated with oil at initial water saturation (point X in Figure 2-42). The spontaneous measurements are carried out by placing the sample in a container containing a known volume of the fluid to be imbibed such that it is completely submerged (steps 1 and 3 in Figure 2-40) for water and oil respectively), and measuring the volume of the fluid displaced by the imbibing fluid ( step 1 and stp 3 of Figure 2-40). The forced measurements are carried out by injection of the ‘imbibing’ fluid through the rock sample and measuring the amount of the displaced fluid (steps 2 and 4 in (Figure 2-40)). The important measurements are the spontaneous imbibition steps of oil and water, and the total (spontaneous and forced) imbibitions of oil and water. Water-wet samples only spontaneously imbibe water, oil-wet samples only spontaneously imbibe oil, and those that spontaneously imbibe neither are called neutrally-wet. The wettability ratios for oil (AB/AC) or water (CD/CA) are the ratios of the spontaneous imbibition to the total imbibition of the each fluid. A water-wet system should have an Amott Index for water close to 1 and its Amott Index for oil is close to zero. An oil-wet system should have an Amott index to water close to 1, an Amott index to oil close to zero.
10
Paul Glover
68
Amott Ternary Wettabillity Diagram 0
Water Wetting Index
1
NEUTRAL WET
Neutrality index
WEAKLY WATER WET
Serie s1
WEAKLY OIL WET
MIXED WET 1
WATER WET
0
0.2
OIL WET
0.4
0.6
0.8
0 1
Oil Wetting Index
Figure 2-41- Amott Index Ternary Diagram
Figure 2-42- Amott Index Calculation
In general use the samples to be measured are centrifuged or flooded with brine, and then flooding or centrifuging in oil to obtain Swi. The standard Amott method is then followed. At the end of the experiment the so called Amott-Harvey wettability index is calculated:
(2-71) (2-72)
69
USBM (U.S. Bureau of Mines) method: This is a macroscopic mean wettability of a rock to given fluids. It is similar to the Amott method but considers the work required to do a forced fluid displacement. As with the Amott method, it has no validity as an absolute measurement, but is industrial standard method for comparing the wettability of various core plugs. It is usually done by centrifuge, and the wettability index W is calculated from the areas under the capillary pressure curves A1 and A2: (2-73) where, A1 is the area under the drainage curve and A2 is the area under the imbibition curve as are shown in Figure 2-43. A water-wet system should have a relatively large positive USBM Index while an oil-wet system should have a relatively large negative USBM index. Note that in this case the initial conditions of the rock are Sw=100%, and an initial flood down to Swi is required (shown as step 1 in Figure 2-43), although either case may be necessary for either the Amott or USBM methods. Figure 2-43 shows typical USBM test curves for water wet, oil wet and neutrally wet cores.
Ca pill ary Pre ssu re
Figure 2-43- USBM Index Calculation
The contact angle measures the wettability of a specific surface, while the Amott and USBM methods measure the average wettability of a core.
Wettability alteration by surfactants A surfactant is a polar compound, consisting of an amphiphilic molecule, with a hydrophilic part (anionic, cationic, amphoteric or nonionic) and a hydrophobic part. As a result, the addition of a surfactant to an oil-water mixture would lead to a reduction in the interfacial tension. In the past time, the surfactants were used to increase oil recovery by lowering IFT. Later on, due to the difficulty of initiating imbibition process in oil-wet carbonate rocks, many researchers have focused on how to alter the oil-wet 70
carbonate to water-wet by using surfactants. The most successful method reported is the surfactant flooding in the presence of alkaline. There are a number of mechanisms for surfactant adsorption such as electrostatic attraction/repulsion, ion-exchange, chemisorption, chain-chain interactions, hydrogen bonding and hydrophobic bonding. The nature of the surfactants, minerals and solution conditions as well as the mineralogical composition of reservoir rocks play a governing role in determining the interactions between the reservoir minerals and externally added reagents (surfactants/ polymers) and their effect on solid-liquid interfacial properties such as surface charge and wettability
CAPILLARY PRESSURE The capillary pressure characteristics of a given reservoir will impact the choice of recovery method(s) and displacement mechanisms. For instance, the displacement of oil by water in a water-wet reservoir requires a totally different process compared to the displacement of oil by water in an oil-wet reservoir. The capillary forces in a petroleum reservoir are the result of the combined effect of the surface and interfacial tensions of the rock and fluids, the pore size and geometry, and the wetting characteristics of the system. Any curved surface between two immiscible fluids has the tendency to contract into the smallest possible area per unit volume. This is true whether the fluids are oil and water, water and gas (even air), or oil and gas. When two immiscible fluids are in contact, a discontinuity in pressure exists between the two fluids, which depend upon the curvature of the interface separating the fluids. This difference existing across the interface is referred to as the capillary pressure (Pc). In other words Capillary pressure pc is defined as the pressure difference between the non-wetting phase and the wetting phase as a function of the (wetting phase) saturation. The displacement of one fluid by another in the pores of a porous medium is either aided or opposed by the surface forces of capillary pressure. As a consequence, in order to maintain a porous medium partially saturated with nonwetting fluid and while the medium is also exposed to wetting fluid, it is necessary to maintain the pressure of the nonwetting fluid at a value greater than that in the wetting fluid. Consider Figure 2-44 that a capillary tube in immersed in a beaker of water weather the oil is the other fluid.
71
Where
= Pressure in oil at point A’ = Pressure in oil at point B = Pressure in water at point A = Pressure in water at point B
Figure 2-44- Pressure Relation in capillary Tube
If the beaker is large the interface at point A’ is flat and capillary pressure is zero. Therefore (2-74) At the free water level in the beaker. The pressures at point B in oil and water phase, according to oil and water density, are: (2-75) (2-76) The pressure difference across the interface at point B is: (2-77) This pressure difference (capillary pressure) is the reason of the curvature at the interface of the two liquid in the capillary tube. By increasing the capillary pressure this curvature increases and vice versa. The greater pressure is always on the concave side of the interface. Therefore the non-wetting phase in a porous material is at a higher pressure than the wetting phase. At the free level of the beaker (point A’) which interface is flat the capillary pressure is neglected. According to the eq.(2-77) the capillary pressure must be in equilibrium with gravitational forces if the fluids are in equilibrium and not flowing. Considering eq.(2-67) and eq.(2-77) the expression of capillary pressure in term of surface force is: (2-78)
72
This equation shows that capillary pressure is inversely proportional to the capillary tube diameter. This is the reason that by decreasing the capillary diameter, water column height and curvature in the interface increase (Figure 2-45.a).
(a)
(b)
Figure 2-45- Dependency of Water Column to (a). Capillary Radius, (b). Wettability.
Figure 2-45.a shows the dependency of the equilibrium liquid column to the capillary tube radius. In addition to change the capillary radius, change in wetting characteristic changes the liquid column height, such that the greater wettability (adhesion tension) the greater equilibrium height obtained. In Figure 2-45.b wettability decreases from ‘a’ to ‘c’ (liquid ‘c’ is non-wet). The interfacial phenomena described above for capillary tubes also exist when bundles of interconnected capillaries of varying sizes exist in a porous medium. The capillary pressure that exists within a porous medium between two immiscible phases is a function of the interfacial tensions and the average size of the capillaries which, in turn, controls the curvature of the interface.
CAPILLARY PRESSURE IN PACKING SPHERES The relation between the pressure difference across an interface and tension can be determined by displacing the interface an infinitesimal distance in the direction of the normal to the interface. When the system is in mechanical equilibrium, the work to stretch (or contract) the interface is balanced by the pressure - volume work done in displacing the interface. The equation for mechanical equilibrium across a fluid interface is the Young - Laplace equation: (2-79)
Where
σ is the surface or interfacial tension H is the mean curvature Δp is the pressure difference across the interface such that the higher pressure is on the concave side of the interface.
The mean curvature can be expressed as a function of the radius of curvatures: 73
(
)
(2-80)
A general expression of capillary pressure as a function of interfacial tension and curvature of the interface can be generated by substitution of H in eq.(2-80) into the Young-Laplace eq. (2-79): (
)
(2-81)
Where and are the principle radii of the curvature of the interface that are measured in perpendicular planes and is the interfacial tension between the fluids. In the case of a curved interface between immiscible fluids the plus sign is used. In the case of pendular rings (Figure 2-46) the minus sign is used.
Figure 2-46- Principle radii for wetting fluid and spherical grain
Comparing eq.(2-81) with eq.(2-78) in capillary tube, it is found that the mean radius (
)
is defined by (2-82)
It is impossible to measure the values of and , so they are generally referred to by mean radius of the curvature and empirically determined from other measurements on a porous medium. For a fluid-fluid interface in a capillary tube: (2-83) Where ‘r’ is the radius of the capillary and is the contact angle. Wettability determines the fluid distribution in a porous medium. Wetting phase has tendency to occupy the smaller pores. In a packing bed porous medium 2 different types of wetting phase distribution can be found, depend on the saturation of the wetting phase. These two types are funicular distribution and pendular-ring distribution (Figure 2-47). By decreasing the wetting phase saturation distribution type changes from funicular to pendular. In funicular distribution the wetting phase completely covers the solid surface and is continues through the system. But by decreasing the
74
saturation this continuity is failed and non-wetting phase is contact with some of the solid surface and wetting phase occupies the smaller spaces (pendular distribution).
(a). Funicular distribution
(b). Pendular-ring distribution
Figure 2-47- Wetting and non-Wetting fluid distribution about inter grain contact of sphere.
By decreasing the saturation and both tend to decrease in size. According to the eq.(2-68) capillary pressure increases by decreasing the principle radii. So, capillary pressure can be expressed as a function of rock saturation when the rock is saturated by two immiscible fluids. Also for this specific medium it is possible to find the pore distribution because capillary pressure would be dependent upon the area of the various pores for any particular value of the saturation.
SATURATION HISTORY We will be able to understand capillary phenomena in porous media much better if we see how the capillary pressure changes in some elementary geometry. Assume there are one wetting and one nonwetting fluid in the system that is axial-symmetric so the two radii of curvature are equal. Consider first Figure 2-48 where a non-wetting fluid enters a circular pore from a bulk reservoir. The capillary pressure will increase with volume until it reaches a limiting value of 2σ/R (R is the principle radius and is equal to , eq.(2-83)). In other word the capillary tube could be displaced completely by the non-wetting fluid if: (2-84) Where, is a very small value.
75
Figure 2-48- Non-Wetting fluid entering the capillary tube.
The next example is a capillary tube with varying diameter. Capillary pressure increases by decreasing the tube diameter, eq.(2-78). So, the capillary pressure increases by decreasing the wetting phase saturation and the required pressure to displace the wetting phase increases step by step (Figure 2-49).
Figure 2-49- Non-Wetting fluid entering the non-uniform capillary tube.
Another example is a non-wetting fluid exiting a circular pore and entering a large reservoir filled with the wetting fluid as in Fig. 5.2. This is similar to blowing bubbles from a glass straw. Suppose the interface is flat at the exit of the pore after the previous bubble detached. If the non-wetting phase is pumped with a controlled volumetric rate, the pressure will first increase as the interface increases in a curvature from zero (flat) to that of a hemisphere. The pressure will then decrease as the interface expands into a surface of a growing sphere. If the non-wetting fluid was a reservoir with an increasing pressure rather than at a constant volumetric rate, the bubble (or drop) will first grow slowly corresponding to the rate of pressure increase until it reaches the maximum pressure upon reaching the hemispherical shape and then it will suddenly grow in size (Figure 2-50). By continuing the non-wetting injection and reducing the interface curvature, capillary increases to reach to its maximum at the entrance of the second capillary tube. 76
Figure 2-50- Non-Wetting Fluid enter to a bubble and exit it.
Capillary Pressure versus wetting phase saturation in rock Figure 2-48 showed the capillary pressure vs. non wetting phase saturation in a capillary tube. As mentioned by increasing the non-wetting phase saturation by
higher than the
the capillary tube will
be totally filled with non-wetting phase. Now assume that there is a bundle of capillary tubes with different radius sizes (r1>r2>…>rn, r: capillary tube radius) that is fully saturated with wetting phase and we want to displace the saturating fluid with non-wetting phase by increasing the pressure (Figure 2-51.a). Non-wetting phase will not enter to the medium till pressure reaches to
, that non-
wetting phase completely swipes the largest capillary tube completely. Decrease in the wetting saturation is equal to the volume fraction of the largest tube in the bundled. Figure 2-51.b that a finite capillary pressure must be applied (i.e., the breakthrough capillary pressure, Pc
) during
primary drainage before any appreciable amount of wetting phase drains out of the sample. Then by increasing the pressure there will not any change in the saturation till the pressure reaches to
,
that the non-wetting phase enters to the second largest capillary tube and the wetting phase saturation decreases by a value equal to the second largest capillary tube volume fraction in the bundle. By continuing this process we will end with the following graph (Figure 2-51.b).when the pressure reach to the
the bundle will be fully saturated with the non-wetting phase. As the Figure 2-51.b shows by
increasing the pressure saturation change amount decreases because of decreasing in the volume fraction of smaller capillary tubes in the bundle.
77
Ca pill ary Pr ess ur e
Ca pill ary Pr ess ur e Wetting Phase Saturation
(a). Bundle of capillary tubes
Wetting Phase Saturation
(b)
(c)
Figure 2-51- Capillary Pressure versus wetting phase saturation
By increasing the number of capillary tubes with different size to have a continuous spectrum of capillary sizes, discontinuity in the capillary curve can be eliminated (Figure 2-51.c). According to this test it is obvious that shape of the capillary pressure curve is affected directly by pore size distribution and wettability of the porous medium. Illustrated process is a named “Drainage” that the wetting phase is displaced by a non-wetting phase. The inverse process is “Imbibition Process” that wetting phase is the displacing fluid. If the non-wetting phase pressure at the final step of the drainage process (in the bundle of tubes test) decrease to
,
wetting phase will swept the smallest capillary tube (rn) completely. Other capillary tubes could be swept by wetting phase by continuing decrease in the non-wetting phase pressure. It could be concluded that the imbibition process starts from flooding the smallest capillary tubes (or pores in the rocks) in contrast to the drainage that starts from the largest ones. It is thus seen that the capillary pressure saturation relationship is depend upon:
Size and distribution of the pores, The fluid and solid that are involved, The history of the saturation process.
Thus in order to use the capillary pressure data properly, these factors should be considered before the data are actually applied to the reservoir calculations. Figure 2-52 is an example of typical oil-water capillary pressure curves. In this case, capillary pressure is plotted versus water saturation for four rock samples with permeabilities increasing from k1 to k4 . It can be seen that, for decreases in permeability, there are corresponding increases in capillary pressure at a constant value of water saturation. This is a reflection of the influence of pore size since the smaller diameter pores will invariably have the lower permeabilities. Also, as would be expected the capillary pressure for any sample increases with decreasing water saturation, another indication of the effect of the radius of curvature of the water-oil interface.
78
50 K = 300 mD K = 100 mD
Capillary Pressure
40
K = 30 mD
30
K = 10 mD K = 3 mD
20 10 0 0
20
40 60 Water Saturation (%)
80
100
Figure 2-52- Variation of Capillary Pressure with Permeability
Effect of Pore Geometry on Capillary Pressure Capillary pressure is affected by the geometry of a given porous medium. The following section discusses the effects of flow into and out of a constriction. In the case of fluid flowing into a constriction (Figure 2-53), pressure is needed to force the non-wetting phase into the constriction and the wetting phase out of the constriction. According to the figure: (2-85) The equation for capillary pressure then becomes: (2-86) Where, R decreases from left to right (in the direction of flow) resulting in an increased Pc, allowing the meniscus to move from left to right into the constriction.
Figure 2-53- Flow into a Constriction (cone).
Figure 2-54 shows an example of a fluid flowing out of a constriction, Pc decreases with position as meniscus moves from left to right in the direction shown in this figure. According to the figure: (2-87)
79
The equation for capillary pressure becomes: (2-88) In order to change the position of the meniscus Pnw, Pw or R may be altered (all other parameters are fixed).
Figure 2-54- Flow out of a Constriction
Dominance of Capillary Forces over Viscous Forces Viscous force is reflected in the pressure gradient generated by the flow through a porous medium. The pressure gradient is proportional to the viscosity and the fluid velocity and inversely proportional to the conductivity of the medium. Viscous force in a circular capillary tube can be found from eq.(2-17) or: (2-89) Where:
= fluid velocity, cm/sec = capillary tube radius, cm = pressure loss over length L, = fluid viscosity, centipoise = length over which pressure loss is measured, cm
As mentioned before for a porous medium, the pressure gradient is given by Darcy’s law. Now, consider the displacement of oil by water from a capillary tube, at velocity simplicity let us assume that
(Figure 2-55). For
80
According to the pressure drop because of viscous and capillary forces we have (2-90) Typical values are: = 1.0 cp and and velocity = 0.3 m/d.
= 30 mN/m Figure 2-55- Flow in a Capillary Tube.
Calculated vales of PB – PA are listed in Table 3 for different r’s . Table 3- Capillary and Viscous Forces for Different Sizes of Pore Radii.
Note that the capillary force is much higher than the viscous force and the downstream pressure is higher in contrast to one phase flow that the downstream pressure is always lower than the upstream pressure.
Capillary Number The capillary number is a dimensionless number that describes the relative importance of viscous forces to capillary forces during the course of an immiscible displacement. Table 4 offers a fairly comprehensive list of the various forms of the capillary number as used by different authors. The most common versions of the capillary number are number 6, 10. Table 4- Capillary Number or “displacement ratio” correlating groups
No.
Author(s)
Year
Porous Media
Correlating Group
1
Fairbrother and Stubbs
1935
Capillary
2
Leverett
1939
Sandstone
3
Brownell and Katz
1947
Sandstone
4
Ojeda, Preston and Calhoun
1953
Sandstone
5
Moore and Slobod
1956
Sandstone
6
Saffman and Taylor
1958
Hell-Shaw Cell
7
Taber
1969
(Berea) Sandstone
⁄
8
Foster
1973
(Berea) Sandstone
⁄
⁄
√ ⁄ ⁄ ⁄ ⁄
⁄
81
⁄
9
Lefebvre duPrey
1973
Teflon, Steel, and Aluminum
10
Melrose and Brandner
1974
Unconsolidated Glass Beads
⁄
11
Ehrlich, Hasiba and Raimondi
1974
Sandstone
⁄
12
Abrams
1975
Sandstone, Limestone
⁄
13
Reed and Healy
1977
Various
⁄
According to the work of Chatzis and Morrow (1984) the range of capillary numbers over which capillary displacement is predominant is Nca < 10-5 to 10-4. When the capillary number exceeds a value of 10-4 then the residual oil is mobilized through a stripping process.
Bond Number The Bond number is the ratio of gravity forces to capillary forces and it is of great importance in vertical displacements. The Bond number is especially useful for gravity assisted displacement processes and in centrifuge core analysis, and is often defined as: (2-91) Where, R is a characteristic length. On occasion the permeability of the medium replaces R.
LABORATORY MEASUREMENT OF CAPILLARY PRESSURE Laboratory experiments have been developed to simulate the displacing forces in a reservoir in order to determine the magnitude of the capillary forces in a reservoir and, thereby, determine the fluid saturation distributions and connate water saturation. Essentially four methods of measuring capillary pressure on small core samples are used:
The Porous Diaphragm (or restored capillary pressure) Method The Centrifugal Method The Mercury Injection Method Dynamic capillary pressure method
Differences in wetting characteristics should be taken into account when applying laboratory data to the field. Thus air-water and mercury-air data obtained on cleaned core will represent a fully water wet system for the drainage case. They may not adequately describe a mixed or oil wet system. The closest one can get to this situation, on a routine basis, is with an actual oil water (oil-brine) system when one would expect to find lower Swi values than a water wet system.
The Porous Diaphragm Method The essential requirement of this method is a permeable membrane of uniform pore size distribution containing pore of such size that the selected displacing fluid will not penetrate the diaphragm when the pressure applied to the displacing phase are below some selected maximum pressure of investigation. Various materials such as fritted glass, porcelain, and cellophane could be used as diaphragm in this test. The membrane is saturated with the fluid to be displaced (Wetting phase). Any combination of the fluids 82
can be used: gas, oil and/or water. In the gas-water system, gas is the non-wetting phase and water is wetting phase. Air is used as the gas fluid sample commonly. As the first step saturate the core sample with wetting phase (water here), then placing the core on a porous membrane which is saturated 100% with water and is permeable to the water only, under the pressure drops imposed during the experiment. Air is then admitted into the core chamber and the pressure is increased until a small amount of water is displaced through the porous, semi-permeable membrane into the graduated cylinder. Pressure is held constant until no more water is displaced, which may require several days or even several weeks. By measuring the exited water volume and using rock pore volume, saturation change in each step could be calculated: (2-92) At each step the applied pressure is actually the capillary pressure which was defined as non-wetting phase pressure (nitrogen applied pressure here) minus wetting phase pressure (water pressure equal to zero), so the pressure data can be plotted as capillary pressure data versus wetting phase (water) saturation. Figure 2-56 shows schematically porous diaphragm capillary pressure device.
Figure 2-56- porous diaphragm capillary pressure device11.
Complete determination curve of capillary pressure with this method is time consuming owning to the vanishing pressure differentials causing flow as the core approaches equilibrium at each imposed pressure. Time to reach the equilibrium increases step by step because of the reduction of displaced fluid relative permeability as a result of saturation decrease. Most determinations of capillary pressure by this method are drainage tests.
11
Welge and Bruce (1947)
83
Centrifugal Method In this method injection pressure (capillary pressure PC) applied through centrifugal force. The high acceleration in the centrifuge increases the field of force on the fluids, subjecting the core, in effect, to an increased gravitational force. In other words it relies upon increasing the ‘g’ term in the equation: (2-77) When the sample is rotated at various constant speeds, a complete capillary pressure curve can be obtained. The speed of rotation is converted to capillary pressure using the following equation: (
)
(2-93)
Where:
= radius of rotation of the bottom of core sample = radius of rotation of the top of the core sample = rotational speed
For the oil-water drainage cycle, water fully saturated core samples are immersed in oil in specially designed core holders. Starting at a low rpm setting, the amount of brine expelled from the plug is noted for a given rate of rotation. A calibrated glass vial is attached to the end of the sample. The volume of fluid being deposited in this vial can be read while the centrifuge is spinning fast. Thus, the saturation can be obtained. The rate is then increased in stages and produced water volumes are recorded for each rotation speed to give the drainage curve. The most important advantage of this method is its speed of obtaining data. A complete curve may be obtained in a few hours.
Figure 2-57- Centrifugal apparatus
Mercury Injection The mercury capillary pressure apparatus was developed to accelerate determination of the capillary pressure vs. saturation relationship. These tests can only be carried out on cleaned, dried test plugs. Mercury is normally a non-wetting fluid. The core sample inserted into a core chamber and evacuated (Figure 2-58). The pressure in the system, effectively the differential across the mercury/vacuum interface, is raised in stages to force mercury enter to the core sample. The volume of mercury which has entered the pores at each pressure (as non-wetting phase saturation) is determined from volumetric 84
readings, and the proportion of the pore space filled can be calculated. This procedure is continued until the core sample is filled with mercury or the injection pressure reach to some pre-determined value. These data gives the drainage curve. Further readings can be taken as the pressure is lowered to provide data for the imbibition case.
(a): Dried Sample is Evacuated (b): Mercury is added while the system remains evacuated (c): Atmospheric Pressure is applied to the sytem allowing the mercury to enter the larger pores (Macropores) (d): pressure is raised to 60,000 psi allowing Hg to ener progressively smaller pores. The drop in the Hg level gives the volume of the pores saturated with Hg at each pressure level Figure 2-58- Mercury Injection Method12
It is possible from the capillary pressure curve to calculate the average size of the pores making up a stated fraction of the total pore space. The volume of mercury injected into the pores at a given pressure is usually expressed as a proportion of the total pore space, and is presented as a pore size distribution, (Figure 2-59) or converted to oil or gas-brine data using appropriate contact angles and interfacial tensions. Typical conversions are given below:
(2-94) (2-95)
(2-96) (2-97)
12
Paul Glover
85
Cumulative Volume os Mercury Injected (Pore Volume Fraction)
1
0.8 0.6 0.4 0.2 0 0.01
0.1
1
Pore throat Radius (microns)
10
100
Fraction of Pore Volume
0.1 0.08 0.06 0.04 0.02 0 0.01
0.1
1
10
100
Pore throat Radius (microns) Figure 2-59- Pore size distribution from mercury injection test.
Two important advantage of this method are
The time for determination is reduced to a few hours The range of pressure investigation increased (between 0 to 60,000 psi), At 60,000 psia mercury is forced through pore throat diameters as small as 36 Å.
Two most important disadvantages of this method are
The data obtained apply to a system containing a fully wetting phase and a fully non-wetting phase. The capillary pressure data obtained will not necessarily apply to pores containing fluids showing partial wetting preferences. Permanent loss of the core sample.
86
Dynamic Capillary Pressure Method Figure 2-60 shows schematically the apparatus used in this method. This involves injecting the two fluids into a rock core simultaneously, and producing one behind a semi-permeable membrane. Simultaneous steady state flow of two fluids is established in the core. By the use of special wetted disks (semipermeable membrane), the pressure of the two fluids in the core is measured and the difference is capillary pressure. The saturation is varied by changing the rate of one fluid.
Figure 2-60- Dynamic Measurement of Capillary pressure13.
CAPILLARY PRESSURE HYSTERESIS It is generally agreed that the pore spaces of reservoir rocks were originally filled with water, after which oil moved into the reservoir, displacing some of the water and reducing the water to some residual saturation. When discovered, the reservoir pore spaces are filled with a connate water saturation and an oil saturation. All laboratory experiments are designed to duplicate the saturation history of the reservoir. The process of generating the capillary pressure curve by displacing the wetting phase, i.e., water, with the non-wetting phase (such as with gas or oil), is called the drainage process. This drainage process establishes the fluid saturations which are found when the reservoir is discovered. The other principal flow process of interest involves reversing the drainage process by displacing the non-wetting phase (such as with oil) with the wetting phase, (e.g., water). This displacing process is termed the imbibition process and the resulting curve is termed the capillary pressure imbibition curve. Figure 2-61 shows typical drainage and imbibition capillary pressure curves. The two capillary pressure-saturation curves are not the same.
13
Brown(1951)
87
Figure 2-61- Capillary Pressure Curve.
As can be seen in Figure 2-61, the reduction of non-wetting phase saturation, Snw, or alternatively an increase in the wetting phase saturation, Sw, is much less over the same incremental change in pressure during drainage, i.e., there is capillary pressure hysteresis: [
]
[
]
(2-98)
On the other hand at the same saturation capillary pressure in imbibition and drainage process are different. This difference in the saturating and de-saturating of the capillary-pressure curves is closely related to:
Contact angle Hysteresis during the Displacement According to the definition ‘hysteresis’ occurs when a measured variable depends on the direction of change of an independent variable. According to eq.(2-78) contact angle hysteresis will cause hysteresis in the capillary pressure. Consider a capillary tube containing two immiscible fluids (Figure 2-62): Case 1: Displacement by drainage R = receding contact angle (i.e. wetting phase receding) Occurs when the non-wetting phase displaces the wetting phase by increasing Pnw.
Case 2: Equilibrium E = equilibrium or static contact angle 88
Case 3: Displacement by imbibition A = advancing contact angle (i.e. Wetting phase is advancing) Occurs when the non-wetting phase is displaced by the wetting phase by a decrease Pnw
Figure 2-62- Contact Angle Hysteresis during the Displacement
In general it can be stated that: (2-99) Therefore at the same saturation: (2-100) Figure 2-63 shows dynamic contact angle behavior
Figure 2-63- Dynamic Contact Angle Behavior.
89
Figure 2.64: Static values of advancing and receding contact angles at rough surfaces versus values at smooth surfaces (where E refers to smooth surface measurements
PHASE TRAPPING As the capillary pressure is reduced to zero, the value of Snw does not decrease to zero, but reach to the non-wetting residual phase saturation, Snwr. As a result, the complete recovery of oil from water-wet oil bearing reservoirs by imbibition capillary forces is impossible. The same can be said with regard to the wetting phase, i.e., complete removal of the wetting phase from the porous medium by drainage displacement mechanism is impossible. Capillarity plays a very important role in the displacement of one immiscible fluid by another immiscible fluid. The detailed pore structure of the porous medium (pore geometry and pore topology) governs the fluid-fluid interfaces established within the porous medium, i.e. the way they interact with one another and the manner of hierarchy of advancement. Entrapment of one fluid by another immiscible fluid during the course of an immiscible displacement is the net result of the interaction of capillary forces at the microscopic pore level. Because pores differ in size and geometry, the presence of a given number of fluid-fluid interfaces does not give rise to the same capillary pressure value. There are different trapping mechanisms:
Snap-off Consider a square capillary into which a non-wetting phase has entered (Figure 2-65). The capillary pressure at which the non-wetting fluid first enters the square capillary is approximately 90
(2-101)
⁄ As the capillary pressure increases above this value, the wetting phase (shaded) will be pushed further into the corners as the radius of curvature of the interface decreases. (When the non-wetting phase enters at the above capillary pressure, the wetting phase will already be pushed into the corners compared to Figure 2-65). Suppose now the wetting phase is allowed to flow back along the corners without the end of the drop exiting the capillary. If the capillary pressure is now then decreased below Pcso, the capillary pressure of a cylindrical filament that just touches the capillary walls,
Wetting Phase Non-Wetting Phase Ph
Figure 2-65- Non-Wetting fluid Enter to a capillary tube with square cross section.
(2-102) The interface will pull away from the walls of the capillary. The non-wetting phase is now a thin filament that is unsupported by the walls. The non-wetting phase will neck down in some places and will swell in other places until it is supported by the walls of the capillary. The necking down is unstable and the non-wetting phase will ‘snap-off’ into droplets or bubbles of disconnected non-wetting phase.
Figure 2-66- Side view after snap-off (Cross section from middle of square)
Now consider a porous media instead of the capillary tube. Suppose the porous medium is made up of pore bodies and pore necks of many different sizes as shown in Figure 2-67. The medium is initially saturated with the wetting phase. The non-wetting phase is first allowed to enter only the largest pores as in Figure 2-67.a and then the capillary pressure reduced to zero. Non-wetting phase will be trapped as in Figure 2-67.b. With additional cycles with increasing initial non-wetting saturation additional trapping will occur as in Figure 2-67.d and Figure 2-67.f. according to the previous definition the process in Figure 2-67.a,c,e are drainage and Figure 2-67.b,d,f are imbibition.
91
Figure 2-67- Trapping in a porous media14.
Such an experiment with increasing initial saturation of the non-wetting phase saturation measuring the residual saturation at each initial saturation generates an initial - residual saturation curve as in Figure 2-68. This curve probes the volume of non-wetting phase trapping sites as a function of entering increasingly finer pores. This curve can be used to determine the saturation of non-wetting phase that is trapped at a given saturation if information is available on the maximum non-wetting saturation attained, i.e., memory of its history is available15. Residual Non Wet Phase Saturation
1
Uniform unconsolidated sand stones Berea Sand Stone
0.8
Vuggy Carbonates Cemented Sand Stone
0.6
100% Trapping 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
Initial N.W.P. Saturation Figure 2-68- Typical non-wetting phase trapping characteristics of some reservoir rocks16.
14 15 16
Stegemeier 1977 L. W. Lake Stegemeier 1977
92
Bypassing According to the Pc equation, at the same condition non-wetting phase tends to invade the larger pores because of the lower capillary pressure. This tendency leads to special trapping mechanism named bypassing. The main trapping mechanism in drainage is bypassing although during the imbibition process non-wetting phase could be trapped by bypassing mechanism but it is not the major trapping mechanism. A reservoir rock consists of a complex network of branching and reuniting pore elements of differing sizes and geometry. Any successful analysis of the displacement process should eventually recognize this complex network. However one can begin by understanding the processes which take place in the simpler elements of the network. The mechanisms of drainage displacement and imbibition displacement are best illustrated by visualization of flow experiments in transparent capillary networks etched on glass plates. The simplest type of pore system used in such displacement studies is the pore doublet (Figure 2-69). Throat Pore
Pore Throat
Figure 2-69- Pore Doublet Model.
93
Smaller throat is invaded first during the imbibition because of higher capillary pressure and larger throat is swept first
during the drainage process.
Figure 2-70 illustrates the drainage and imbibition mechanisms for the pore doublet model.
(a) Drainage Process, No Trapping (b) Drainage Process, Bypassing in smaller throat (c) Imbibition Process, Bypassing in larger throat (d) Imbibition Process, Snap off (e) Imbibition Process, Snap off in smaller throat and Bypassing in larger throat
94
Figure 2-70- Imbibition and Drainage mechanisms in a pore doublet model
The condition that bypassing happen in the larger pore during the imbibition could be found by using the capillary and viscous forces formula. Assume a pore doublet model as in Figure 2-71. For pore #1:
(2-103) (2-104)
Figure 2-71- Pore doublet model for illustration for displacement and trapping of oil17.
(2-105) Assume
and defining
, we get (2-106)
Similarly for second pore:
17
Willhite (1986)
95
(2-107) The overall pressure change is same for both pores, therefore: (2-108) This equation gives a relationship between v1 and v2: (
)
(2-109)
For v2 to be positive (no trapping by bypassing), we must satisfy (
)
(2-110)
The values of v1 needed to make v2 positive are listed in Table 5 for a typical case. At reservoir type velocities, v2 will not be positive and oil will be displaced only from the smaller pore. Once the water breaks through at the other end of the pore doublet, the oil in the larger pore is trapped and very difficult to mobilize.
96
Table 5- Pore doublet model, required velocity in small pore to maintain zero velocity in large pore (r1=2.5
⁄
v1, (ft/Day)
2
5315
4
7940
10
9580
20
10090
40
10600
)
Other pore space models used to understand displacement mechanisms are:
bundle of tubes network contact angle
Jamin Effect18 The Jamin effect is defined as that resistance to liquid flow through capillaries which is due to the presence of bubbles. Presence of bubbles can retard the flow of a liquid as it progresses through a
capillary tube of small diameter. The Jamin effect may be defined as that resistance to flow under pressure through a capillary tube which is encountered by liquid globules interspaced with large bubbles. This effect or action is a phenomenon quite apart from frictional resistance and is due to difference of capillary pressure between two sides of the trapped globule. This effect can be described more easily by analyzing a trapped oil droplet or gas bubble in a preferentially water wet capillary tube. In Figure 2-72 assume the system is static with different pressure existence between point A and B because of capillary forces. The static pressure difference must be exceeded for flow to occur, in other words the static pressure difference, PA-PB, must be overcome to initiate flow. In Figure 2-72.a the capillary tube size varies and therefore the radius is smaller on one side of the drop than on the other. Figure 2-72.b shows a situation that the contact angle is different on the two sides of the drop, which could result, for example, if the drop displaced in one direction that there will have one Advancing contact angle and one Receding. In Figure 2-72.c a gas drop is trapped between water and oil. An assumption is made that the pressure within the oil or gas drop is constant from one end to the other end of drop. With this assumption we have (
) (
( )
) (
(2-111) )
(2-112)
Subscripts A and B are the values are determined for the interfaces at point A and B.
18
Green and Willhite (1998)
97
(a)
(b)
(c)
Figure 2-72- Trapping of a droplet in a capillary tube.
Application of this equation in Figure 2-72.a, b, c yields the following forms. Figure 2-72.a: Assume
the pressure difference at static condition is (
)
(2-113)
If rA>rB then PA>PB and a pressure drop exists in the direction from point A to point B. this gradient would have to be exceeded to induce flow into the narrower part of the capillary constriction. Figure 2-72.b: (2-114) For an advancing contact angle at point B and a receding at point A, A > B and . Again PA>PB and a pressure gradient exist in the potential direction of flow at static, or trapped, condition. Figure 2-72.c: (
)
(2-115)
In this case IFT and contact angles are different at the two interfaces because of different fluid systems. If a pressure drop exists from point A to point B when this system is static.
Averaging Capillary Pressure Data (Leveret J-Function) Capillary pressure data are obtained on small core samples that represent an extremely small part of the reservoir and, therefore, it is necessary to combine all capillary data to classify a particular reservoir. The fact that the capillary pressure-saturation curves of nearly all naturally porous materials have many features in common has led to attempts to devise some general equation describing all such curves. Leverett (1941) approached the problem from the standpoint of dimensional analysis. At first, Leverett made an attempt to convert all capillary pressure data to a universal curve. But, a universal capillary pressure curve does not exist because the rock properties affecting capillary pressures in reservoir have extreme variation with lithology (rock type). Realizing that capillary pressure should depend on the porosity, interfacial tension, and mean pore radius, Leverett defined the dimensionless function of saturation, which he called the J-function, as 98
√
(2-116)
Where, c is constant. In fact, The Leverett J-function is an attempt at extrapolating capillary pressure data for a given rock to rocks that are similar but with differing permeability, porosity and wetting properties. It assumes that the porous rock can be modelled as a bundle of non-connecting capillary tubes, where the factor √ characteristic length of the capillaries' radii and Leverett interpreted √
is a
as being proportional to the
square of a mean pore radius. For the same formation, however, this dimensionless capillary-pressure function serves quite well in many cases to remove discrepancies in the pc versus Sw curves and reduce them to a common curve. Some authors altered eq. (2-116) by including the
:
√
(2-117)
Field Scale example: Browns (1951) considered the j-function (eq. (2-116)) as a correlating device for capillary pressure data of samples from Edwards formation in the Jourdanton field. () shows the correlation obtained for all samples available from the field. There is considerable dispersion of data points, although the trend of correlation is good. If core samples are divided to two different groups (limestone and dolomites) according to their texture, the correlation will be improved. () shows the correlations for the limestone and dolomite samples respectively. The dolomite samples show a good correlation while the limestone sample data shows a scattering of data in the range of lower water saturation. If the limestone samples are subdivided to micro-granular and coarse-grained samples, the correlation for each of them will be improved (). Scattering of data for the coarse-grained limestone samples are greater than other groups of samples. This is to be expected , as the coarse-grained limestone contains solution cavities, vugs and channels which are not capillary in size, hence deviation from trends established in capillary-pressure data.
(a)
(b)
99
(c)
(d)
(a). J curve for all cores (b). J curve for Limestone cores (c). J curve for dolomite cores (d). J curve for micro-granular limestone cores (e). J curve for coarse-grained limestone cores
(e) Figure 2-73- J-function correlation of capillary pressure data in Edwards Jourdanton field.
Example 2-7 A laboratory capillary pressure test was conducted on a core sample taken from the Nameless Field. The core has a porosity and permeability of 16% and 80 md, respectively. The capillary pressure-saturation data are given as follows: Sw
Pc (psi)
1.0
0.50
0.8
0.60
0.6
0.75
0.4
1.05
0.2
1.75
The interfacial tension is measured at 50 dynes/cm. Further reservoir engineering analysis indicated that the reservoir is better described at a porosity value of 19% and an absolute permeability of 120 md. Generate the capillary pressure data for the reservoir.
100
Solution Calculate the J-function using the measured capillary pressure data and eq. (2-116). √
√
Sw
Pc (psi)
J(Sw)
1.0
0.50
0.048
0.8
0.60
0.058
0.6
0.75
0.073
0.4
1.05
0.102
0.2
1.75
0.169
At the next step, Using the new porosity and permeability values, solve eq. (2-116) for the capillary pressure Pc. ⁄ √ √ Sw
J(Sw)
Pc (psi)
1.0
0.048
0.441
0.8
0.058
0.533
0.6
0.073
0.671
0.4
0.102
0.938
0.2
0.169
1.553
RELATIVE PERMEABILITY The absolute permeability is a property of the porous medium and is a measure of the capacity of the medium to transmit fluids. The absolute permeability is referred to the rock condition where single phase saturation is considered in other words the rock is fully saturated with one homogeneous fluid. Fluid flow in reservoirs typically involves more than one fluid, which means that the general ability of one fluid to flow is affected by the presence of other fluids in the reservoir. In order to develop estimates of fluid behavior in reservoirs, this phenomenon needs to be quantified in some way. The concept of describing this multiphase flow in reservoirs is known as relative permeability, which is defined as the ratio of the effective permeability of a fluid to the absolute permeability of the rock. The effective permeability is a relative measure of conductance of the porous medium for one fluid phase in the presence of other fluid phases.
101
Where,
kro = relative permeability to oil krg = relative permeability to gas krw = relative permeability to water k = absolute permeability ko = effective permeability to oil for a given oil saturation kg = effective permeability to gas for a given gas saturation kw = effective permeability to water for a given water saturation
In the definition of effective permeability each fluid phase is considered to be completely independent of the other fluids in the flow network. The fluids are considered immiscible, so that Darcy’s law can be applied to each individually: ( ( (
)
(2-118) )
)
(2-119) (2-120)
Where, s is direction. Since the fluid is not completely saturating the rock, other factors come into play other than the pore size distribution. According to experimental evidence, effective permeability is a function of dominant fluid saturation, saturation history, wettability characteristic and pore geometry. So to determine the relative permeability for one phase it is necessary to specify the fluid saturation. In contrast to absolute permeability, relative permeability for each phase is not a unique value and there is a special relative permeability value for each saturation amount. Since the effective permeabilities may range from zero to k, the relative permeabilities may have any value between zero and one. It should be pointed out that when three phases are present the sum of the relative permeabilities (kro + krg + krw) is both variable and always less than or equal to unity.
Relative Permeability Curves When a wetting and a non-wetting phase flow together in a reservoir rock, each phase follows separate and distinct paths. The distribution of the two phases according to their wetting characteristics results in characteristic wetting and non-wetting phase relative permeabilities. 102
So for a water-oil system different relative permeability curves exist for:
Water-wet systems Oil-wet systems, and Intermediate wettability.
Many reservoir systems fall between the two extremes, which does nothing to make laboratory waterflood data easier to interpret. However, knowledge of the two extreme cases allows misinterpretation of intermediate data to be minimized. It should be remembered that in water-wet systems capillary forces assist water to enter pores, whereas in the oil wet case they tend to prevent water entering pores. Because of the capillary forces, the wetting phase occupies the smaller pore openings at small saturations, and these pore openings do not contribute materially to flow, it follows that the presence of small wetting phase saturation will affect the permeability of the non-wetting phase only to a limited extent. On the other hand, since the nonwetting phase occupies the central or larger pore openings which contribute materially to fluid flow through the reservoir, the relative permeability to the wetting phase is characterized by a rapid decline in value for small decreases in the wetting phase saturation from original saturation. The relative permeability to wetting phase normally approaches zero or vanishes at relatively large wetting phase saturation. This is because the wetting phase preferentially occupies the smaller pore spaces, where capillary forces are the greatest. The saturation of the water at this point is referred to as the irreducible water saturation Swir or connate water saturation Swi (Figure 2-74). Irreducible water saturation originally is higher in consolidated rocks than unconsolidated rocks. Another important phenomenon associated with fluid flow through porous media is the concept of residual saturations. As when one immiscible fluid is displacing another, it is impossible to reduce the saturation of the displaced fluid to zero. At some small saturation, which is presumed to be the saturation at which the displaced phase ceases to be continuous, flow of the displaced phase will cease. This saturation is often referred to as the residual saturation. This is an important concept as it determines the maximum recovery from the reservoir. Conversely, a fluid must develop a certain minimum saturation before the phase will begin to flow. The saturation at which a fluid will just begin to flow is called the critical saturation. Theoretically, the critical saturation and the residual saturation should be exactly equal for any fluid; however, often they are not identical. Critical saturation is measured in the direction of increasing saturation, while irreducible saturation is measured in the direction of reducing saturation. Thus, the saturation histories of the two measurements are different. Figure 2-74 is a typical relative permeability curve for a water-oil system in a water-wet medium. According to the figure, the non-wetting phase begins to flow at the relatively low saturation of the nonwetting phase. The saturation of the oil at this point (point A) is called critical oil saturation Soc. Some references called this point as the “equilibrium saturation”, which the non-wetting phase becomes mobile. This saturation may vary between zero and 15% non-wetting phase saturation. As Figure 2-74 shows attribution of oil phase to flow reach to 100% in saturation less than 100%. This is the result of capillary pressure. The capillary pressure force the wetting phase to occupy the smallest pores at low saturation that have an ignorable contribution to the flow.
103
Figure 2-74- Typical relative permeability curve.
The curve shapes shown in Figure 2-74 are typical for wetting and non-wetting phases and may be mentally reversed to visualize the behavior of an oil-wet system. Note also that the total permeability to both phases, krw + kro, is less than 1, in the region of two phase flow. The following example uses capillary tubes and the HP equation to illustrate the effective reduction in permeability caused by introduction of second phase.
Example 2-8 Consider five capillary tubes of length L and diameters of 0.0002, 0.001, 0.005, 0.01, and 0.05 cm. the total pore volume of the four capillary tubes are:
Total flow rate through these capillary tubes because of applying a pressure difference could be calculated, using the HP equation:
Now assume fluid number 2 with the same viscosity occupied the largest capillary tube, the same as the non-wetting phase occupied the largest pores at first. It is possible to express the conductive capacity 104
when the two fluids are saturating the system to the conductive capacity when only one fluid saturates the system. According to eq.() the conductance capacity for fluid number 2 is:
And the ratio of conductive capacities is:
And from Darcy’s law:
In this example the relative permeability values for the two fluids sum up to one. This behavior is not true in actual porous media. In real porous medium, there is a minute film of wetting phase that wet the surface. If consider this film in this example, it would decrease the diameter of the larger tube available for fluid number 2 to flow, thus reduces the flow capacity for the second fluid, and yet the film itself would contribute no flow capacity to the wetting phase. Thus the total fluid capacity of the tubes would be decreased. This is a rather normal feature of most relative permeability curves where sum of relative permeability values is less than one. The above discussion may be also applied to gas-oil relative permeability data, (Figure 2-75). Note that this might be termed gas-liquid relative permeability since it is plotted versus the liquid (water + oil) saturation. This is typical of gas-oil relative permeability data in the presence of connate water. Since the connate (irreducible) water normally occupies the smallest pores in the presence of oil and gas, it appears to make little difference whether water or oil that would also be immobile in these small pores occupies these pores. Consequently, in applying the gas-oil relative permeability data to a reservoir, the total liquid saturation is normally used as a basis for evaluating the relative permeability to the gas and oil. Note that the relative permeability curve representing oil changes completely from the shape of the relative permeability curve for oil in the water-oil system. In the water-oil system, as noted previously, oil is normally the non-wetting phase, whereas in the presence of gas the oil is the wetting phase. Consequently, in the presence of water only, the oil relative permeability curve takes on an S shape whereas in the presence of gas the oil relative-permeability curve takes on the shape of the wetting phase, or is concave upward (Figure 2-75). Note further that the critical gas saturation Sgc is generally very small.
105
Figure 2-75- Typical Gas-Oil Relative Permeability Curve
It is important to note that relative permeability experiences hysteresis effects, so it is critical that the curves be specified as either imbibition or drainage. Hysteresis effect means that the value of relative permeability at a given saturation will be different depending on which direction the value is approached from (increasing or decreasing saturation). Drainage relative permeability refers to a saturation change where the wetting phase is decreasing from the reservoir rock, while imbibition refers to a saturation change where the wetting phase is increasing. An example of drainage would be the displacement of oil by gas during primary depletion. An example of imbibition would be water flooding a reservoir that is water wet.
(a)
(b)
Figure 2-76- Relative Permeability Curve, (a) Drainage, (b) Imbibition
106
Figure 2-76 show typical drainage and imbibition relative permeability curves for a water/oil system. The arrows indicate the direction of saturation change for each phase.
Laboratory Measurements of Relative Permeability There are essentially five means by which relative permeability data can be obtained:
Direct measurement in the laboratory by a steady state fluid flow process Direct measurement in the laboratory by an unsteady state fluid flow process Calculation of relative permeability data from capillary pressure data Calculation from field performance data Theoretical/empirical correlations
Values obtained through laboratory measurements are usually preferred for engineering calculations, since they are directly measured rather than estimated. Steady state implies just that, values are not measured until the tested sample has reached an agreed upon level of steady-state behavior. Subsequently, unsteady-state measurements are taken while the system is still changing over time.
Steady-State Techniques Steady-State techniques of estimating relative permeability are often considered the most reliable source of relative permeability data. Since steady state is achieved in these tests, it is possible to use Darcy’s law to determine the effective permeability for each phase at a given saturation. In the experimental procedure, two phases are injected simultaneously into the test core at constant rates and pressures. Once the measured pressure drop across the core remains relatively constant, the system is assumed to be at steady state. At this point the outlet flow rate of each phase and the pressure drop is measured, and the subsequent values are used in Darcy’s Law (eq. (2-118, 2-119, 2-120)) to calculate the effective permeability of the fluid at that saturation. The inlet flow rate ratio is then changed, and the process repeats itself. In this way, relative permeability curves for each phase can be obtained. The number of data points can be controlled simply through the number of steady-state levels reached. Once again, it must be noted that the saturation changes must be all in one direction to avoid hysteresis effects. The following is a simple schematic diagram of a typical steady-state test apparatus:
107
Figure 2-77- Hafford Relative Permeability Apparatus19
The situation in Figure 2-76 is a 2-phase relative permeability test with gas and oil being the fluids of interest. Both the gas and oil enters the core simultaneously. A gas meter and oil burette on the other end record the outlet flow rates. The gas pressure gauge is used to measure the pressure drop across the core. Following is a simplified diagram indicating the fluid saturation during a typical steady-state test.
19
Richardson (1952)
108
Figure 2-78- Fluid Saturation during Steady-State Test
In this situation, the test is imbibition displacement for a water-oil case, with water being the wetting phase. Initially, the core is completely saturated with water. The test begins by injecting just oil until steady state is reached. At this point the water is at irreducible saturation and theoretically only oil should be flowing through the core. However, there is evidence that an extremely small amount of water is still mobile in the system. At this point the flow rate of oil and the pressure drop is measured and these values are used in Darcy’s law to calculate the effective permeability to oil at the irreducible water saturation. Next, water is injected into the core simultaneously with oil at point A. A new steady state level is reached and the outlet flow rates for each phase are measured along with the pressure drop, and effective permeabilities of each fluid at the given saturation are calculated. The inlet flow rate ratio is changed (increasing the amount of water) and the process is again repeated. In this stepwise manner relative permeability curves can be developed for as many points as desired. To obtain the last end point value for effective water saturation, only water is flowed through the core until irreducible oil saturation is reached at point D. The main problem associated with steady state tests, is that it can take an extremely long time for steady state to be reached at a given saturation level. It may take hours or even days for a single saturation level to be tested. As such, these tests can be uneconomical to run. However, they are definitely the most accurate and reliable technique for estimating core permeability. There are numerous steady-state methods available. Examples of these include the Penn State, Modified PennState and Hassler methods. The only significant difference between them is in how they account for negating capillary end effects. Capillary end effects are discussed later.
109
Unsteady State Techniques Experimentally and computationally the unsteady state technique is far more complicated. Following is a schematic diagram of an unsteady-state test apparatus.
Figure 2-79- Unsteady state apparatus.
As can be seen, there is only one inlet since only one phase is injected into the core. The major difference in unsteady state techniques is that saturation equilibrium is not achieved during the test. As well, fluids are not injected simultaneously into the core. Instead, the test involves displacing in-situ fluids with a constant rate/pressure driving fluid. The outlet fluid composition and flow rate is measured and used in determining the relative permeability. Figure 2-80-b shows an example of such a curve from an unsteady state waterflood experiment. At the beginning of the experiment, the core is saturated with 80% oil, and there is an irreducible water saturation of 20% due to the water wet nature of this particular example. Point A (Figure 2-80-a) represents the permeability of oil under these conditions. Note that it is equal to unity, because this measurement has been taken as the base permeability. Point B represents the beginning water permeability. Note that it is equal to zero because irreducible water is, by definition, immobile. Water is then injected into the core at one end at a constant rate. The volumes of the emerging fluids (oil and water) are measured at the other end of the core, and the differential pressure across the core is also measured. During this process the permeability to oil reduces to zero along the curve ACD, and the permeability to water increases along the curve BCE. Note that there is no further production of oil from the sample after kro=0 at point D, and so point D occurs at the irreducible oil saturation, Sor. Figure 2-80-a is a simplified schematic diagram detailing the fluid saturation in the core during a typical unsteady-state test.
110
(a)
(b)
Figure 2-80- (a) Unsteady State Water Flood Procedure, (b) Typical Relative Permeability Curve
Since steady state is not reached, Darcy’s Law is not applicable. The Buckley-Leverett equation for linear fluid displacement is the basis for all calculations. The Johnson-Bossler-Naumann (JBN) solution is used most often for calculating relative permeabilites from unsteady-state displacement tests. It must be stressed, however, that these curves are not a unique function of saturation, but are also dependent upon fluid distribution. Thus the data obtained can be influenced by saturation history and flow rate. The choice of test method should be made with due regard for reservoir saturation history, rock and fluid properties. The wetting characteristics are particularly important. Test plugs should either, be of similar wetting characteristics to the reservoir state, or their wetting characteristics be known so that data can be assessed properly. JBN Analysis: The experimental data generally recorded includes: Qi
: Quantity of displacing phase injected
P
: Pressure differential
Pi : Pressure differential at initial conditions Qo
: Volume of oil produced
Qw
: Volume of water produced
111
These data are analysed by the technique described by Johnson, Bossler and Nauman, which is summarised below20. Three calculation stages are involved:
The ratio kro/krw. The values of kro and hence krw. The value of Sw.
The method is aimed at giving the required values at the outlet face of the core which is essentially where volumetric flow observations are made. (a).
:
The average water saturation (Swav) is plotted against Qi (Figure 2-81-a). Fractional flow of oil, at the outlet face of the core sample is: (2-121) Also we have:
(2-122)
(a)
(b)
Figure 2-81- (a) Average Water saturation vs. Water Injection, (b) Injectivity Ratio
(b). kro: A plot of
against Qi is used to obtain injectivity ratio (Figure 2-81-b). (2-123)
20
Honarpour, M., Mahmood, S.M (1988) 112
kro is obtained by plotting
versus
:
and using the relationship: ⁄ ⁄
(2-124)
krw can be calculated from eq. (2-121), or: (2-125)
Unsteady state tests are popular because they require much less time and money than steady state tests to operate. However, it must be noted that the values obtained from these tests are generally less reliable. This is for a number of reasons. For one, the system is not at steady state when measurements are taken. Therefore, properties are still changing at that estimated saturation level. This leads to repeatability concerns. The relative permeability calculated during one un-steady state run at a certain saturation level could be quite different from a run performed earlier. For this reason lower confidence is placed on the actual value determined from displacement tests. Another concern is that the interpretation techniques introduce many simplifying assumptions. For example, the Welge method (utilized in the JBN solution) was developed with respect to a homogeneous reservoir, so it might not be entirely accurate if applied to a heterogeneous one. Also, the Buckley-Leverett equation was developed for incompressible/immiscible fluids and assumes completely linear displacement. For all these reasons, values obtained from unsteady state tests should be considered as qualitative, and not representative of the reservoir. As a note, viscous fingering is a definite concern with the majority of unsteady-state tests, but it has been found that the centrifuge method can be used to eliminate these problems.
113
Determining Fluid Saturations The accuracy of relative permeability measurements (regardless of measurement technique) depends largely on accurate determination of fluid saturation. There are generally two methods to determine the fluid saturation in the core: external and internal techniques. External techniques involve inferring saturation from fluid production. A material balance calculation is performed (fluid in – fluid out = fluid retained in core). The saturation obtained from material balance is an average value for the core. Another external technique is the gravimetric method, in which the weight of a saturated core is compared with the weight of a dry core, and the saturation is estimated from the difference in the weights. Problems associated with external techniques that can affect the material balance are the possibility of dead volume, evaporation in the core, or difficulties separating the fluid at the outlet. Internal techniques involve measuring the quantity of fluids in the core directly. This method offers greater reliability than external techniques. A primary advantage is that it allows for different saturation levels to be measured along the length of the core, rather than just returning an average value. The basic premise of internal techniques is that a stimulus is applied to the core, and is then compared against the response from a completely dry and a completely saturated core. Saturation is then inferred from this measurement. The most common forms of in-situ fluid saturation determination include x-ray absorption and electrical resistivity.
Experimental Considerations Aside from the actual testing of the cores, there are numerous other experimental considerations that must be accounted for in order to minimize the error associated with measuring core data.
Capillary End Effects The most common source of error associated with laboratory measurements of relative permeability is capillary end effects. This problem comes as a result of the saturation of the wetting phase being much higher at the inlet and outlet of the core. This is caused by the tendency of the wetting phase to remain in the capillaries, rather than enter a non-capillary space. There have been numerous methods developed to negate this problem. One of the most common is the Hassler technique21, in which porous plates of similar wettability to the core sample are placed at the inlet and outlet of the core. The wetting phase is passed through these plates, while the non-wetting phase is introduced just to the core. The Penn-State method is quite similar to the Hassler method, the major difference being that both the wetting and non-wetting fluid are passed through the porous plates.
Sampling Considerations The accuracy of the laboratory measurements depends greatly on whether or not the core sample has been damaged in any way. In order to get a representative sample of the reservoir, it is important to keep the core as close to reservoir conditions as possible. An example of this would be to use a drilling fluid that does not adversely affect the core sample. As well, taking larger diameter cores (although 21
Honarpour, M., Mahmood, S.M (1988)
114
more expensive) can potentially minimize flushing concerns. Weathering is a definite problem, so once the cores have been brought to surface they should be sealed and covered as quickly as possible. Once the core gets to the lab, all handling and cleaning should also be kept to a minimum. It is recognized that it is completely unrealistic to expect that a core will not be introduced to non-reservoir conditions from the ground to the lab. However, these damaging effects should be kept to a minimum if accurate results are desired.
Steady or Unsteady State? The majority of literature acknowledges that steady state techniques provide more accurate values than unsteady state techniques. However, the main advantage to using unsteady state techniques is that they are much faster and more economical to run. Therefore, if time and budget constraints are not a deciding factor in the test, it is recommended that the steady state technique be performed in order to obtain the most reliable relative permeability data.
Empirical Correlations of Relative Permeability As a result of the difficulties and cost involved in measuring relative permeability values, empirical correlations and calculations are often employed in order to estimate the values. This is typically done in areas where no core data is available, or if economics dictate that running laboratory permeability tests is not feasible. Estimating relative permeability values through calculations is extremely fast, however the accuracy of the results is debatable. There are numerous methods that are available to estimate 2phase relative permeability curves. Since it is such an important topic, numerous individuals have devoted their lives to developing reasonable methods to estimate relative permeability values. Two of the more common methods will be discussed here. These are the well-known Corey relations (an entirely theoretical approach to the problem) and the empirical Hornarpour correlations.
Corey Relations The often-used Corey relations are actually an extension of equations developed by Burdine et al. (1953), for normalized drainage effective permeability. The equations shown here are the original Burdine equations modified for relative permeability calculations: (2-126) ( (
) (
)
)
(2-127) (2-128) (2-129)
Where,
krw = krn =
wetting phase relative permeability non-wetting phase relative permeability 115
kro = Sw* = 𝛌 = Sm = Sw = Siw =
non-wetting phase rel. perm. at irreducible wetting phase saturation normalized wetting phase saturation pore size distribution index 1 - Sor (1 - residual non-wetting phase saturation) water saturation initial water saturation
The major difference between the Burdine solutions and the equations shown here is in the non-wetting phase equation, eq. (2-127). The kro term is added to account for the fact that the non-wetting phase solution must be at irreducible wetting phase saturation. The other modification is the Sm term proposed by Corey in order to represent the point where the non-wetting phase first begins to flow. This is known as the critical saturation point. What this means, is that for a period at the beginning of the non-wetting phase curve, there exists a period where there is no connectivity. At the critical saturation, a minimum number of pores are connected, at which point flow is possible and the first relative permeability value can be determined. The Sm term describes the saturation at which flow is first possible and is necessary in order to calculate realistic relative permeability values.
Determining Pore Size Distribution Index The λ value (pore size distribution index) seen in equations (2-126) and (2-127) is critical in calculating relative permeability. The actual number represents how uniform the pore size is in the sample/reservoir. A low value of λ (i.e. 2) indicates a wide range of pore sizes, while a high value represents a rock with a more uniform pore size distribution. Using a λ value of 2 in equations (2-126) and (2-127) results in the well-known Corey equations. This value is considered a general value, and is thought to represent a wide range of pore sizes. Since a λ value of 2 is so general, it is often used when nothing else is known about the reservoir. Using a λ value of 2.4 or infinity results in Wyllie’s equation for 3 rock categories22:
λ = 2 (cemented sandstones, oolotic and small-vug limestones) λ = 4 (poorly sorted unconsolidated sandstones) λ = infinity (well sorted unconsolidated sandstones)
Wyllie’s equations are used when there is some general knowledge of the geology of the reservoir. The Corey and Wyllie equations are sufficient for approximation purposes, but in order to obtain a more accurate value of the pore size distribution index, λ can be determined empirically from capillary pressure data. The equation shown here was developed by Brooks and Corey (1964 & 1966), and relates capillary pressure to normalized wetting phase saturation: (2-130) Where,
22
Pc = capillary pressure (2)
M.B. Standing (1975)
116
Pe = minimum threshold pressure Sw* = normalized water saturation, eq. (2-128)
If capillary pressure data is available, eq. (2-130) can be used to determine the pore size distribution index. A log-log plot of the capillary pressure vs the normalized water saturation should result in a straight line with a slope of –1/ λ and an intercept of Pe. This method of determining λ from experimental data is preferable to using the Wyllie or Corey relations, since the value obtained with eq. (2-130) can actually be backed up with hard data.
Example 2-923 For a water wet reservoir with Swi=0.16 and residual gas saturation equal to 0.05, use the following capillary pressure data to find the relative permeability data. Pc (Sw)
Sw
Pc (Sw)
Sw
0.5
0.965
8
0.266
1
0.713
16
0.219
2
0.483
32
0.191
4
0.347
300
0.16
Solution Step 1: Calculate Normalized Water Saturation (Sw*), using eq. (2-130) Pc (Sw)
Sw
Sw*
Pc (Sw)
Sw
0.5
0.965
0.958
8
0.266
0.126
1
0.713
0.658
16
0.219
0.070
2
0.483
0.385
32
0.191
0.037
4
0.347
0.223
300
0.16
0.000
Sw*
Step 2: Deterimne λ by Plotting LogPc vs. LogSw*
log (Pc)
100
10
1 0.01
0.1 log(Sw*)
1
23
International Petroleum Consultants, “Fundamentals of Relative Permeability”, Mobil Oil Indonesia Course Notes, 1985.
117
Recall eq.(2-130), Slope of the graph is –1/ λ = -1.25, Therefore, λ=0.8 Step 4: Calculating Non-Wetting Phase Relative Permeability at Irreducible Wetting Phase Saturation (kro), o
Recall eq. (2-129), kr =0.919 and Sm=0.95=1-Srg. Step 5: Calculating Relative Perm Values, Recall equations (2-126) and (2-127) to find the relative permeability of the respective phases at various water saturations: Sg 0.05 0.08 0.11 0.14 0.17 0.23 0.26 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0.5 0.53 0.56 0.59 0.62 0.65 0.68 0.71 0.74 0.77 0.8 0.83
Sw 0.95 0.92 0.89 0.86 0.83 0.8 0.74 0.71 0.68 0.65 0.62 0.59 0.56 0.53 0.5 0.47 0.44 0.41 0.38 0.35 0.32 0.29 0.26 0.23 0.2 0.17
Sw* 0.940476 0.904762 0.869048 0.833333 0.797619 0.761905 0.690476 0.654762 0.619048 0.583333 0.547619 0.511905 0.47619 0.440476 0.404762 0.369048 0.333333 0.297619 0.261905 0.22619 0.190476 0.154762 0.119048 0.083333 0.047619 0.011905
krg 0 0.000389 0.002044 0.005592 0.011529 0.020236 0.046977 0.065306 0.08702 0.112108 0.140514 0.172151 0.206906 0.244653 0.285259 0.328595 0.374538 0.422983 0.473843 0.527057 0.582591 0.640441 0.700632 0.763217 0.828271 0.895881
krw 0.714846005 0.578419139 0.464053818 0.368874458 0.290282915 0.225941893 0.131869218 0.098622548 0.072565567 0.05242819 0.037108863 0.025660479 0.017276687 0.011278614 0.007102021 0.004284901 0.002455538 0.001321065 0.000656515 0.000294425 0.000115008 3.69371E-05 8.79412E-06 1.24991E-06 5.8541E-08 2.97984E-11
Then plot krg and krw vs. Sw in order to display the information in its most typical form: Relative Perm Curves
1
kr
0.8 0.6
krg krw
0.4 0.2 0 0
0.2
0.4
Sw
0.6
0.8
1
118
Hornarpour Correlations A different technique was approached by Hornarpour et al. in order to come up with correlations to estimate relative permeability. Instead of attempting to solve the problem theoretically, Hornarpour developed an entirely empirical solution to the problem. Data from numerous different fields around the world was gathered, and stepwise linear regression analysis was used to come up with mathematical descriptions to match the actual data. Data sets came from Canada, U.S., Alaska and the Middle East. The data sets were classified as either carbonate or non-carbonate and also broken up into wettability and property ranges. In this manner, equations were developed for numerous different reservoir conditions. In order to use the Hornarpour correlation, a general understanding of the reservoir is required. Namely the fluids in the system, a rough description of the geology, the wettability and the range of rock properties and fluid saturations in the reservoir. This information is then used to determine which set of equations are to be utilized in calculating the wetting and non-wetting phase relative permeability. Hornarpour’s paper24 provides all the necessary details (tables and equations) that are required for this method.
Comparison of Empirical Methods Out of the methods discussed here, the most accurate results are obtained from the method in which the pore size distribution index is determined experimentally, and is then utilized in equations (2-126) and (2-127) for drainage relative permeability. The other methods (Corey, Wylie and Hornarpor) rely a great deal on general estimates of the reservoir conditions. For example, in the Hornarpor correlations the only choices available for geology are either carbonate or non-carbonate. Since the description of the reservoir in these methods is so vague, it is uncertain how accurate the results obtained from them can be with respect to the actual reservoir conditions. Conversely, the pore size distribution index is determined experimentally and there is hard data to back up the results.
Other Methods Since relative permeability is such an important topic, numerous individuals have devoted their lives to determining methods to estimate relative permeability values. As such, there are numerous different techniques and methods that are used to calculate relative permeability. For example, relative permeability can be calculated through field data and analysis of pressure data. Some of the other methods not discussed in detail include theoretical models such as Naar-Wygal’s and Naar-Henderson (1961), which are for imbibition processes. The Burdine equations discussed here are for drainage calculations only.
Overall Comparison of Methods As discussed, there are many different ways, in which relative permeability values can be estimated, from rigorous laboratory measurements, to “quick and dirty” calculations. With all of these methods available, the question as to which method should be used becomes extremely important. Like most 24
Honarpour et al (1982)
119
things in the reservoir-engineering world, all of the options have their advantages and disadvantages. The only way to make a sound decision is to see what is best for the given situation, and weigh the options accordingly. For example, if relative permeability data is required for a major discovery in the North Sea, then chances are good that you would decide to spend the money to get an excellent core sample in order to run a rigorous steady-state test on it. Conversely, if data for a mature oil field is required then a quick calculation or displacement test would be sufficient for your purposes. As always, economics and practicality will be the overriding factor in deciding what sort of method to use in order to estimate relative permeability values.
Capillary Pressure and Wettability Wettability is measured, in practical terms, through the relative saturation changes in spontaneous drainage and imbibition. The ratio of the displaced volume through a spontaneous displacement to the volume displaced by the sum of the spontaneous and forced displacement is called the Amott Index. For a water/oil system, there can be an Amott Index for oil and an Amott Index for water. Figure 2-82 shows the Amott ternary wettability diagram, where the different wettability indexes are presented. The log of the area under the drainage curve to the area under the imbibition curve is called the USBM Index. A water-wet system should have an Amott Index for water close to 1, an Amott Index for oil close to zero and a relatively large positive USBM Index. An oil-wet system should have an Amott index to water close to 1, an Amott index to oil close to zero and a relatively large negative USBM index. Centrifuge technology is often used for the determination of the capillary pressure and the wettability characteristics of a given reservoir, this technology will be covered in greater detail in the next chapter. 0
1
Neutral Wet
Fractional Wet 1
0 0
Oil Wetting Index
1
Figure 2-82: Amott Ternary Wettability Diagram
120
Relative Permeability Chatenever and Calhoun (1952) found that in a system at steady state with two immiscible fluids that flow simultaneously in a porous medium will establish their own pathways. In their work the simultaneous injection of oil and water into a porous medium resulted in the creation of pore pathways fully occupied by one fluid and fluid-fluid interfaces such that the flow channels followed by one fluid were not influenced by the flow behaviour of the other fluid. This was used as the basis for extending Darcy’s Law to apply for each of the immiscible fluids: ⃑
(
) (⃑
)
(2-131)
⃑
(
) (⃑
)
(2-132)
Where, ke1 = effective permeability of fluid phase 1 ke2 = effective permeability of fluid phase 2 V1 = Darcy velocity of fluid phase 1, as defined by Q1/A (linear flow conditions) V2 = Darcy velocity of fluid phase 2, as defined by Q2/A (linear flow conditions)
1 = viscosity of fluid phase 1 2 = viscosity of fluid phase 2 1
= density of fluid phase 1
2
= density of fluid phase 2
The effective permeability is a function of the saturation and the saturation history of the sample being examined. It is customary to express effective permeability as a fraction of some base permeability; such as the single phase (absolute) permeability, k, of the medium. These fractional permeabilities are also known as relative permeabilities, kr1, and kr2, and are defined by: (2-133) (2-134) Using the relative permeabilities, the individual phase velocities can then be written as follows: ⃑
(
) (⃑
)
(2-135)
⃑
(
) (⃑
)
(2-136)
The pressures values (Pi, where i = 1, 2) of each of the phases at any macroscopic point in the porous medium are assumed to be related to each other via the capillary pressure, Pc. Let P1 denote the 121
pressure of the wetting phase and P2 denote the pressure of the non-wetting phase. The gradients P1 and P2, are related to each other by the capillary pressure gradient Pc. (2-137)
In the case of uniform saturation along the length of a core sample, Pc = 0. Some behaviours to note regarding relative permeability include:
For a given water saturation (Sw) value under water-wet conditions, (kw/ko) is higher for a higher permeability sample in comparison to (kw/ko) in a low permeability sample.
For a given Sw value under oil-wet conditions, the (kw/ko) ratio is higher for low permeability samples in comparison to high permeability samples.
The relative conductivity of large pores in a low permeability system is larger than that in a high permeability system.
Comparison between Capillary Pressure and Relative Permeability Similarities and analogs between capillary pressure and relative permeability curves for a two phase system is shown in Figure 2-83.
122
Primary Drainage Secondary Drainage
Capillary Pressure: Pc = Po - Pw
Primary Imbibition
Swir
1-Sor
1
Water Saturation
Relative Permeability
Secondary Drainage
Drainage
Imbibition
Drainage
Imbibition
Water Saturation
Pressure
For Uniform Saturation, ΔPc = 0
Pc
Length, X Figure 2-83: Comparison between capillary pressure and relative permeability curves
Fluid Distribution in Two Phase Flow There are several important points regarding the distribution of fluids in porous media under two-phase flow condition. The wetting phase seeks to maintain its continuum in the network of pore space involving the relatively small pore throats connecting pore bodies. While the non-wetting phase seeks to maintain a continuum in a pore
123
network involving the large pore throats. As the non-wetting phase saturation increases, the wetting phase is “trapped” in the pore spaces accessible through relatively small pore throats. However, as the non-wetting phase saturation decreases, krnw decreases (i.e. imbibition displacement) the wetting phase imbibes into relatively small pore throats and pore bodies in a hierarchical manner. This results in the non-wetting phase being trapped in the relatively large pore bodies and large pore throats connecting trapped oil in pore bodies. Also, in a water-wet formation at S*or conditions, only water is flowing and kro(S*or) is equal to zero. However, if the formation is oil-2 wet, at S*or conditions, kro(S*or) is very small (10 ) but never becomes zero because of wetting “film” flow. Additionally, if the IFT is very small, there is a significant change in the relative permeability characteristics.
Relative Permeability Plots and Characteristics The following summarises some important points about relative permeability and relative permeability curves for water-wet and oil-wet systems: Water-wet system:
krw = kro at Sw > 0.5, krw at S*or is less than 0.3 (S*or is the waterflood residual oil saturation), The amount of connate water saturation, Swc, is usually greater than 20% (Swc 20 to 25% PV) and has an effect on relative permeability behaviour, (kro)primary drainage > (kro)imbibition for the same Sw, The krw curve exhibits no hysteresis (e.g., (krw)pd = (krw)imb), As permeability increases, (krw/kro) becomes higher for a given saturation.
Oil-wet system:
krw = kro at Sw < 0.5, krw (S*or) > 0.5, Swc has no effect on relative permeability behaviour, if Swc < 20%, krw(Sw)|primary drainage > krw(Sw)|imbibition, kro curve exhibits no hysteresis, As permeability decreases, (krw/kro) becomes higher for a given water saturation.
For both oil and water-wet systems, the sum of (krw + kro) is always less than 1, when the same base permeability is used for normalization (i.e., (krw + kro) < 1). The main reasons for this are: 1) dendritic to flow pore space; 2) the connectivity of the network of each of the phases in the occupied pores is much smaller than that of the porous medium under single phase flow; and 3) pressure of immobile (trapped) fluids.
TWO PHASE RELATIVE PERMEABILITY LITERATURE SURVEY. Experimental Research and Relative Permeability Predictions In the last forty years, there have been many experimental investigations on relative permeability measurements. Shankar (1979) gives a detailed discussion on the experimental techniques that apply for these measurements. He also gives a review of the experimental work done and of some attempts to predict relative permeability up until the early seventies. McCaffery (1973) also gives a detailed discussion on the proposed methods of prediction of relative permeability characteristics up until that time.
124
There are certain factors that influence the experimental measurements of permeability and the relative permeability characteristics in general. These factors are:
The Boundary or End Effects The Hysteresis Effect The Migration or Partial Water Saturation Effect The Flow Rate Effect The Effect of Gas Expansion
as they were discussed by Geffen et al (1951)and Shankar (1979). In later research works, the effect of many parameters on relative permeability and/or effective permeability characteristics was examined. These parameters are:
Wettability Capillary number Pore geometry Heterogeneity of the system Anisotropy of the system Fluid viscosity Interfacial tension (IFT) Temperature.
One parameter of major importance in relative permeability characteristics is wettability. McCaffery (1973) examined its effect on the relative permeability characteristics of porous media made from Teflon using different fluids in various wetting conditions. Shankar (1979) examined the effect of wettability in sandstones and he reported experimental results that cover a range of wetting conditions as well. Some results are shown in Figure 2-83. The effect of interfacial tension on relative oil/water permeabilities for the case of consolidated porous media has been investigated by Amaefule and Handy (1982). The samples used were fired Berea cores. They also tested temperature effects both in steady and unsteady state cases. They have found that oil/water relative permeabilities are greatly affected at IFT values less than 0.1 mN/m. They provide some empirical equations that fit their data. Keelan (1976) examined the gas-water relative permeability characteristics during imbibition. He reported plots of relative permeability as a function of saturation and its variations due to pore geometry. He correlated his results using the end-point relative permeability and the trapped gas saturation. Early studies by Johnson et al (1959) provided the first calculations of relative permeability based on data from displacement experiments. Honarpour et al (1982) report empirical equations for the estimation of two-phase relative permeability in consolidated rocks. Torabzadeh and Handy (1984) report data for relative permeability for high and low tension systems at temperatures ranging from 220C to 1750C. They state that temperature affects relative permeability especially at low IFTs and describe the effects of temperature at low and high IFTs. They also state that the relative permeability of water and oil will increase with decreasing IFT for constant temperature, while the rate effect is not important after a critical minimum point. 125
Figure 2-84 Relative permeability curves of Berea Sandstone. Strongly Water-wet conditions (Sankar 1979)
Yadav et al (1984A) (1984B) studied the distribution of two immiscible fluids in the pore space of sandstones. This distribution can give us information about the percentage of each pore that is occupied by each phase and therefore the microscopic information that is necessary for relative permeability studies at the pore level. Naar and Henderson (1961) correlated relative permeability characteristics during imbibition to the ones from drainage or drainage capillary curves for two-phase flow. A similar work for three-phase flow is given by Naar and Wydal (1961). Downie and Crane (1961) and Odeh (1959) examined the effect of viscosity on relative permeability. Although their results contradict each other, it is shown that viscosity must be taken into account when the relative permeability values are calculated. However, this value is a property of the medium and it is not a function of the fluid viscosity. Morgan and Gordon (1970) investigated the effect of pore geometry on water-oil relative permeability. They grouped porous media in categories according to the volume and the interconnectivity of the pores and they provide characteristic curves for each different case. 126
Morrow and McCaffery (1977) present results of experimental studies of the effect of wettability on relative permeability, capillary pressure and spontaneous imbibition. They classify the uniformly wetted systems in Wetted (contact angle O0-620), Intermediately-Wetted (contact angle 620-1330) and NonWetted (contact angle 133 0-1800). Sigmund and McCaffery (1979) introduced a nonlinear least square procedure to analyse the recovery and pressure response to two-phase laboratory displacement tests. They state that two-parameter relative permeability curves determined by a least squares one-dimensional simulation history match could describe quantitatively the dynamic two-phase flow characteristics of various rock types with differing degrees of heterogeneity. They determined the relative permeability characteristics in the form of two saturation exponent parameters. They made simulator calculations and found out that capillary forces significantly affect the pressure and recovery response data obtained from dynamic displacement test. Capillary forces are most likely to have influence on low rate displacements in the drainage direction. Batycky et al (1981 B) presented a procedure to test the results of the above work. They performed tests in both high-rate displacement and low-rate displacement and they found out that for low-rate displacement there has to be consideration of effects due to capillary forces and the "end-effects". These effects can extend over significant portions of the core in systems with strong wetting characteristics. They also found that there is high possibility of a significant error in the simulation of relative permeability curves if the capillary forces are neglected. Thomeer (1983) proposes a relationship to determine air permeability from the mercury/air capillary pressure data. Similar relationships are reported by Swanson (1981) for brine permeability. Bardon and Longeron (1980) studied the influence of IFTs, flow rates and viscosity ratios on Fontainebleau sandstone. They found out that if the lFT is greater than 0.04 mN/m, the relative permeability can be predicted through a simulation but, for very low IFTs, there are great changes in relative permeability values and the prediction is not satisfactory. The macroscopic anisotropy and the heterogeneities of a porous medium are a common phenomenon in oil reservoirs. The terms of directional, vertical and horizontal permeabilities appear in many papers that describe the effect of heterogeneity in permeability and reservoir behaviour. Warren and Price (1961), Parsons (1972), Rose (1982), and Aguilera (1982) (for the case of naturally fractured reservoirs), give descriptions and methods on the aspects that govern those systems. Another area of great interest is the one that deals with low permeability cores. Such cores can be provided from tight gas reservoirs, but the measurements must take into consideration the Klinkenberg effect as well as the effect of confining pressure. There have been quite a few papers in this area although no modeling has been attempted, as far as we know. Reports have been given by Thomas and Ward (1972), Jones and Owens (1980), Walls (1982) (who suggests that a network approach might be the most appropriate modelling approach for this kind of media), Kassemi (1982), Keighin and Sampath (1982) and Freeman and Bush (1983). Finally, Jones and Roszelle (1978) suggest that some graphical techniques can be used for simple and accurate determination, of relative permeability. 127
Relative Permeability Simulations During the past forty years, many investigators tried to simulate the capillary and transport phenomena in porous media. Many different methods were used. These methods are classified as macroscopic approaches or pore level (microscopic) approaches. Greenkom (1981) gives a review of the theoretical advances in the study of flow through porous media and discusses the phenomenological approaches for phenomena that occur in a porous medium under steady state conditions. He also provides an extensive list of references. According to Dullien (1979) and Chatzis (1980), the early models that deal with the porous media are classified as:
Phenomenological models (i.e. Carman-Kozeny's 'Hydraulic Radius Theory'), Geometrical models (i.e., serial type, parallel type, and serial-parallel type capillaric models), and Statistical models, "cutting and random rejoining models", (Haring and Greenkom (1970)).
Macroscopic approaches published include Slattery (1968), Casulli and Greenspan (1982), Gilman and Kassemi (1983) for the case of naturally fractured reservoirs, and Ramakrishnan and Wasan (1984) in connection with the effect of capillary number. The pore level or network model approaches are based on the pioneering work ofFatt (1956), who stated that a network of tubes is a valid model of porous media. He also was the first to state that the relative permeability characteristics of porous media are a direct consequence of the network structure of these media. Later, the network principles were combined with the aspects of percolation theory and gave the stochastic network models. This approach is the basis of the work of Chatzis (1976) (1980), Chatzis and Dullien (1978) (1982) (1982B), Larson et al (1981), Mohanty et al (1980), Koplik (1982), Heiba et al (1982) (1984), Mohanty and Salter (1982), Diaz (1984), and Kantzas (1985). The various models have many similarities but some crucial differences. Details on the above models are given below.
Theoretical Developments Pertinent to Relative Permeability Studies the Network Approach for Modelling Porous Media A porous medium can be represented as a random network'of pores. The construction of such a network must follow some theoretical considerations (e.g., porosity, pore to pore coordination number, pore shapes and pore sizes, etc.). Many of these aspects are based on the percolation theory and its applications to flowing porous media. Some introductory information about percolation theory concepts is given in the sections that follow. For more details on percolation theory and its applications in porous media, see Shante and Kirkpatrick (1971), Kirkpatrick (1973), Chatzis (1976), (1980), Larson (1977), and Larson et al (1981).
Percolation Theory There are many physical phenomena in which a fluid is "spreading" randomly in a medium. The terms "spreading", "fluid" and "medium" are not necessarily used in their strict sense. Except for the spreading itself, external causes such as gravity forces, for example, may control the process and affect the 128
random mechanism. The random mechanism may be ascribed to either the medium or the fluid, depending on the nature of the particular problem. Broadbent and Hammersley (1957) introduced the term "Percolation Process" for the process that ascribes the random mechanism to the medium. This term came to distinguish the above mathematical analysis from the ones that are confined to the random mechanism of a process generally ascribed to the fluid which are labeled as "diffusion processes". Percolation theory has been a very important tool in the theoretical development of the conductivity of random mixtures of conducting and non-conducting materials (Kirkpatrick 1973, Shante and Kirkpatrick 1971). In percolation theory a new terminology is introduced. The "medium" is defined to be an infinite set of abstract objects called "atoms", or "nodes", or "sites". A fluid flows from the source site along paths connecting different sites. These paths are called "bonds", and can be oriented or unoriented. A bond is defined as oriented when it permits the flow only in a specified direction. The fluid that flows along a bond will wet its two end points. The latter is used in porous media simulations. The coordination number, z, of a network of sites connected by bonds is defined to be the weighted average number of bonds leaving a node in a network (Chatzis 1980). The random mechanism can be assigned to the medium in two distinct ways, leading to two different percolation problems. The first is the bond percolation problem in which each bond has a constant probability of transmitting fluid and the bonds "transmitting" fluid are assigned at random everywhere in the network. This probability is independent of the existence of other bonds at the level of a site. The second is the site percolation problem in which a site. A, has a certain probability of allowing, fluid reaching A to flow on, along bonds leaving A. In this case^every bond between two open sites is to flow and the random mechanism is assigned to the sites. When a site is missing, all bonds connecting it to its neighbours are missing too. There are quite a few works that are based on either problem (e.g., Chatzis (1976), (1980)). In the field of porous media the conventional methods treated drainage as a bond problem and imbibition as a site problem. Chatzis (1980) used the site problem for drainage by taking the bonds into account. This is known as the bond correlated site problem in which the sites are assigned at random and the event of a bond being open is correlated with the event of two adjacent sites being open. Network models of pore structure with pore body sizes randomly distributed over the sites and the pore throat sizes-assigned according to a correlation scheme have been found to be sound models of simulating capillary pressure curves in sandstones (Chatzis and Dullien (1982), Diaz (1984)). Network models of pore structure obeying the bond percolation problem have been found to be unrealistic in simulating pore structure and flow behaviour (Chatzis (1980)). The most important finding of percolation theory is the critical percolation threshold. This is defined as the minimum fraction of bonds (bond percolation threshold, Pbc)' or tne minimum fraction of sites (site percolation threshold, Pgc), that must be present in the network so that the "medium" is conducting to flow. A background to the approach taken in applying percolation theory to random network models of pore structure at the University of Waterloo is given next.
129
Network Modeling at Waterloo (Waterloo Network Model) WATNEMO is based on the work done by Chatzis (1976), (1980), the extensions given by Diaz (1984) and Kantzas (1985). A random network involves three key parameters: the average coordination number z, the pore body size distribution, and the pore throat size distribution. A 3-D random model is generated using the cubic lattice as a complete topological representation of an irregular network. The case of 2-D networks is not considered in this work since, as was pointed out by Chatzis (1976), 2-D networks cannot be used as physically sound models of real porous media. The main reason for this is that, in a 2-D network, it is impossible to have nodes and bonds fully occupied by one phase to form a continuum, while the other phase fully occupies nodes and bonds that a continuum as well (i.e., bicontinua cannot exist). Using the fact that the real network of pores in a porous medium is in 3-D, one can create many different types of 3-D networks characterized by different values of the coordination number. It has been found that it is very convenient to work-with regular networks that have a cubic lattice arrangement and a coordination number of six. This simplification introduces the "physical' assumption that the pores of the medium are well connected, and facilitates straightforward computer algorithms. Taking the above fact into account, the other parameters to be defined are the size distributions of pore throats, assigned to the bonds, and of the pore bodies assigned to the nodes of the network. The network models generated using bond correlated site percolation have been found to bemore suitable models of pore structure in sandstones (Chatzis (1980), Diaz (1984)). The principle of generating such networks is simple. The nodes of the network are assigned with indices 1,2 ....n at random. The nodes assigned the index 1 represent the largest pore bodies and the nodes assigned index n are the smallest pore bodies. A pseudo-random number generation technique can be used to provide a random setting of those indices in the sites of the network. The next step is to define the bonds by assigning an index equal to the larger index of the two nodes it connects. The accessibility properties of random network models using the bond correlated site percolation scheme have been investigated at Waterloo with special attention given to cubic networks (Chatzis (1980), Diaz (1984), Kantzas (1985)). Generalized number based accessibility functions have been obtained and applied to model capillary pressure saturation relationships for mercury-air systems and water-oil systems with entrapment. Having specified the number based pore body size distribution, the pore throat size distribution and pore geometry, there are algorithms available to convert the number based accessibility data to volume based capillary pressure curve data (Chatzis (1980), Chatzis and Dullien (1982), Diaz (1984), Kantzas (1985)). In addition to the simulation of capillary pressure curves, the WATNEMO has been applied successfully to model the "dendritic" non-wetting phase saturation during primary drainage in mercury-air experiments (Chatzis and Dullien (1982)) and relative permeability characteristics in mercurypermeametry experiments. The work of Diaz (1984) expanded the capability of WATNEMO to model fluid distributions of the non-wetting phase as well as the wetting phase, as a function of saturation and saturation history, while Kantzas (1985) used the same principles for the simulation of two phase relative permeabilities as described below.
130
The conductivity properties of random network models of interest to us for the simulation of relative permeability behaviour in oil-water systems have not been explored in detail. Results presented by Kirkpatrick (1973) for the cubic networks are limited by the fact that all bonds in the network allowed to conduct flow had the same conductivity value. There are no analytical methods for the calculation of the conductivity properties of random networks with size distribution of conductivities or with variable coordination numbers. Effective medium theory approximations (Kirkpatrick 1973, Koplik 1982) have been applied by Larson et al (1981 A), Heiba et al (1982) to two phase flow problems of relevance to relative permeability behaviour. Mohanty and Salter (1982) have also looked at the application of the conductivity properties of random networks to simulate relative permeability behaviour, but their simulation results were not in good agreement with experimental results. Moreover, the information provided in their paper is not sufficient to warrant a detailed criticism. In all of the past studies involved with the simulation of relating permeability characteristics, several tacit assumptions are made. These include: 1) The nodes in the network simulation have no resistance to fluid flow; 2) Only the bonds in the network carry the information of the resistance to flow in pore networks; and 3) With the exception of Chatzis and Dullien (1982), a node in the network can be simultaneously part of the non-wetting phase network as well as part of the wetting phase network (e.g., Fatt (1956C), Heiha et al (1992), Winterfeld et al (1981)). In addition to the above assumptions, several inconsistencies arise in transforming the relative conductivities of random networks into the form of relative permeability curves. For example, for the conductivity of a pore with diameter D, g(D) may be taken to be proportional to D cubed or D to the power of four, while the volume of such a pore V(D) is proportional to D to the power of 0.84 instead of V(D) proportional to D cubed or D squared as it should be (e.g., Heiba et al (1982), Soo and Slattery( 1983)). Part of this work is devoted to clarify some of these inherent inconsistencies in past studies. To do so, however, the capability of WATNEMO to model relative permeability behavior had to be developed. This was the main objective of Kantzas (1985) - the development of software for the investigation of the conductivity properties of random networks of the bond correlated site percolation type. The following sections form the basis for understanding the development of this software for simulating immiscible two-phase flow problems in porous media with applications to predictions of relative permeability behavior.
Conductivity and Permeability, the Main Algorithm The term conductivity is one of the most commonly used terms that appear in all the texts and studies that deal with flow problems. It describes the difficulty of flow through a certain medium, where the flow refers to momentum, energy, mass, electricity, etc. The most well known conductivity is the electrical conductivity, which is described by Ohm's law, (2-138) Where, 131
I is the electric current (in Amperes), V is the voltage drop (in Volts), G is the electricalconductivity of a resistor (in I/Ohm). Eq. (2-138) can be rewritten in terms of current density J and charge density F as follows: (
)
(2-139)
Thermal conductivity is also another well-known conductivity and it is defined by Fourier's law, (
)
(2-140)
Where, The heat flux, The thermal conductivity, ( )
The temperature gradient along the x direction.
In porous media, the conductivity to flow (or permeability) is given by Darcy's law. Darcy's law and equations (2-139) and (2-140)are obviously similar. Many researchers who have seen this similarity tried to give solutions for the permeability simulation problems using algorithms originally designed for the electrical conductivity analogs. This strategy is very reasonable. The same strategy is used in Kantzas (1985).
First, let us consider the case of calculating the overall conductivity of a network of resistors. This can be done by applying Kirchhoffs' laws ∑
(2-141)
at any node i, and for any closed loop of resistors: ∑
∑
(2-142)
where m is the number of bonds that form a closed loop. The second law applies in such cases where we have many sources of electricity within a loop. If we consider a network that contains only resistors then application of the first law can give ∑
∑
(2-143)
This says that we have one equation for every node in the network. A group of equations can be formed that describes the whole network, such that: 132
[ ][ ]
[ ]
(2-144)
Where, [ ]
The conductivity matrix,
[ ]
The voltage drop vector,
[ ]
The current vector.
This way of approaching the problem is different from Dodds' approach (1968) that used the resistances instead of the conductivities resulting in the inverse set of equations: [ ][ ]
[ ]
(2-145)
where R is the resistance matrix. For the optimum numerical solution, eq. (2-145) is preferred because of less memory space and less execution time requirements (George and Liu (1981)). So, for a network that consists of n nodes, a set of n linear equations is created in which all the elements of the current vector have the value of zero. The linear set however, is not positively definite. Therefore, it is not that easy to get a solution for eq. (2-145), and the solution most probably will not be unique. Also, this solution will not provide any information for the overall conductivity of the network, which is the main aim of this work. So, it is necessary to have an additional independent equation that relates the overall conductivity of the network, Gt, the overall voltage drop, Vt, across the network and the overall current, It. This set up provides the extra information that relates It, Vt and Gt. ∑
∑
(2-146)
where e denotes the resistors that connect the network to the external node. If the value of the overall current (It) is defined, then a linear and positive definite set of equations is obtained which can be used for obtaining the solution for the voltage vector. If, in addition to the above, It = 1, then the overall conductivity of the network can be automatically calculated, since (2-147) Therefore, if the conductivity of each resistor is known, the overall conductivity of the network can be calculated. This calculation was performed by Kantzas (1985) and the result was a set of relative permeability curves that provided a good agreement with experimental results in Berea Sandstone.
Three Phase Relative Permeability Three-phase flow situations occur when gas is injected into a reservoir or when a reservoir is produced at a pressure below its bubble point. The flow of one phase relative to the other phases is represented by the relative permeability of that phase. Therefore, relative permeability is an essential variable in the prediction of reservoir performance in reservoir simulation studies.
133
Relative permeability is a direct measure of the ability of the porous medium to conduct flow of one fluid when two or more fluids are present. The flow of the fluids in the medium is controlled by the pore geometry, wettability, fluid distribution, and saturation history. Wettability is a controlling factor in determining three-phase relative permeability characteristics through its effect on the fluids distribution and flow of the three phases. The fluids distribution in water wet and oil wet systems is quite different. When a system is water wet, water fills all the small pores and exists as a film in the larger pores. When the medium is oil wet, the reverse is true; oil will occupy the small pores and exists as a film in the larger pores. It is very rare for a reservoir to be gas wet, so gas wet systems will not be discussed. It is generally believed that reservoirs are originally water wet. As oil migrates into the reservoir, crude oils come in contact with the rock surface and adsorption can occur which alters the wettability of the rock1. This can lead to many different forms of wettability. The wettability of a system can range from strongly water wet to strongly oil wet. When the rock has equal preference for both oil and water, the system has neutral or intermediate wettability. Fractional wettability is a condition when different areas of the core have different wetting preferences. There is also a special case of fractional wettability, called mixed wettability, where the small pores are water wet and the large pores are oil wet. There are two methods of evaluating the relative permeability of each phase: steady state and unsteady state. Each method has its own advantages and disadvantages, however most researchers say that the unsteady state method should not be used. The experimental procedure to evaluate relative permeability of two-phase flow is easy, so a lot of relative permeability data were collected. However, for three-phase flow, the procedure is quite complicated so not many experiments were carried out. Therefore, relative permeability characteristics of three-phase system are not fully understood.
Generally, the relative permeability can be obtained for imbibition (wetting phase displacing nonwetting phase) or drainage (nonwetting phase displacing wetting phase). At the same wetting phase saturation, the fluid distribution in the pores will be different, thus it is expected that relative permeability in drainage and imbibition will also be different. Many researchers published contradictory findings, so hysteresis between drainage and imbibition in three-phase flow is still being studied. Most researchers have seen that in strongly wetted mediums, the wetting phase and nonwetting phase show very little hysteresis. However, for the intermediate wetting phase significant hysteresis was seen. Due to the complex nature of three-phase relative permeability experiments and the lack of agreement between the limited data, many models were developed to generate three-phase relative permeability values. These models have often not compared with all the available experimental data. There were many experiments performed with two-phase flow to study the effects of interfacial tension, temperature, flow rate, and viscosity on the relative permeability characteristics. Some researchers found that these parameters affect relative permeability while some reported otherwise. Experiments on three-phase flow to study the effects of these parameters are very rare to non-existent. The effects of these parameters on two-phase relative permeability are included in the discussion, since it is likely that these parameters will affect three-phase flow in a similar manner. 134
The flow of fluid in carbonates is also investigated. Most carbonates are usually oil wet, this means that relative permeability should be different than that of sandstones, which are usually water wet.Relative permeability in heavy oil systems is also being studied.
Relative Permeability in Two-Phase Systems The wetting fluid in a uniformly wetted system is located in the smaller pores and as a thin film in the larger pores, while the non wetting phase is located in the centers of the larger pores. In general, at a given saturation, the relative permeability of a fluid is higher when it is the non wetting fluid1. This means that the water relative permeability is larger in an oil wet system than it would be in a water wet system. This occurs because the wetting fluid usually travels through the small pores, while the nonwetting fluid moves through the larger pores, which results in better flow of the nonwetting fluid. Also, at low nonwetting phase saturation the nonwetting phase will be trapped as discontinuous blobs in the larger pores which block the flow of the wetting fluid, thus decreasing its relative permeability. On the other hand, the relative permeability of the nonwetting phase at low wetting phase saturation is very high since it travels through the center of the larger pores. This nonwetting phase relative permeability could be as high as the absolute permeability, which indicates that the wetting phase does not significantly restrict the flow of the nonwetting phase. In a water-oil system, as the system becomes more oil wet, relative permeability of water (krw) increases, relative permeability of oil (kro) decreases, and the cross-over point occurs at a lower water saturation (see Figure 2-85)1. 1
Relative Permeability
Water Wet Oil Wet
0
Oil Water
Oil
Water
0
Water Saturation
1
Figure 2-85 Relative permeability of water wet and oil wet systems.
These special characteristics allowed Craig to generalize a few rules of thumb1, which indicate the differences in the relative permeability of strongly water wet and strongly wet oil cores. These rules are presented in Table 1 below.
135
Table 6. Craig’s Rules of Thumb for Determining the Wettability
Water wet
Oil wet
> 20-25 % of PV
< 10-15% of PV
Sw when krw = kro
> 50%
< 50%
krw at max Sw
< 30%
> 50% (can be 100%)
Swc
Exceptions have been found to these rules, therefore wettability experiments should be performed separately to evaluate the wettability of the reservoir instead of relying only on Craig’s rules of thumb. Also, Craig’s rules of thumb can only distinguish water wet conditions from oil wet; they cannot identify the degree of wettability, or neutrally wet, fractionally wet or mixed wet conditions. Pore geometry can also have a strong effect on the measured relative permeability curves, affecting the cross over point in two-phase flow and the irreducible water saturation1. If the pore medium consists of a significant number of small pores, the irreducible water saturation is relatively large. Anderson has also mentioned the significant differences in Swc found in rocks with large, well interconnected pores compared to rocks containing smaller and less well connected pores. In the water wet cores, the smaller pores are filled with water, thus a larger number of small pores increases the irreducible water saturation, but if these pores are less well connected, the water flow is not better. When comparing these two samples, the rock with many smaller pores has a larger irreducible water saturation and the cross-over point for the relative permeability occurs at higher water saturation.
Relative Permeability in Three-Phase Systems In three-phase systems, there is no obvious distinction in relative permeability of water wet and oil wet systems. Most researchers can identify the characteristics of the wetting phase and the nonwetting phase. However, the pattern for the intermediate wetting phase is not fully identified yet. For fractionally wet, intermediate and mixed wet system, special characteristics of relative permeability are not known at all. Most researchers agree that the relative permeability of the wetting phase depends on its saturation only. Anderson mentioned that many researchers found that in a water wet system, there was a good agreement between the wetting phase relative permeability of two-phase systems (water in the water oil tests and oil in gas-oil tests)1. However, there have been some experiments which show that the wetting phase relative permeability depends on saturations of the other phases. Some researchers even found that the trapped gas saturation could affect both the water and oil relative permeabilities. There are other experiments which show that the wetting phase relative permeability was affected by the nonwetting phase saturation in water wet systems. Most researchers still insist however that the relative permeability of the wetting phase should be a function of its saturation only. They suggest that some of these systems that showed significant difference in the relative permeability of the wetting phase were not strongly water wet.
136
Three-phase relative permeability data for water wet sandpacks have been reported by several people. However, there is very limited three-phase data for oil wet cores. The data for water wet cores and sandpacks show generally consistent behavior; gas and water primarily depend on the gas or water saturation respectively, and are weak functions of the saturations of other fluids present. These data indicate that a strongly wetting phase (water) and a nonwetting phase (gas) are not affected by the other phases. However, Schneider et al. reported that the three-phase relative permeability of gas (krg) values are smaller than the two-phase krg in drainage2. This is because of flow interference between the gas (nonwetting) phase and the oil (intermediate wetting) phase in a water wet system containing three phases. Corey et al., however, saw from their experiments that gas relative permeability was essentially the same in two-phase and three-phase systems3. The intermediate wetting phase (oil) appears to be influenced by interactions with the other phases, but the nature of the interaction is not very distinctive. In some cases the dependence of oil relative permeability on the saturations of the other phases is apparent and in some cases it is not. The experimental data often shows so much scattering that no conclusion can be reached. Even though the evidence is not as overwhelming as for the wetting phase, many researchers agree that the relative permeability of the nonwetting phase is a function of its own saturation only. In water wet systems with trapped gas present, Schneider et al. found that the maximum value of kro is larger than the value in the case without gas2. They explained that this is because gas is the nonwetting phase with respect to oil. Also, the trapped gas saturation resulted in lower krw. This is not consistent with the general belief that for water wet system; water relative permeability is a function of water saturation only.
DiCarlo et al. performed three-phase experiments and they have seen other characteristics4:
The permeabilities of the most wetting fluid are always similar. In both cases of water wet and oil wet media, krw and kro can be described by a power law, kr S, where = 5. However, a lower saturation is reached in the oil-wet medium than in the water-wet medium. This is consistent with Craig’s generalization that for oil wet systems Swi is lower than Swi in water wet system. For the strongly and uniformly water wet and oil wet media, it is expected that the configuration of the most wetting fluid should be similar in both systems at the same saturation, thus they should have the same relative permeability characteristics. At high saturations, the permeabilities of oil (nonwetting phase) in water wet media and water (nonwetting phase) in oil wet media are similar. Both can be described by a power law, kr S, where = 4. DiCarlo et al. believed that at high saturations the arrangement of oil filled pores in a waterwet medium and the water filled pores in oil wet medium are quite similar. At low saturations, the permeabilities of oil in water wet media and water in oil wet media are very different. The oil relative permeability remains finite at low saturations while the water relative permeability drops off quickly and approaches zero. At low saturation phases may remain connected through wetting layers in crevices in the pore space. It is this connectivity which controls relative permeability at low saturation. It is possible that the pore scale configuration and connectivity of oil and water is very different for water wet and oil wet media. 137
The gas relative permeability for the oil-wet medium is approximately half its value in a water wet medium. In water wet media, the water phase will occupy the smallest pores and crevices while the gas phase occupies the large pore spaces since gas is more nonwetting than oil. In oil wet media, the water and gas phases compete for the largest pores. It is possible that at a specific gas saturations in the oil wet system, the gas is in smaller pathways leading to a lower permeability.
For the fractionally wet sand, the oil, water and gas relative permeabilities are between the oil, water and gas relative permeabilities in the water wet and oil wet sands.
Schneider et al. mentioned that for systems having no strong wetting preference, characteristics of three-phase flow can be very different than for two-phase flow2. This suggests that the relative permeability might be a function of other phases as well.
Drainage and Imbibition Relative Permeabilities Many people believe that in strongly wetted systems, the relative permeability of the wetting phase is usually a function of its own saturation, which means that it is not a function of saturation history. Anderson reported that the wetting phase relative permeability is very similar for both two and threephase in strongly wetted systems at a specific wetting phase saturation. This implies that the wetting phase distribution in two-phase is very much similar to the one in three-phase systems. Therefore the hysteresis between the drainage and imbibition of the wetting phase is very small. In three-phase systems, many people assumed that gas is almost always a nonwetting phase. There were other experiments, which show that there was hysteresis for the wetting phase in both water wet system and oil wet system. It is possible that for the cases where hysteresis was seen in the relative permeability of the wetting phase, the medium was not strongly wet. In general, the majority of the results show that there is little or no relative permeability hysteresis in the wetting phase. Most of the results that show significant hysteresis in the wetting phase relative permeability were obtained from unsteady state methods. Craig and others believe that there are problems with the unsteady state relative permeability measurement in strongly wetting systems1. As mentioned before, for the nonwetting phase in a strongly wet medium, many people agree that its relative permeability is a function of its saturation only, thus there is no hysteresis. However, there was significant hysteresis seen for the relative permeability of the intermediate wetting phase.
138
Methods of Relative Permeability Measurements Honarpour and Mahmood provide a detailed assessment of the steady state and unsteady methods of measuring relative permeability5. They believe that steady state provides the most reliable relative permeability data. In this method, two or three fluids are injected simultaneously at a constant rate or pressure drop. The system is then allowed to reach equilibrium. At this point, the saturations, flow rates, and pressure gradients are measured. Darcy’s law is then used to calculate the effective permeability of each phase. The injection rate or pressure drop is then changed and the cycle is repeated again until enough points are collected to plot the relative permeability curve. This method is very time consuming, because equilibrium at each flow rate may take several hours to days to achieve. Unsteady state is the quickest method of obtaining relative permeability. In this method, the in-situ fluids are displaced at a constant rate with the effluent volumes being monitored continuously. Equilibrium is not achieved, so the entire set of relative permeability curves can be obtained in a few hours. However, there are many difficulties associated with this method, such as capillary end effects, viscous fingering and channeling in heterogeneous medium, which cannot be properly accounted for. Capillary end effect is a phenomenon in which the saturation is high at the inlet or outlet of the core. To minimize the boundary effect, the fluids can be flowed through the core at high rates to make the capillary forces insignificant compared to the viscous forces due to the flow of fluids. The fluid dispersion in the porous media increases with the flow rate, thus minimizing the boundary effect at the input end6. However, the use of high flow rate can aggregate viscous fingering problems. Also, in this method viscous oils are usually used to prolong the period of two-phase production (at flow breakthrough, no more information on relative permeability can be obtained). Again, this will further enhance the viscous fingering problem. Craig and other researchers recommend that the unsteady state method not be used with strongly water wet cores. They believe that the combination of high velocities and high viscosities will cause a strongly water wet medium to behave as an oil wet system during waterflood. During a waterflood, the injected water will move rapidly through the larger pores, bypassing a large number of pores that are filled with viscous oil, causing early breakthrough. Usually, in waterflooding of an oil wet system, it was seen that breakthrough recovery is low and additional oil can still be recovered. Thus, the performance of a waterflood with viscous oil is very similar to a waterflood of an oil wet system. Also, in comparison with the steady state relative permeability of two-phase, the calculated unsteady state relative permeabilities measured on the same core appear to be more oil wet. During the process of core cleaning and handling, the wettability of the core can be altered, which can significantly affect the relative permeability. The cleaned core is usually more water wet than the actual native state wettability, thus the most accurate relative permeability measurements are obtained on native state core when the core’s wettability is reserved. When these cores are not available, restored state core could be used. The process of core wettability restoration must be handle with care since any error can be resulted in significant wettability alteration. There are other problems associated with obtaining data from core. Heaviside et al. said that the core sample might not truly represent the reservoir because the core samples are only a small fraction of the 139
reservoir, and may therefore be statistically unrepresentative7. Also, the sample may not be in the same state or have the same properties as when it was in the undisturbed reservoir. When the core plug is brought up to the surface, the light ends hydrocarbon can be liberated, causing the oil to be more heavy7. Some components can be deposited on the pore surface, making the rock more oil wet.
Three-Phase Relative Permeability Correlations Due to the complex nature of the experimental process to obtain three-phase relative permeability data, many models were developed to predict these values instead. These models can be divided into three catagories: those that are based on bundle of capillary tubes theory, those that are based on channel flow theory (Stone’s methods and their modifications), and other miscellaneous models. Bundle of capillary tubes theory There are many models which used this theory in the derivation of the relative permeability model. The models that will be discussed are Corey et al., Naar and Wygal, and Land. Corey, Rathjens, Henderson and Wyllie: The first model of three-phase relative permeability was published by Corey et al. in 19563. The model assumed that fluid flow in reservoirs can be represented by a bundle of capillary tubes. The flow paths through the medium can be approximated by the equivalent hydraulic radius of the capillary tubes. A tortuosity correction was included to account for the differences in path length of tubes of different sizes. Corey’s model also assumed that the wetting and nonwetting phase relative permeabilities are independent of the saturations of the other phases. In other words, relative permeability of the wetting phase is a function of wetting phase saturation and relative permeability of nonwetting phase is a function of nonwetting phase saturation. In this model, the intermediate wetting phase (oil) occupies the pores between those occupied by water and gas which are intermediate in size. Its flow can be interfered with by both the wetting and nonwetting fluids. Relative permeability of the intermediate wetting phase is proportional to the area of the pores occupied by this phase relative to the pores occupied by other phases. Corey also assumed that the relationship between saturation and drainage capillary pressure is represented by, (
)
(2-148)
(
)
(2-149)
Where, Pb = fitting parameter Pc = capillary pressure S = saturation Slr = residual liquid saturation for gas oil displacement (obtain from experiment) The model for the intermediate wetting phase (oil) is given as:
140
(2-150) Where, Sl = total liquid saturation Sw = water saturation The derivation for this model is based on the hypothesis that the residual oil saturation of a system containing water >= Slr will be zero. Corey compared the values calculated from his model to the experimental values which he collected from experiment on Berea sandstone. This is shown on Figure 2-86.
Figure 2-86 Comparison between experimental and predicted values (after 3).
Figure 2-86 shows that the match is quite good. One advantage of this method is that it requires only the value of residual liquid saturation to predict the oil relative permeability. However, this model does not allow for the adjustment of the end points of relative permeability in oil-gas and oil-water system. Therefore, its estimation of these end point relative permeabilities may be very different from the values obtained from the experiments.
Naar and Wygal: This model for imbibition was published in 19618. It is also based on the concept of flow in straight capillaries. It has random interconnections of pores and storage capacity, which makes trapping of the nonwetting phase by the invading wetting fluid possible. Again, relative permeabilities to the wetting and nonwetting phases are considered to depend only on their own wetting and nonwetting phase saturations, while the intermediate phase relative permeability depends on all threephase saturations. The relative permeability to water (wetting phase) can be obtained by assuming that oil and gas combine to form a single nonwetting phase; thus the system is two-phase. To compute the 141
relative permeability to gas, the system can be treated as a two-phase system with water and oil forming the single wetting phase. The relative permeability to oil is more complex. When the water invades the porous medium, part of the oil is blocked and the rest invades the pores that are full of gas. However, some of the gas phase is blocked while the remainder is pushed out. Because some of the oil is trapped, the relative permeability of oil decreases. In this model, the oil phase is more wetting than the gas phase, thus oil exists in smaller pores than gas does. So oil will invades gas in slightly larger pores first. Since some oil moves from small pores into larger pores, the permeability of oil increases. This model was derived by assuming that the capillary pressure curve is approximated by (2-151) Where C is a constant and S* is the reduced saturation. The relative permeability of the intermediate wetting phase can be predicted by:
k ro S of*3 (S of* 3S fw*) S of*
S *fw
S ob
(2-152)
S o S ob 1 S wi
(2-153)
S w S wi S ob
(2-154)
1 S wi
S S 1 1 S wi w wi 2 1 S wi
2
(2-155)
Where, So = oil saturation Swi = initial water saturation Sw = water saturation At the time of the model’s development, Naar and Wygal did not compare the model predictions with any experimental data. Land: Land has seen some evidence showing that the initial gas saturation controls the residual gas saturation, which is trapped after imbibition9. Residual gas saturation was seen to increase with increasing initial gas saturation. When the initial gas saturation is unity (ie. the reservoir contains only gas), the residual gas saturation is a maximum. Land published a model for imbibition in 1968, which takes into account the effect of initial saturations. This model assumes that the maximum residual hydrocarbon saturation is constant, whether the hydrocarbon is gas or oil, and that the residual hydrocarbon is related to the initial hydrocarbon saturation. Land assumed that in three-phase systems the gas relative permeability is the same as for two-phase systems. He reasoned that this is because the gas is a nonwetting phase, thus it occupies the same 142
pores regardless of the nature of the liquids present. Also, it was assumed that for a water wet medium, the imbibition relative permeability to water is the same function of water saturation for both two- and three-phase systems. Land’s model is very similar to that of Naar and Wygal. However, Land believed that Naar and Wygal wrongly placed trapped the gas in the largest pores rather than the smaller pores which are likely to be invaded first by the wetting phase. Land’s model also includes the dependence of relative permeability on saturation, saturation history and the capillary pressure. *2 kro Sof 1 S gf
* * Shr (max) 1 S hr (max) S gf ln * * * 2 2 Shr(max) 1 Shr(max) S gi * * 2 S w Sob 2 C 1 1 * * * * * S* hr(max) 1 Shr(max) S gf Shr(max) 1 Shr(max) S gi
S gf*
1 * * S g S gc 2
S w*
S w S wc 1 S wc
S g* C
S
* g
* S gc
2
4 * * S g S gc C
* S gc S gi*
(2-157)
(2-158)
(2-159)
1 S wc
1 (S )
(2-156)
Sg
1
(2-160)
1 * S gi 2
(2-161)
* gc max
S gi*
S gi
(2-162)
1 S wc
Where * ( S gc ) max =residual gas saturation after imbibition process which started with 100% gas
S
* hr max
=normalized maximum trapped hydrocarbon saturation
*
( S ob is defined earlier in Naar and Wygal) If water saturation increases and oil saturation decreases while the gas saturation remains constant (or is increasing), the main equation simplifies to
k ro S of*3 2(S w* S or* ) S of*
(2-163)
When all gas in the porous medium is trapped,
143
k ro S
*3 of
2S
* w
S
* of
* *2 2 * 1 S gr S S gr S gr ln * C C S gl *2 of
(2-164)
Land did not compare this model’s prediction with any experimental values. Parker, Lenhard, and Kuppusamy: This model was developed in 1987 and it is very similar to Corey’s10. However, in this model the flow path is assumed to be proportional to the square of the mean hydraulic radius of the pores occupied by that phase. In this model,
k ro S o*1 / 2 1 S w*1 / m
1 S m
2 *1 / m m l
(2-165)
With
S l* = normalized liquid saturation The fitting parameter n can be obtained by a curve fit of capillary pressure or by fitting two-phase relative permeability. Therefore this model will tend to give good results for cases where there is a satisfactory fit of the two-phase data. Stone’s methods and their modifications: The methods proposed by Stone are the ones most commonly used for prediction of three-phase relative permeability. Many people have attempted to improve the estimations of Stone’s methods. Theory and assumptions used in Stone’s models: Stone’s method 1 and 2 are probability based models11,12. Both methods are based on the channel flow theory, which states that in any flow channel there is at most only one mobile fluid. As a consequence, the wetting phase is located in the small pores and the nonwetting phase in the large pores, and the intermediate phase occupies the pores in between. Thus at equal water saturations the fluid distributions will be identical in a water-oil system and in a water-oil-gas system, as long as the direction of change of water saturation is the same in both. This implies that water relative permeability and water-oil capillary pressure in the three-phase system are functions of water saturations alone. Also, they vary the same way in the three-phase system as in the two-phase water oil system. The gas phase relative permeability and gas oil capillary pressure are the same functions of gas saturation in the three-phase system as in the two-phase gas oil system.
These two models require two sets of two-phase data: water oil displacement and gas oil displacement. Hysteresis can be taken into consideration by using the appropriate two-phase data. Stone’s method 1: This method was published in 197011. It assumes that the flow of oil is interfered with by the presence of gas and water, and that the effects of gas and water are independent. Oil relative * permeability is a function of normalized oil saturation S o , w and g , which depend on the water and
gas saturation, respectively. 144
k k rog S o* w g k ro S o* row* * 1 S w 1 S g S g*
Sg 1 S wc S om
(2-166)
(2-167)
S o*
S o S om 1 S wc S om
(2-168)
S w*
S w S wc 1 S wc S om
(2-169)
Where, w = probability that an oil filled pore is not blocked by water g = probability that an oil filled pore is not blocked by gas krow = relative permeability of oil in oil water system krog = relative permeability of oil in oil gas system Som = minimum residual oil saturation Sg = gas saturation So = oil saturation Sw = water saturation Swc = connate water saturation = Swir = Swi Stone mentioned that Som can be expressed as a function of fluid saturation but since sufficient data is not available to correlate this relationship, a constant value is set for Som instead. Stone compared the simulated values from the model with the data of Corey et al. and Dalton et al. (see Figure 2-87 and Figure 2-88).
145
Figure 2-87 Compare with results of Corey et al. (after 11)
Figure 2-88 Compare with results of Dalton et al. (after 11)
In Figure 2-87 and Figure 2-88, the dashed curves are obtained with Som = 0.1, while the solid curves correspond to Som = 0. From the figures it can be seen that the value of Som only has an affect on the small values of relative permeability. Also, data obtained by Corey et al. has significant scattering, so it is hard to say whether the match is good or not. Overall, Stone’s method 1 fits the data of Dalton et al. better than Corey et al.’s data. Stone’s method: In the three-phase system, the sum of all three relative permeabilities is always less than or equal to unity. Stone assumes that it achieves its maximum value in the cases where So = 1 - Swc, Sw = Swc and Sg = 0 and So = 1 - Sgc, Sw = 0, Sg = Sgc. For these cases, kro = 1 and krg = krw = 012.
k ro k row k rwo k rog k rgo k rwo k rgo
(2-170)
Where, krow = relative permeability of oil in oil-water system krwo = relative permeability of water in oil water system krog = relative permeability of oil in oil gas system krgo = relative permeability of gas in oil gas system Stone claimed that this method predicts better values than method 1, especially at low oil saturations. It is also not required to estimate Som in this method; the model can predict the value of Sor. Comparison of Stone’s method 2 predictions with Corey et al.’s data is shown in Figure 2-89.
146
Figure 2-89 Stone’s Method 2 predictions and Corey et al.’s experimental data (after 12)
At low water saturation, the agreement is good. However, at higher water saturation, the predicted values are too low. Stone also compared the predicted Sor with Holmgren-Morse residual oil data of sandstone, see Figure 2-90.
Figure 2-90 Stone’s prediction of Sor and Holmgren-Morse’s data (after 12)
These data shows that as the initial free gas saturation increases, residual oil saturation decreases and residual gas saturation increases. Stone’s method 2 consistently predicts higher Sor. Dietrich and Bondor: It has been observed in experiments on water-wet systems that the water relative permeability is approximately a function of water saturation only, and that the gas relative permeability is a function of the gas saturation. However, the oil relative permeability seems to be a function of both the water and gas saturation. This model (proposed in 1976) takes into account the reduction in oil relative permeability caused by the presence of water and gas13. 147
Dietrich and Bondor realized that the oil permeability at connate water saturation predicted by Stone’s method 2 would be applicable only if krow and krog can be one. However, this is not always possible. Thus, they rewrite the Stone’s model as,
k ro
k row k rwo k rog k rgo k rocw
k rwo k rgo
(2-171)
Where krocw is the relative permeability of oil at connate water in oil water system. Hirasaki: Hirasaki modified Stone’s method 1 in 197313. Hirasaki’s model assumes that both water and gas may be flowing simultaneously in the pore space with the oil. He derives a three-phase oil relative permeability expression by considering the total reduction in oil relative permeability caused by the water and gas.
k ro k rowk rog S g 1 k row 1 k rog
(2-172)
Aziz and Settari: This model was published in 197910. Aziz and Settari suggested the use of the absolute permeability as the basis for calculating relative permeability and use krocw as a partial normalizing factor. For method 1:
k / k k rog / k rocw k ro k rocwS o* row rocw * * 1 S w 1 S g
(2-173)
k ro k rocwS o* w g
(2-174)
For method 2,
k ro k row k rwo k rog k rgo k rwo k rgo
(2-175)
Aziz and Settari said that Stone’s method 2 usually predicts too low oil permeabilities and method 1 usually predicts too high oil permeabilities. They believed that the use of as a free parameter is a convenient way to improve the predicted permeability. Fayers and Mathews: This model was published in 198414. Fayers and Mathews believed that the value of Som should be better predicted to improve the estimations of Stone’s method 1 (Som is a constant in Stone’s method 1). They suggested the use of a saturation dependent value of Som for Stone’s method 1,
S om S orw 1 S org 1
Sg 1 S wc S org
(2-176) (2-177)
In the presence of trapped gas,
S om S orw 0.5 S g
(2-178)
148
Where, Sorg = residual oil saturation in gas oil system Sorw = residual oil saturation in water oil system Baker said that this provides an improved fit to the residual oil data. Aleman: In 1986, Aleman suggested another approximation of Som for Stone’s method 110.
S om
Sg S w S wc S orw S org 1 S wc S orw 1 S wc S org
(2-179)
The free parameters and are used to fit the curvature of the zero oil permeability isoperm Aleman et al. later developed a statistical structural model for prediction of two-phase relative permeability based on a local volume average approach to a bundle of capillaries model15. In this model, pores are randomly distributed but there are no pore interconnections. It assumed that the saturation change is in the direction of decreasing intermediate wetting phase saturation. All other assumptions are the same as the ones made by Stone in the development of his methods. Aleman expanded this model for three-phase flow with the intermediate wetting phase relative permeability:
k ro k ro(I )
(2-180)
Where kro(I) is the relative permeability to oil predicted by Stone’s method 1 and is a correction term.
* * * * * * S *o k row 1 k rog 1 k row k rog k rwo k rgo
k
* rgo
1 k
* Where k row
* row
k
* rwo
k
* row
1 k
* rgo
k
* rog
(2-181)
k row * * * , and other parameters such as k rog , k rwo , k rgo are defined in a similar way. * 1 Sw
This model is sensitive to the value of Som, and may predict incorrect oil permeabilities for values of Som which are too small. The problem may be due to assumptions made in the model about the distributions of fluids. Larger values of Som give more reasonable predictions. Aleman recommended that this model should be used only if the predicted isoperms shows reasonable match with experimental data. Other methods Other methods included in this report are the models that were proposed by Baker, Pope, Blunt, and Moulu et al. Saturation-weighted interpolation: This method was proposed by Baker in 198810. This is a simple model of oil relative permeability that is based on saturation-weighted interpolation between water-oil and gas-oil data.
k ro
S w S wc k rwo S g S gr k rog S w S wc S g S gr
(2-182)
149
For the wetting and nonwetting phase,
k rw
S o S or k rwo S g S gr k rwg S o S or S g S gr
k rg
S o S or k rgo S w S wc k rgw S o S or S w S wc
(2-183)
(2-184)
These equations assume that the end points of the three-phase relative permeability isoperms coincide with the two-phase relative permeability data. The saturation-weighted interpolation method may give erroneous results if the two-phase relative permeability curves being interpolated between are very different. This is especially a problem if the end point saturation (Sorw and Sorg) of the oil-water and oil-gas curves differ significantly. Pope: Pope published his model in 198816. This model does not require two-phase data:
k ro k rocw a S o 1 S w
1 a S o 1 S g
(2-185)
Where, So Sw
Sg
1 S w S g S or 1 S wc S gr S or S w S wr 1 S wc S gr S or
S g S gr 1 S wc S gr S or
(2-186) (2-187) (2-188)
Where Sgr is the residual gas saturation. The parameters , , , and a can be obtained by matching the two-phase data, such that = + = eow, = + = eog, and a=1/2. eow and eog are obtained by curve fitting the oil-water and oilgas relative permeability data sets into exponent functions. Empirical model: This model was developed by Blunt and published in 200017. It is based on the saturation-weighted averages of the two-phase relative permeabilities, which includes gas and oil trapping. The model also allows for drainage of oil at low saturations. In this model, six relative permeability sets are required: krwo and krow from oil-water displacement, krog and krgo from gas-oil displacement, and krgw and krwg from gas-water displacement. If the medium is water wet or mixed wet and oil is spreading then layer drainage is also considered. In this case, the bulk oil relative permeability, kob, is obtained by:
S g S o* S ol Min * , S o S g
(2-189)
150
k ol
* k rg k rog * rgo
S ol2
*2 o
k S
(2-190)
k ob S ob k rog S o k ol S ol
(2-191) *
Sol represents the oil saturation in layer. S o is the saturation of oil when the layer drainage regime * starts, and k rog is the corresponding oil relative permeability. If experimental data does not indicate
* * * layer drainage, it is assumed that S o S orw . S g is the gas saturation when So = S o (all oil resides in * * layers). Note that S o and S g in this model are defined differently than before.
If the medium is not at the condition described above, then kob = krog(So). When trapping is considered: S gr
S gmax
1
Cg
(2-192)
1 C g S gmax S grw
1
(2-193)
S gmax
S gf
1 S g S gr 2
S hr
S hmax 1 C h S hmax
S
S gr 2
g
4 S g S gr Cg
(2-194) (2-195)
S hmax is the largest hydrocarbon saturation during displacement S hf
1 S h S hr 2
S h S hr 2
4 S h S hr Ch
(2-196)
S S k S b S S b S S k S S S S S S S S a k S b k S S S S S
S ofb min S o S ol , max S hf S gf S ol ,0
k rw
a S o
k ro
S
w
o
S oi a g
g
S wi ao k row
gr
ofb
w
rwo
wf
o
o
oi
g
g
gr
wi
g
o
ob
o
oi
g
gr
rwg
wf
(2-198)
gr
ofb
o
rgo
ofb
gr
So k ol S ol ao k row S hf bo k rgw S hf So S g k rg
g
(2-197)
(2-199)
S w S wi S o S oi S w S wi a g k row S gf bg k rgw S gf S o S oi a g k ob S gf bg k rgo S ogf
Sg So S g
a k S b k S g
row
hf
g
rgw
(2-200)
hf
151
ai max min
i
g0 0
bi max min o 0 0
i
,1,0
(2-201)
,1,0
(2-202)
crit N cap ,1,0 N cap
max min
max min 1
crit N cap ,1,0 N cap
(2-203)
(2-204)
Where, o0
= reference oil density
g0
= reference gas density
0
= reference density different between oil and gas
crit N cap = critical capillary number
Ncap = capillary number Ch is the Land trapping constant for hydrocarbon. In strongly water wet medium, it can be assumed that Ch = Cg = Co. In other cases, it can be assumed that Ch = min ( Co , Cg). Mathematical model: Moulu et al. developed a mathematical model to simulate the three-phase flow in porous media in 199918. The model assumes that the reservoir consists of fractal pores with gas in the center, and in the case of water wet pores, the water is residing near the wall. A fractal is defined as a shape made of parts similar to the whole in some way. This model assumes that krw is a function of Sw only and krg is a function of Sg and Sw. Essentially in this model:
The fluids flow together in the same fractal pore with the wetting phase along the rock walls, the gas phase in the center and the third phase in between The flow of each fluid is given by their relative permeabilities, which are analytical functions of a linear fractal dimension DL Capillary pressure is taken into account when calculating the relative permeabilities by using DL
DL is the slope obtained from log-log plot of the capillary pressure curve. This plot will give two different slopes. The sharp slope is used for the wetting phase flow (krw) at the wall and for the intermediate phase (kro). The other slope is use for gas flow (krg). For a water wet medium:
k rw S
4 DL 2 DL w
S
4 DL 2 DL wi
(2-205)
152
42 DDL 4 DL k ro k ro (2 Ph ) S L L S w S or 2 DL
k rg k rg max 1 S L
4
1 2 DL
(2-206)
(2-207) (2-208)
For cases when the medium is not water wet,
m
WI 1 with WI = wettability index 2
k rw mS w S wi
(2-209)
k ro 1 mS o S oi m k row S L S w S or
(2-210)
Where Soi = initial oil saturation The expression for gas relative permeability can still be used in this case since gas always shows its strongly nonwetting phase behavior. It is important to note that the values of DL used in krw (and kro) and krg are different. Comparison There are many correlations available for the prediction of three-phase relative permeability. These include the models of Stone, Hirasaki, Corey et al, Naar and Wygal, Land, Aleman, and Parker et al. The comparison by Baker shows that the models are often not very good predictors of the experimental data10. This means that there is a need for better relative permeability models of three-phase flow. Baker has shown that in most cases, saturation weighted interpolation between the permeabilities at the two-phase data set provided a better fit of the experimental data than other models. Baker has seen that most of the prediction methods fit the data sets equally well. This shows that each of the methods is capable of representing three-phase relative permeabilities in the high oil permeability region. Stone’s method 1, using saturation-weighted interpolation fits better than Stone’s method 2 for most cases. Hirasaki’s model is one of the worse. Parker’s model is generally equal to or better than Corey’s, Land’s, Naar and Wygal’s, and Aleman. Stone’s method 1 usually predicts too high oil permeability at low oil saturation, while Stone’s method 2 predicts too low oil permeability in the same region. With better values Som or the predictions of Stone’s method can be improved. However, the value of Som or are not always easily determined. Thus, saturation-weighted method seems to be a good model due to its simplicity and its ability to yield comparable results with Stone’s methods. Delshad and Pope claim that the model proposed by Pope agrees with several sets of data, which makes it more superior than other methods (Corey, Naar-Wygal, Land, Stone’s, Baker and Parker).
153
A comparison of all the methods presented is not seen in the literature, thus it is not known which method is most superior16. Effects of Low IFT or Nca on Relative Permeability In the literature, many researchers agree that as the interfacial tension (IFT) is reduced, relative permeabilities increases. There are people who believe that the relative permeability relationship with saturation is a linear function and there are people who report otherwise. Harbert noticed that the flow of low IFT fluids differs from that of conventional gas-oil-water systems in both sandstones and carbonates. He found that both water and oil relative permeability curves were found to shift upward, indicating that the two phases interfere less with each other as IFT is reduced19. This is shown in Figure 2-91.
Figure 2-91 Effect of capillary number on relative permeability (after 9)
In this figure, kro appears to be a straight line while krw still maintains its curvature (N in the figure is the capillary number, which is the same as Nca). Also, the change in krw as IFT changes is not significant. However, Sor did change; in this case it becomes less than 10%. For the lower capillary number, the water relative permeability curve is a function of saturation (no hysteresis). Harbert also mentioned that at the same Nca and IFT, a less permeable core has a higher residual saturation. This is expected because a core that was less permeability has pores that are not well connected, and the invading fluid cannot easily displace oil from these pores. Foulser et al. proposed a model in 1992 for relative permeability at high capillary number for the flow of two or more phases20. This model allows the residual oil saturation to be reduced as the capillary number is increased. As mentioned before, many people conventionally assumed that relative permeabilities are linear functions of saturations for low IFT fluids. However, this model assumes that relative permeability is non-linear, as shown in some reports. It also assumes that the flow of two or three phases can be
154
represented by the flow of droplets of fluid through the pore network. The geometry of the pore network was assumed to consist of capillary tubes. The relative permeability can then be derived as a function of saturation according to this equation,
k r k ro S* / S *
2-211
S* f ( N 1 / f ) / N
2-212
And
Where, kr=relative permeability of phase
k ro = end point relative permeability of phase
= viscosity of phase f = fractional flow of phase N = number of droplets N defines the droplet size relative to the capillary tube length. When N = 1 flow is segregated and the relative permeabilities are linear functions of saturation. As N approaches infinity, a “mixed flow” regime is created, in which the droplet size is very small compared to the tube length, giving the relative permeabilities as a function of saturation. In this model the relative permeabilities are a function of phase viscosities as well as saturations.
Effects of Temperature on Relative Permeability Even though there are contradictory reports about the effects of temperature on relative permeability, many researchers agreed that as temperature changes, relative permeability also changes. However, they have not agree on the effect of temperature on the adsorption mechanisms. Handy et al Handy et al.’s experimental results on two-phase flow in Berea sandstone indicate that relative permeability curves are affected by temperature, especially for low IFT cases21. As temperature increases, relative permeability to oil increases and relative permeability to water decreases at a given saturation while residual oil saturation decreases and irreducible water saturation increases. This is shown in Figure 2-92 and Figure 2-93 below.
155
Figure 2-92 Low IFT systems (after 21)
Figure 2-93 High IFT systems (after 21)
Figure 2-92 and Figure 2-93 show the effects of temperature on high and low-tension systems respectively. In both figures, the cross-over point shifts to higher Sw as temperature increases. In general, as temperature increases oil is able to flow much easier than water can. These results suggest an increase in water wettability of sandstone with temperature. The figures also show that temperature effects are more significant for low tension than for high tension systems. The residual oil saturation decreases significantly at higher temperature, but only small changes in irreducible water saturation (Swir) are seen in the high-tension systems. At any temperature, Swir for the low-tension system is smaller than that observed for the high-tension system. Relative permeability to oil and water both increase with increasing temperature up to 100°C for the low-tension system. For the low tension system, the changes in Swir with temperature are not significant, but Swir values for low tensions were lower than those at high tensions. This implies that the increase in temperature and decrease in IFT have opposite effects on Swir; increasing temperature increases Swir but as IFT is decreases, Swir decreased. Therefore it is possible that Swir in the low-tension system at high temperatures is controlled by both the IFT and wettability changes due to temperature.
156
For both low IFT and high IFT systems, krw/kro decreases with increasing temperature at a given saturation (as temperature increases, krw decreases and kro increases, thus krw/kro decreases). Thus krw/kro shifts toward higher water saturations with increasing temperatures. Sufi et al. Sufi et al. found that in the temperature range of 20°C to 85°C, the relative permeability of oil and water obtained from unconsolidated clean Ottawa sand does not change22. In a separate experiment, they saw that as temperature increases, Sor decreases. This observation is consistent with Handy et al. Sufi et al. also said that as temperature increases, the viscosity ratio of the oil water system decreases. This reduction in the viscosity ratio is responsible for the change of the fractional flow curve as seen in the second experiment. Thus a change in Sor is not necessarily due to a change in wettability. They also found that the irreducible water saturation increases as the temperature of the system increases. This observation is similar to that of Handy et al. The changes in irreducible water saturation can be illustrated to depend only on the change in the viscous force, irrespective of whether it is caused by a temperature or a rate change. Figure 10 demonstrates this:
Figure 2-94 Effects of viscous force on Swir (after 22).
From this figure, it can be seen that the change in Swi due to temperature or flow rate follows the same pattern. This indicates that changes in irreducible water saturation with temperature can be caused by changes in viscous forces not necessarily changes in wettability.
Other theories Nakornthap et al. believed that the adsorbed layer of polar components in crude oil on rock surfaces is thermodynamically unstable, thus it can be desorbed at high temperature and the rock may become more water wet23. From the analysis of three-phase relative permeability data, Maini and Okazawa also found that the studies of cleaned sand shows that relative permeability does not change as temperature changes. They 157
mentioned that Polikar et al. reported that the temperature effects could be controlled by the characteristics of the porous medium24.
Effects of Flow Rate on Relative Permeability Theoretically, relative permeability is not a function of flow rate. However, there has been experimental data which shows that the relative permeability changes as the flow rate changes. In general, the relative permeabilities for both phases were found to increase with increasing flow rate. Leverett et al. were the first to report about the influence of flow rate on relative permeability, however they later attributed this to capillary end effects. Crowell et al. and Greffen et al. found that injection rate has no affect on the viscous flow of water and gas. Also, Labastie et al. reported that relative permeabilities were independent of flow rate except near residual oil saturation25. However, there are many others who reported that relative permeability does change as flow rate changes. Heaviside et al. believed that for an intermediate wettability medium, at low flow rates the oil segments would exist in the center of pores, and blocks the flow of water. However, at high pressure gradients or high flow rates water would be forced through the throats with the oil coating the pore surface. Thus relative permeability can change with flow rate7. Heaviside et al. also found that the viscous force does not affect residual oil saturation for strongly water wet chalk. In the experiment, the pressure gradient is increased but no change in production rate is seen7. Handy et al. reported that rate affects relative permeability below 120 cc/hr. Above this rate, no significant rate effect was observed21. Also, they have seen that the rate effect does not significantly affect Sor or end point relative permeabilities Sandberg et al., however, reported that the relative permeability of the water phase increased very slightly with flow rate. In contrast, the oil phase relative permeability increases significantly. He reasoned that the rate effect on the oil phase may be a result of some tendency for the oil phase to flow in slugs6. Sufi et al. performed a more in depth study regarding the effects of flow rate on relative permeability. They noticed a trend in the relative permeability curves from the experimental results, which is shown in Figure 2-95.
158
Figure 2-95 Effects of flow rate on relative permeability (after 22)
This figure shows that the relative permeabilities for oil remain unchanged, while the water relative permeability curves are low at low flow rate, and become independent of rate when the flow is high (greater than 240 cc/hr). Therefore, it was assumed by Sufi et al. that the rate of 240 cc/hr represents the minimum rate required for front stability. As mentioned before, Handy said that in their experiment when the flow rate is larger than 120 cc/hr, no effect is seen. Thus Sufi and Handy disagree about the value of the critical stability flow rate. It was found by Akin et al. that gas relative permeability increases with increasing total flow rate. They have also seen that the effects on brine and hexane relative permeability are much more compared to gas relative permeability. The oil and water capillary pressure values for higher flow rates were also greater at high brine saturation data. Thus it is expected that relative permeability should change with flow rate25. Heaviside et al. reported that for strongly wetted systems, at high IFT, there is a negligible rate dependency, even at very high flow rate. However, in cases of low IFT, rate dependency can be seen7.
Effects of Viscosity on Relative Permeability Researchers still haven’t reached a conclusion regarding the effect of viscosity on relative permeability. A number of researchers report that viscosity does not affect relative permeability at all, while others say that it does. Sandberg and Gournay studied the effects of oil viscosity on relative permeability of sandstone outcrop. They reported that the relative permeability of both phases is not a function of the nonwetting phase viscosity6. Odeh mentioned that Leverett had shown earlier that the wide range of viscosity had essentially no effect on relative permeability26.
159
Odeh developed a two-phase flow model which assumed that the porous medium consists of straight circular capillaries of different radii. He also assumed that there are no interconnections among the capillaries and no mass transfer across the oil water interface. During waterflooding, a layer of water is left between the oil and the walls of the capillary26. The model is as follow:
n 4 o 1 n n rn3 1 r k ro S o f k , x w m 4 ro mrm
(2-213)
1
n
r S o f k , x ro
nrn
4
1 m mrm4
(2-214)
1
Where, kro = relative permeability of oil o = viscosity of oil w = viscosity of water m = total number of capillaries in porous sample n = number of capillaries through which one phase flow k = a constant n = thickness of water film = rc - rn rc = radius of any capillary rn = radius of oil phase in the capillary
r r In Odeh’s model, f k , x is a series in x where rx is the radius of the smallest capillary which is filled ro ro with oil at an oil saturation So. This equation indicates that relative permeability to oil is a function of saturation as well as the viscosity ratio, the thickness of the molecular layer and pore sizes. Odeh performed experiments to investigate the effects of viscosity ratio on the relative permeability of the nonwetting phase (kro). The results are shown in Figure 12.
160
Figure 2-96 Effect of viscosity on relative permeability (after 26)
The figure shows that the maximum differences in relative permeability values due to viscosity ratio variation occur at the point of minimum brine saturation. This increase tends to a limit as the viscosity ratio becomes larger. As mentioned before, Sufi et al. found that as temperature increases, viscosity ratio (oil to water) decreases, which changes the fractional flow of water and oil22. This in turn changes the relative permeability curves. Thus it is possible that viscosity ratio does affect the relative permeability curves.
Carbonates Sandstones are usually being investigated in laboratory, thus the majority of the relative permeability data shows the effects that are specific to sandstones only. Very little data were obtained on carbonates in literature. From the limited literature, Schneider et al. showed that for oil wet carbonate samples, they found no effect on the oil (wetting phase) relative permeability when comparing two- and three-phase measurements2. This means that relative permeability of oil is a function of its saturation only. This observation is consistent with what other people have seen. Also, the water relative permeability was lowered by the trapped gas, showing the interaction between the two nonwetting fluids. Schneider et al. also reported that the flow behavior of the uniform porosity carbonate samples tested was similar to that of consolidated sandstones2. Thus rock type does not seem to influence the flow relationships other than through its wetting preference. However, due to the surface minerals most of the time carbonate rocks are oil wet, while sandstones are usually water wet. Also, carbonates usually have vugs. When large vugs exist in the reservoir, the core sample used in the experiment might not be representative of the reservoir. When laboratory data of carbonates is being used to predict threephase flow, a greater uncertainly has been added. Also, most of the relative permeability models were derived from the assumption that the medium if water wet. Attention must be paid to the assumptions of the model to select the appropriate one for carbonates. 161
It is expected that IFT will affect relative permeability and recovery in the same way regardless of the rock properties since IFT is a fluid property. The effects of temperature on relative permeability in carbonates have not been reported in literature. It can be expected that a change in temperature will change the characteristics of relative permeability of fluids in carbonates. However, no data is available to identify what those changes are. Again, nothing can be said about the effects of flow rate and viscosity on the flow of fluids in carbonates since there is no data available.
Relative Permeability of Heavy Oil Systems Relative permeability includes contributions from a number of different variables, each causing some resistance to flow. The resistance to flow of a given phase in a multiphase situation depends primarily on how this phase distributes itself within the porous medium in the presence of other fluids. The variables that affect fluid distribution in a two-phase system include: pore structure and pore size distribution, wettability, saturation history, interfacial tension, interfacial viscosity, viscosity ratio, density ratio, and flow rate24. For heavy oil systems, it may no longer be safe to assume that the local fluid distribution at a given saturation depends only on the first three factors listed above, and is independent of the viscosity ratio and fluid velocity involved as many people have assumed. Furthermore, while relative permeability of a fluid depends on its own distribution within the pore space of the medium, this may not be the case in heavy oil systems. Assuming that the residual oil is distributed within the porous medium in the form of small globules, if the viscosity of the oil is very high, these globules can behave like solid particles and may plug pore throats more efficiently than globules of a low viscosity of oil24. Also, the use of relative permeability model must also be carefully chosen since most of this model does not take viscosity into account. More experiments with heavy oil are required to shed more light regarding the effect of each parameter on relative permeability. It cannot be assumed that these parameters will affect relative permeability of oil in the same way described earlier.
Conclusions Relative permeability for water wet and oil wet systems is distinctively different for two-phase flow. In water oil system, as the system becomes more oil wet, relative permeability of water increases, relative permeability of oil decreases and the cross over point occurs at smaller water saturation. In three-phase systems where the medium is strongly wetted, the relative permeability of the wetting phase is a function of its saturation only. There is also some evidence that the relative permeability of the wetting phase and the non-wetting phase is a function of its own saturation only in strongly wetted medium. This indicates that the relative permeability of the wetting and non-wetting phase is a function of its own saturation only. This means that there is little or no hysteresis between drainage and imbibition relative permeability of the wetting phase and the non-wetting phase. However, for the intermediate wetting phase, its relative permeability has been seen to be a function of other saturations, as well as saturation history. Thus significant hysteresis in relative permeability of this phase is seen. The steady state and unsteady state method can be used to evaluate relative permeability of each phase. However, the steady state method yields more reliable data. Most people recommend that this method should always be used. 162
The experiment to find relative permeability of three-phase systems is very complex, thus many models were developed. Corey et al. published the first three-phase relative permeability model in 1956. This model was based on the assumptions that the pore space in the medium can be represented by a bundle of capillary tubes. Other methods that made the same assumption include: Naar and Wygal, Land, Parker et al. Stone’s models are based on the channel flow theory. He published method 1 in 1970 and method 2 in 1973. Since then many people have modified these models. These people include: Dietrich and Bondor, Hirasaki, Aziz and Settari, Fayers and Mathews, Aleman, and Parker et al. There were also other models published by Pope, Baker, Blunt and Moulu et al. Baker compared the models of Stone, Hirasaki, Corey et al., Naar and Wygal, Land Aleman and Parker et al. He found that Stone’s method 1 could fit data better if the estimation of Som is good. However, he found that his own method, which is a simple interpolation between the various set of data, yields the best fits. Delshad and Pope evaluated the predictions of Corey, Naar and Wygal, Land, Stone, Baker, Parker and Pope. They found that Pope’s model fits better than the rest. Up to this point, an extensive comparison of all the models with all the experimental data is not seen in literature, so it cannot be concluded which model is best. However, Stone’s methods are the most commonly used. There are some parameters that researchers have seen to affect relative permeability characteristics (of two-phase systems). Many have seen that as interfacial tension decreases, the relative permeability of water and oil increases. The reduction in IFT reduces the interference between two phases, making them able to flow better. The relationship between relative permeability and saturation of low IFT system is still a point of debate; many say that this relationship is linear, while others say that it is not. When temperature changes, the relative permeability also changes. Handy et al. reported that the effect of temperature is more significant in low IFT systems. In both high and low IFT, Sor decreases and Swir increases with increasing temperature. The cross over point also shifts to higher Sw values. Thus Handy et al. believed that as temperature increases, the system becomes more water wet. Nakornthap et al. explained that this increase in water wetness is due to the breakdown of the organic layer on the surface of the rock. Sufi et al. disagree with this. They believe that the change in relative permeability is due to the change in viscous forces, not a change in wettability. Flow rate was seen to affect relative permeability. Handy et al., Sandberg et al. and Sufi et al. reported that as the flow rate increases, relative permeability of water increases, while relative permeability of oil decreases. However, Handy and Sufi disagree with the critical stable flow rate. The effect of viscosity on relative permeability is also investigated. Sandberg said that oil viscosity has no effect on relative permeability. However, Odeh reported that the effect of viscosity is most significant at connate water saturation. Also, Sufi et al believed that changes in viscosity lead to changes in relative permeability. The studies of relative permeability and the effects of other parameters were mainly done on sandstones (or sandpacks) with conventional oil. For carbonate systems, if the rock properties are uniform, the same flow characteristics seen before can be expected. However, carbonates usually are oil wet while the majority of the models assumed that the medium is water wet. Thus care must be taken when choosing a model to predict three-phase flow. Also, carbonates have vugs, so the core sample evaluated might not be representative. With heavy oil systems, the flow characteristics might be different. The effects of these parameters on relative permeability may be different as well. More 163
experiments must be conducted to investigate the effects of each parameter on relative permeability. The selection of an appropriate model to predict relative permeability of viscous oil should be made with care.
164
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
Anderson, W.G., “Wettability Literature Survey – Part 5: The Effects of Wettability on Relative Permeability”, SPE 16323. Schneider, F.N. and Owens, W.W., “Sandstone and Carbonate Two- and Three-Phase Relative Permeability Characteristics”, Trans., AIME (1970) 249, 75-84. Corey, A.T., Rathjens, C.H., Henderson, J.H., “Three-Phase Relative Permeability”, Trans., AIME (1956) 207, 349-351. DiCarlo, D.A., Sagni, A. And Blunt, M.J., “The Effect of Wettability on Three-Phase Relative Permeability”, SPE 49317. Honarpour, M. and Mahmood, S.M., “Relative-Permeability Measurements: An Overview”, SPE 18565. Sandberg, C.R. and Gournay, L.S., “The Effect of Fluid-Flow Rate and Viscosity on Laboratory Determinations of Oil-Water Relative Permeabilities”, Trans., AIME (1958) 213, 36-43. Heaviside, J., Brown, C.E., and Gamble, I.J.A., “Relative Permeability for Intermediate Wettability Reservoirs”, SPE 16968. Naar, J. and Wygal, R.J., “Three-Phase Imbibition Relative Permeability”, Trans., AIME (1961) 222, 254258. Land, C.S., “Calculation of Imbibition Relative Permeability for Two- and Three-Phase Flow From Rock Properties”, SPE 1942. Baker, L.E., “Three-Phase Relative Permeability Correlations”, SPE 17369. Stone, H.L., “Probability Model for Estimating Three-Phase Relative Permeability”, SPE 2116. Stone, H.L., “Probability Model for Estimating Three-Phase Relative Permeability and Residual Oil Data”, SPE 19439. Dietrich, J.K. and Bondor, P.L., “Three-Phase Oil Relative Permeability Models”, SPE 6044. Fayers, F.J. and Mathews, J.D., “Evaluation of Normalized Stone’s Methods for Estimating Three-Phase Relative Permeabilities”, Society of Petroleum Engineers Journal (April 1984) 24, 225-232. Aleman-Gomez, M., Rmamohan, T.R. and Slattery, J.C., “ A Statistical Structural Model for Unsteady-State Displacement in Porous media”, SPE 13265. Delshad, M. and Pope, G.A., “Comparison of the Three-Phase Oil Relative Permeability Models”, Transport in Porous Media 4 (1989), 59-83. Blunt, M.J., “An Empirical Model for Three-Phase Relative Permeability”, SPE 67950. Moulu, J.C., Vizika, O., Egermann, P. and Kalaydjian, F., “A New Three-Phase Relative Permeability Model For Various Wettability Conditions”, SPE 56477. Harbert, L.W., “Low Interfacial Tension Relative Permeability”, SPE 12171. Foulser, R.W.S., Goodyear, S.G. and Sims, R.J., “Two- and Three-Phase Relative Permeability Studies at High-Capillary Numbers”, SPE 24152. Torabzadeh, S.J. and Handy, L.L., “The Effect of Temperature and Interfacial Tension on Water/Oil Relative Permeabilities of Consolidated Sands”, SPE 12689. Sufi, A., Ramey, H. And Brigham, W., “Temperature Effects on Relative Permeabilities of Oil-Water Systems”, SPE 11071. Nakornthap, K. And Evans, R.D., “Temperature-Dependent Relative Permeability and Its Effect on Oil Displacement by Thermal Methods”, SPE 11217. Maini, B. and Okazawa, T., “Effects of temperature on heavy oil/water relative permeability of sand”, Journal of Canadian Petroleum Technology, May-June 1987, 33-41. Akin, S. And Demiral, B., “Effects of Flow Rate on Three-Phase Relative Permeabilities and Capillary Pressures”, SPE 38897. Odeh, A.S., “Effect of Viscosity Ratio on Relative Permeability”, Trans., AIME (1959) 216, 346-353.
165
BIBLIOGRAPHY Aguilera, R.. Relative Permeability Concepts for Predicting the Performance of Naturally Fractured Reservoirs. JCan PetT,.22,41, (1982) Amaefule, J.O. and Handy, L.I., The Effect oflnterfacial Tensions on Relative Oil/Water Permeabilities of Consolidated Porous Media. Soc Pet E J, 371, (1982) Bardon, C. and Longeron, D., Influence of Very Low Interfacial Tensions on Relative Permeability. Soc Pet EJ, 20, 391,(1980) Batycky, J.P., Mirkin, M.I., Besserer, G.J., Jackson, C.H., Miscible and Immiscible Displacement Studies on Carbonate Reservoir Cores. JCanPetT,2L, 104,(1981A) Batycky, J.P., McCaffery, F.G., Hodgins, P.K., Fisher, D.B., Interpreting Relative Permeability and Wettability from Unsteady-State Displacement Measurements. Soc Pet E J, 21, 296, (198IB) Bemsten, R.G., Using Parameter Estimation Techniques to Convert Centrifuge Data into a Capillary Pressure Curve. SPE paper 5026 presented at the SPE-AIME 49th Annual Meeting, Houston, Texas, (Oct. 6-9 1974) Broadbent, S.R. and Hammersley, J.M., Percolation Processes: 1. Crystals and Mazes. Camb Phil Soc, 53, (3), 629,(1957) Burden, R.L-, Faires, J.D., Reynolds, A.C...Numerical Analysis. Second Edition, Prindle, Weber and Schmidt, 1981 Casulli, V. and Greenspan, D., Numerical Simulation of Miscible and Immiscible Fluid Flows in Porous Media. Soc Pet EJ, 20, 635, (1980) Chandler, R., Koplik, J., Lerman, K., Willemsen, J.F., Capillary displacement and Percolation in Porous Media. J Fluid Mec, .119. 249, (1982) Chatzis, I., Dullien, F.A.L., Mercury Porosimetrv Curves of Sandstones. Mechanisms of Mercury Penetration and Withdrawal. Powd Tech, 29,117, (1981) Chatzis, I., Morrow, N.R., Lim, H., Magnitude and Detailed Structure of Residual Oil Saturation. SPE paper No. 10681 presented at the 1982 SPE/DOE Third Joint Symposium of Enhanced Oil Recovery of the Soc. ofPetr. Engineers held in Tulsa, OK, (April 4-7 1992) Chatzis, I., Dullien, F.A.L., Mise en Oeuvre de la Theorie de la Percolation Pour Modeliser Ie Drainage des Millieux Poreux et la Permeabilite Relative au Liquide non Mouillant Injecte. Revue de L'lnstitut Francais du Petrole 37, 183, (Mars-Avril, 1982) Chatzis, I., Dullien, F.A.L., Modelling Pore Structure by 2-D and 3-D Networks with Application to Sandstones. Pet. Soc. ofCIM paper No 7607 presented at the 27th Annual Technical Meeting of the Pet. Soc. ofCIM, Calgary, (June 711 1976B) Chatzis, I., Morrow, N.R., Correlation of Capillary Number Relationships for Sandstones. Soc Pet E J, 24, (Nov. 1984) Chatzis, I., A Critical Study of the Applicability of 2-D and 3-D Networks of Capillary Tubes to Simulate the Drainage Capillary Pressure Curve of Sandstones. M.A.Sc. Thesis, University of Waterloo, (1976)
166
Chatzis, I., A Network Approach to Analyse and Model Capillary and Transport Phenomena in Porous Media. Ph. D. Thesis, University of Waterloo, (1980) Chatzis, I., Dullien, F.A.L., A Network Approach to Analyse and Model Capillary and Transport Phenomena in Porous Media. Presented at Symposium sur les effets D'echelle en Millieux Poreux, Thessaloniki, Greece, (Aug.-Sept. 1978) Chatzis, I., Dullien, F.A.L., Dynamic Immiscible Displacement Mechanisms in Pore Doublets: Theory Versus Experiment. J Coll I Sc, 91, 1, (1983) Chatzis, I., Visualization of Oil Displacement Mechanisms and Enhanced Oil Recovery. Proceedings of the 27th annual conference of the Ont. Petr. Institute held in London Ont., (Oct. 1983). Coats, K.H. and Smith, B.D., Dead End Pore Volume and Dispersion in Porous Media. Soc Pet E J, 4, 73, (1964). Core Laboratories Inc., A Course in the Fundamentals of Core Analysis. Da Prat, G., Ramey, Jr. H.J., Cinco-Ley, H.. A Method to Determine the Permeability-Thickness Product for a Naturally Fractured Reservoir. J P T, .34, 1364, (1982) Diaz, C.E., Simulation of Fluid Distributions and Oil-Water Capillary Pressure Curves in Water Wet Porous Media Using Network Modeling Techniques. M. A. Sc. Thesis, University of Waterloo, (1984). Dodds, J.A., The Pore Size Distribution and Dewatering Characteristics of Packed Beds and Filter Cakes. Ph. D. Thesis, Loughborough University of Technology (1968). Downie, J.and Crane, F.E., Effect of Viscosity on Relative Permeability. Soc Pet E J, i, (2), 59, (1961) Dullien, F.A.L-, Chatzis,I., and El-Sayed, M.S., Modelling of Transport Phenomena in Porous Media by Networks Consisting of Non-Uniform Capillaries. Presented at the 57th Annual Fall Technical Conference and Exhibition of The Society of Petroleum Engineers of AIME, New Orleans, (Oct. 3-6, 1976). Dullien, F.A.L., Wood's Metal Porosimetrv and its Relation to Mercury Porosimetry. Powd Tech, 29,109, (1981) Dullien, F.A.L., Porous Media: Fluid Transport and Pore Structure. Academic Press, (1979) Fatt, I., The Network Model of Porous Media I. capillary Pressure Characteristics. Pet. Trans., AIME, 207 114, (1956A) Fatt, I., The Network Model of Porous Media II. Dynamic Properties of a Single Size Tube Network. Pet. Trans., AIME, 207 160, (1956B) Fatt, I., The Network Model of Porous Media III. Dynamic Properties of Networks with Tube Radius Distribution. Pet. Trans., AIME, 207. 164, (1956C) Freeman, D.L. and Bush, D.C., Low Permeability Laboratory Measurements by Non-Steady State and Conventional Methods. Soc Pet E J, 23, 928, (1983) Geffen, T.M., Owens, W.W-, Parrish, D.R., Morse, R.A., Experimental Investigation of Factors Affecting Laboratory Relative Permeability Measurements. Pet. Trans., AIME, j92, 99, (1951) George, A., Liu, J. W-H, Computer Solution of Large Sparse Positive Definite Systems. Prentice Hall, (1981) Gilman, J.R. and Kasemi, H. Improvements in Simulation of Naturally Fractured Reservoirs. Soc Pet E J, 23,695,(1983) Gladwell, G.M.L-, Mansour, W.M-, Matrices and linear equations. University of Waterloo, (1970)
167
Greenkom, R.A. Steady Flow Through Porous Media. AICHEJ, 27, (4), 529, (1981) Hagoort, J.. Oil Recovery bv Gravity Drainage. Soc Pet E J, 20, 139, (1980) Haring, R.E. and Greencom, R.A. A Statistical Model of a Porous Medium with Nonuniform Pores. AICHEJ, 16, (3), 447, (1970) Heiba, A.A., Davis, H.T. and Scriven, L.E., Statistical Network Theory of Three-Phase Relative Permeabilities. SPE paper No 12690 presented at the SPE/DOE Fourth Symposium of Enhanced Oil Recovery held in Tulsa OK, (April 15-18,1984) Heiba, A.A., Sahimi, M., Scriven, L.E., Davis, H.T., Percolation Theory of Two Phase Relative Permeability. SPE paper No 11015, presented at the 57th. Annual Fall Technical Conference of SPE-AIME, New Orleans, (Sept. 26-29, 1982) Honarpour, M., Koederitz, L.F., Harvey, H.A.I, Empirical Equations for Estimating Relative Permeability in Consolidated Rock. JPT, 34, 2905, (1982) Hoshen, J., Kopelman, R., Percolation and Cluster Distribution I. Cluster Multiple Labelling Technique and Critical Concentration Algorithm. Phys Rev B., .14, 8, 3438, (1977) Jha, K.N., Enhanced Oil Recovery - An Introduction. Chem Can, (Sept. 1982). Johnson, E.F., Bossier, D.P., Naumann, V.O., Calculation of Relative Permeability from Displacement Experiments. Pet. Trans., AIME, 216, 370, (1959) Jones, P.O. and Owens, W.W., A Laboratory Study of Low-Permeability Gas Sands. JPT. 32. 1631. (1980) Jones, S.C. and Roszelle, W.O., Graphical Techniques for Determining Relative Permeability from Displacement Experiments. J P T, 31, 807, (1979) Kantzas, A., Computer Simulation of Relative Permeability Properties of Porous Media. M.A.SC. Thesis, University of Waterloo, (1985) Kassemi, H., Low Permeability Gas Sands. J P T, 34, 2229, (1982) Keelan, D.K., A Practical Approach to Determination of Imbibition Gas-Water Relative Permeability. J P T,28, 199,(1976) Keighin, C.W. and Sampath, K., Evaluation of Pore Geometry of Some Low-Permeability Sandstones-Unita Basin. J P T, 34, 65, (1982) Kirkpatrick, S., Percolation and Conduction. Rev M Phys, 45, (4), 574, (1973) Koplik, J. and Lasseter T.J., Two-Phase Flow in Random Network Models of Porous Media. Society of Petroleum Engineers paper No. 11014. Koplik, J., Creeping Flow in Two-Dimensional Networks. J Fluid Mec, 119,219, (1982). Lai, F.S.Y., Reduction of the So Called 'Irreducible' Wetting Phase Saturation - A Study Using Water- Wet and OilWet Transparent Micromodels and Sandstone Samples. Ph. D. Thesis, University of Waterloo, (1984). Larson, R.G., Percolation in Porous Media with Application to Enhanced Oil Recovery. M.S. Thesis, University of Minnesota (1977)
168
Larson, R.G., Scriven, L.E. and Davis, H.T., Percolation Theory of Two Phase Flow in Porous Media. Chem Engng Sci, 36, 57, (1981) Larson, R.B. and Morrow, N.R., Effects of Sample Size on Capillary Pressures in Porous Media. Powd Tech,30,lZ3,(1981). Larson, R.G., Scriven, L.E. and Davis, H.T., Displacement of Residual Non-Wetting Fluid From Porous Media. Chem Engng Sci, 36, 75, (1981). Lenormand, R., Zarcone, C. and Sarr, A., Mechanisms of the Displacement of One Fluid by Another in a Network of Capillary Ducts. J Fluid Mec, 135, 337, (1983) Leverett, M.G., Flow of Oil-Water Mixtures Through Unconsolidated Sands. Trans AIME, 132. 149, (1939) Levine, S. and Lowndes, J., Application of Percolation Theory to Capillary Pressure-Saturation Effects in a Porous Medium. Unpublished paper, (1983) Lin, C. and Slattery, J.C., Three-Dimensional Randomized. Network Model for Two-Phase Flow Through Porous Media. AICHEJ,28, 2,311, (1982) Loren, S.D., Permeability Estimates from NML Measurements. J P T, 24,923, (1972) Lucia, F.S., Petrophvsical Parameters Estimated from Visual Descriptions of Carbonate Rocks: A Field Classification of Carbonate Pore Space. JPT, 35, 629, (1983) Marle, C.M.. Multiphase Flow in Porous Media. Institute Francais Du Petrole, Gulf Publishing Company, (1981) McCaffery, F.G., The Effect of Wettability on Relative Permeability and Imbibition in Porous Media. Ph. D. Thesis, University of Calgary, (1973). Mann, R., Androutsopoulos, G.P. and Golshan, A., Application of a Stochastic Network Pore Model to Oil Bearing Rock with Observations Relevant to Oil Recovery. Chem Engng Sci, 36, 337, (1981) Mohanty, K.K.and Salter, S.J-, Multiphase Flow in Porous Media: II. Pore Level Modeling. SPE paper No. 11018, presented at the 57th Annual Fall Technical Conference and Exhibition of SPE of AIME held in New Orleans in LA (Sept. 26-29 1982) Mohanty, K.K., Scriven, T.H. and Davis, E.L., Physics of Oil Entrapment in Water Wet Rock. SPE paper No. 940, presented at the 55th Annual Fall Technical Conference of the Society of Petroleum Engineers of AIME, Dallas, Texas, (Sept. 21-24 1980) Moran, J.H. and Papaconstantinou, C.M., A Novel Dynamic Measurement of Permeability. Soc Pet E J, 2L 670, (1981) Morgan, J.T. and Gordon, D.T., Influence of Pore Geometry on Water-Oil Relative Permeability. J P T, 22, 1199,(1970) Morrow, N.R., Chatzis, L, Siegel, D., Taber, J.J., Corrections to In-Situ Measurements of Residual Oil for Flushing at the Wellbore. Pet. Soc. of CIM paper No 83-34-26 presented at the 34th. Annual Technical Meeting of the Petroleum Society of CIM, Banff, (May 10-13 1983) Morrow, N.R., and Chatiudompunth, Application of Hydraulic Radii Concept to Multiphase Flow. P.R.R.C. Report, 82-85, (1982)
169
Morrow, N.R., Chatzis, I. and Lim, H., Relative Permeabilities at Reduced Residual Saturation. Pet- Soc. of CIM paper No 83-34-28, Presented at the 34th Annual Technical Meeting of the Petroleum Society of CIM, Banff, (May 1983) Morrow, N.R. and McCaffery, F.G., Displacement Studies in Uniformly Wetted Porous Media. Wetting, Spreading and Adhesion, J.F. Padday, ed. Academic Press (1978) 289-319 Naar, J. and Henderson J.H., An Imbibition Model - Its Application to Flow Behaviour and the Prediction of Oil Recovery. Soc Pet E J, 1, (2), 61, (1961) Naar, J. and Wygal, R.G., Three-Phase Imbibition Relative Permeability. Soc Pet E J, 1, (5), 254, (1961) Nakamura, M.. A Method to Improve the Effective Medium Theory Towards Percolation Problem. Jphys C, .15, 749, (1982) Nakamura, M., Evaluation of the Effective-Medium Theory Towards Percolation Problem. Phys Rev B, 28, 4,2216,(1983) Nakamura, M., Conductivity for the Site-Percolation Problem by an Improved Effective-Medium Theory. Phys Rev B, 29, 3691, (1984) Nash J.C., Compact Numerical Methods For Computers: linear algebra and function minimization. Adam Hilger Ltd, (1979) Ng, K..M., Payatakes, A.C., Stochastic Simulation of the Motion. Break-up and Standing of Oil Ganglia in Water-Wet Granular Porous Media During Immiscible Displacement. AICHEJ,26, 3, 419, (1980) Odeh, A.J., Effect of Viscosity Ratio on Relative Permeability. Trans AIME, 216. 346, (1959) Parsons, R.W., Directional Permeability Effects in Developed and Unconfined Five Spots. J P T, 24, 487, (1972) Pathak, P., Davis, H.T. and Scriven, L.E., Dependence of Residual Non-Wetting Liquid on Pore-Topology. SPE paper No 11016, presented at the 57th Annual Fall Technical Conference, New Orleans., (Sept. 26-29,1982) Payatakes, A.C., Ng, K.M. and Flumerfelt, R.W., Oil Ganglion Dynamics During Immiscible Displacement: Model Formulation. AICHEJ,26, 3,430, (1980) Peaceman. D.W.. Fundamentals of Numerical Reservoir Simulation. Elsevier, (1977) Quiblier, J.A., A New Three-Dimensional Modeling Technique for Studying Porous Media. J Coll I Sc, 98, 1, 84,(1984) Ramakrishnan, T.S. and Wasan, D.T., The Relative Permeability Function for Two-Phase Flow in Porous Media: Effect of Capillary Number. SPE paper No 12693 presented at the SPE/DOE Fourth Symposium of Enhanced Oil Recovery held in Tulsa, OK, (April 15-18 1984). Rose, W.D., A New Method to Measure Directional Permeability. J P T, 34, 1142, (1982) Salter, S.J. and Mohanty, K.K., Multiphase Flow in Porous Media": I. Macroscopic Observations". SPE paper No. 11017, presented at the 57th Annual Fall Technical Conference and Exhibition of SPE ofAIME held in New Orleans in LA (Sept. 26-29 1982) Sehwartz, H.R., Rutishanser, H., Stiefel, E., Numerical Analysis of Symmetric Matrices. Prentice Hall, (1973) Shankar, P.K., Experimental Investigation of Two-Phase (Oil-Brine) Relative Permeability Characteristics in Mixed Wet Sandstone Systems With Reference to Oil Recovery Efficiency. Ph.D. Thesis, University of Waterloo, (1979) Shante, V.K.S. and Kirkpatrick, S., An Introduction to Percolation Theory. Adv Phys, 42, 385, (1971)
170
Sigmund, P.N. and McCaffery, F.G., An Improved Steady State Procedure for Determining the Relative-Permeability Characteristics of Heterogeneous Porous Media. Soc Pet E J, 19,15, (Feb. 1979) Slattery. S.C. Multiphase Viscoelastic Flow Through Porous Media. AICHEJ,14 50, (1968) Soo, G. and Slattery, J.C., Interfacial Tension Required for significant Displacement of Residual Oil. Soc Pet E J, 19, 83, (April 1979). Swanson, B.F., A Simple Correlation Between Permeability and Mercury Capillary Pressures. J P T, 33, 2498,(1981) Talash, A.W., Experimental and Calculated Relative Permeability Data for Systems Containing Tension Additives. SPE paper No 5810 presented at the Improved Oil Recovery Symposium held in Tulsa, OK, (March 22-24, 1976) Thomas, R.D. and Ward, D.C., Effect of Overburden Pressure and Water Saturation on Gas Permeability of Tight Sandstone Cores. J P T, 24, 120, (1972) Thomeer, J.H., Air Permeability as a Function of Three-Pore Network Parameters. J P T, 35, 809, (1983) Torabzadeh, S.J. and Handy, L.L., The Effect of Temperature and Interfacial Tension on Water/Oil Relative Permeabilities of Consolidated Sands. SPE paper No 12689 presented at the SPE/DOE Fourth Symposium of Enhanced Oil Recovery held in Tulsa, OK, (April 15-18, 1984) Walls. J.D., Tight Gas Sands-Permeability. Pore Structure, and Clay. J PT, 22, 2708, (1982). Wardlaw, N.C., Pore Geometry of Carbonate Rocks as Revealed by Pore Casts and Capillary Pressure. Am Ass Pet G Bull, 60, (2), 245, (1976). Wardlaw, N.C., The Effects of Geometry. Wettability, Viscosity and Interfacial Tension on Trapping in Single PoreThroat Pairs. J Can Pet T, 22, 21, (1982). Warren, J.E. and Price, H.S., Flow in Heterogeneous Porous Media. Soc Pet E J, 1, 3, 153, (1961). Westlake, J.R., A Handbook of Numerical Matrix Inversion and Solution of Linear equations. John Wiley and Sons, (1968). Winterfeld, P.H., Scriven, L.E. and Davis, H.T., Percolation and Conductivity of Random Two-Dimensional Composites. J PhysC, 14,2361, (1981). Yadav, G.D., Chatzis, I. and Dullien, F.A.L., Microscopic Distribution of Two Immiscible Fluids in Pore Space of Sandstone. Chem Engng Sci, 39. in press, (1984A). Yadav, G.D., Dullien, F.A.L., Chatzis, I. and Macdonald, I.F., Microscopic Distribution of Wetting and Non-wetting Phases in Sandstones During Immiscible Displacements. SPE paper No 13212 presented at the 59th Annual Technical Conference and Exhibition held in Houston, Texas, (Sept. 16-19 1984B). Yadav, G.D. and Mason, G., The Onset of Blob Motion in a Random Sphere Packing of the Surrounding Liquid. Chem Engng Sci, 38,9,1461, (1983).
171
3 .
C H A P T E R
3
MOLECULAR DIFFUSION Introduction If a few crystals of a colored material like copper sulfate are placed at the bottom of a tall bottle filled with water, the color will slowly spread through the bottle. At first the color will be concentrated in the bottom of the bottle. After a day it will penetrate upward a few centimeters. After several years the solution will appear homogeneous. The process responsible for the movement of the colored material is molecular diffusion that often called simply diffusion, which is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. In gases, diffusion progresses at a rate of about 5 cm/ min; in liquids, its rate is about 0.05 cm/min; in solids, its rate may be only about 0.00001 cm/min. This slow rate of diffusion is responsible for its importance. In many cases, diffusion occurs sequentially with other phenomena. When it is the slowest step in the sequence, it limits the overall rate of the process. For example, diffusion often limits the efficiency of commercial distillations and the rate of industrial reactions using porous catalysts. It limits the speed with which acid and base react and the speed with which the human intestine absorbs nutrients. The result of diffusion is a gradual mixing of material. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration, but it is important to note that diffusion also occurs when there is no concentration gradient. In gases and liquids, the rates of these diffusion processes can often be accelerated by agitation. For example, the copper sulfate in the tall bottle can be completely mixed in a few minutes if the solution is stirred. This accelerated mixing is not due to diffusion alone, but to the combination of diffusion and stirring. Diffusion still depends on random molecular motions that take place over smaller distances. The agitation or stirring is not a molecular process, but a macroscopic process that moves portions of the fluid over much larger distances. After this macroscopic motion, diffusion mixes newly adjacent portions of the fluid. In other cases, such as the dispersal of pollutants, the agitation of wind or water produces effects qualitatively similar to diffusion; these effects, are called dispersion.
Fick’s Law of Binary Diffusion25 The description of diffusion involves a mathematical model based on a fundamental hypothesis or ‘‘law’’. Imagine two large bulbs connected by a long thin capillary (Figure 3-1). Both of bulbs are at the
25
“Diffusion, Mass Transfer in Fluid Systems”, E. L. Custler.
172
same pressure and temperature but are filled with two different gases (N2 upper bulb and CO2 lower one). The concentration of the carbon dioxide in the initially N2 filled bulb is measured with time to find how fast these two gases will mix. The concentration of CO2 varies linearly with time (Figure 3-1). So the amount of CO2 transferred could be determined from this graph at each time step. Carbon dioxide flux is defined as follows:
Figure 3-1 - Simple diffusion experiment
We can assume that the flux is proportional to the gas concentration difference and we can recognize that increasing the capillary tube length will decrease the flux, so:
(
)
3-1)
The new proportionality constant D is the diffusion coefficient. Its introduction implies a model for diffusion, the model often called Fick’s law. There is a similarity between Fick’s law and Ohm’s law for flux of electrons: (
)(
)
The diffusion coefficient D is proportional to the reciprocal of the resistivity. So the general form of Fick’s law is (3-2) Where, j is the diffusion flux, and the minus appears because of the opposite directions of diffusion flux and concentration gradient. ⁄ . Since Fick's law is derived for molecules moving in From eq. (3-2), we see that D has units Brownian motion, D is a molecular diffusion coefficient, which is called Do to be specific. The intensity (energy and freedom of motion) of these Brownian motions controls the value of D. Thus, D depends on the phase (solid, liquid or gas), temperature, and molecule size. It should be considered that the Fick’s Law could not be applied when the capillary is very thin or two gases react. Parallel to Fourier’s law for heat conduction Fick’s second law is developed as; 173
(
)
(3-3)
for one dimensional unsteady state diffusion, and for the constant area, A, it becomes the more known Fick’s second law equation: (
)
(3-4)
This equation can be applied only for isotropic media and when the potential for diffusion is only given by concentration gradients. The diffusion coefficient is also independent of concentration.
Example 3-1 Find the diffusion flux and concentration profile in a steady diffusion across a thin film.
Solution The objective is to determine how much solvent moves across the film and how the solvent concentration changes within the film. On each side of the film is a well-mixed solution of one solvent. The solvent diffuses from the fixed higher concentration, located at x<0 on the left-hand side of the film, into the fixed, less concentrated solution, located at x>l on the right-hand side. As the first step mass balance on a thin layer located at some arbitrary position x within the film is written: (
)
(
)
C0
(
)
x
Cl x
Figure 3-2 – Diffusion across a thin film
Because the process is in steady state, the accumulation is zero. The diffusion rate is the diffusion flux times the film’s area A. Thus (3-5) Divide this equation by the layer volume: →
(3-6)
Combining eq. (3-6) with Fick’s law equation (3-2) yields the following equation: (3-7)
174
Where C = solvent concentration in the layer. There are two boundary conditions for this differential equation: (3-8) (3-9) Analytical solution of eq. (3-7), according to these two boundary conditions, will be the concentration profile: (3-10) The resulted solution for the concentration profile shows that the profile is independent of the diffusion coefficient. Base on the Fick’s law the diffusion flux can be found by differentiation of concentration profile: (3-11)
Diffusion Coefficient The diffusion coefficient was introduced as a proportionality constant, the unknown parameter appearing in the Fick’s law. Often Do is used as the molecular diffusion nomenclature. Mass fluxes and concentration profiles in many situations can be found using Fick’s law equation and, most of the time, the results contained the diffusion coefficient as an adjustable parameter. There exist four important definitions of the diffusion coefficient depending on the nature of the diffusion process26: Overall Diffusion Coefficient (Mutual Diffusivity): It is denoted by DAB, and it refers to the diffusion of one constituent in a binary system (A and B). For liquids, it is common to refer to the limit of infinite dilution of A in B using the symbol, D°AB. Assume a bitumen-solvent vertical system where solvent is on the top of the bitumen. The diffusion rate of solvent through a unit area is described by eq. (3-2) as follows: (3-12) Where, = Diffusion coefficient of solvent, Cs = Solvent concentration, y = Vertical direction. As solvent diffuses into the bitumen, a diffusion flux occurs for oil molecules toward the solvent: (3-13)
26
Perry et al., 1999.
175
Where, = Diffusion coefficient of solvent, Cb = Solvent concentration. According to the volume conservation in the system, the diffusion flux of bitumen should be equal to the solvent flux, but in opposite direction: (3-14) Also: (3-15) From eqs ((3-13) to (3-15)): (3-16) The overall diffusion coefficient Dbs is a material property that describes the mobility of either component in the mixture. Self Diffusion Coefficient: It is denoted by DA’A and is the measure of mobility of a species in itself. Also when we have a binary system where A and B are the less mobile and more mobile components respectively, their self-diffusion coefficients can be used as rough lower and upper bounds of the mutual diffusion coefficient in some systems. That is, DA’A < DAB < DB’B. Intrinsic Diffusion Coefficient: When there is a considerable difference between the molecule size of solvent and oil, for example in the previous system of solvent-bitumen, during diffusion of solvent into bitumen (in the previous system), the small solvent molecules tend to diffuse faster than the large molecules of bitumen. The intrusion of solvent molecules into bitumen causes a buildup of pressure in bitumen part. This pressure gradient causes a bulk flow of bitumen solution toward the solvent-rich part. In this situation the net flow of solvent is the sum of the diffusional flux of solvent (
) and
the flux of solvent displaced by bulk flow ( ), which ‘u’ is measured in the direction of ‘y’. According the volume balance solvent, net flow should be equal to the bitumen net flow with diffusion and bulk flow: (3-17) Where, Bitumen Intrinsic Diffusion Coefficient, Solvent Intrinsic Diffusion Coefficient, Bulk Flow Velocity Intrinsic diffusion coefficient is dependent on the concentration as the overall diffusion coefficient. Bulk flow velocity could be found from eq. (3-17):
176
(3-18) When there is not much difference between the molecule sizes of solvent and oil the net flow velocity is almost zero, because of the equality between intrinsic diffusion coefficient of oil and solvent. Although for solvent-bitumen system which intrinsic diffusion coefficient of solvent is higher than bitumen, because of smaller molecular size, bulk flow will be most significant. Eq. (3-19) describe the volume conservation of solvent in a solvent-bitumen system: (
)
(3-19)
But the net flux of solvent was found with eq. (3-12). Equal eq. (3-12) with eq. (3-19), with attention to eq. (3-16) to find the following equation: (3-20) Substituting eq. (3-18), for ‘u’, into the eq. (3-20) will be ended with the following relation between the overall and intrinsic diffusion coefficient: (3-21) So the overall diffusion coefficient value is equal to the intrinsic diffusion coefficient of solvent in pure bitumen (Cs = 0) and intrinsic diffusion coefficient of bitumen in pure solvent (CB = 0). Ds and Db are concentration dependent as Dbs, but there is no way to find the equation of this dependency other than with experiment. Most of the time a linear relationship between the intrinsic and concentration is assumed. The minimum value of the solvent intrinsic diffusion coefficient is the overall diffusion coefficient when solvent concentration is zero and the maximum is solvent self-diffusion coefficient, and the same for bitumen. By knowing the minimum and maximum of the intrinsic diffusion coefficient of bitumen and solvent and assumption of a linear relationship between these coefficients and concentration a relation between overall diffusion coefficient and concentration could be found using eq. (3-21). Figure 3-3 shows a result for such calculation.
177
Diffusivity (cm2/s * 1e6)
6
DS DBS
4
2
0
D
0.25
0.50
CB
0.7 5
1.00
Figure 3-3 – Prediction overall diffusion from intrinsic diffusion
Regarding eq. (3-21) and Figure 3-3, the overall diffusivity reaches a maximum at an intermediate concentration. During the experimental measurement of diffusion coefficient, what we find is the overall diffusion coefficient. Tracer Diffusivity: Denoted by DA’B is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach selfdiffusivities at the pure component limit. In some cases the diffusion coefficient can be reasonably taken as constant (e.g. dilute solutions) while, in some others it depends very markedly on concentration.
The diffusion coefficient as a function of concentration27 The diffusion coefficient can be considered constant in many occasions especially when we refer to the diffusion of gases. However when we move to complex and denser fluids the assumption of constant diffusion coefficient may become unreal. The assumption of constant diffusion coefficient for the solvent/heavy oil or bitumen systems should be based on three important conditions that need to be fulfilled in order to support the hypothesis. If one of the three conditions is not fulfilled it is highly expected that the diffusion coefficient will be function of concentration. The three conditions are: Dimensions and shape: The molecular diameter and molecular shape should be similar for the diffusing components. That means in this case that the molecular diameter and shape of the solvent should be similar to those of the heavy oil and bitumen. We know that large hydrocarbon chains are present in heavy oil and bitumen. Thus the assumption of similar size and shape is not valid especially when we
27
“The Diffusion Coefficient of Liquid and Gaseous Solvents in Heavy Oil and Bitumen”, U. E. Guerrero-Aconcha
178
take into consideration that the solvents used in the recovery process are light gases and small hydrocarbons molecules. Molecular interactions: The molecular interactions between the diffusing components should be negligible. That means the attraction and repulsion forces should not interfere in the diffusion process. However, it has been shown that the repulsive forces play the most important role in the diffusion process. Non reacting environment: There should be a non-reacting environment in the system. That means no transformations of any kind due to the components on the system and/or the system conditions (pressure and temperature). However the interaction of solvent with heavy oil or bitumen may in some cases cause organic deposition, mainly asphaltenes. Based on the concentration dependency of the diffusion coefficient Fick’s second law is modified as follows: (
)
(3-22)
Example 3-2 Proof eq. (3-22)
Solution Assume a control volume like as Figure 3-4. There is no source or sink in the control volume. Solvent diffuses in from left hand side and go out by diffusion from right hand side. Write material balance for solvent in over the control volume:
(
)
(
)
(3-23)
Where: V = Volume, A = Area available for diffusion, C = Solvent concentration, D = Diffusion coefficient, = incremental time.
( 𝐷
𝑉
𝜕𝐶 𝐴 ) 𝜕𝑥 𝑥
𝐴
𝑥 ( 𝐷
x
𝐴
𝜕𝐶 ) 𝜕𝑥 𝑥
𝑥
x+ 𝑥
Figure 3-4 – Diffusion process in a control volume with a concentration dependent diffusion coefficient
179
According to the dependency of diffusion coefficient on the concentration , because we have a diffusion process and the potential for this process is concentration difference between x and x+ . With a constant area: . Now divide both side of eq.(3-23) by : (
→
)
(
(
)
)
Effective Diffusion Coefficient Experiments and field data show that the diffusion process in porous media is slower than that of two liquids adjacent to each other in a vial. This results from the fact that the diffusion coefficient in porous media is smaller than the bulk diffusion coefficient; therefore, an effective diffusion coefficient is proposed, which is based on the average cross-sectional area open to diffusion and the distance traveled by molecules in porous media.
For a bundle of straight capillary tubes, the effective diffusion coefficient and the bulk diffusion coefficient are the same. However, the straight capillary model is not a very good representation of a porous rock. A lot of work has been done to predict the apparent diffusion coefficient in porous media. Carman (1939) has shown that, in porous rock, fluids must move on the average at about 45o to the direction of flow (Figure 3-5). Hence, when fluid has traveled a net distance, L, it has traveled an actual average distance of about √
. On the other hand he assumed a tortuosity ( )
equal to √ for the porous medium. Carman proposed the following ratio between the bulk and effective diffusion coefficients:
√
(3-24)
Where, Deff = Effective Diffusion Coefficient, D = Molecular Bulk Diffusion Coefficient. Here 0.707 is a correction for the actual diffusion process length. On the other hand in the porous medium the cross section area available for diffusion is not the total cross section of the medium. By assumption of the equality between areal and volume porosity it can be concluded that only a portion (equal to the porosity) of the total cross section of the medium is available for diffusion process. According to this assumption Penman (1940), after some experiments on a packing of spherical particles, proposed the next equation to relate the bulk and effective diffusion coefficient to each other:
180
(3-25) Where, is the porosity. In this formula, 0.66 is the correction for real length of diffusion process and is the correction for actual area available for diffusion.
Net Path Real Path Figure 3-5 – In a porous medium fluid generally flowing at about 45o with respect to average direction of flow
On the other hand grain shapes and sizes of a porous medium are not homogeneous all the time so the assumption of constant values such as 0.707 or 0.66 is not reasonable to use as the inverse of the tortuosity, so it could be more sophisticated to use the tortuosity in the formula instead of a constant value: (3-26) A more sophisticated and comprehensive approach was suggested by Brigham, Reed et al. (1961) and van der Poel (1962). These investigators recognized that there is an analogy between diffusion and the electrical resistivity factor with the following formula: (3-27) Where, F is the formation resistivity factor. Diffusion coefficient in the absence of the porous media, is sometimes called bulk diffusion coefficient in contrast to the effective diffusion coefficient at the presence of the porous medium.
Importance in petroleum engineering In 2004, Alberta’s Oil Sands were recognized by international media, for the first time, as part of global oil reserves. This established Canada as second only to Saudi Arabia in the hierarchy of potential oil producing nations. While oil sands extraction is more expensive than conventional sources, continuing technological advances are reducing the importance of those cost differences. Moreover, conventional oil production in Canada is declining, underscoring the importance of the oil sands as a vital source of North American supplies (Timilsina et al., 2005). To recover these resources steam injection is widely used for heavy oil and bitumen reservoirs. The advantage of the process is its high recovery factor and its high oil production rate. However, the high 181
production rate is associated with excessive energy consumption approximately 1 million BTU/barrel, CO2 generation, and expensive postproduction water treatment. Additionally steam injection has operational restrictions that do not allow its application in all types of reservoirs. In order to overcome the problems associated to steam injection additional techniques have been developed to recover the heavy oil and bitumen. Among those techniques the vapor injection process (VAPEX), the cyclic solvent injection and the co-injection of steam and solvent (SAS, SAP, ES-SAGD and LASER) are the ones with the most promising future, thanks to the viscosity reduction of the oil phase, the change in absolute and relative permeability and the upgrading of the oil phase. The above processes involve the injection of solvent into the oil reservoir. The objective of the solvent is mainly to reduce the viscosity of the heavy oil or bitumen by mixing with it. This mixing process is a mass transfer process and its velocity is controlled by the diffusion coefficient. Therefore the diffusion coefficient is one of the most important parameters for the proper characterization of the solvent based recovery processes. Accurate diffusion data for these processes are necessary to determine: -
The amount and flow rate of solvent required to inject into a reservoir, The portion of reserves that have been affected by the solvent undergo viscosity reduction, The time required by the reserves to become less viscous and more mobile as desired, The rate of live oil production from the reservoir.
To find the diffusion coefficient value we must depend largely on experimental measurements of these coefficients, because no universal theory permits their accurate a-priori calculation. Unfortunately, the experimental measurements are unusually difficult to make, and the quality of the results is variable.
Measurement techniques It is noteworthy that there is no well-established and universally applicable technique for measuring the molecular diffusion coefficient. Unlike the measurements of viscosity or thermal conductivity, for which standardized techniques and equipment are readily available, the measurements of mass transfer characteristics are often more difficult due to difficulties in measuring point values of concentration and other issues which complicate this transport process. Considerable efforts were made to determine diffusion coefficient for diffusion of solvent in the oil experimentally. Different experimental methods can be classified into direct methods and indirect methods. Also there are some empirical correlations based on the experimental results that could be used under some condition to find the dispersion coefficient.
Direct Methods: Direct methods evaluate the diffusion coefficient by measuring concentration of the diffusing species (solvent) as a function of depth of penetration. Such methods are often more reliable and include the wide variety of physicochemical methods like mass spectrometry, radio-active tracer technique, spectrophotometry etc. The diffusivity is estimated by using compositional analysis techniques (Schmidt 1989). The drawbacks of direct methods are expense, time consumption and many of them are systemintrusive.
182
Indirect Methods: Indirect methods measure the changes of one of the system parameters that depend on the diffusion rate. These parameters could be the rate of change of solution volume or movement of the gas-liquid interface, rate of pressure drop in a confined cell which is known as pressure decay method, rate of gas injection from the top to a cell in which the pressure and solution volume are kept constant, magnetic field characteristics, computed tomography (CT) analysis and dynamic pendant drop analysis. The advantage of these methods is that they do not need to determine the change in composition. In this section, the more common and modern methods will be introduced: Pressure Decay Method: Among indirect methods, the pressure decay method has attracted more attention due to its simplicity in terms of experimental measurements. In this method, gas (as a solvent) and oil are injected into a cell (Figure 3-6). The cell content is initially at a non-equilibrium state. As the experiment progresses, the gas dissolve into the oil and the pressure inside the cell decreases as a result. By recording the pressure and the level of the liquid in the cell, the amount of gas transferred into the oil can be determined. From this, the diffusion coefficient is calculated. In cases involving complex hydrocarbon mixtures with possible multiphase behaviors, the pressure decay method fails. This method, however, had been discredited by Luo and Gu28. They showed that, minor changes to assumptions related to boundary conditions led to orders of magnitude differences in reported values. This method was first applied by Riazi29 and for dissolution of methane in n-pentane. He took the nonequilibrium gas into contact with n-pentane in a sealed container at a constant temperature. He determined the final state by thermodynamic equilibrium. However, the time which was required to reach the final state was determined from the diffusion process in each phase. He assumed that at the gas-liquid interface, thermodynamic equilibrium exists between the two phases at all times. However the position of the interface as well as the pressure may change with time. The rate of change of pressure and the interface position as a function of time depends on the rate of diffusion in each phase and therefore on the diffusion coefficients. P
T = const. Lg
Gas LT
Lo
Oil
Figure 3-6 – Pressure decay test cell
28 29
Luo, P. and Y. Gu, Fluid Phase Equilibria, 2009 Riazi, M. R., J. Pet. Sci. Eng. 1996
183
Other researchers have proposed and developed different mathematical solutions by modeling the interface boundary condition differently. Modeling the physics of the interface (when the pressure is declining) often requires complex mathematical solutions, and it is known that more simplified analysis based on assumption of constant equilibrium concentration at the interface introduces significant error in the estimation of the diffusion coefficient. To overcome some of these shortcomings different researcher used different models. Upreti and Mehrotra30 improved the pressure decay method to find concentration dependent diffusion coefficient. Their experimental apparatus primarily consists of a closed cylindrical pressure vessel used to hold gas over a layer of bitumen. A pump supplied the gas as a solvent at desired pressure into the vessel. The vessel was submerged in a constant temperature water bath, the pressure vs. time data were recorded by a pressure sensor. Using the logged pressure versus time data, the experimental mass of a given gas diffused into the bitumen was determined. The gas diffusivity was then calculated by fitting the calculated mass of the gas diffused into bitumen (given by a mass transfer model) to the experimental mass. The tuning parameter was the diffusion coefficient. Additionally a correlation was provided for the average diffusivity as a function of temperature. Refractive Index Method: Refractive index is the ratio of the velocity of wave propagation in a reference phase to that in the phase of interest. Normally, the refractive index used in diffusion measurement is taken as the ratio of the velocity of light in vacuum to the velocity of light in the relevant phase: (3-28) Where, c = velocity of light in vacuum,
= velocity of light in the relevant phase, and n = refractive index. With the velocity of light in vacuum chosen as the reference, the refractive index is always greater than 1. For example, the refractive index of water is 1.33, meaning that light travels 1.33 times as fast in vacuum as it does in water. As light moves from a medium, such as air, water, or glass, into another it may change its propagation direction in proportion to the change in refractive index.
Figure 3-7 - Refraction of light at the interface between two media 30
Upreti, S., Lohi, A., Kapadia, R. and El-Haj, R., 2007
184
Eq. (3-29), Snell's law (also known as the Snell–Descartes law, and the law of refraction), is a formula used to describe the relationship between the angles of refraction and refraction index, when referring to light or other waves passing through a boundary between two different isotropic media, such as two fluid with different concentration of a solvent. (3-29) Where,
is the angle of refraction.
It is noted that, for a solution, different concentrations of a sample substance will lead to different refractive indices. As a result, from the angle of refraction, the concentration of a solution phase can be determined. Normally, in experimental measurements of refractive indices, a laser light is emitted through the diffusion cell and, according to the concentration of the solution at each elevation, the corresponding refractive angle of the laser beam is determined. As a result, the point at which the laser beam is captured by a CCD camera will represent the concentration inside the diffusion cell at that specific elevation. Figure 3-8 shows a sample picture of CCD during the diffusion process.
(a)
(b)
Figure 3-8 - Sample of light refraction results a) initial time b) after diffusion occurred
It is noted that the above method is suitable only for transparent fluids. As heavy oil, even when diluted, is opaque, such a method cannot be employed. NMR Method: Nuclear Magnetic Resonance (NMR) was mainly developed for chemical-physical-medical use. The principle of this method is to calculate the density of hydrogen protons. NMR is a phenomenon, which occurs when the nuclei of certain atoms are immersed in static magnetic field and exposed to a second oscillating magnetic field. Some nuclei experience this phenomenon, and others do not, depending on atomic composition. Based on the nature of magnetic resonance, NMR measurement can be made on any nucleus that has an odd number of protons or neutrons or both, such as the nucleus of hydrogen (H), carbon (C), and sodium (Na). For most of the nuclei found in earth formations, the nuclear magnetic signal induced by external magnetic fields is too small to be detected with NMR device such as borehole NMR logging tool. However hydrogen as the main atom of water and other hydrocarbon molecules produces a strong signal. So the strength of the signal could be used as a scale of the existing hydrogen.
185
Principles of NMR and Processing31,32 Polarization Process: The nucleus of the hydrogen atom is a proton, which is small, positively charged particle with an associated angular momentum or ‘spin’. The spin of this proton causes the proton to behave like a tiny magnet with a north and south poles (Figure 3-9, a). In the absence of an external magnetic field, the hydrogen nuclear spin axes are randomly aligned (Figure 3-9, b). This results in a net magnetization of zero. In the presence of an external magnetic field, the nuclear spins attempt to line up with the field either parallel or anti parallel to the net magnetic field (B0). According to quantum mechanics, the proton in a net magnetic field is forced into one of two energy states, high-energy or low-energy state. The protons that their processional axes are parallel to the net magnetic field are in the low energy state, which is the preferred state. On the other hand the protons are in the high-energy state when their processional axes are anti-parallel to the magnetic field.
+
N
S (a)
(b)
Figure 3-9 - (a) Hydrogen nuclei behave as a tiny bar magnets aligned with the spin axes of the nuclei. (b) Spinning protons with random nuclear magnetic axes in the absence of an external magnetic field.
The difference between the numbers of protons with high and low energy level produces the bulk magnetization ‘M’, which provides signal measured by NMR devices (Figure 3-10). The bulk (Macroscopic) magnetization ‘M’ is defined as the net magnetic moment per unit volume.
31 32
Behnaz Afsahi, M.Sc. Thesis, 2007 NMR logging, Principle and Application, G.R. Coates, L. Xiao, and M. G. Prammer, Halliburton Energy Services
186
+ + +
+
+ + + + + + + +
+ M=0
+ +
+ + +
+ + +
+
B0 M
+ +
Figure 3-10 – Line up nuclear spins in an external magnetic field
M is measurable and is proportional to the number of protons, the magnitude B0 of the applied magnetic field, and the inverse of the absolute temperature. After the protons are exposed to the static external magnetic field (B0), they are said to be polarized. Polarization does not occur immediately but rather grows with a time constant, which is the longitudinal relaxation time, T1: (
)
(3-30)
Where, t = the time that the protons are exposed to the B0 field, The magnitude of magnetization at time t, when the direction of B0 is taken along the z axis The final and maximum magnetization in a given magnetic field T1 is the time at which the magnetization reaches 63% of its final value, and three times T1 is the time at which 95% polarization is achieved. Figure 3-11 is a T1 relaxation or polarization curve. Different fluids, such as water, oil, and gas, have very different T1 relaxation/polarization times. Relaxation definition will be illustrated later. Tipping Process: After exposing the protons to the net static magnetic field (B0), apply an oscillating magnetic field (B1) perpendicular to B0, therefore, the magnetization M will precess farther and farther away from the zaxis. This process is called ‘Tipping’. According to the quantum mechanics point of view, if a proton is at the low-energy state (its processional axes are parallel to the net magnetic field), it may absorb energy provided by B1 and jump to the high-energy state. The application of B1 also causes the protons to precess in phase with one another. This change in energy state and in-phase precession caused by B1 is called nuclear magnetic resonance.
187
M(t)/M0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1
2
3
4
5
6
PolarizationTime / T1 Figure 3-11 – Polarization/Relaxation curve
The magnetization vector M can be thought of as having a component MZ along the z-axis (direction of B0), which called longitudinal magnetization, and a component Mxy perpendicular to the field named transverse magnetization. At equilibrium Mxy = 0 and MZ = M. when the protons are exposed to the oscillating magnetic field B1, MZ decreases and Mxy increases. The angle of deflection or rotation of sample’s net magnetization depends primarily on the product of amplitude (energy) of the B1 and the length of time that B1 is applied: (3-31) Where, Tip angle, B1 = Amplitude (energy) of the oscillating field, Time over which the oscillating field (B1) is applied The gyromagnetic ratio (measure of the strength of the nuclear magnetism) Each nucleus has a fixed
value. For hydrogen: ⁄
MHz/tesla.
Angular pulse terms such as π pulse (180o pulse) or a π/2 pulse (90o pulse), refer to the angle through which magnetization is tipped by B1 (Figure 3-12).
B1
M
𝜃
M
θ
𝑜
𝑜
𝜃
M
M M
τ
o
90 Pulse
M
o
180 Pulse
Figure 3-12 – the Tipping process
188
Relaxation Process: When the B1 field is turned off, the net magnetization decreases and system gradually returns to equilibrium. During this process, the protons gradually lose their extra energy and return to equilibrium by emitting radio waves and by transferring energy to surrounding molecules. The processes by which nuclei transfer energy to their surroundings to return to equilibrium state are called relaxation. The relaxation processes are exponential in time and are described by two time constants, T1 as the longitudinal magnetic relaxation time constant and T2 as the transverse magnetic relaxation time constant. These two constant values are seldom equal. Transverse relaxation is always faster than longitudinal relaxation; consequently, T2 is always less than or equal to T1. In general, for protons in solids, T2 is much smaller than T1. T1-Longitudinal relaxation time As the protons absorbed energy from B1, lift up to the high-energy state during T1 relaxation, any given spin can return to the ground state only by dissipating excess energy to the surrounding (lattice) (Figure 3-13). Therefore this process also called spin-lattice relaxation. During the T1 time the z-component of M returns to 63% of its original value.
(1)
(2)
(3)
(4)
Figure 3-13 – Net magnetization return to equilibrium by turning off the B1, (the arrow represent net magnetization)
T2-Transverse relaxation time During T2 relaxation, no energy is exchanged from the nuclei to lattice. Exchange of energy happens among nuclei. Therefore T2 also called spin-spin relaxation. Transfer relaxation corresponds to the loss of phase coherence or randomization of spins in transverse direction (x-y) direction, which causes the loss of transfer magnetization. T2 refers to the time required for the transvers component Mxy to decay to 37% of initial value. A 90o pulse B1 gives energy to the protons and M rotates entirely into the x-y plane (Figure 3-14). Coherence now exists in this transverse plane at the end of the pulse. The protons are all synchronized and precess at the same frequency. A transfer of energy can occur between these protons. Spin-spin relaxation refers to this energy transfer from an exited proton to a nearby unexcited proton. This energy exchange produces a gradual loss of phase coherence to the spins. As the coherence gradually disappears, the value of Mxy decreases toward zero (Figure 3-14). This loss of coherence is a consequence of T2. T2 relaxation is more efficient in large molecules since they reorientate more slowly than small molecules.
189
Mxy
(1)
(2)
(3)
(4)
Figure 3-14 – de-phasing (loss of phase coherence) during T2
When a wetting fluid fills a porous medium, such as a rock, both T2 and T1 are dramatically decreased, and the relaxation mechanisms are different from those of the protons in either the solid or the fluid. There are many different properties of the fluid and porous media that could be measured or explained by using the relaxation process and (T1, T2) values. Spin-Echo and CPMG pulse sequence Once the 90o B1 pulse is turned off, the proton begin to de-phase or lose phase coherency in B0 (Figure 3-14). As the net magnetization in the transverse plane decreases, a receiver coil that measures the magnetization in the transverse direction could detect a decay signal in this situation. If the magnetic field was really homogeneous (the amplitude is not a function of x, y or z), the signal would decay with a time constant T2. However, since the B0 has some inhomogeneity, the signal actually decays faster with the time constant T2*, which called Free Induction Decay (FID). The FID is very short, which is lasting a few milliseconds. Consequently in the small time interval between the two pulses, very little T1, some T2 de-phasing and substantial T2* occurs. The de-phasing resulting from T2* occurs at a constant rate since it arises from the spatial inhomogeneity of the magnetic field. T2 de-phasing on the other hand fluctuates randomly since it results from the interaction among the nuclei themselves. This type of de-phasing provides valuable sample information. In order to measure T2, the signals must be recombined. It can be done by applying an 180o pulse after the 90o pulse (after τ ms) to re-phase the proton magnetization vectors in the transverse plan (Figure 3-15). In effect, the phase order of the transverse magnetization vectors is reversed, so that the slower vectors are ahead of the faster vectors. The faster vectors overtake the slower vectors, rephrasing occurs, and a signal is generated that is detectable in the receiver coil. This signal is called spin echo. The echo time (TE) defined as the time between the 90o pulse and the re-phasing completion, which is 2τ.
Mxy o
Applying 90 B1 pulse at time 0.
190
De-phasing after turning off the B1
At time τ ms, a 180° B1 pulse is applied to reverse the phase angles and thus initiates re-phasing.
Re-phasing is complete, and a measurable signal (a spin echo) is generated at time 2τ ms.
Figure 3-15 – Spin-echo sequence
Only a single echo decay very quickly. One way for determining T2 from spin echo amplitudes is by repeating the spin echo method several times with very time τ. In CPMG method a series of 180o pulse are applied at intervals τ, 3τ, 5τ, 7τ, etc., following the 90
o
pulse. Echoes are then observed to form at times 2τ, 4τ, 6τ, 8τ, etc. because the de-phasing resulting from molecular interactions the protons can no longer be completely refocused, and the CPMG spin-echo train will decay. On multiple repetitions of the 180o pulse, the height of the multiple echoes decreases successively as a consequence of T2 de-phasing (Figure 3-16).
191
Received Signal Amplitude
TE = 2τ
o
90
o
180
2τ
o
4τ
180
6τ
o
8τ
o
180
180
180
o
10τ
Time (ms)
Figure 3-16 – CPMG pulse sequence
Amplitude
M0
0
50
100
150
200
250
Time (ms) Figure 3-17 - The amplitudes of the decaying spin echoes yield an exponentially decaying curve with time constant T2
Typical NMR Experiment: The above theory for an entire NMR experiment involves these procedures: the first step is to polarize the hydrogen protons in an external magnetic field, B0. The second step is to tip the magnetization from the longitudinal direction to a transverse plane by applying an oscillating magnetic B1 perpendicular to B0. The last is to measure their true relaxation back to equilibrium polarized direction. There are four important parameters to control these procedures: the length of the echo time (TE), the number of 1800 pulse (NE), the waiting time (TW) to re-polarize protons along B0 and the number of trains. As TE decreases, spin echoes will be generated and detected earlier and more rapidly, and effective signal-to-noise ratio is increased because of the greater density of data points. As TE increase, spin echoes will be generated and detected longer, but more B0 power is required. The first two parameters determine the length of a single NMR experiment. Waiting time is the time between the ends of one CMPG sequence to the start of the next CMPG sequence. The set up the waiting time depend on the sample. In general, if we simply try to capture the full signal from all the protons present in the sample, the wait time must be set long enough that all protons returns back to their equilibrium position. Then number of trains simply as how many times the entire process is repeated. All four parameters can be 192
controlled manually. Generally the more times experiment is repeated, the better the results should be. Usually for pure fluid, such as water, which only contains a single relaxing species, it is not necessary to use too many trains. For the fluid, such as crude oil, which contains several different relaxation hydrocarbon species, it is good to run more trains to get better results. One of the most important steps in NMR data processing is to determine the T2 distribution that produces the observed magnetization. This step is called echo-fit or mapping, is mathematically inversion process, the total measured NMR signal is inverted from a decay curve by Echo-fit software to give T2 spectrum (Figure 3-18).
Porosity
Echo Amplitude
In general, the strength of a received signal is directly proportional to the number of hydrogen protons present, which can be correlated to the amount of fluid present. The relaxation time T2 is a function of the viscosity of the fluid, or the confinement of the space where molecule is relaxing. For example heavy oil and bitumen are made of complex chains containing branches and rings, so the amount of hydrogen present in a given mass of oil will generally be less than the hydrogen present in the same mass of water. Generally the total amplitude (At) of a bitumen sample will be less than that of water of the same mass. On the other hand T2 value for heavy oil is much less than its value for water because of high viscosity of heavy oil.
Echo Time (ms)
T2 Distribution
Figure 3-18 - The echo train (echo amplitude as a function of time) is mapped to a T2 distribution (porosity as a function of T2)
In a NMR experiment, several parameters of interest are measured. The first is the total signal amplitude, which is the amplitude of both oil and solvent components. Amplitude refers to the measurable hydrogen protons in the sample and it is proportional to the sample mass. The second is amplitude index, which is simply the measurable NMR amplitude per gram of bulk fluid. Knowing the amplitude index allows for the amplitude of any given sample to be correlated to the sample’s mass: (3-32) Where, AI = amplitude Index, AP = amplitude of the fluid signal, m = mass of the fluid.
193
Not all NMR machines are operated at the same signal gain, so the value of oil amplitude index could vary from machine to machine. In order to normalize the measured amplitude index, the term relative hydrogen index is defined33: (3-33) Where, RHI = relative hydrogen index. Since oil and bitumen consist of different components, each with different relaxation rate, their signal will be detected with a broader relaxation range compared to pure substances with a single relaxation time like water. In other words the protons in crude oil do not all relax with a single value of T2, yielding a variety of relaxation times. The characteristic time for the oil relaxation is the geometric mean T2gm, of the oil spectrum34: (
∑ ∑
)
(3-34)
Where, The geometric mean T2 value (ms), The T2 value of a component in the mixture (ms), The amplitude of a component at a time constant T2i. For a fluid like oil, which consist of multiple components, all the components.
represents the mean relaxation rate for
Application of Low Field NMR in Diffusion Measurements35 The basis for using NMR is the fact that the relaxation spectra of a heavy oil or bitumen are distinctly different than those of a solvent. Figure 3-19 shows the spectra of pure viscous oil, of pure solvent, and of a mixture of solvent with bitumen that was prepared by mild heating and stirring. As mentioned before the amplitude indices of the bitumen signals (AIb) are lower than the amplitude indices of the signals from the solvent (AIS). It is because of the large molecular size of bitumen that lead to lower amount of hydrogen in unit volume compare to the solvent with smaller molecular size. On the other hand relaxation time for bitumen is much faster than the relaxation time for solvent because of high viscosity of bitumen. It can be seen that the spectrum of the mixture is distinctly different than the spectra of the individual components. This distinction forms the basis of the methodology used to study the diffusion of solvent in heavy oil using NMR.
33 34 35
Y. Wen, M. Sc thesis, 2004. B. Afsahi, A. Kantzas, 2007. Y. Wen, J. Bryan, A. Kantzas, 2005.
194
T2c
Figure 3-19 – Typical NMR spectrum for pure bitumen, pure solvent, and a mixture of them
From Figure 3-19, it is evident that during the diffusion of a solvent in bitumen the solvent spectrum shifts to faster relaxation times by increasing its viscosity during the mixing, while the bitumen spectrum shifts to longer relaxation times by decreasing of its viscosity. Two separate peaks of pure heavy oil/bitumen and solvent become one continuous multi-peak spectrum. It is assumed that the amount of solvent in bitumen is proportional to the expansion of the spectrum of the bitumen component (see Figure 3-19). If this assumption is correct, then there will be a critical relaxation time that will split the mixture spectrum into two components. Spectrum peaks with relaxation times faster than the critical relaxation time will correspond to bitumen and diffused solvent. Spectrum peaks with relaxation times slower than the critical relaxation time will correspond to pure solvent that has not yet diffused in bitumen. It then becomes important to define this critical relaxation time, T2c. T2c is defined as the maximum relaxation time observed in a thoroughly mixed solvent/bitumen mixture, which is shown in Figure 3-19. Usually, the spectra of heavy oil and bitumen are less than 10 ms, the spectra of solvents are beyond 1,000 ms, and the spectra of the mixtures of solvent with heavy oil or bitumen extend to some intermediate value. For bitumen-solvent system of Figure 3-19, the signal under the 100 ms was considered to be solvent diffusing in the bitumen, in other words the signal amplitude change up to 100 ms was attributed to solvent mass transfer. The signal beyond the 100 ms was considered to be pure solvent. Prior to starting the experiment, mixtures of known percentages of bitumen and solvent are prepared and the correlation between the NMR parameters and the concentration of bitumen in the mixture is determined (NMR parameters calibration with bitumen content). These correlations were used to calculate the concentration of the solvent that diffused in the heavy oil or bitumen during the test (Figure 3-20). (3-35) Where, Mass of diffused solvent, Bulk solvent signal amplitude, Pure solvent amplitude Index. 195
1
Ln[1/(RHI * T2gm)]
1/(AI * T2gm)
100 10 1 0.1 0.01 0.001
-1
y = 4.5954x2 + 2.5996x - 7.3702
-3 -5 -7 -9
0
0.2
0.4
0.6
0.8
1
Bitumen Fraction
0
0.2
0.4
0.6
0.8
1
Bitumen Fraction
Figure 3-20 – two samples of NMR calibration for bitumen-solvent mixture36,37
The mass of diffused solvent could be calculated as: (3-36) Where, Mass of initial pure solvent, Mass of diffused solvent. The total amplitude of diffused solvent is calculated using the following relation: (3-37) Where, Total pure solvent amplitude, Total bulk solvent amplitude. The combined amplitude index for the diffused solvent and bitumen/oil is calculated, using amplitude index definition: (3-38) Where, Signal amplitude of diluted bitumen, Mass of pure bitumen, Mass of diffused solvent, Amplitude index of diluted bitumen.
36 37
Y. Wen, J. Bryan, and A. Kantzas, 2003. D. Salama and A. Kantzas, 2005.
196
As the next step, geometric mean relaxation time for the diluted bitumen, T2gm, is calculated according to eq. (3-34). The solvent content (concentration) could be calculated form the generated correlation between NMR properties and bitumen content (such as Figure 3-20) at different times. The concentration of solvent determined through NMR spectra change is the overall concentration, ̅ , in the mixture area, which is a function of time and diffusion coefficient. The correlation between concentration ̅ , time t, and D is described in the following equations: (3-39) ∫
∫
(3-40)
At given t: ∫ ̅
∫
(3-41)
∫
As a result of eq. (3-41), one can construct models of ̅ with D and t, and pursue estimates of the diffusion coefficient. In the NMR test, it is assumed that the D value is constant during each small time interval. So the following equation could be used for the determination of the diffusion coefficient: ̅
(
√
)
(3-42)
Where, Starting concentration, t = the time of measurement, x = the equilibrium distance for the early times (for example first two days) of measurements, D = the diffusion coefficient that gives the best fit. Since there is no ability to obtain the correlation between concentrations with distance from NMR spectra, an overall constant diffusion coefficient is considered from the analysis of NMR data. x is determined independently using X-Ray CAT scanning. The reason for using early time data is because the interface between solvent and heavy oil/bitumen can only be maintained stable for a short time. That was the observation made from CAT scanning experiments38. Figure 3-21 shows a sample result of diffusion coefficient calculated using NMR experiment. Afsahi and Kantzas (2007) used the same method to find the diffusion coefficient in sand-saturated sample. They mixed the bitumen with sand thoroughly in a way that the final mixture of sand and bitumen had almost 35 to 40% porosity (40% bitumen, 60% sand). Then the bitumen-sand mixture was placed in the vial and compacted completely. Solvent was placed on top of oil-saturated sand in the vial. In order to keep the sand in place after dilution of heavy oil by the solvent they used a nylon mesh to separate the oil-saturated sand from the solvent. NMR measurements were taken frequently and 38
Y. Wen, A. Kantzas, and G.J. Wang, 2004.
197
Diffusion Coefficient, 106 cm2/s
changes in the spectra were related to the change of oil and solvent properties as solvent diffused into the oil-sand matrix. Another step of the procedure was the same as for the bulk diffusion that explained earlier. 100 80 60 40 20 0 0
6
12
18
24
30
36
42
Time, hr Figure 3-21 – Diffusion coefficient as a function time, NMR experiment result
Another interesting usage of NMR method, such as pore size distribution or wettability determination, will be illustrated later. Computer-Assisted Tomography: Computer-assisted Tomography (CAT) scanning using X-ray has been extensively used in research laboratories around the world for reservoir rock characterization and fluid flow visualization. Principles of CAT Scanning and Processing39 X-rays lose their energy as they pass through a medium, and this reduction depends on the density of the substance and the path length through that substance. CAT is based on emitting x-rays from a source which revolves around the object in consideration while one-dimensional projections of attenuated x-rays are collected by a detector on the other side of the source. These projections are collected as the sample travels through the scanner longitudinally and are used to reconstruct a twodimensional image of the object. Intensity values of attenuated x-rays are collected from small volumetric elements, called pixels. These elements are typically 0.40×0.40 mm in area and 3 cm in depth (along the direction of the x-ray beam) for a second generation CAT scanner. Once these elements are all assigned an intensity values after a complete radial and longitudinal scan, these data are processed by a computer. This processing constitutes the major part of the CAT. The inlet intensity and the outlet intensity are related through the following relationship: (3-43) Where: I = The intensity remaining after the X-ray passes through a thickness (kV),
39
L. Song, A. Kantzas, J. Bryan, 2010.
198
Io = The incident X-ray intensity (kV), µ = Linear attenuation40 coefficient, L = Path Length. This relationship applies only for a narrow mono-energetic beam of x-ray photons which travels across a homogeneous medium. If the medium in consideration is heterogeneous, the above equation holds true while replaced by the line integral of the linear attenuation coefficients. The modified form is: ( )
∫
(3-44)
The following equation relates the linear attenuation coefficients to the number stored in computer (known as the CT numbers or CTn), (3-45) Where: CTn = CT number, x-ray linear attenuation coefficient of the object scanned, x-ray linear attenuation coefficient of water. Linear attenuation coefficient (µ) is a function of the bulk density and the effective atomic number of the sample, given by: (3-46) Where: Bulk oil density (kg/m3) a = Energy-independent coefficient called Klein- Nishina coefficient b = Constant Z = Effective atomic number of the sample E = Mean photon energy (kV) When exposing a medium to x-rays, gathering the exiting x-rays from the medium (Figure 3-22), and averaging the intensity at each cross section, a transmitted intensity vs. elevation curve can be constructed (Figure 3-23). The resulted curve could be converted to a density curve. According to the relation between the x-ray intensity and density.
40
In physics, attenuation (in some contexts also called extinction) is the gradual loss in intensity of any kind of flux (X-Ray) through a medium.
199
Figure 3-22 - Schematic view of CAT scanning using x-ray
A series of calibration tests for liquid and solid samples of known densities are performed in order to correlate the CT numbers (as the intensity of the detected x-ray), generated by the scanner, to densities. In Figure 3-23 the calibration curves for liquid samples and liquid/solid samples are shown respectively. Using these calibration curves, the densities of the scanned samples can be back calculated.
(a)
(b)
Figure 3-23 – Calibration curves for the CAT scanner, (a) Liquid calibration curve, (b)Liquid-solid calibration curve41
In contrast to the refractive index method, this method has the ability to operate with opaque solutions such as bitumen + pentane. Diffusivity investigation using CAT scanning Wen and Kantzas42 used this method to monitor the concentration profiles at a bitumen-solvent interface. Solution of the solvent in the oil dilutes the oil and change the linear attenuation coefficient. This is the basic idea of using the CAT scanning for diffusion process study. For a diffusion study, as the first step heavy oil is adjacent to the solvent in a fixed volume cell (Figure 3-24). Because of higher oil gravity, solvent is on the top of the heavy oil in the cell. A fixed vertical-sectional position of the diffusion cell is scanned at fixed frequency during the diffusion process. Figure 3-24 illustrates the typical CT scan image of solvent diffusing into the heavy oil.
41 42
D. Salama and A. Kantzas, 2005. Y. Wen, A. Kantzas, 2005
200
Figure 3-24 – Image sample of diffusion process
The two-dimensional CT image shows the diffusion process along the vertical direction (x axis) of length of the diffusion cell. A central area (region of Interest, ROI) is cut as shown by a dotted line in Figure 3-24. The ROI is used for diffusion coefficient calculation and analysis. This study considers the diffusion process as a one-dimensional vertical diffusion process. Therefore, only an average CT number in the horizontal direction is calculated to represent the CT number in the center, and then the profile of CT numbers change with vertical distance for each “x” value is obtained. The changes in the CT numbers are related to the changes of oil densities as a solvent diffuses into the heavy oil, and CT number has a linear relationship with density. Thus, the CT number profiles can be converted to the density profiles. In addition, the mixture (solvent and heavy oil) density has a linear relation with the solvent content in the mixture. Therefore, using the following equation (eq. ((3-47)), normalized concentration profiles could be obtained from the densities. (3-47) Where: Normalized Concentration (volume fraction), Bulk oil density (kg/m3), Density of the solvent close to the interface (kg/m3) Initial Density of the oil (kg/m3) By assumption of linear relationship between concentration and diffusion coefficient43 the Fick’s second law equation (eq. (3-22)) could be converted to the following equation: (
)
(3-48)
Using CAT scanning we have density at each point for any measurement time. This was the base idea for Guerrero-Aconcha and Kantzas44 to find diffusion coefficient as a function of concentration. They assume that the diffusion process is in the x direction and at each cross section there is a uniform 43 44
Upreti, S.R., Mehrotra, A.K., 2000. Guerrero-Aconcha, U., Kantzas, A., 2009.
201
concentration. They divided the diffusion length to several control volumes and discretized eq. (3-48) explicitly and apply discretized equation on the control volumes, as follows: (
⁄
(
))
(
⁄
⁄
(
))
(
(3-49)
)
In this equation only the diffusion coefficients are the unknown values. To solve this equation for the domain, two boundary conditions are needed. Guerrero-Aconcha and Kantzas assumed a constant concentration at the interface between oil and solvent (point A) as the first boundary condition and a no flow boundary surface at the end of medium (point B). Solvent Diffusion Direction
A
i+1
i
i-1 A
i
i-1 x/2
B
i+1 x/2
x Figure 3-25 – Medium Domain
According to these two boundary conditions a discretized equation for the first and last control volumes are as follows: Boundary A: (
⁄
(
))
(
(
⁄
))
(
)
(3-50)
Boundary B: (
⁄
(
⁄
))
(
)
(3-51)
By arranging the discretization equations within the medium domain and at the boundary surfaces, the Equations ((3-49), (3-50) and (3-51)) can be written in matrix form as Ax=b. The components of vector x are the unknown diffusion coefficients. The diffusion coefficients can be obtained by solving the system of linear equations. Effective Diffusion Coefficient Using CAT scanning give us the opportunity to find effective diffusion coefficient for a porous media. The procedure is the same as the procedure of finding bulk diffusion coefficient except that the vessel is half filled with oil saturated sand and topped with solvent. In contrast to the bulk diffusion coefficient that the region of interest (ROI) was in both solvent and oil region (Figure 3-24), here ROI includes only the liquid volume in the solvent region because of the complexity of obtaining a smooth concentration profile that could be used in the calculation of diffusion coefficients when including the sand volume 202
(Figure 3-26). The diffusion calculations were made based on the assumption that the amount of oil that diffused in the solvent is equal to the amount of solvent that diffused in the oil/sand mixture but in opposite direction. The calculation procedure is the same as for bulk diffusion coefficient.
Figure 3-26 - Sample of diffusion in sand saturated with oil
Diffusion Coeficient, 106 cm2/s
Figure 3-27 shows the result of CAT scanning experiment for bulk diffusion coefficient of three different solvents in heavy oil. Each point is the average diffusion coefficient over the height of the region of interest (ROI).
12
Pentain
10
Hexane
8
Octane
6 4 2 0 0
0.5
1
1.5
2
2.5
3
3.5
Time, Day Figure 3-27 - Average diffusion coefficients for pentane, hexane and octane in heavy oil10
Figure 3-28 compares the bulk diffusion coefficient and effective diffusion coefficient of Pentane in heavy oil. Figure 3-28 provide evidence that the diffusion coefficients of hydrocarbon solvents in bulk oil are higher than in presence of sand.
203
Diffusion Coefficient, 106 cm2/s
15
D in bulk
12
D in presence of sand 9 6 3 0 0
1
2
3
4
5
6
7
Time, hr Figure 3-28 - Comparison of the diffusion coefficients of pentane in heavy oil in absence/presence sand10
204
4 .
C H A P T E R
4
Immiscible Displacement Introduction Fluid displacement processes require contact between the displacing and the displaced fluid. The movement of the interface between displacing and displaced fluids and the breakthrough time associated with the production of injected fluids at producing wells are indicators of sweep efficiency. This chapter shows how to calculate such indicators using the Buckley-Leverett theory.
Buckley-Leverett Theory45,46 One of the simplest and most widely used methods of estimating the advance of a fluid displacement front in an immiscible displacement process is the Buckley-Leverett method. The Buckley-Leverett theory [1942] estimates the rate at which an injected water bank moves through a porous medium. The approach uses fractional flow theory and is based on the following assumptions:
Flow is linear and horizontal
Water is injected into an oil reservoir
Oil and water are both incompressible
Oil and water are immiscible
Gravity and capillary pressure effects are negligible
In many rocks there is a transition zone between the water and the Oil zones. In the true water zone, the water saturation is essentially 100. In the oil zone, there is usually present connate water, which is essentially immobile. Only water will be produced from a well completed in the true water zone, and only oil will be produced from the true oil zone. In the transition zone both oil and water will be produced, and at each point the fraction of the flowrate that is water will depend on the oil and water saturations at that point. Frontal advance theory is an application of the law of conservation of mass. Flow through a small volume element () with length ∆x and cross-sectional area “A” can be expressed in terms of total flow rate qt as:
(4-1)
45 46
“Principle of applied reservoir simulation”, John R. Fanchi “Applied Petroleum Reservoir /engineering”, B.C. Craft, M. Hawkins, 1991
205
Where q denotes volumetric flow rate at reservoir conditions and the sub-scripts {o,w,t} refer to oil, water, and total rate, respectively and fw and fo are fractional flow to water and oil (or water cut and oil cut) respectively: (4-2) (4-3)
(4-4)
4-5)
is a function of saturation. So for constant viscosity fw is just a function of saturation. Figure 4-1 is a plot of the relative permeability ratio, range of
, versus water saturation. Because of the wide
values, the relative permeability ratio is usually plotted on the log scale of semi-log paper.
Like many relative permeability ratio curves, the central or main portion of the curve is quite linear. As a straight line on semi-log paper, the relative permeability ratio may be expresses as a function of the water saturation by: (4-6) The constants “a” and “b” may be determined from the graph, such as Figure 4-1, or determined from simultaneous equations from known data of saturation and relative permeability. Relative Permeability Ratio
1000 100 10 1 0.1 0.01 0.001 0
0.2
0.4 0.6 Water Saturation
0.8
1
Figure 4-1 – Semilog plot of relative permeability ratio versus saturation
Substituting eq. (4-6) into eq. (4-5) will end with:
206
(4-7)
Fractional Flow of Water, fw
If the water fractional flow is plotted versus water saturation, an S-shaped curve will result that is named fractional flow curve. 1 0.8 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
Water Saturation Figure 4-2 – Fractional flow curve
Assume that the total flow rate is the same at all the medium cross section. Neglect capillary and gravitational forces that may be acting. Let the oil be displaced by water from left to right. The rate the water enters to the medium element from left hand side (LHS) is: (4-8) The rate of water leaving element from the right hand side (RHS) is: (4-9) The change in water flow rate across the element is found by performing a mass balance. The movement of mass for an immiscible, incompressible system gives: Change in Water Flowrate = water entering – water leaving = =
(4-10)
This is equal to the change in element water content per unit time. Let Sw is the water saturation of the element at time t. Then if oil is being displaced from the element, at time (t + t) the water saturation will be . So water accumulation in the element per unit time is: (4-11) Where,
is porosity. Equating equations (4-10) and (4-11) results:
207
(4-12) In the limit as ∆t → 0 and ∆x → 0 (for the water phase): (
)
(
)
(4-13)
The subscript x on the derivative indicates that this derivative is different for each element. It is not possible to solve for the general distribution of water saturation Sw(x,t) in most realistic cases because of the nonlinearity of the problem. For example, water fractional flow is usually a nonlinear function of water saturation. It is therefore necessary to consider a simplified approach to solving Eq. ((4-13)). x
A Figure 4-3 – Horizontal bed containing oil and water.
For a given rock, the fraction of flow for water fw is a function only of the water saturation Sw, as indicated by Eq. (4-13), assuming constant oil and water viscosities. The water saturation however is a function of both time and position, which may be express as fw = F(Sw) and Sw = G(t,x). Then: (
)
(
)
( (
) )
(4-14) 4-15)
Now, there is interest in determining the rate of advance of a constant saturation plane, or front ( ) , where Sw is constant and dSw = 0. So from eq. (4-14): (
)
(
)
(4-16)
Substituting eqs (4-13) and 4-15) into eq. (4-16) gives the Buckley-Leverett frontal advance equation: (
)
(
)
(4-17)
208
The derivative (
) is the slope of the fractional flow curve and derivative ( ) is the velocity of the
moving plane with water saturation Sw. Because the porosity, area, and flowrate are constant and because for any value of Sw, the derivative (
)
is a constant, then the rate dx/dt is constant.
This means that the distance a plane of constant saturation, Sw, advances is proportional to time and )) at that saturation, or:
to the value of the derivative (( (
)
(
)
(4-18)
Where, is the distance traveled by a particular Sw contour is the cumulative water injection at reservoir conditions. In field units: (
)
(
)
(4-19)
Example 4-1 Assume a cubical reservoir under active water drive with oil production of 900bbl/day. The flow could be approximated as a linear flow. The cross sectional area is the product of the width, 1320 ft, and the true formation thickness, 20 ft, so that for a porosity of 0.25, eq. (4-19) becomes: (
)
Consider that because we assume the fluids are completely incompressible, so the oil production rate is equal to the total flowrate in the different cross sections of the reservoir. 0
x
Flow Direction
1320 ft
20 ft Water
Transition zone
Oil + Connate Water
Figure 4-4 – Cubic reservoir under active water drive
If we let x=0 at the first point of the transition zone, then the distances the various constant water saturation planes will travel in, say, 60, 120, and 240 days are given by:
209
(
)
(
) (
(4-20) )
The value of the derivative (
) may be obtained for any value of water saturation, Sw, by plotting fw
from eq. (4-7) versus Sw and graphically taking the slopes at various values of Sw. Assume you find a=1222 and b=12 from Figure 4-1 (intercept = 1222 = ‘a’ and slope of the straight line = 13 = ‘b’) for eq. (4-7). For example at Sw = 0.4, fw = 0.129. The slope taken graphically at Sw = 0.4 and fw = 0.267 is 1.66. The derivative ( ( (
) may also be obtained mathematically using eq.(4-7):
) (
)
(4-21)
)
Figure 4-5 shows the water fractional flow curve and also the derivative (
) plotted against water
saturation from eq. (4-21). Since Eq. (4-7) does not hold for the very high and for the quite low water saturation ranges (see Figure 4-1), some error is introduced below 30% and above 80% water saturation. Since these are in the regions of the lower values of the derivatives, the overall effect on the calculation is small. Fractional Flow of Water, fw
0.8
3
0.6
2 0.4
1
0.2
Derivative, dfw/dSw
4
1
0
0 0
0.2
0.4
0.6
0.8
1
Water Saturation Figure 4-5 – Water fractional flow ant its derivative
A plot of Sw versus distance using Eq. (4-20) and typical fractional flow curves leads to the physically impossible situation of multiple values of Sw at a given location. For example Figure 4-6 shows water saturation distribution according to eqs (4-20) and (4-21). For example, at 50% water saturation, the value of the derivative is 2.87; so by eq. (4-20), at 60 days the 50% water saturation plane will advance a distance of:
210
(
)
This distance is plotted as shown in Figure 4-6 along with the other distances that have been calculated using eqs (4-20) and (4-21) for other time values and other water saturations. These curves are characteristically double-valued or triple valued. For example, Figure 4-6 indicates that the water saturation after 240 days at 400 ft is 20, 39, and 69%. The saturation can be only one value at any place and time. What actually occurs is that the intermediate values of the water saturation have the maximum velocity (Figure 4-5 and eq. (4-17)), will initially tend to overtake the lower saturations resulting in the formation of a saturation discontinuity or shock front. Because of this discontinuity the mathematical approach of Buckley-Leverett, which assumes that Sw is continuous and differentiable, will be inappropriate to describe the situation at the front itself. The difficulty is resolved by dropping perpendiculars at point Xf (as flood front position) so that the areas to the right (A) equal the areas to the left (B), as shown in Figure 4-6. In other words a discontinuity in Sw at a flood front location Xf is needed to make the water saturation distribution single valued and to provide a material balance for displacing fluid.
Xf
Xf
A B Initial Water Saturation Initial Water Saturation
(a)
(b) Figure 4-6 – (a) Fluid Distribution at 60, 120, 240 days (b)Triple-valued saturation distribution (after Buckley and Leverett, 1942)
211
A more elegant method of achieving the same result was presented by Welge in 1952. This consists of integrating the saturation distribution over the distance from the injection point to the front, thus obtaining the average water saturation behind the front Sw, as shown in Figure 4-747.
Figure 4-7 - Water saturation distribution as a function of distance, prior to breakthrough
The situation depicted is at a fixed time, before water breakthrough, corresponding to an amount of water injection. At this time the maximum water saturation, Sw = 1 - Sor, has moved a distance X1, its velocity being proportional to the slope of the fractional flow curve evaluated for the maximum saturation which, as shown in Figure 4-5, is small but finite. The flood front saturation Swf is located at position x2 measured from the injection point. Applying the simple material balance: (̿̿̿̿
)
(4-22)
So: ̿̿̿̿
(4-23)
Where,
is cumulative water injection.
Using eq. (4-18): (
)
(4-18)
)
(4-24)
At breakthrough time: ( Where, Breakthrough time, Total injection rate, Medium length From eq. (4-24): 47
“Fundamentals of Reservoir Engineering”, L.P. Lake, 1978.
212
(
)
(4-25)
Where PVI is the pore volume injected. So: (
)
(4-26)
The average water saturation in the reservoir at the time of breakthrough is given by material balance as: ̿̿̿̿̿̿
(4-27)
From eqs (4-26) and (4-27): ̿̿̿̿̿̿ (
(4-28)
)
Therefore: (
)
4-29)
̿̿̿̿̿̿
i.e. the slope of the fractional flow curve at conditions of the front is given by eq. 4-29). To satisfy eq. (4-29) the tangent to the fractional flow curve, from the point Sw = Swc, where fw = 0, must have a point of tangency with co-ordinates Sw = Swf; fw = fwf, and extrapolated tangent must intercept the line fw = 1 at the point (Sw = ̿̿̿̿̿̿; fw = 1). See Figure 4-8.
̿̿̿̿̿ 𝑆𝑤𝑏𝑡 Swf, fwf
Figure 4-8 - Tangent to the fractional flow curve from Sw = Swc
The use of either of these equations ignores the effect of the capillary pressure gradient, ∂Pc/∂x.
213
This simple graphical technique of Welge has much wider application in the field of oil recovery calculations. As eq. (4-19) shows the velocity of every saturation front is constant, the graph of saturation location vs. time is set of straight lines starting from the origin. This graph is often plotted in dimensionless form. The equation can be made dimensionless by defining:
(4-30)
Where Normalized distance Pore volumes injected Eq. (4-19) becomes: (4-31) Figure 4-9 is a graph of dimensionless distance vs. dimensionless time for the movement of water saturation predicted by the frontal advance equation. Saturation Siw
Sw =Swi
Figure 4-9 – xD vs. tD for a linear waterflooding.
Saturation profiles or saturation histories can be constructed by making cross sections through the time/distance graph. A saturation profile is a graph of the locations of all saturations along a cross
214
section of fixed time, as illustrated by the continuous line at tD=0.28 in Figure 4-9. Figure 4-10 displays the saturation profile at tD =0.28 that was obtained from Figure 4-9.
Swf Flood Front Swi
Figure 4-10 – Saturation Profile at tD = 0.28
The saturation history is the graph of saturation vs. time at a particular value of xD. A plot of water saturation vs. tD for xD = 1, shown in Figure 4-11, illustrates the arrival of water saturations at the end of the linear system.
Swf
Figure 4-11 – Saturation History at xD = 1, producing face of the medium
Figure 4-12 represents the initial water and oil distributions in the reservoir unit and also the saturation distributions after 240 days, provided the flood front has not reached the produced face of the cubic reservoir. The area to the right of the flood front in Figure 4-12 is commonly called the oil bank and the area to the left is sometimes called the flooded or drag zone. The area above the 240-day curve and below the 90% water saturation curve represents oil that may yet be recovered, or dragged out of the high-water saturation portion of the reservoir by flowing large volumes of water through it. The area above the 90% water saturation represents unrecoverable oil since the critical oil saturation is 10%. This presentation of the displacement mechanism has assumed that capillary force is negligible. Figure 4-12 also indicates that a well in this reservoir will produce water-free oil until the flood front approaches the well. Thereafter, in a relatively short period, the water cut will rise sharply and be 215
followed by a relatively long period of production at high, and increasingly higher, water cuts. For example, just behind the flood front at 240 days, the water saturation rises from 20% to about 60%-that is, the water cut rises from zero to 66% (see Figure 4-5). When a producing formation consists of two or more rather definite strata, or stringers, of different permeabilities, the rates of advance in the separate strata will be proportional to their permeabilities, and the overall effect will be a combination of several separate displacements, such as described for a single homogeneous stratum. Sor = 10%, Unrecoverable oil Recoverable oil after infinite flooding
Recovered oil after 240 Days
Flood Front
Initial Water Saturation
Figure 4-12 - Saturation distribution after 240 days
Water Injection Oil Recovery Calculations Before water breakthrough (t= tbt) in the producing face/well, eq. (4-18) can be applied to determine the positions of planes of constant water saturation, for Swf < Sw < 1 – Sor, as the flood moves through the reservoir, and hence the water saturation profile. At the time of breakthrough and subsequently, this equation is used in a different manner, to study the effect of increasing the water saturation at the producing well. In this case x = L, the length of the reservoir block, which is a constant, and eq. (4-18) can be expressed as: (
(
)
)
(4-32)
Where, Injected Pore Volume (dimensionless) Water saturation at the end (producing) face 216
Using equation (4-19), someone can find the saturation at the producing face after the breakthrough time. Before this time Swe = Swi, and at the breakthrough time it is equal to the Swf and then it increased with time. Before breakthrough occurs the oil recovery calculations are trivial. For incompressible displacement the oil recovered is simply equal to the volume of water injected, there being no water production during this phase. At the time of breakthrough the flood front saturation, Swf = Swbt, reaches the producing well and the reservoir watercut increases suddenly from zero to a phenomenon frequently observed in the field and one which confirms the existence of a shock front. At this time (breakthrough time) produced pore volume of the oil can be calculated as follows: (̿̿̿̿̿̿
)
(
(4-33)
)
Where, Produced pore volume of oil, Produced pore volume of oil at breakthrough time. After breakthrough, L remains constant in eq. (4-32) and Swe and fwe, the water saturation and fractional flow at the producing well, gradually increase as the flood moves through the reservoir. During this phase the calculation of the oil recovery is somewhat more complex and requires application of the Welge equation, for average saturation of a segment of porous medium: Welge Equation – Average saturation in a segment of porous medium The average saturation between two points, x1 and x2, behind the front in a linear system with uniform A and is: ̅̅̅̅
∫ ̅̅̅̅
∫
(4-34)
The integral in the numerator of this equation can be evaluated using the method of integration by parts, i.e. ∫
∫
to give: ∫
[
]
∫
[
]
∫
(4-35)
The value of x in the last term of the above equation can be related to Sw by the frontal advance equation:
217
(
)
(4-18)
Substituting eq. (4-18) for x in eq. (4-35) gives: ∫
∫ (
∫
)
∫
(4-36)
(4-37)
Average saturation between two cross sections at x1 and x2, could be found by substituting eq. (4-37) into eq. (4-34): ̅̅̅̅
(4-38)
The above equation can be used to relate the average saturation in any segment of the linear system to the saturations at the two ends of that segment. If we consider the situation after breakthrough and take x1 to be the injection face (x1 = 0, fw1 = fwin) and x2 to be the production (end) face (x2 = L, Sw2 = Swe, fw2=fwe), we can write an expression for the average saturation in the system as: ̅̅̅̅
4-39)
In the above equation, fwin is the fractional flow of water at the injection face, which would always be equal to 1. Hence: ̅̅̅̅
4-40)
Frontal advance equation can be used to relate the value PVI to saturation at the outflow end. The axial position of the saturation Sw2, which is now at the producing end, is given by eq. (4-18): (
)
(4-41)
Or:
(
)
(4-42)
Substituting eq. (4-42) in eq. (4-40), we get: ̅̅̅̅ (
)
(4-43)
218
This is the Welge equation that relates the saturation at the end (producing) face to the average saturation in the linear system. Subtracting Swc = connate water saturation, from both sides of eq. (4-43), and using eq. (4-33) gives the oil recovery equation: ̿̿̿̿ Where,
(4-44) is dimensionless pore volume oil production.
The following steps show the procedure for calculating waterflood performance using the frontal advance equation:
Draw the fractional flow curve (fw vs. Sw), using eq. (4-7) and appropriate relative permeabilities and viscosities (as mentioned previously capillary pressure is neglected in developing this equation).
To find the front saturation and water fraction, draw the tangent to this curve ( the point:
. As described in the previous section, the point of tangency has the
co-ordinates:
) from
,.
Determine the average saturation behind the front (and at breakthrough time) by extrapolation of tangent line at front point to
.
Using eq. (4-33) to calculate breakthrough recovery.
Use eqs. (4-32) and (4-40) to calculate the values of average saturation (̿̿̿̿) and
values
corresponding to a bunch of selected Swe values that are higher than Swf.
Calculate the corresponding recovery using eq. (4-44). 1
(a)
Fractional Flow of Water, fw
fw-Sw
0.8
Tangent line from (Swi, 0) to (Swf, fwf)
0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
Water Saturation
219
(b)
Fractional Flow of Water, fw
1
0.9
fw vs. Sw Tangent line from (Swe, fwe) to (Ave.Sw,1)
0.8 0.6
0.7
0.8
0.9
1
Water Saturation Figure 4-13 - Application of the Welge graphical technique to determine: (a) The front saturation, (b) Oil recovery after breakthrough
Mobility Ratio Effect48 The basic mechanics of oil displacement by water can be understood by considering the mobilities of the separate fluids. The mobility of any fluid is defined as: (4-45) Where, k is absolute permeability and
is relative permeability.
The manner in which water displaces oil is illustrated in Figure 4-14 for both an ideal and non-ideal linear horizontal waterflood.
Figure 4-14 - Water saturation distribution as a function of distance between injection and production wells for (a) ideal or piston-like displacement and (b) non-ideal displacement
48
“Fundamentals of Reservoir Engineering”, L.P.Dake, 1978.
220
In the ideal case there is a sharp interface between the oil and water. Ahead of this, oil is flowing in the presence of connate water (
=), while behind the interface water
alone is flowing in the presence of residual oil (
). This favorable
type of displacement will only occur if the ratio ⁄ ⁄ Where, M is known as the end point mobility ratio and, since both and relative permeabilities, is a constant.
(4-46) are
the end point
If M ≤ 1 it means that, under an imposed pressure differential, the oil is capable of travelling with a velocity equal to, or greater than, that of the water. Since it is the water which is pushing the oil, there is therefore, no tendency for the oil to be by-passed which results in the sharp interface between the fluids. The displacement shown in Figure 4-14(a) is, for obvious reasons, called "piston-like displacement". Its most attractive feature is that the total amount of oil that can be recovered from a linear reservoir block will be obtained by the injection of the same volume of water. This is called the movable oil volume where: (4-47) The non-ideal displacement depicted in Figure 4-14(b), which unfortunately is more common in nature, occurs when M > 1. In this case, the water is capable of travelling faster than the oil and, as the water pushes the oil through the reservoir, the latter will be by-passed. Water tongues or fingers develop leading to the unfavorable water saturation profile. Ahead of the water front oil is again flowing in the presence of connate water. This is followed, in many cases, by a waterflood front, or shock front, in which there is a discontinuity in the water saturation. There is then a gradual transition between the shock front saturation and the maximum saturation Sw = 1−Sor. The dashed line in Figure 4-14(b) depicts the saturation distribution at the breakthrough time. In contrast to the piston-like displacement, not all of the movable oil will have been recovered at this time. As more water is injected, the plane of maximum water saturation (Sw = 1−Sor) will move slowly through the reservoir until it reaches the producing well at which time the movable oil volume has been recovered. Unfortunately, in typical cases it may take five or six MOV's of injected water to displace the one MOV of oil. At a constant rate of water injection, the fact that much more water must be injected, in the unfavorable case, protracts the time scale attached to the oil recovery and this is economically unfavourable. In addition, pockets of by-passed oil are created which may never be recovered. An even more significant parameter for characterizing the stability of Buckley Leverett displacement is the shock front mobility ratio, Ms, defined as:
221
(4-48)
in which the relative permeabilities in the numerator are evaluated for the shock front water saturation, Swf. Hagoort49 has shown, using a theoretical argument backed by experiment, that Buckley-Leverett displacement can be regarded as stable for the less restrictive condition that Ms < 1. If this condition is not satisfied there will be severe viscous channeling of water through the oil and breakthrough will occur even earlier than predicted using the Welge technique.
Figure 4-15 - (a) Microscopic displacement (b) Residual oil remaining after a water flood
Example 4-250 A thin reservoir zone (8 ft thick) is to be opened in several wells to production. The oil from this zone is expected to have a viscosity of 2.36 cp at reservoir temperature. Because the oil is essentially dead, it will be necessary to supplement the reservoir energy to produce oil at economical rates. A waterflood is under consideration. Relative permeability data from an adjacent interval have been correlated with water saturation by use of eqs (4-49) and (4-50). Log calculations indicate that the initial water 49
“Displacement Stability of Water Drives in Water Wet Connate Water Bearing Reservoirs”, Hagoort, J., 1974. Soc.Pet.Eng.J., February: 63-74. Trans. AI ME. 50 “Enhanced Oil Recovery”, Don, W. Green, G. Paul Willhite, 1998
222
saturation should be 0.136. A residual oil saturation of 0.325 was obtained on cores from a geologically similar interval. (
)
(4-49) (4-50)
Where, (
)
(4-51)
Calculate the waterflood recovery (PV) when the saturation at end face is equal to 0.54. The viscosity of water at reservoir temperature is 0.63 cp.
Calculate the oil recovery when water oil ratio (WOR) =27.57.
Calculate post breakthrough performance
Solution The first step is drawing fw-Sw curve according to eq. (4-7). Eqs (4-49) and (4-50) are used to find relative permeabilities data that are needed for fw calculations: Sw
SwD
kro
krw
fw
Sw
SwD
kro
krw
fw
0.1500
0.0260
0.7058
0.0000
0.0000
0.4500
0.5826
0.0577
0.0663
0.8115
0.2000
0.1187
0.4322
0.0000
0.0003
0.5000
0.6753
0.0331
0.1063
0.9232
0.2500
0.2115
0.2943
0.0007
0.0086
0.5500
0.7681
0.0163
0.1528
0.9724
0.3000
0.3043
0.2035
0.0044
0.0746
0.6000
0.8609
0.0058
0.2034
0.9924
0.3500
0.3970
0.1391
0.0150
0.2876
0.6500
0.9536
0.0008
0.2558
0.9992
0.4000
0.4898
0.0921
0.0354
0.5905
0.6750
1.0000
0.0000
0.2821
1.0000
223
1
1
0.9
0.9 kro
0.8
krw
Relative permeability, fraction
Relative permeability, fraction
0.8 0.7 0.6 0.5 0.4 0.3
0.7 0.6 0.5 0.4
0.3
0.2
0.2
0.1
0.1
0
fw
0 0
0.2
0.4
0.6
0.8
1
0
Water saturation, fraction
0.2
0.4
0.6
0.8
1
Water saturation, fraction
(a)
(b)
4Figure 4-16 – (a) Relative Permeability Curves, (b) Fractional Flow Curve Swbt=0.516
1
To find the water saturation at the front and average water saturation at breakthrough time, a tangent is drawn from initial water saturation point (Swi, 0) = (0.15 , 0) to the (fw vs. Sw) curve. Line touches the curve at the front saturation point. According to Figure 4-17, the front point coordination is: (Swf , fwf)=(0.46 , 0.85).
0.9
(Swf,fwf)=(0.46,0.85)
Relative permeability, fraction
0.8
fw
0.7 0.6 0.5 0.4
Average saturation at breakthrough time could be found by continuing the tangent line to the line fw=1. According to the Figure 4-17 average water saturation at breakthrough time is: ̿̿̿̿̿̿ .
0.3
0.2 0.1 0 0
0.2
0.4
0.6
0.8
1
Water saturation, fraction
Figure 4-17 – Graphical determination of front saturation and water fractional flow.
The pore volume injected of water (PVI) up to breakthrough time that is equal to the produced oil until this time (fluids are assumed to be incompressible), could be calculated using either eq. (4-40) or (4-44): ̅̅̅̅̅̅
(
)
̅̅̅̅̅̅ (
)
4-40) (4-52)
224
(̿̿̿̿̿̿
)
(4-53)
The difference between two results (from eqs (4-52) and (4-53)) is resulted from the graphical reading of the front saturation and fractional flow. So the water flood recovery at breakthrough time is around 37%. To calculate oil recovery after breakthrough time when producing WOR is, using the following equation to convert WOR to fw: (4-54) According to Wedge equation, we should pass a tangent through the (fw-Sw) curve at point Sw=0.54, fw=0.965), see Figure 4-18.
Fractional Flow of Water, fw
1.00
0.98 (Swe, fwe) = (0.54,0.965)
0.96
0.94
0.92
0.90 0.45
0.50
0.55
0.60
0.65
0.70
Water Saturation, Sw Figure 4-18 –
The tangent line cut the fw=1 line at Sw = 0.578, which is the average saturation in the medium when Sw at the producing (end) face is 0.54. Npd could be found using eq. (4-53): (̿̿̿̿
)
The following calculations are done to find post breakthrough performance:
( ̅̅̅̅
(4-42)
) ̅̅̅̅
225
Using the fractional flow curve and these equations, the following table is generated for saturation higher than the front saturation (Swf = 0.46): ̿̿̿̿
⁄
WOR
0.47
0.8668
2.1854
0.4576
0.5310
0.3810
6.5050
0.49
0.9073
1.6538
0.6047
0.5460
0.3960
9.7918
0.52
0.9480
1.0440
0.9578
0.5698
0.4198
18.2301
0.55
0.9724
0.6232
1.6046
0.5943
0.4443
35.1883
0.57
0.9827
0.4275
2.3390
0.6105
0.4605
56.8196
0.59
0.9898
0.2856
3.5012
0.6257
0.4757
97.0241
0.62
0.9962
0.1471
6.7995
0.6458
0.4958
262.0579
0.65
0.9992
0.0615
16.2719
0.6631
0.5131
1239.4319
Npd, Pore Volume Produced
0.55 0.50 0.45 0.40 0.35 0.30 0.00
4.00
8.00
12.00
16.00
PVI, Pore Volume Pore Injection Figure 4-19 – dimensionless pore volume oil recovery vs. dimensionless pore volume water injection
226
Example 4-3[48] Oil is being displaced by water in a horizontal, direct line drive under the diffuse flow condition. The rock relative permeability functions for water and oil are listed in following table: Sw
krw
kro
Sw
krw
kro
0.20
0.000
0.800
0.55
0.100
0.120
0.25
0.002
0.610
0.60
0.132
0.081
0.30
0.009
0.470
0.65
0.170
0.050
0.35
0.020
0.370
0.70
0.208
0.027
0.40
0.033
0.285
0.75
0.251
0.010
0.45
0.051
0.220
0.80
0.300
0.000
0.50
0.075
0.163
Pressure is being maintained at its initial value for which: ⁄
⁄
Compare the values of the producing watercut (at surface conditions) and the cumulative oil recovery at breakthrough for the following fluid combinations. Case
Oil Viscosity (cp)
Water Viscosity (cp)
1 50 2 5 3 0.4 Assume that the relative permeability and PVT data are relevant for all three cases.
0.5 0.5 1.0
Solution 1) For horizontal flow the fractional flow in the reservoir is 4-5) While the producing watercut at the surface, fws, is ⁄ ⁄
(4-55)
⁄
Where the rates are expressed in the surface watercut as:
(
⁄ . Combining the above two equations leads to an expression for
)
4-5)
The fractional flow in the reservoir for the three cases can be calculated as follows:
227
Fractional Flow (fw) Case 1 ⁄ 0.20
0.000
0.800
0.25
0.002
0.610
0.30
0.009
0.35
⁄
Case 2
Case 3
⁄
⁄
0
0
0
305.000
0.247
0.032
0.001
0.470
52.222
0.657
0.161
0.008
0.020
0.370
18.500
0.844
0.351
0.021
0.40
0.033
0.285
8.636
0.921
0.537
0.044
0.45
0.051
0.220
4.314
0.959
0.699
0.085
0.50
0.075
0.163
2.173
0.979
0.821
0.155
0.55
0.100
0.120
1.200
0.988
0.893
0.250
0.60
0.132
0.081
0.614
0.994
0.942
0.394
0.65
0.170
0.050
0.294
0.997
0.971
0.576
0.70
0.208
0.027
0.130
0.999
0.987
0.755
0.75
0.251
0.010
0.040
0.999
0.996
0.909
0.80
0.300
0.000
0.000
1.000
1.000
1.000
1 = 0.35 = 0.55
Fractional Flow, fw
0.8 (Swf, fwf) = (0.297,0.65)
0.6
(Swf, fwf) = (0.80,1)
(Swf, fwf) = (0.45,0.7)
= 0.80
0.4
Case 1 Case 2
0.2
Case 3 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Water Saturation, Sw Figure 4-20 - Fractional flow plots for different oil-water viscosity ratios
Fractional flow plots for the three cases are shown in Figure 4-20, and the results obtained by applying Welge's graphical technique, at breakthrough, are listed in the following table: ̿̿̿̿̿̿
Case 1
0.297
0.65
0.71
0.35
0.15
2
0.45
0.85
0.88
0.55
0.35
3
0.80
1.00
1.00
0.80
0.60 228
The results show an increment in breakthrough recovery of water injection process by increasing the water-oil viscosity ratio ( ⁄ ). Values of M and Ms for the three cases are listed in the following. Using these data: Case
⁄
1
100
0.297
0.006
0.520
1.40
37.50
2
10
0.45
0.051
0.220
0.91
3.75
3
0.4
0.80
0.300
0.000
0.15
0.15
Recovery performance of water injection increases by increasing water to oil viscosity ratio, Figure 4-21.
(a)
(b) Figure 4-21 –(a) Water Saturation Distributions in Systems for Different Oil/Water Viscosity Ratios (b) Areal sweep efficiency at breakthrough, five-spot pattern (Craig, 1980)
The typical injection/production well configurations and associated flooding patterns are shown in 4-22, including the five-spot pattern used for the results shown in 4-21.
229
Production Well
Injection Well
Pattern Boundary
Two Spot
Three Spot
Five Spot
Inversed Five Spot
Inversed Nine Spot
Regular Nine Spot
Regular Four Spot
Inversed Four Spot
Direct Line Drive
Staggered Line Drive
Figure 4-22 - Typical injection/production well configurations and associated flooding patterns
230
Vertical and Volumetric Sweep Efficiencies Reservoirs are formed over long periods of time in a variety of depositional environments. After deposition, physical, biological, and chemical reorganization occur. As none of these processes necessarily occur uniformly in time or space, it is understandable that reservoirs are generally very heterogeneous. All intensive properties of the reservoir such as permeability, porosity, wettability, connate water saturation, crude oil properties, and pore size distributions are likely to be non-uniform. Of these, permeability variations are considered most frequently in the literature. Craig (1980)51 gives a detailed review of reservoir heterogeneity and considers three general types of reservoir heterogeneities. These are: areal permeability variations, vertical permeability stratification and reservoir scale fractures. Others (Allsop, 198852) use alternative terminology: microscopic, mesoscopic, macroscopic or megascopic heterogeneities. One consequence of reservoir heterogeneities is that the displacement front of any injected fluid will move as an irregular front and must be properly considered in reservoir calculations. For instance, a measure of the uniformity of water invasion is termed vertical sweep efficiency (EI) and is defined as the cross sectional area enclosed in all layers behind the injected fluid front. As such, the vertical sweep efficiency is a measure of the two dimensional (i.e. vertical cross-section) effect of reservoir nonuniformities (Figure 4-23). The volumetric sweep efficiency, Ev, is a measure of the three-dimensional effect of reservoir heterogeneities. It is defined as the product of the pattern areal sweep and the vertical sweep as shown in the following equation: (
)
Where, areal sweep efficiency, Ad = area of displacement, AR = area of reservoir,
= porosity, S = gas or oil saturation.
51 52
Craig, F.F., SPE of AIME, 3rd printing, 1980 Allsop, H.A., Ph.D. Thesis, University of Waterloo, 1988
231
(a)
(b)
Swept Zone
Non Swept Zone
Figure 4-23 - Schematic representation of the two components of the volumetric sweep: (a) areal sweep; (b) vertical sweep in stratified formation
Review of Gravity Related Oil Recovery Studies There are a number of studies in the literature that deal with the effect of gravity forces on displacement behavior. Emphasis has been given to gravity drainage, i.e. the downward self propulsion of oil in the reservoir rock (Lewis, 1944). This phenomenon was studied extensively from 1940 to 1965; results from the first gravity drainage study were presented by Stahl et al. (1943). In this study, air was used to displace various liquids from a column containing Wilcox Sand. The results showed the dependence of liquid saturation on column height at both equilibrium and dynamic conditions. They also showed that the drainage rates were temperature dependent. Lewis (1944) gave an extensive review of the general aspects of gravity drainage and discussed conditions that favor gravity drainage. He also reported results from some field studies. Terwilliger et al. (1951) performed gravity drainage experiments on silica sand using brine and gas. The main difference between the work of Terwilliger et al. (1951) and that of Stahl et al. (1943) is that Terwilliger et al. (1951) conducted experiments with continuous production of the wetting phase. Stahl et al. (1943) however, halted the experiment momentarily for sampling. As a result, the work of Terwilliger et al. (1951) yielded more typical Buckley-Leverett plots. They applied the Buckley-Leverett approach in order to model their displacement tests and succeeded in matching their experimental results to this model.
232
Marx (1956) describes a method for predicting the complete gravity drainage characteristics of arbitrary, long columns from centrifuge drainage measurements on reconstituted core samples. Marx claims that oil residuals corresponding to hundreds of years of normal gravity depletion can be obtained in a few hours on the centrifuge. The flow rate due to gravity at any stage of the depletion process may be determined from the time correlation obtained in this study. Craig et al. (1957) used scaled reservoir models to study gravity segregation in frontal drives. The scaling criteria were as follows: ( )
(
)
4-56)
( ) 4-57)
( )
(4-58)
(
)
(4-59)
Where, = linear injection rate, = length, = thickness,
= density difference kx, ky = effective permeabilities,
d = fluid viscosities, = interfacial tension,
= contact angle, g = gravitational constant, Ra = directional permeability constant, Rb = mobility ratio, Rc = viscous to capillary forces constant, Rd = viscous to gravity forces constant. The tests included unconsolidated and consolidated media in either a five spot pattern or linear pattern and variable length. Unconsolidated stratified systems were also used. In these studies, the
233
experiments were discontinued after breakthrough of the injected fluid. From this Craig et al. (1957) concluded:
For linear gas or water injection operation in flat formations of uniform rock texture, segregation of the fluids, due to gravity effects, can result in oil recoveries at breakthrough as low as twenty percent of those otherwise expected. In five spot injection operations in flat uniform systems, the oil recoveries at breakthrough can be as low as forty percent of those predicted by methods that assume negligible gravity effects. In secondary recovery operations in stratified rock formations, the oil recovery at breakthrough may be affected to a greater degree by fluid segregation due to variations in rock properties, rather than by gravity effects. The magnitude of segregation of the fluids due to gravity is influenced by the average injection rate, rather than day-to-day or week-to-week variations.
Hovanessian and Fayers (1961) used a finite difference approach to solve the one-dimensional displacement equation for a homogeneous porous medium, including the effects of gravity and capillary forces. The authors state that the inclusion of capillary effects eliminates the triple valued BuckleyLeverett saturation profiles. Templeton et al. (1962) examined the gravity counter-flow segregation in a closed system. They completed immiscible displacement studies in glass beads and tried to calculate relative permeabilities by applying Darcy's law. Moreover, they observed saturation distributions as a function of time with a method similar to the one of Terwilliger et al. (1951). They concluded that Darcy's equations, if modified for the separate phases, are generally valid for counter-flow due to density differences. Cook (1962) performed a mathematical analysis of gravity segregation process during natural depletion conditions. He developed and solved the differential equations for two types of flows: 1) A "distributed flow" where vertical permeability in the reservoir is zero and fluids flow only in the dip direction while uniformly distributed over a hypothetical sand thickness, and 2) A "segregated flow" where vertical permeability is sufficient to permit gas to segregate against the sand top before proceeding up dip. Gardner et al. (1962) performed a series of experiments on gravity segregation of miscible fluids in linear models with both rectangular and circular cross-sections. They concluded that: • A sharp interface between fluids with different densities and different viscosities moving in a closed horizontal linear model tends to become straight and tilts in accordance with the following equation:
(
)
( ) (
)
(4-60)
4-61) 4-62) 234
Where, x = distance of leading edge of the viscous fluid from initial position of the interface, t = time measured from the instant when the interface is perpendicular to the length of the model, kH = permeability of the model in x and y directions, kv = permeability of the model in the z direction,
= fluid viscosity, av = average viscosity [(1 + 2) / 2], = density,
= density difference, h = height of rectangular models or diameter of cylindrical models. Their experiments indicated that the effect of diffusion becomes significant, starting from a sharp interface, when the following is true, 4-63) Where, De = effective diffusion coefficient of fluid = 0.95 x 10-5 cm2/s, b = width of rectangular models. • The effect of gravity on the displacement of a fluid from a horizontal linear model by a fluid of equal viscosity but different density is described by the following equation: 4-64)
Where, U = total fluid flow, = distance from outlet to initial position of interface, Ebt = breakthrough efficiency (volume of original fluid recovered at the breakthrough divided by the volume of the injected fluid). • When steady-state circulation is established in a closed vertical model, the advance of the more viscous fluid is given by the equation given below. However, the measured velocity may be influenced by fluid mixing. (4-65)
235
• The maximum stabilized zone length in an inclined model is 2xcrsin, where is the inclination of the x-axis with the horizontal and xcr is the critical distance. The stabilized zone length may exceed the minimum by a substantial amount if the initial interface is sharp. • When calculating the effect of diffusion, the three-dimensional nature of the motion can be an important factor. This is of particular importance when the height is small compared to the width (presumably always true in radial systems). Slobad and Howlett (1964) examined the effects of gravity segregation in laboratory studies of miscible displacement in vertical unconsolidated porous media. Fluids of varying viscosity and density were injected into the top of the core and moved at constant rates of twenty-five, fifty or one hundred feet per day using a positive displacement pump. The refractive index was measured to determine the composition of the efflux. The following cases were examined: 1.
1/2 favourable, favourable
2.
1/2 favourable, unfavourable
3.
1/2 unfavourable, favourable
4.
/2 unfavourable, unfavourable
They conducted their experiments in a vertical core of unconsolidated sand, measuring four feet in length and two inches in diameter ( = 0.37, k = 18 Darcies), and they demonstrated the effects of the four cases on the mixing zone length. They also discussed the effect of other parameters on the mixing zone length, including: molecular diffusion, permeability variations, convective mixing, and fingering and adverse gravity segregation. For their system, they showed the length of the mixing zone can be as short as 0.06 - 0.07 PV under favorable conditions. It was found that gravity will tend to increase the length of the mixing zone under favorable density conditions. In the case of a favorable viscosity ratio (viscosity of displaced fluid less than viscosity of displacing fluid) a favorable density difference was found to reduce the mixing zone length and with sufficient time, allowed gravity segregation to sharpen the zone. When the viscosity ratio was unfavorable the mixing zones were longer due to fingering, while favorable density differences drastically reduced the mixing zone length. Overall, the mixing zone was found to be a function of the relative magnitude of the viscous forces to the gravity forces. Hiatt (1968) outlined a general fluid displacement equation for two-phase, incompressible vertical flow through porous media and discussed the effect of fluid drive, gravity and capillary pressure. The equations of this report are not mathematically solved and the results are not verified experimentally. Thompson and Mungan (1969) performed miscible displacement tests on artificially fractured sandstones in order to examine the effect of various reservoir and process parameters on oil recovery. They discussed the importance of sub-vertical fractures, matrix permeability, displacement rate and fracture density in oil recovery and they stated that the magnitude of the fractured permeability, the fracture orientation, the core length and the connate water have little effect on the process. Sheffield (1969) used matrix techniques and computer algorithms to solve the differential equations for one and two-dimensional reservoir performance predictions with all forces present. He did not provide 236
any experimental support for his results. Kuo et al. (1970) gave a theoretical prediction of oil production by gravity drainage for a steam soak process. However, their results were not experimentally verified. Spivak (1974) used a reservoir simulator to examine the effect of gravity segregation in two-phase displacement processes. His work dealt with a wide variety of problems but it also lacked experimental verification. Dumore and Schols (1974) performed capillary pressure experiments in order to examine whether mercury porosimetry curves can be transformed to drainage capillary pressure curves. They found that with the aid of measured surface tension, permeability and porosity, various fluid combinations could give a unique dimensionless capillary pressure curve. They also found that the mercury-air interfacial tension is time dependent. For both capillary drainage and gravity drainage they observed the following:
The presence of connate water in the permeable medium led to very low residual oil saturations in the rock-gas-oil systems used by the authors. The low oil saturations obtained were independent of whether or not the oil phase spreads on water in the presence of gas. The drainage rate of oil in the presence of connate water and gas is possibly film flow, after the relatively short period in which the main oil production occurs. The contribution of film flow to oil production in secondary gas caps is thought to be generally negligible in the lifetime of the reservoir. It may play a role, however, when film flow takes place over short distances in very permeable rock. In primary gas caps, very low oil saturation may occur where film flow of oil may have been active for geological periods of time,
Lefebvre Du Prey (1978) examined gravity and capillary effects on imbibition in porous media. He used a scale-up procedure and expressed his results as functions of the scale-up parameters. Hagoort (1980) stated that gravity drainage could be a very effective way of production. In his theoretical discussion he covered the importance of relative permeability to oil. Additionally, he performed relative permeability studies in a centrifuge in order to get an exact representation of field conditions. Joshi and Threlkeld (1985) conducted an experimental study of a thermally aided gravity drainage mechanism using a horizontal well. The laboratory apparatus was a glass bead packed rectangular box with one transparent wall, allowing for flow visualization. For comparison reasons, various combinations of vertical and horizontal wells were tested. Results indicated that the horizontal well pair (both injecting and producing wells being horizontal) yielded maximum oil recovery. Their experimental data was used to justify the theoretical and experimental work of Butler et al. (1981) and Butler and Stephens (1981) where steam was used to assist the gravity drainage of heavy oil. The temperature distribution in steam-assisted gravity drainage using horizontal wells was also investigated in this work and by Chung and Butler (1988).
237
Coning53 One important example of interface encroachment is the local deformation of the interface (G/O or O/W) near a producing well. Draw off caused by a pressure difference between the well and the interface caused the interface to be distorted and to approach the well. If the well is in production and the medium is homogeneous and isotropic, then the flow follows symmetry of revolution. However, if the flow is not exactly radial-cylindrical and the well is not perforated along the entire thickness of the bed, then the stream lines in the lower portion of the bed straighten upwards to reach the base of the well (O/W interface). This upward flow of the oil and water distorts the water/oil contact surface, which then assumes a roughly conical shape. Several types of coning can occur according to the input and distribution conditions of the fluids. Figure 4-24 illustrates the two main types: bottom coning and edge coning.
(a)
(b)
Figure 4-24 – a) Bottom coning at oil-water or gas-oil contact, b) Edge coning at oil-water or gas-oil contact
Figure 4-25 shows a borehole that has been drilled to depth, hp, in a layer of thickness, H. This layer is impregnated with oil to height, ho, and by water to height (H - ho). The ratio hp/ho is the penetration of the borehole. The greater the draw off effect of the well, i.e. the greater the penetration and flow rate, the higher the "cone". Conversely, gravitational forces exert a stabilizing effect.
Figure 4-25 - Stable Cone
Infra-Critical Flow For a set of given values of the parameters shown in Figure 4-25, and if the flow rate is sufficiently low with respect to penetration, the cone will form a fixed surface in space. Such a cone is stable and does 53
Cossé, R., Houston: Gulf Publishing Company, 1993.
238
not reach the base of the perforations. Also, no water encroachment occurs in the well shown in Figure 4-25. Note that at critical flow a further increase in pressure differential will not increase the flow rate. Supercritical Flow This section includes only a discussion of bottom coning as edge coning requires very complex analysis and is studied by simulation. Investigations conducted by Bournazel and Jeanson (IFP) clarify the correlations established by Sobocinsky (Esso) concerning bottom coning. The critical flow rate is given by the following equation, in US field units (barrels per day, pounds per cubic foot, millidarcys, feet, centipoise): (
)
4-66)
Where,
= 1.41x10-5 (Sobocinsky) and = 1.15x10-5 (Bournazel and Jeanson correlation results). If metric units are used, = 1.52x10-3 and = 1.24x10-3, respectively. This formula is derived from the expression of the forces in Figure 4-25:
Using Darcy’s Law: ( ) If hc (ho – hp ), by definition Qo Qc and, ( ) With, The coefficient is obtained from an average correlation including ln(R/rw) and the partial penetration effect. It is important to note that the critical flow rate is dependent on the horizontal permeability kh. Additionally, the well results often give a critical flow rate higher than the one given by this equation. This is partly a result of the assumption that the formation of the cone is considered “statically" whereas the vertical movement of water also occurs, and partly because little is known about the permeabilities, kh and kv, near the well.
239
Bournazel's study also provided a methodology to calculate the breakthrough time and the change in the WOR after breakthrough, and that the WOR reaches a limit for horizontal input and steady-state flow. Although coning has been investigated methodically since the 1950's, it is so complex and unstable that two basic questions remain unanswered: 1.
How to drill a well subject to coning?
2.
What is the correct production rate?
Controversies exist on these subjects. The cautious solution involves perforating over a short stretch of pay zone and adopting a low flow rate to delay the arrival of the undesirable fluid at the well for as long as possible. This method is adopted if Qo it is not much higher than Qc, see (1) in Figure 4-26.
Figure 4-26 – Flow rate versus time
From this standpoint, it would be better to avoid drilling a well crossing the O/W, G/O or G/W interfaces with a large thickness of undesirable fluid. However, the more recent "full pot" solution involves perforating widely and withdrawing at maximum flow rate (see (2) in Figure 4-26), with considerable production of the second fluid, if the second fluid is mostly comprised of water. In fact, at largely supercritical conditions, the final recovery can be improved with high flow rates. It should be added that empirical laws serve to make production forecasts for coning in supercritical conditions after breakthrough. Included are the methods of Hutchinson and Henley, who established correlations between recovery and WOR as a function of the mobility ratio, the estimated recovery area and the penetration or the flow rate. However, numerical models adapted for coning are widely employed to simulate these problems. A vertical subdivision is used to represent the horizontal and vertical permeability variations of the reservoir, providing a better approximation of the potential results. Also, various assumptions regarding heterogeneities serve to achieve some sensitivity in the results. Studies are also under way to prevent water influxes by injecting polymers that are designed to plug the water influx zones. 240
References Allsop, H.A., Ph.D. Thesis, University of Waterloo, 1988. Buckley, S.E. and Leverett, M.C., Trans AIME, 146, 107-116, 1942. Butler, R.M., McNab, G.S., and Lo, H.Y., Can J. of Chem. Eng., 59, 455, 1981. Butler, R.M. and Stephens, D.J., J. Can. Pet. T., 59, 90, 1981. Chung, K.H. and Butler, R.M., J. Can. Pet. T., 28(1), 36, 1988. Cook, R.E., Soc. Pet. Eng. J., 2, 261, 1962. Cossé, R., Houston: Gulf Publishing Company, 1993. Craig F.F., Sanderlin, J.L., Moore, D.W. and Geffen, T.M., Petr. Trans. AIME, 210, 275, 1957. Craig, F.F., SPE of AIME, 3rd printing, 1980. Dumore’, J.M. and Schols, R.S., Soc. Pet. Eng. J., 14, 437, 1974. Forrest, F. and Craig, J.R., Monograph 3, Society of Petroleum Engineers, Dallas, TX 1971. Gardner, G.H.F., Downie, J. and Kendall, H.A., Soc. Pet. Eng. J., 2, 95, 1962. Haitt, W.N., Soc. Pet. Eng. J., 8, 225, 1968. Hagoort, J., Soc. Pet. Eng. J., 20, 139, 1980. Hall, H.N., Soc. Pet. Eng. J., 1, 927, 1961. Hovanessian, S.A. and Fayers, F.J., Soc. Pet. Eng. J., 1, 32, 1961. Joshi, S.D. and Threlkeld, C.B., AOSTRA Jour. of Res., 2(1), 1985. Kuo, C.H., Shain, S.A. and Phocas, D.M., Soc. Pet. Eng. J., 10, 119, 1970. Lefebvre Du Prey, E., Soc. Pet. Eng. J., 18, 195, 1978. Leverett, M.C., Trans AIME, 142, 149, 1941. Lewis, J.O., Trans AIME, 155, 133, 1944. Marx, J.W., Petr. Trans. AIME, 207, 88, 1956. Sheffield, M., Soc. Pet. Eng. J., 9, 255, 1969. Slobad, R.L. and Howlett, W.E., Soc. Pet. Eng. J., 4, 1, 1964. Spivak, A., Soc. Pet. Eng. J., 14, 619, 1974. Stahl, R.F., Martin, W.A. and Huntington, R.L., Trans. AIME, 151, 138, 1943. Templeton, E.E., Neilsen, R.F. and Stahl, C.D., Soc. Pet. Eng. J., 2, 185, 1961. Terwilliger, P.L., Wilsey, L.E., Hall, H.N. and Bridges, P.M., Petr. Trans. AIME, 192, 285, 1951. Thompson, J.L. and Mungan, N., Soc. Pet. Eng. J., 9, 247, 1969.
241
242
5 .
C H A P T E R
5
Miscible Displacement Introduction In an immiscible displacement process, such as water flooding, the microscopic displacement efficiency, eq. (5-1), is generally much less than unity. (5-1) Where, Initial oil saturation, Residual oil saturation after immiscible displacement, Microscopic displacement efficiency Part of the crude oil in places contacted by the displacing fluid is trapped as isolated drops, stringers, or pendular rings, depending on the wettability. At this condition relative permeability to oil would be almost zero and no more oil will be produced by continuing displacing fluid injection. In this situation capillary pressure prevent the oil drops to move and pass through constrictions in the pore passages. This limitation to oil recovery may be overcome by the application of miscible displacement processes in which the displacing fluid is miscible with the displaced fluid at the condition existing at the interface between the displaced and displacing fluids two fluids that mix together in all proportions within a single fluid phase are miscible. If the two fluids do not mix in all proportions to form a single phase, the process will be immiscible54. Most practical miscible agents exhibit only partial miscibility toward the crude oil itself, so some times “solvent flooding” term is used instead of “miscible flooding”. It should be mentioned that there is difference between “miscibility” and “solubility”. In contrast to the “miscibility”, “solubility” is defined as the ability of a limited amount of one substance to mix with another substance to form a single homogeneous phase while miscibility is defined as the ability of two or more substances to form a single homogeneous phase when mixed in all proportions. The physics of miscible and immiscible displacement are significantly different. Therefore, different factors dominate these displacements at the pore level, and different phenomena are considered in modeling them. In immiscible displacements in porous media an interface separates the fluids. One fluid is never completely displaced by another, immiscible fluid. There will be irreducible or residual saturations after the displacement has reached steady state. Typical oil residual in displacements by water (Sorw) can be 25-40% of the original oil in place (OOIP). In a capillary system, such as a porous medium, the interfacial tensions associated with the immiscible fluid interfaces play a significant role in determining the fluid 54
“Enhanced Oil Recovery”, D. W. Green and G. P. Willhite, 1998.
243
distributions within the porous medium, as covered under the subject of capillarity. The displacement front in an immiscible displacement will be sharper at higher flow rates, or when capillary forces are neglected. On the other hand, in a miscible displacement, no interface exists between miscible fluids of different composition. In the absence of fluid/fluid interfaces, capillary forces are absent. Steady state is reached when one fluid has completely displaced the other fluid; the concept of irreducible or residual saturations does not apply. This means that by using miscible displacement process we could achieve very high “pore scale” recovery efficiencies. Once the solvent front has moved through a volume of the porous medium containing the original oil, little to no residual oil is left behind. Miscible displacement can be used as a secondary recovery process just after primary recovery of the oil or as a tertiary recovery method at the end of water injection process. The main oil recovery mechanisms during miscible flood are extraction, dissolution, vaporization, solubilization, condensation, or other phase behavior change involving the crude oil, viscosity reduction, oil swelling and solution gas drive, but the primary mechanism must be extraction55.
Oil & LPG (Miscible zone)
LPG (Primary Slug)
LPG & Lean Gas (Miscible Zone)
Chase Gas
(a)
Water
Water
Chase Gas
In practice, solvents that are miscible with crude oil are more expensive than water or dry gas, so instead of continues solvent injection a slug of the solvent (such as LPG56 ), with a size of approximately 5% of reservoir pore volume is injected that is followed by a larger volume of a less expensive fluid (chase fluid), such as water or a lean or flue gas (Figure 5-1). To have an efficient displacement of the primary (solvent) slug, ideally this secondary slug should be miscible with the primary (solvent) slug. To improve the overall sweep efficiency by LPG process, the hydrocarbon slug I displaced by altering the chase gas with water slug and finally with continuous water injection (Figure 5-1)
Oil Bank
Connate Water
Residual oil Oil Bank
Oil & LPG (Miscible zone)
LPG (Primary Slug)
LPG & Lean Gas (Miscible Zone)
Chase Gas
Water
(b)
Chase Gas
Water
Flood Direction
Injected water from water flooding
Connate Water
Flood Direction Figure 5-1 – Miscible Displacement, a) secondary recovery, b)Tertiary recovery.
55 56
“Enhanced Oil Recovery”, L. W. Lake, 1989 Liquefied Petroleum Gas
244
The miscible displacement process can be classified as first-contact miscible (FCM) or multi-contact miscible (MCM) on the basis of the manner that miscibility developed. In FCM the injected solvent forms only a single phase upon first contact when mixed in all proportion with the crude oil, propane injection is a sample of this process. In the MCM process (dynamic miscibility) miscible conditions are developed in situ through composition alteration of the injected fluid or crude oil (by in situ mass transfer) as the fluid move through the reservoir. These processes are categorized into: -
Vaporizing lean gas drive, also called ‘high pressure lean gas injection’, Condensing rich gas drive, Combined vaporizing-condensing drive.
The high pressure gas and the enriched gas drive are member of the MCM process. Various gases and liquids are suitable for use as miscible displacement agents in either FCM or MCM processes. These include low-molecular weight hydrocarbons, organic alcohols, ketones, refined hydrocarbons, condensed petroleum gas (LPG), liquefied natural gas (LNG), carbon dioxide, air, nitrogen, exhaust gas, flue gas and mixture of these. It should be mentioned that because there is only one phase in the miscible region, the wettability of the rock and relative permeability lose their importance since there is no interface between fluids. However the mobility ratio, which is defined as the viscosity ratio between miscible solution and the displaced oil, has a significant effect on the recovery efficiency. For a miscible flood to be economically successful in a given reservoir several conditions must be satisfied57: -
57
An adequate volume of solvent must be available at a rate and cost that will allow favorable economics, The reservoir pressure for miscibility between solvent and reservoir oil must be attainable, Incremental oil recovery must be sufficiently large and timely for project economics to compensate for the associated added cost.
“Advanced Resevoir Management and Engineering”, T. Ahmed, N. Meehan, 2011.
245
FLUID PHASE BEHAVIOR Various methods exist to represent the vapor-liquid phase behavior of multicomponent systems. These methods include the use of pressure-temperature, pressure-composition, and ternary diagrams that provide a convenient way to present the boundaries of the single and multiphase regions typically determined from experimental data or from calculation with an EOS.
Pressure-Temperature diagram (P-T diagram) Figure 5-2 shows a P-T diagram for a pure component. The line connecting the triple point and critical points is the vapor pressure curve; the extension below the triple point is sublimation point. As this figure shows in pure materials, by decreasing the pressure at a fixed temperature, phase change happens just at a point (vapor pressure curve is a line). According to Figure 5-2, the phase boundary between liquid and gas does not continue indefinitely. Instead, it terminates at a point on the phase diagram called the critical point. This reflects the fact that, at extremely high temperatures and pressures, the liquid and gaseous phases become indistinguishable.
Figure 5-2 - A typical phase diagram for a pure component
The phase behavior of a multi-component system is more elaborate than that of a pure compound. the complexity generally compounds as components with widely different structure and molecular sizes comprise the system. Reservoir fluids are mainly composed of hydrocarbons with similar structure so their phase behavior is not generally highly complex. Figure 5-3 shows an idealized P-T diagram for a multi component with a fixed overall composition. As it shows, there is a transition zone between the complete liquid phase and complete vapor phase. In other words in contrast to the pure substance, phase change from liquid to vapor happens, by decreasing the pressure at a fixed temperature, on a line. So there is a region that two phases are at equilibrium. Two phases region that is bounded by the bubble point and dew point curves is called “phase envelope”. The bubble point and dew point curves meet at the critical point. Two phases can exist at a pressure greater than critical pressure and at a temperature greater than critical temperature, unlike a pure component system. Cricondonbar for a multicomponent system is defined as the maximum pressure that two phases (vapor-liquid) can exist in equilibrium and cricondontherm is the maximum temperature that two phases can exist in equilibrium.
246
Figure 5-3 shows the general behavior for a fixed composition of a system consisting of two or more components. If the composition were changed then the position of the phase envelope on the P-T plot would shift. Cricondenbar Critical Press., Pc
Critical temp., Tc
Cricondentherm
Figure 5-3 – Typical P-T diagram for a multicomponent system
Figure 5-4.a shows the phase diagram of Ethane-normal Heptane system. As it shows, the critical temperature of different mixtures lies between the critical temperatures of the two pure components. The critical pressure, however, exceeds the values of both components as pure, in most cases. The locus of critical or “plait” points is shown by the dashed line (Figure 5-4.a). The difference between the critical pressure of two components system and each pure component critical pressure increases by increasing the difference between the critical points of the two pure components (Figure 5-4.b). No binary mixture can exist as a two-phase system outside the region bounded by the locus of critical points.
(a)
(b)
Figure 5-4 – (a). Phase diagram of ethane-normal heptane58, (b) Critical loci for binary mixtures
58
“PVT and Phase Behavior of Petroleum Reservoir Fluids”, A. Danesh, 2003
247
Extracted data from the graphs such as Figure 5-4 indicate conditions at which mixture of binary pairs are miscible. To have a miscibility of two compounds of ethane and normal pentane with any composition, at each temperature the pressure should be higher than the pressure indicated by the locus of critical pressure line for that specific temperature. At lower pressure there are possible concentrations at which the system will separate into two phases. So a minimum miscible pressure for each temperature could be determined according to graph’s data such as Figure 5-4. For systems containing a large number of components no maps of critical points are available.
Example 5-1[50] A FCM miscible displacement is to be designed in which a slug of butane is the primary displacing solvent. The butane slug is to be displaced by dry gas consisting essentially of methane. Assume that the crude oil could be represented by n-decane. The reservoir conditions are 160oF and 2500 psi. Use Figure 5-4.b to determine whether miscible conditions would exist at the front and back of the solvent (butane) slug. Is the miscible condition would be exist at p=1500 psi and T=160oF?
Solution The primary slug is butane displacing oil. Refer to Figure 5-4.a. the locus of critical points for C4-C10 is not given, however at 160oF and 2500 psi C4 and C10 are in liquid state. C4 and C10 will be miscible. The secondary slug is methane displacing butane. In Figure 5-4.a at 160oF, a pressure of 2600 psi is well above the locus of critical points for C1-C4 mixtures. Therefore C1 and C4 will be miscible. So at 160oF and P=2500 psi proposed displacement will be miscible at both the leading and trailing edges. If the reservoir pressure was 1500 psi then the displacement of butane by methane would be at conditions below the critical locus. At this condition, at some points in the process two phases would form and the dry gas would displace butane immiscibility at reduced microscopic displacement efficiency. So some amount of the butane may be bypassed during the displacement by dry gas and trapped in the reservoir. This would result in a relatively rapid degradation of the integrity of the primary butane slug and drastically influence oil production.
Pressure-composition diagram Pressure-composition diagram is another way to present the phase behavior information. Usually the composition is expressed as a mole fraction of the more volatile component (Figure 5-5).
248
T = 90oF T = 140oF T = 200oF
P T = 260oF
25
50
75
100
Mole % Methane in Methane-Ethane Mixture Figure 5-5 – Typical P-X diagram for the Methane-normal Butane system.
For a binary system at temperature higher than the critical temperature of volatile component, two phases do not form over all compositions of the more volatile component. For the system that shown in Figure 5-5 at 90oF two phases are not formed above the methane composition of about 90%. The composition of methane above which only one phase exists decease with increasing temperature. The p-x diagram also shows the cricondenbar pressure for each temperature. For each temperature cricondenbar is the maximum pressure that two phases can exist. Above this pressure all methaneethane mixtures will be a single phase, in other words complete miscibility exist. Figure 5-6 presents a px diagram for a three components system consisting of methane, n-butane and decane at a fixed temperature of 160oF59. The graph is pressure versus mol% of methane adding to the reservoir mixture containing methane, butane, and decane. A single phase exists above the phase-boundary line, while two phases exist below this line. Miscibility condition of added methane (C1) and reservoir fluid could be find from the graph. For example, if incremental amounts of C1 were added to the reservoir fluid at a constant pressure of 2000 psia and temperature of 160oF, the mixture of fluid would be single phase as long as the concentration of added C1 was less than about 25 mol%. Further C1 addition would cause a gas phase separates from the liquid phase. However miscibility would be reached if the concentration of added C1 exceeds about 97%. At low-gas concentration end, gas goes into solution in the liquid, while at the high-gas concentration end, the liquid is vaporized into the gas. If the pressure on the system were raised to a value equal or higher than cricondenbar pressure, then addition of C1 to the reservoir fluid would not cause two phases to form and the gas C1 and reservoir fluid would be single phase over all possible concentration of added C1.
59
“Enhanced Oil Recovery”, D. W. Green and G. P. Willhite, 1998
249
Figure 5-6 – Pressure-composition diagram for mixture of C1 with a liquid mixture of C1-nC4-C10
Adding some heavier hydrocarbons, such as C4, to the injected gas causes shifting down the cricondenbar. So miscibility condition reaches at lower pressure.
Ternary diagram: Ternary or triangular phase diagrams can be used to plot the phase behavior of systems consisting of three components by outlining the composition regions on the plot where different phases exist. The advantage of using a ternary plot for depicting compositions is that three variables can be conveniently plotted in a two-dimensional graph and mixture of different components can be easily represented. A ternary diagram for hypothetical components A, B and C is shown in Figure 5-7.The phase behavior on ternary diagram is plotted at fixed pressure and temperature and If these three components were miscible at that special temperature and pressure then no multiphase region would appear on the diagram. Every point on a ternary plot represents a different combination of the three components. Compositions could be represented in ternary diagram as weight, mole or volume percentage. The volume percentage is used only when there is not significant volume change during the mixing. The vertexes represent the pure components, and the sides of equilateral triangle are scaled to represent the binary compositions of the three possible pairs. There are three common methods used to determine the ratios of the three species in the composition. The first method is an estimation based upon the phase diagram grid. The concentration of each species is 100% (pure phase) in its corner of the triangle and 0% at the line opposite it. The percentage of a specific species decreases linearly with increasing distance from this corner, as seen in Figure 5-7. By drawing parallel lines at regular intervals between the zero line and the corner (Figure 5-7.a), fine divisions can be established for easy estimation of the content of a species. For a given point, the fraction of each of the three materials in the composition can be determined by the first. For phase diagrams that do not possess gridlines, the easiest way to determine the composition is to set the altitude of the triangle to 100% (that is equal to the sum of all altitudes from the mixture point to three sides) and determine the shortest distances from the point of interest to each of the three sides. The distances (the ratios of the distances to the total height of 100%) give the content of each of the species, 250
as shown in Figure 5-7.b. The third method is based upon a larger number of measurements, but does not require the drawing of perpendicular lines. Straight lines are drawn from each corner, through the point of interest, to the corresponding side of the triangle. The lengths of these lines, as well as the lengths of the segments between the point and the corresponding sides, are measured individually. Ratios can then be determined by dividing these segments by the entire corresponding line, by the socalled inverse-lever-arm rule, as shown in the Figure 5-7.c. In Figure 5-7.c, point M is a mixture of (
) of component ‘B’ and (1-
) of binary mixture ‘b’ of ‘A’ and
‘C’. These 3 rules could be applied to the combination of any two mixtures represented on the ternary. Figure 5-8 shows two different mixtures represented by M1 and M2. Any mixture of M1 and M2 would be along line ̅̅̅̅̅̅̅̅̅ . The point representing the composition of the final mixture would again be determined by applying the inverse-lever-arm rule along the line.
251
(a)
Component
Weight% in mixtur (M)
A
50
B
30
C
20
Component
Weight% in mixture (M)
A
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅
A
c b M
(b)
B C
B
C
a
A c
b M
Component
Weight% in mixture(M)
A
̅̅̅̅ ̅̅̅̅
B
̅̅̅̅ ̅̅̅̅
C
̅̅̅̅ ̅̅̅
(c)
B
a
C
Figure 5-7 – Ternary phase diagram for a system consisting of components A, B, and C which are miscible in all proportions
252
Phase relationship may also be represented on a triangular phase diagram Figure 5-11. The diagram shows the phase condition at equilibrium at constant pressure and temperature. The plot is typical for hydrocarbon systems in which vapor-liquid equilibrium exists over regions of the concentration domain.
Figure 5-8 – All mixture of M1 and M2 would be along line ̅̅̅̅̅̅̅̅̅
Example 5-2[50] Figure 5-9.a shows the phase behavior of mixtures of chemicals A, B and C. the diagram is on the pound mass basis and for a fixed T and P. three different mixtures of components A, B and C are contained in three vessels as shown in Figure 5-9.b. The contents of three vessels are mixed together in forth vessel, which is held at the P and T corresponding to the ternary diagram. Describe the resulting mixture as follows: a) Find the final mixture composition, and show the overall composition on the diagram, b) Indicate whether the final mixture is one phase or two phase at specified T and P, c) If the mixture is two phase, specify the composition and amount of each phase.
(a)
#1 1.0 lbm
#2 1.0 lbm
#3 2.0 lbm
85% A
60% A
0% A
5% B
30% B
0% B
10% C
10% C
100% C
(b)
Figure 5-9 – Example 5-2 a) Ternary diagram, b) Chemicals to be mixed
Solution For the first step we should determine the mass of each components (A,B, and C) in each vessel using the composition and mass of the chemicals. Table 5-1 contains the results for this step and sample calculation:
253
Table 5-1 – Composition calculation, example 5-2.
Component
A
B
C
#1
0.85 * 1 = 0.85 lbm
0.05 lbm
0.10 lbm
#2
0.60 lbm
0.30 lbm
0.10 lbm
#3
0.00 lbm
0.00 lbm
2.00 lbm
0.85 + 0.60 + 0.00 = 1.45 lbm
0.35 lbm
2.20 lbm
Chemical
[[
Final Mixture
To find the composition of the final mixture, mass of each component should be divided by total mass:
Using methods explained before (Figure 5-7), the final mixture could be shown in the diagram: The composition of the final mixture located in the two phase region of the diagram. V
To find the amount of each phase using the lever law:
Final Mixture
36% C
M
To find the composition of each phase using the methods explained in Figure 5-7: A
L
B
C
Vapor (V) 9% B
55% A
Liquid (L)
Figure 5-10 – Liquid vapour phase diagram for example 5-2.
Reservoir fluids are complex mixture of hydrocarbons with components ranging from methane to C40+. In miscible displacement process a fluid miscible with the fluid reservoir during FCM or MCM process. This injected fluid alters thermodynamic properties of the reservoir fluid by changing the chemical composition of the reservoir fluid. Rigorous thermodynamic analysis needs identification of all chemical constitutes and their composition. But it is not practical. The experiences have shown that complex hydrocarbon systems can be represented with groups of hydrocarbons that preserve many of the important properties of the system. The combined groups of the hydrocarbons are called 254
“pseudocomponents”. A decomposition of crude oil into CH4-N2, C2-C6, and C7+ components is a common example of “pseudocomponents”. A typical representation of miscible displacement process is a “pseudoternary” diagram with C1, C2-C6, and C7+as pseudocomponents as shown in Figure 5-11.b. generally possible grouping that is used frequently includes the following mixed components: -
The first component represents a volatile pseudo component and most of the times contains C1, N2, CO2, The second pseudocomponent is a mixture of intermediate hydrocarbon components such as C2 through C6, sometimes CO2 is grouped in this group. The last group is essentially C7+ fraction of the reservoir hydrocarbon.
An important assumption in using pseudocomponents and a pseudoternary diagram is that the composition of the pseudocomponents does not change in the different phases. Within the two phases region there are tie lines whose ends represents the composition of equilibrium phases (Figure 5-11). Tie lines length shrinks toward the critical (plait) point where the properties of two phases are indistinguishable. The position of plait point changes with temperature at a fixed pressure. As Figure 5-11.b shows, all composition represented by points (M1-M5) inside the two-phase envelop would separate into two phases (V1-V5 as vapor and L1-L5 as liquid), the relative amount of two phases can be calculated using inverse-lever-arm rule. The points outside the two phase envelop are representative of a single phase composition. The critical tie line is the fictitious tie line tangent to the bimodal curve at the critical point. The critical tie line is the limiting case of the actual tie line as the plait point is approached. A
C1
T = constant P = constant
Two phase region
Two phase region Immiscible
Critical or Plait point
Tie lines
V1 V2 V3
Critical Tie Line V4 V5
Critical Point
M1
Critical Tie Line
M2
M3 M4
One phase region Miscible
M5
2-Phase envelope L5
L1 L2
One phase region
B
C
L3
L4
C2-C6
C7+
(a). ternary phase diagram for a system of components A, B, C with limited miscibility
(b). Pseudoternary diagram
Figure 5-11 – Ternary Phase Diagram.
As the pressure increases, the two phase region shrinks, or in other words light-heavy miscibility increases (Figure 5-12). No general statement is possible about the effect of temperature though the two phase region generally growth with increasing temperature.
255
C1
C1
C1 Two phase region Immiscible
Two phase region Immiscible
One phase region Miscible
One phase region Miscible
C7+
C2-C6 T = T1, P = P1
C7+
C2-C6
Miscible with all proportion
C7+
T = T1, P = P2 > P1
C2-C6 T = T1, P = P3 > P2
Figure 5-12 – Pressure effect on the miscibility, P1
A Two phase region Immiscible
M1 One phase region Miscible
M3
B
M2
C
Figure 5-13 – Determination of miscible condition from ternary diagram.
As mentioned in Error! Reference source not found., in ternary iagram the mixture results of any combination of two components will lie on a straight line connecting two components to each other. According to this, Figure 5-13 shows that any combination of components A and C, and any combination of B and C, form a single phase, in other word in this specific pressure and temperature A and C, B and C are miscible. A and B are not miscible because the straight line between them pass through the two phase region, so mixing them with especial compositions will end with a two phase mixture. And with the same manner M1 and M2 are miscible the same as M2 and M3. But M3 and M1 are not miscible, because straight line between them passes through the two phase region (Figure 5-13).
First Contact Miscibility Process Two fluid phases are said to be first contact miscible (FCM) if, when mixed together in any proportion at given conditions, they form a single phase. In oil field practice a FCM process normally consists of injecting a relatively small primary slug that is miscible with the reservoir oil, followed by injection of a larger, less expensive secondary slug. Economic issues determined the primary slug size. To avoid trapping of the injected solvent in the reservoir the secondary slug tried to be miscible with the solvent in the first slug. Therefore phase behavior should be attended for the leading and trailing edges of the primary slug. Ternary diagram could be used to illustrate FCM, MCM and immiscibility condition. If a straight line dilution path between the solvent and the crude does not intersect the two-phase region, the displacement will consist of a single hydrocarbon phase that change in composition from reservoir oil to solvent through the solvent oil mixing zone. There is a range of solvent composition that will be first contact miscible with the crude at specific temperature and pressure (Figure 5-14).
256
C1 D Two phase envelop P=P1 A Range of solvent composition to have FCM with reservoir oil at P=P2>P1
Two phase envelop P = P2 > P1 B
Range of solvent composition to have FCM with reservoir oil at P=P1 C
C7+
C2-C6 Figure 5-14 – First Contact Miscibility.
First contact miscibility can be achieved for highly rich gas, or at very high pressure for lean gas. As Figure 5-14 shows, higher-molecular-weight hydrocarbons, such as propane or LPG, are miscible with crude oil, at most reservoir conditions. Reservoir porous medium could be assumed as a series of finite well-mixed cells. Figure 5-15 typically illustrate the miscible and immiscible oil flooding by using a wellmixed cell. Methane is a gas at reservoir condition and it doesn’t mix with the oil in all proportion and two phase exist. So crude oil displacement with methane is an immiscible flood (Figure 5-15.a).propane is gas at atmospheric condition however it would be a liquid phase at reservoir condition. Liquid propane and liquid crude oil mix together completely. The displacement of oil with propane at the specified condition would be a miscible process (Figure 5-15.b). Propane and butane exist as a gas at atmospheric condition. At the reservoir condition (such as P=2000 psi and T=150oF) propane is liquefied but the methane remains as a gas phase. At this condition they mix together in all proportions. So displacement of propane with methane at the specified condition would be a FCM displacement (Figure 5-15.c). To use LPG products such as ethane, propane and butane, pressure is required to liquefied the LPG products and achieve FCM with the reservoir oil. On the other hand LPG products are injected as a slug that is followed by a chase gas such as lean gas or flue gas. To decrease LPG residual saturation after displacing the primary slug with chase gas the pressure should be high enough to achieve FCM between chase gas and LPG as well as between reservoir oil and LPG. The required pressure to achieve miscibility between the chase gas and the LPG slug is much higher than the pressure required liquefying LPG hydrocarbons. So the controlling pressure during the miscible flooding of the oil with LPG slug is the minimum pressure required to have FCM between LPG slug and chase gas.
257
Gas Phase Atmospheric condition
Methane
Liquid Phase Atmospheric condition
P = 2000 psi T = 150oF
Methane
Propane
Propane Methane
Mixture Reservoir oil
Reservoir oil
(a)
Atmospheric condition
P = 2000 psi T = 150oF
P = 2000 psi o T = 150 F
Methane
Mixture Reservoir oil
Mixture
Reservoir oil Propane
(b)
Propane
(c)
Figure 5-15 – Miscibility and immiscibility illustration using well-mixed cells [1].
Multiple Contact Miscibility Processes An injected solvent which is not miscible with oil at first contact may achieve miscibility during multiple contacts and mass transfer between reservoir oil and injected fluid. In-situ mass transfer of intermediate molecular weight components between gas and oil phases results in mixtures that are miscible with either the injected solvent or initial oil phases. The multiple contact miscibility are categorized as ‘vaporizing gas drive’ (VGD), ‘condensing’, ‘condensing/vaporizing-gas’ (enriched gas) and CO2 displacement. The condensing and vaporizing gas drive for enriched gas injection was first proposed and verified by experiments and numerical simulations by Zick in 198660. Interphase mass transfer of the intermediate components is the key process of the mechanism. Later, analytical theory for the combined condensing/vaporizing mechanism was developed. It can be argued that condensing/vaporizing gas drive is the most common mechanism developed in miscible injection field projects as injected solvents usually contain somewhat light- and heavier-intermediate components.
Vaporizing Gas Drive In the vaporizing gas drive, the injected fluid is generally a relatively lean gas that contains mostly methane and other low molecular weight hydrocarbons. Sometimes the injected gas in this process contains inert gas such as nitrogen. The intermediate weight hydrocarbons from the reservoir oil vaporize into the injected solvent. Under proper conditions this enrichment can be such that the injected fluid of modified composition will become miscible with oil at some point in the reservoir. From that point under idealized conditions, a miscible displacement will occur. In this process miscibility can be achieved with natural gas, flue gas, CO2 or nitrogen, provided that the reservoir pressure is above the minimum miscibility pressure.
60
“Introduction to Chemical Engineering Thermodynamics”, J. M. Smith, H.C. Van Ness, M. M. Abbot
258
Usually a pseudoternary diagram is used to describe MCM displacements. In the process description using ternary diagram it is assumed that thermodynamic equilibrium exists between the different phases. This assumption is generally thought to be valid at reservoir displacement conditions where advance rates are very low. Suppose that the injected gas (pint ‘S’ in Figure 5-16) that is mostly consist of light component C1. The point ‘O’ represents the reservoir oil. As mentioned before oil in-place and the injected gas are not miscible at the first contact as the dilution straight line between them pass through the two phase envelop. The miscibility development operates conceptually as follows: -
-
-
The injected gas ‘S’ after contacting the oil ‘O’ forms a mixture ‘M1’ that is split into two equilibrated phases of liquid L1 and gas V1, determined by the equilibrium tie line. It should be mentioned that the gas phase, V1, is the original solvent gas, S, after it was enriched with some intermediate and heavy fractions from the oil phase. The gas V1 will have much higher mobility than L1 and moves forward and makes further contact with fresh oil to form mixture M2. The mixture M2 splits into gas V2 and liquid L2. The gas V2 is richer particularly in the intermediates. For the next time V2 passes L2 because of higher mobility and contacts to the fresh oil to form mixture M3 that is split to L3 and V3, and so far. After some steps the gas phase will no longer form two phases when contacts with fresh oil. In other words the dilution straight line between ‘O’ and gas phase does not pass through the two phase region and the gas become miscible with oil at point ‘C’, that is, where the tangent line at the critical point, which is the critical tie line with zero length, goes through the oil composition ‘O’.
It is quite evident, that the dilution path must go through the critical point, as it is the only condition that equilibrated phases lose distinctions, and a continuous transition from gas to oil can be achieved without any phase boundary. C1 S = Injected Solvent Composition
S
O = Reservoir Oil Composition V1 V2 M1
Tie Lines
M2
V3 V4
C = Critical Point L1
L2 L3 L4
O
C7+
C2-C6 Figure 5-16 – schematic of vaporizing gas drive process
In the vaporizing gas drive there is a transition zone, the miscibility is achieved at the front of the advancing gas, the gas composition varies gradually from that of the injected gas till reaching the ‘critical 259
point composition’. Then it miscibility displaces the original reservoir oil in a piston-type manner. No phase boundary exists within the transition zone. As long as the reservoir oil composition lies on or to the right of the critical tie line (that shows it is rich in intermediate components) miscibility can be attained by vaporizing gas drive process with natural gas that is lean and lie on the left side of the critical tie line. The requirement that the oil composition must lie on the right side of the critical tie line implies that only oils that are under saturated with respect to C1 can be miscibly displaced by methane or natural gas. C1
S = Injected Solvent Composition O = Reservoir Oil Composition
S V1 V2
V3 V4
Tie Lines M1
M4 L1
M2 M3
L2
Critical Point
O L3
L4
Critical Tie Line
C7+
C2-C6
Figure 5-17 – Schematic of an immiscible displacement
Figure 5-17 shows a schematic of an immiscible displacement. The injection gas (solvent) ‘S’ does not achieve multiple contact miscibility with oil ‘O’. The initial mixture ‘M1’ is the first mixture after contacting of the gas ‘S’ and oil ‘O’. The mixture is split to gas V1 and liquid L1. The gas phase will flow forward to form mixture M2, and so forth. This gas is being enriched in intermediate components at the leading edge of solvent-oil mixing zone as discussed before. But enrichment cannot proceed beyond the gasphase composition given by the tie line whose extension passes through the oil ‘O’ which is called Limiting tie line. In other word enrichment of the advancing gas is limited by the tie line (V4L4 here) which if extended goes through oil ‘O’.
It was explained that the miscibility cannot be achieved when the oil composition and the injection solvent composition are at the same side of the critical tie line. The MCM can be achieved for oil ‘O’ (Figure 5-18) by rising the pressure sufficiently to shrink the phase envelop (dotted curve). The pressure at which the critical tie line extension goes through the oil (Figure 5-18.b) or injection gas (Figure 5-18.a) is the minimum required pressure to achieve miscibility which called Minimum Miscibility Pressure (MMP). It is the minimum pressure at which in-situ miscibility can be achieved in the MCM process for a specified fluid system. At MMP, the limiting tie line becomes the critical tie line as the gas phase enriches through multiple contacts with the original oil attaining the critical composition.
Example 5-3 Using Figure 5-4.a, find the minimum miscibility pressure of the n-decane and methane at T = 160oF.
Solution Refer to Figure 5-4.a, at T=160oF the critical pressure is about 5200 psi. at the pressure higher than 5200 C1-C10 system would be single phase over the total concentration range from 100% C1 to 100% C10 and thus would be miscible. Miscibility by the vaporizing gas drive mechanism can be achieved using nitrogen or flue gas that contains about 88% CO2 and 12% N2. flue gas is generated by burning hydrocarbons with air.
260
C1
Two phase envelop P=P1
C1
S = Injected Solvent Composition O = Reservoir Oil Composition P2 = Minimum Miscible Pressure
S
Two phase envelop P = P2 > P1
S
Two phase envelop P=P1
Critical Tie Line at P = P2 > P1
S = Injected Solvent Composition O = Reservoir Oil Composition P2 = Minimum Miscibility Press.
Critical Tie Line at P=P2>P1 Two phase envelop P = P2 > P1
Critical Point
Critical Point
O O
Critical Tie Line at P=P2>P1
C7+
C2-C6
Critical Tie Line at P=P2>P1
C7+
(a)
C2-C6 (b)
Figure 5-18 – Minimum Miscibility Pressure a) Critical tie line pass through the injection solvent composition, b) Critical tie line pass through oil composition
The gas composition appears to have no effect on achieving the miscibility state on vaporizing gas drive as it is fully controlled by the oil phase. Relatively high pressure is required for most oil for the vaporizing gas drive process. So this process is limited to reservoir depth that will permit the use of pressure in excess of 3000 psi. reservoirs with under saturated oil with a high concentration of intermediate hydrocarbons (C2-C6) are the first candidate for this process.
Condensing Gas Drive In this process dynamic miscibility result from the in situ transfer of intermediate molecular weight hydrocarbons, predominantly ethane through butane, from the injected fluid into the reservoir oil. The modified oil then becomes miscible with injected fluid. So the injected fluid should contain significant amount of intermediate components, rather than being a dry gas.
261
C1 S = Injected Solvent Composition O = Reservoir Oil Composition Tie Lines V1 V2
S
V3 M2
V4
M1
Limiting tie line coincides with critical time line
M4
O
L1 L2
C7+
L3
Critical Point M3
L4
C2-C6 Figure 5-19 – Schematic condensing gas drive
Suppose crude oil ‘O’ and injection solvent ‘S’ (Figure 5-19) are on opposite side of the critical tie line but reversed from the vaporizing gas drive (Figure 5-16). Oil and injected fluid are not miscible initially as the dilution straight line between them pass through the two phase region. M1 is the first mixture resulted after first contact of ‘S’ and ‘O’. M1 will split to liquid L1 and gas V1 that are in equilibrium at this point in the reservoir. The liquid phase L1 is richer in intermediate components than the original oil ‘O’. Gas phase V1 move faster because of its higher mobility and leaves oil phase L1 to mix with the fresh fluid injected ‘S’ and form mixture M2. The new mixture will split to liquid L2 and gas V2. Liquid L2 lies closer to the critical (plait) point than L1 and is richer in intermediate components. The gas passes the liquid phase and L2 contact with the fresh solvent to form M3 and so forth. By continuing injection of the solvent ‘S’ the composition of the liquid phase is altered progressively in a similar manner along the bubble point curve until it reached the critical point. The plait point fluid is directly miscible with injection fluid ‘S’. the limiting tie line in this process pass through the solvent composition ‘S’ (in contrast to the VGD that pass through the oil composition), so the MMP in this process is defined as the pressure at which the critical tie line coincides with limiting tie line and its extension passes through the solvent composition (Figure 5-18.a).
262
C1
S = Injected Solvent Composition O = Reservoir Oil Composition
Tie Lines V1 V2 V3
S Limiting tie Line
M2
Critical Tie Line
M1
O
L1
M3 L2 L3
Critical
C7+
C2-C6
Figure 5-20 – Schematic of an immiscible displacement
For dynamic miscibility to be achieved by the condensing-gas drive method with an oil whose composition lies to the left of the critical tie line, the enriched gas composition must lie to the right of the critical tie line. If a gas injected contains less intermediate hydrocarbon so that both oil and solvent compositions located on two phase side of the critical tie line the oil cannot be enriched to the point of miscibility. In Figure 5-20 enrichment of the liquid phases (L1, L2,…) continue till a point that resulted mixture lie on the tie line that pass through the injected solvent composition point ‘S’. Enrichment will stop at this point. For this system miscibility can be achieved by raising the pressure to shrink the phase envelop so that the limiting tie line coincide with the critical tie line (Figure 5-18.a).
The original oil composition has no effect on achieving the miscibility state in the condensing gas drive. This process (CGD) is controlled by the injection gas composition. Two variables can be adjusted in condensing gas drive to achieve miscibility: reservoir pressure and injection solvent composition. The effect of reservoir pressure and MMP concept was explained before (Figure 5-18). There is an alternative to achieve miscibility in the condensing gas drive process. The injection gas composition can be enriched to achieve miscibility. Minimum Miscibility Enrichment (MME) is defined as the minimum enrichment level which the critical tie line passes through the injection gas composition (Figure 5-21) C1
Two phase envelop P=P1
S = Original Injected Solvent Composition O = Reservoir Oil Composition
S Minimum Miscibility Enrichment (MME)
Critical Point
O Critical Tie Line
C7+
C2-C6
Figure 5-21 – schematically concept of Minimum Miscibility Enrichment (MME)
263
Example 5-4 [50] A miscible displacement is to be considered for the system described on the pseudoternary diagram in (Figure 5-22), which is based on mole fraction. The oil composition is shown on the diagram. The injection fluids being considered are C1 and enriched C1 with C2 through C6 components. The injectedfluid cost increases with increasing amount of C2 through C6. Pressure is fixed.
Figure 5-22 – Pseudoternary diagram
Specify the composition of the minimum cost injection fluid that could be used in an MCM process. Use ternary diagram to describe the mechanism for development of miscibility.
Solution As described before to have a MCM the injection gas and original oil composition points should be on the different sides of the critical tie line. So the miscibility will not improve between pure C1 and reservoir oil, VGD or CGD. So the injected the C1 should be enriched with C2-C6 components till the injection gas and reservoir oil compositions lie on different side of the critical tie line. The less expensive injection gas that could be miscible with oil is a gas that its composition is located exactly on the critical tie line (Figure 5-23.a). This gas has the minimum C2-C6 concentration that is needed to reach miscibility with oil during MCM. The oil composition is on the left hand side of the critical tie line, so it is not rich in intermediate components to vaporize to the injected gas (VGD). Therefore miscibility would be reached during a condensing gas drive that the intermediate components would condense to the oil (Figure 5-23.b).
264
(a)
(b) Figure 5-23 - Example 5-4
Actual dynamic or multicontact phase behavior may be more complicated than shown in the simple pseudoternary diagram of Figure 5-19. As discussed before if the composition of pseudo component at different phases changes (for example the pseudocomponent C7+ content of C6 is different in gas phase and liquid phases) then the pseudoternary diagram for a multicomponent system may be incorrect. So using the ternary diagram for the conceptual discussion of multiple contact miscibility processes is not strictly valid for real reservoir oil. However the basic idea of multiple contact miscibility through mass transfer between the phases and the requirement of achieving the critical composition are valid for real fluid system. Zick (1986) explained the miscible displacement process during the enriched gas injection as follows. The oil/gas system is assumed to be composed of four groups of components: Lean components such as C1, N1, and CO2, 2. Light intermediate components like as C2 through C4, which are the enriching components, 3. Middle intermediate components which may range from C4 through C10 on the low-molecular weight side up to C30 on the high side that are generally not in the injected gas but in the reservoir oil and may be vaporized from the oil to the gas phase. 4. High molecular weight components that cannot be vaporized from the oil in significant amount. 1.
According to Figure 5-19 explanation when the enriched gas containing the components from group 1 and 2, contacts reservoir oil, light intermediates condense into the oil, making it lighter. The gas moves faster than the oil, so the depleted gas moves ahead. Additional fresh injected gas contact the oil, continuing enriched the oil with intermediate components. This process continues to reach miscibility. But there is a counter effect. The middle intermediate in the oil are stripped from the oil into the gas because this components are not originally in the injected gas. Thus the reservoir oil in contact with the fresh gas initially become lighter, but as it contact more gas and loses only some of its lighter heavies, overall it tends to enriched in very heavy fractions and thus becomes less similar to the injection gas. Figure 5-24 shows the variation of measured components groups in the oil phase at the injection point for a sample oil. According the previous explanation of the condensing gas drive the concentration of C7+ 265
should be decreased step by step to reach miscibility while experiments shows that an examination of heavy end, e.g. C20+ has increased distinctly due to vaporization of the lighter heavies. This prevents the development of miscibility between injected gas and reservoir oil. If this were all that occurred, the process would not be very efficient. However, downstream of the injection point is a positive mechanism. By continuing injection of the gas upstream oil becomes saturated with the light intermediates and therefore these components do not condense out of the gas into the oil, thus the downstream oil contacts a gas that is rich both in light and middle intermediates. So this gas vaporizes less middle intermediates from the downstream oil while losing intermediate to the downstream oil that is not rich in the intermediate oil. As this process continue and move downstream at a favorable condition the combined vaporizing/condensing process in a state within the transition zone where the compositional path goes through the critical point. In other worlds a gas is resulted that is almost miscible with reservoir oil. This process can be imagined as a combination of the condensing process at the downstream and vaporization at the upstream. This process is called ‘condensing/vaporizing gas drive’. The reservoirs containing the oil without high concentration of intermediate hydrocarbons are the first nominate for this type of recovery. However this process may be applied to a wide range of crude oil and to reservoirs with pressure of about 1500 psi and greater. 50 Light
Mole %
40
Middle intermediate Intermediate
30
C20+ 20 10 0 0
2
4 6 Vol Gas/Vol Oil
8
10
Figure 5-24 – Vaporization of component groups in contacted oil at injection point with the ratio of injected volume to contacted oil volume
Experimental investigation of the miscibility effect on the final oil recovery Metcalfe and Yarborough (1979)61 conducted an experiment to investigate the effect of phase behavior on the final recovery. Their experiment showed the manner in which the development of miscibility affects oil recovery for a relatively simple system. They used CO2 as the injection solvent and a mixture of butane and decane (40 mol% and 60 mol%, respectively) as the displaced fluid. So they could use 61
“Effect of Phase Equilibria on the Co2 Displacement Mechanism”, SPEJ, Aug. 1979, 242-52
266
ternary diagram to study the miscibility. The porous medium consisted of an 8 feet long, 2 inch diameter Berea sandstone core. The core was fully saturated with oil sample (butane + decane) without any initial water saturation. The core was flooded at temperature of 160oF and with 3 different pressures: 1900 psi, 1700 psi and 1500 psi. For the first test at 1900 psi, the injected gas (CO 2) is miscible with the oil over all composition. Thus the displacement was first contact miscibility displacement. As shown in Figure 5-25.a the reduced liquid concentration followed the ideal mixing line from the original oil composition point ‘O’ to the pure CO2. Oil recovery was 99% of the original oil in place (OOIP). The same experiment was run at 1700 psi. All other parameters such as temperature and porous sample held constant. As Figure 5-25.b shows at this pressure a small two phase region exists, but the oil composition lie to the right of the critical tie line. The produced oil composition was measured after detecting the CO2 in the effluent fluid. The composition measurement of produced oil showed that the composition changed in essentially a linear manner from the initial oil composition to the vicinity of critical point. Composition then moved around the two phase envelop and finally to pure CO2. Thus the concentration change followed the description of an MCM process. Oil recovery for this test was 90% of OOIP. The third experiment was conducted at 1500 psi. According to the Figure 5-25.c achievement of miscibility would not be expected, because both the oil and CO2 composition points lie on the same side of the critical tie line. The produced fluid concentration changed almost linearly from the initial composition to a point on the two phases envelop. The effluent then become two phase, indicating an immiscible displacement. Recovery was 81% of OOIP. o
T = 160 F
Path of Produced oil Composition CO2
CO2
CO2 Critical Point
Critical Tie Line
O: original oil
Critical Tie Line
O: original oil C10
C4
(a). P = 1900 psi
C10
C4
(b). P = 1700 psi
C10
C4
(c). P = 1500 psi
Figure 5-25 – Displacement of a C4/C10 mixture by pure CO2, representation on a ternary diagram [61]
Almost all the oil was displaced during the first contact miscibility process. Although final recovery during MCM process is high but it is less than the first contact miscibility. There are factors that contribute to the efficiency reduction, such as -
Miscibility must be developed in-situ, Complete miscibility may not exist across the entire miscible zone, Dispersion may lead to a temporary or permanent loss of miscibility in some part of the medium.
267
Thus the recovery in a MCM process is usually less than in a FCM process. The experiment showed that immiscible displacement has a much less efficiency than miscible displacement.
Determination of Miscibility Condition Determination of miscibility condition as the condition that miscibility is achieved during FCM or MCM process is very important for designing a miscible displacement process. Temperature, pressure and solvent injection composition are three major parameters that determine the miscibility condition. The temperature of the process is set by the reservoir temperature but the pressure could be changed in a certain interval as well as the injection solvent composition, so most of the methods have been designed to directly measure or predict the minimum miscibility pressure (MMP) or minimum miscibility enrichment (MME) at which miscibility will be achieved in a first or multiple contact process for a specific reservoir fluid composition and the reservoir composition. There three different methods to predict the miscibility conditions (MMP or MME): 1. Experimental Measurements 2. Empirical correlations based on experimental results 3. Phase-behavior calculations based on an equation of state and computer modeling While the experimental methods measure directly MMP and MME other ones predict these values.
Experimental Measurements Experimental tests can be used for some specific studies such as oil vaporization by gas injection or evaluation and tuning of phase behavior models for numerical simulation of the reservoir performance. In this section determination of miscibility conditions (MMP or MME), using these methods, will be explained.
Slim Tube Test It is a laboratory test used to estimate the minimum miscibility pressure (MMP) or minimum miscibility concentration (MMC) of a given injection solvent and reservoir oil. The first slim tube test has been run in the early 1950s. Figure 5-26 shows a schematic typical slim tube test apparatus. The slim tube is a narrow stainless-steel tube with an internal diameter (ID) of about ⁄
in. and a length between 5 and
40 meters. The tube is packed with glass beads of a size on the order of 100 mesh or sand of a specific mesh size (the ratio of particle size to tubing diameter is sufficiently small (less than ⁄ ) that wall effect can be neglected,), so it is a one dimensional model of the reservoir. “The sand packed tube displacement apparatus is a device for bringing about multiple equilibrium contacts between simultaneous flowing fluids. It is not intended to simulate reservoir conditions. Hence, slim-tube tests should not be indicative of ultimate recovery, macroscopic sweep efficiency, transition zone length, etc., to be achieved in actual reservoir”62. The actual displacement in reservoir is influenced by various 62
“Parametric Analysis On The Determination Of The Minimum Miscibility Pressure In Slim Tube Displacements”, Flock, D.L. and Nouar, A., JCPT, 1984
268
mechanisms cause dispersion, such as gravity override caused by gravity effect and viscous fingering caused by unfavorable viscosity ratio and porous medium heterogeneity, while an ideal slim tube should provide a one dimensional dispersion free displacement of oil. At this condition the MMP, MME and the final recovery of the displacement process are only depend on the thermodynamic phase behavior of the system. The tube is coiled in a manner so that flow is basically horizontal and the gravity effects are in significant. The coiled tube is placed inside a constant temperature bath. Packed Coiled Tube
Oil
Solvent
Injection Pump
Sight Glass Separator
Gas Meter
Gas Chromatograph
Back Pressure Pump Collection Cylinder
Figure 5-26 – Schematic Diagram of Slim Tube Apparatus
To do the test, the porous medium in the tube is initially saturated with reservoir sample oil. The coiled tube is maintained at the reservoir temperature using the bath and the pressure is set at a pressure higher than the bubble point pressure. The oil then displaced by injecting solvent into the tube at a constant inlet, or often outlet pressure, that is controlled by back pressure regulator. The pressure drop across the coiled tube is generally a small fraction of the applied pressure on the system, so the entire displacement pressure is considered to be at a single constant pressure. Effluent production, density and composition are measured as functions of the injected volume. As mentioned before by eliminating the dispersion final recovery if a function of thermodynamic phase behavior. Therefore the miscibility conditions are determined by conducting the displacement at various pressure, or injection solvent enrichment levels and monitor the oil recovery. The recoveries are then plotted as a function of displacement pressure, or solvent enrichment level. By increasing the pressure (using back pressure regulator) final recovery increases. This recovery increment is considerable for the pressure lower than the MMP, but for the pressure higher than MMP incremental oil recovery by increasing the pressure is not significant, in other worlds additional recovery above MMP is generally minimal. Based on this observation, MMP could be determined as the pressure at the “break” in the curve (Figure 5-27.a). In other words MMP has been chosen as the pressure for which incremental oil 269
recovery per incremental pressure increase is less than some arbitrary value. Some researchers define MMP as the pressure when the oil recovery is 90 or 95 percent. The same observation could be seen during the displacement of oil with the solvent at different enrichment levels. The final oil recovery increases significantly by increasing the injection solvent enrichment level. But the recovery increment ceased after a specified enrichment level (MME), see Figure 5-27.b. Enrichment level is changed by altering the intermediate hydrocarbon component concentration such as C2+ in the displacing solvent. Visual observation of the flow through a sight glass placed at the coiled tube outlet (Figure 5-26) could help in determination of miscibility condition. The achievement of miscibility is expected to accompany a gradual change of color of the flowing fluid from that of the oil to clear gas. On the other hand, observation of two phase flow is symptomatic of an immiscible displacement. Effluent fluid is cooled and the separated gas is evaluated by a gas chromatograph (Figure 5-26). 100
Oil Recovery, % Oil inplace
Oil Recovery, % oil inplace
100 90 80 70 60
MMP
95 90 85 80
Displacement at 2500psi
75
Displacement at 3000psi
70
50 20
24
28
32
36
40
Pressure, MPa
(a)
44
48
52
0.2
0.4
0.6
C2+ Mole Fraction
(b)
Figure 5-27 –a) Slim tube recovery versus applied pressure b) Slim tube recovery as a function of solvent enrichment63
Using a long tube has an additional rationale. The gravity segregation has an adverse effect on the displacement recovery results, and unstable flow occurs in the early stage of the displacement. Overly stabilized flow could be achieved by increasing the length of the model, so the use of longer slim tubes yields the best estimate of the MMP and MME. Experiments show that final recovery increases by increasing the slim tube length for any injection rate. As previously explained a multi contact miscibility starts as an immiscible displacement. Miscibility achieved through multi contact between the injection and in place fluids which implies that the solvent at the front has to contact enough fresh oil or travel a minimum distance before miscibility is achieved. Dispersion caused by viscous fingering or permeability heterogeneity increases the minimum length requirement to accomplish miscibility. Total recovery is the sum of the recovery obtained while solvent travels the immiscible portion of the tube and the recovery obtained while solvent travels the miscible portion of the tube64:
63
“An Interpretation of the Displacement Behavior of Rich Gas Drives Using an Equation-of-State Compositional Model”, Mansoori, J. and Gupa, S.P., SPE No. 18061, 1988 64 “A Laboratory Investigation of Miscible Displacement by Carbon Dioxide”, Rathmell, J.J., Stalkup, F.I., SPE No. 3483, 1971
270
(
)
(
)
5-2)
Where, A = Cross section area of the slim tube, Average porosity, Total recovery, Recovery in the immiscible portion of the displacement, Recovery in the miscible portion of the displacement, Immiscible portion length, Miscible portion length, Total slim tube length. The Lim is independent of the overall length of the tube so according to eq. (5-2) the final recovery approaches to the miscible recovery by increasing the slim tube length. As mentioned increasing the length stabilizes displacement and this stabilizing effect is rate independent. So as long as the length of the system is sufficiently long the rate of injection is not critical, however a moderately high injection rate is preferable in order to obtain a better differentiation between immiscible and MCM displacement. It can be concluded that except some criteria there is no standard design, no standard operating procedure, and no standard criterion for determining MMPs and MMEs with the slim tube. Also slim tube test is a time consuming method. While each displacing run under specific pressure takes about one day, one or two days are needed to clean and re-saturate the slim tube. So determination of MMP or MME for an oil-solvent system takes at least one week.
Example 5-5 A CO2 displacement test is conducted in a slim tube apparatus. A pseudoternary phase diagram for the fluid system is shown. The oil composition is indicated on the diagram. A displacement test is conducted with a fluid that is 96% CO2 and 4% C2-C6. -
Will the miscibility achieved in this displacement? When the injection gas first break through at the exit end of the slim tube what will its composition be? Consider a location near the entrance of the slim tube. Several PV,s of the gas will have flowed through this position by the end of the run. What will be the final oil composition at this position near the entrance?
271
Solution […]
Rising Bubble Apparatus (RBA) Christiansen and Haines, designed and made the RBA as a reliable, fast alternative to a slim tube for measuring MMP65. The observation of a gas bubble behavior that rises in a visual high pressure cell filled with reservoir oil is the base of this method. The major part of the RBA is a flat glass tube stands vertically in a high pressure sight gauge in a temperature controlled bath. The glass tube is flat for better viewing of bubbles rising in opaque oils. The sight gauge is backlighted for visual observation and photography of rising bubbles in the oil. The gas bubble is injected into the glass tube through a hollow needle that is mounted at the bottom of sight gauge. As the first step of the test the sight gauge and glass tube are filled with distilled water. In the next step, enough oil is injected into the glass tube to displace all but a short column of water in to the tube’s lower end (Figure 5-28). Next, a small bubble of gas of the desired composition is launched into the water. The buoyant force on the bubble caused it to rise through the column of water then through the water-oil interface. As the bubble rises through the oil, its shape and motion are observed and photographed. After that the bubble rises through the oil the used oil is replace with fresh oil and the test run again at different pressure or with different injection gas composition. Pressure Gauge Sight Gauge Glass Tube Oil
Gas
Oil Water
Injection Pump Gas Hollow Needle
Figure 5-28 – Oil, water and gas in RBA at start of test
It is assumed that the mass transfer process that occurs as the gas bubble rises through the oil in the glass tube is similar to the MCM process that occurs in a slim tube test. During the rising of the bubble through the oil at each height it faces fresh oil and the gas bubble enriched with intermediate components as rise through the column, same mechanism that we have during vaporizing gas drive. 65
“Rapid Measurement of Minimum Miscibility Pressure with the Rising-BubbIe Apparatus”, Christiansen R. L. and Haines H. K., SPE No. 13114, 1987
272
According to Figure 5-17, if the compositions of the gas bubble and oil are on the same side of the critical tie line, miscibility cannot be achieved. By increasing the pressure two phase region in the ternary diagram shrinks and for pressure above the MMP with some oil-gas pairs, it is possible for the two phase region to be so small that the gas and oil are first contact miscible.
Bottom
Middle
Top
Bubble behavior during rising through the oil filled tube varies significantly over a range of pressures and can be divided into three distinct patterns:
(a): P<
(b): P~MMP
Figure 5-29 – Bubble Behavior of Vaporizing Gas Process66
1. At pressure far below MMP, the bubble retains its almost spherical shape, but its size is reduced as the gas is partially dissolved in the oil, Figure 5-29.a. As the pressure approaches MMP, a bubble still remains nearly spherical on top, but the bottom interface of the bubble changes from spherical to flat or “wavy”. 2. At or slightly above MMP tail-like features quickly developed on the bottom of a rising bubble, which remain spherical on top. The gas-oil interface vanishing from the bottom of the bubble, Figure 5-29.b. This behavior is a result of multi contact miscibility process. During this MCM process the volume of the bubble is almost constant until the interface starts to deteriorate. 3. At pressure higher than MMP, the bubble disperses very rapidly and disappears into the oil. It should be consider that a gas bubble could disappear into an undersaturated oil without achieving miscibility, but at this condition it will not disperse before disappearing.
Bottom
Middle
Top
In addition to injection of lean gas and simulate the vaporizing gas drive with RBA, the condensing gas process could be happened during a RBA test by injecting several enriched gas bubbles sequentially to the oil filled glass tube. The evolution of bubble shape during the condensing gas process is different compare to the vaporizing gas process:
(a) P<
(b)
(c) P>=MMP
Figure 5-30 – Bubble behavior for
Far below the MMP, the shape evolution of injected bubbles are similar and indicate low interfacial tension when the bubbles first contact the oil (they are enriched gas bubbles with a considerable intermediate hydrocarbon components), after rising a short distance through the oil, the bubble shape evolves to indicate high interfacial tension (the bubbles lost a major part of their intermediate hydrocarbons), Figure 5-30.a. At or above MMP bubbles at different injection sequence shows different shape evolution. The first bubble show the same shape evolution as in P < MMP. It evolves from a shape indicating low interfacial tension to a shape indicating high interfacial tension. But, with each additional bubble injected (the oil sequentially is enriched with intermediate hydrocarbon that condensed from the gas bubble) the size of the bubble which emerges from the swirl of gas and oil at low interfacial tension, shrinks, Figure 5-30.b. After of injecting several bubbles the injected bubble disappears as injected to the oil (the oil is
66
“Measuring Minimum Miscibility Pressure: Slim-Tube or Rising-Bubble Method?”, Elsharkawy A.M., Suez Canal U., Poettmann F.H. and Christianse R.L., SPE no. 24114, 1992.
273
Condensing Gas Drive [66]
enriched in intermediate hydrocarbon and become miscible with the injected gas), Figure 5-30c.
This test is completely qualitative in nature and the miscibility is simply inferred from visual observations. Hence, some subjectivity is associated with the miscibility interpretation of this technique, as it lacks quantitative information. Therefore, the results obtained from this test are somewhat arbitrary, however the RBA approach to measuring MMP’s is much more rapid ( it requires less than 2 hours to determine miscibility) than the commonly accepted slim tube techniques. This method is also cheaper and requires smaller quantities of fluids, compared to slim-tube. The main disadvantages associated with this technique are67:
The subjective interpretation of miscibility from visual observations Lack of any quantitative information to support the results Some arbitrariness associated with miscibility determination
While there are many more slim tube apparatus in industry, a larg portion of MMPs reported in the literature were measured using RBA method.
Slim Tube versus RBA Method Elsharkaway et al.[67] measured MMP of carbon dioxide for twelve different oils using the slim tube and RBA methods to compare these two methods reliability. They showed that depending on the criteria chosen for MMP in the slim tube test, MMP varies. During their test they used three different usual criteria for MMP: Pressure at 90% oil recovery, pressure at 95% recovery and the break over pressure in the oil recovery plot (the pressure for which oil recovery did not increase more than 1% per 100 psi pressure increase). Their results showed that a slim-tube oil recovery criterion of 90% will usually yield MMPs lower than those from the criteria for the slim tube. They mentioned that the oil recovery break over criterion produced an MMP from the slim tube that was in excellent agreement with MMP from the RBA. The pressure drop across the slim tube complicates interpretation of slim tube results. As mentioned before it is assumed that the pressure different between input and output of the slim tube is ignorable compare to the MMP. Because the final report of a slim tube experiment is the oil recovery at a given pressure, the pressure drop becomes an uncertainty in pressure at which the oil recovery is measured, so improper design and operation of a slim tube can give excessive pressure drop. Asphaltene deposition is a phenomenon that occurred during the solvent injection. Elshalkaway et al. defined the blind spot of the slim tube method as the measuring of the MMP for systems with asphaltene deposition problem during the dilution process. Precipitation of the asphaltene could block the slim tube more or less and increase required displacement pressure. Sometimes deposited asphaltenes completely plug the tube and the test fails. In RBA it does not appear to be a problem. Specks of asphalt can be seen precipitating out on the walls of tube; however bubble behavior can still be observed.
67
“Measurement and Modeling of Fluid-Fluid Miscibility in Multicomponent Hydrocarbon Systems”, Ayirrala S. C., 2005
274
Sometimes there is not enough test fluids (oil and solvent) to run slim tube test several times to find the MMP or MME. RBA could work at this time because it requires considerably less fluid to determine the MMP or MME. RBA could be a good chose when the time factor is important. The RBA takes 1-2 hours per MMP determination (excluding preparation time), while the slim tube takes one to two weeks per MMP determination.
Contact Experiment Slim tube and RBA methods were introduced as two direct methods for MMP and MME measurements. There are other experimental tests such as contact experiments, that do not measure directly MMP or MME but their results are used to evaluate and calibrate of phase behavior models that can predict these values indirectly; for example when used with 1D compositional simulators to model slim tube displacements, to estimates MMP and MME. In contact experiments the solvent injection processes are simulated in batch type tests in PVT cells. Some of the most useful bath type solvent injection experiments are discussed in the following sections: Single Contact: a known value of solvent is charged into a visual, high pressure cell containing a known amount of oil. After applying a specific temperature and pressure and equilibrium establishment, a small amount of each phase is withdrawn. The compositions represent the ends of an equilibrium tie line. Single contact experiments are useful for measuring p-x diagrams, since the pressure can be changed at fixed overall composition by changing the cell volume. If the experiment is repeated for various amounts of solvent the single contact experiment traces a dilution path on a ternary diagram between the solvent and oil. The single contact experiment does not simulate the continuous contact between the solvent and oil during a solvent-oil displacement. The static equilibrium tests which closely simulate continuous contact of solvent and reservoir oil are multiple contact experiments. Multiple Contacts: the visual high pressure cell is filled with a known amount of solvent and crude oil, the same as in a single contact test. After reaching equilibrium at the applied pressure the upper phase is removed from the cell and mixed in a second cell with fresh oil, forward contact (the same as the process in downstream contact of a solvent-oil mixing zone in multi contact miscibility process, or the vaporizing gas drive process). The lower phase in the cell is similarly mixed with fresh solvent, reverse contact (the same as what happen in the upstream contact of solvent-oil mixing zone in a multi contact miscibility displacement, or condensing gas drive). The upper phase is repeatedly removed from the cell and adjacent to the fresh oil in another cell. All contacts are at fixed temperature and pressure. After several steps solvent enrichment in the forward contacts or the rude enrichment in reverse contacts can cause one of the phases to vanish. If the original cell is single phase for all combinations of solvent and crude oil, the process is first contact miscible.
275
Forward Contact (Vaporizing gas drive)
After Equilibrium Fresh Oil
After Equilibrium Fresh Oil
After Equilibrium
Fresh Oil Fresh Solvent
Fresh Solvent
Fresh Solvent Fresh Solvent
After Equilibrium
After Equilibrium
After Equilibrium
After Equilibrium
Reverse Contact (Condensing gas drive)
Fresh Oil Figure 5-31 – Schematic of Multiple Contact Experiment.
MMP Prediction The values of MMP or MME may be estimated from: 1. Empirical correlations based on experimental results. 2. Phase behavior calculations based on an EOS and computer modeling. The first approach is easy to apply but it could leads to wrong result values especially if the correlation is applied for a condition that is completely different from the experimental conditions on which the correlation is based. So usually they are used to obtain a rough estimate value. Phase behavior models, which provide information on thermodynamic miscibility, are more reliable to predict the miscibility conditions. The phase behavior models are tuned with experimental data.
276
Empirical Correlations The developed empirical correlation can be developed base on the miscibility process that happens during the solvent injection. When the vaporizing gas drive is the main miscibility process during the solvent injection, the developed empirical correlations are just based on the original oil composition regardless of the injection solvent or gas composition. One of the most reliable empirical correlations of this class (estimation of MMP for Vaporizing gas drive process) is a model that is developed by Firoozabadi and Aziz68 based on experimental slim tube data: [
]
(5-3)
Where, MMP, MPa, Mole fraction of intermediates in oil, ethane to pentane inclusive, Molecular weight of heptane plus, Temperature, K. Eq. (5-3) was found as the most reliable MMP correlation for lean gas and nitrogen injection. This correlation, as mentioned before, relies on the vaporizing gas drive concept which is controlled by the original oil composition only. The oil methane content has no effect on the MMP according to this correlation. The MMP for methane is generally lower than MMP for nitrogen. When nitrogen is used as a displacement solvent it extracted a lot of methane from the oil insofar as the advancing gas drive is very much dominated by methane instead of nitrogen. Therefore the oil methane content is an important parameter for achieving miscibility in nitrogen injection. Based on this fact, Hudgins et al.69, proposed the following correlation for MMP estimation: (5-4)
Where, 5-5) (
) (
)
(5-6)
MMP, MPa, Mole fraction of intermediates in oil, ethane to pentane inclusive, 68
“Analysis and Correlation of N2 and Lean-Gas Miscibility Pressure”, Firoozabadi, A. and Aziz, K., SPE Res. Eng., 575-582, (Nov., 1986)
69
“Nitrogen miscible displacement of light crude oil, a laboratory study”, Hudgins, F.M., Liave, F.M. and Chung, F.T., SPE Res. Eng., 100-106, (Feb., 1990)
277
Molecular weight of heptane plus, Temperature, K. In addition to these two well-known correlations many other correlations has been developed by various investigators. The condensing-vaporizing miscibility is another miscibility process that is achieved at lower pressure compare to the VGD process. Several correlations has been developed to predict the MMP for this process. Glaso70 introduced the following formulas based on the MMP correlation curves for rich gas injection that had been developed by Benham et al.71: For MC2-C6 = 34 (
)
5-7)
(
)
5-8)
(
)
(5-9)
For MC2-C6 = 44 For MC2-C6 = 54 Where, MMP, MPa, Molecular weight of C2 to C6 fraction in the injection gas, Mole fraction of methane in injection gas, Temperature, K, ⁄
5-10)
Specific gravity of C7+ fraction. In this correlation the effect of oil and gas compositions are included for condensing-vaporizing process. There are many other correlations that are used for MMP prediction depend on the process condition and oil and solvent composition.
Fluid properties in miscible displacement The performance of a miscible displacement process depends on fluid physical properties that affect flow behavior in a reservoir. Fluid properties changes should be considered to have a reliable prediction of a miscible displacement performance. The following sections explain some fluid properties and their alteration during miscible displacement. 70
“Generalised Minimum Miscibility Pressure Correlation”, Glaso, O., SPE 927-934 (Dec., 1985)
71
“Miscible flood displacement, prediction of miscibility”, Benham, A.L., Dowden, W.E. and Kunzman, W.J., Trans. AIME, 219, 229-37 (1960)
278
Fluid Density Knowledge of relative density of the fluid and fluid mixtures is important for miscible displacement design. Gravity override or underride and fingering are usual phenomena during displacement process that are results of density difference between displaced and displacing liquids. Detailed description of these phenomena will be explained in the subsequent sections. Crude oils tend to have specific gravities in the 0.80 to 0.95 range at 60oF. Density is a function of pressure and temperature and can be estimated at the reservoir condition. During the solvent injection two different fluids (oil and solvent) with different densities mix together. McCain72 explained how an ideal solution behavior could be used to estimate the density of an oil-solvent mixture: An “ideal liquid solution” is a hypothetical mixture of liquids in which there is no special force of attraction between the components of the solution and for which no change in internal energy occurs on mixing. Under these conditions no change in the character of the liquids is caused by mixing, merely a dilution of one liquid by the other. So when liquids are mixed to give an ideal solution the properties are strictly additive. When the liquids mix as an ideal solution there is no shrinkage or expansion in the volume, so the final volume is the sum of the liquids volume. Other physical properties of the solution (such as viscosity and vapor pressure) can be calculated directly by averaging the properties of the fluids that mix together. There is no ideal liquid or gas solution but when the chemical and physical properties of two fluids that mix together are similar, the resulting solution behaves like as an ideal solution. McCain mentioned that because most of the liquid mixtures encountered by petroleum engineers are mixtures of hydrocarbons with similar characteristics, ideal-solution principle can be applied to find the densities of these mixtures. To find the density of a mixture that is assumed as an ideal solution it is enough to calculate the mass and volume of each of the components of the mixture and sum them together to find the mass and volume of the mixture. And then find the mixture density using these mass and volume values. An estimate of liquid density can be calculated on the basis of a mole-averaged sensitivity of pure components as follows: (5-11) Where, Xi = mole fraction of component I in the mixture, Mole density of component i, Mole density of the mixture. Mole density could be converted to mass density by multiplication by the average molecular weight of the mixture. Gas density can be calculated with the EOS: (5-12)
72
“The properties of Petroleum Fluids”, McCain Jr., W.D., PennWell Publishing Co., Tulsa, OK (1973)
279
Where, Pressure, Specific volume, Temperature, Gas constant, 8.314
⁄
Compressibility factor that shows the deviation from ideal gas law. Compressibility factor is a function of reduced temperature (Tr) and reduced pressure (Pr) that are defined as 5-13) 5-14) For a mixture of gases using the ideal gas mixture assumption critical pressure and temperature are defined as follow and are named pseudocritical pressure (Ppc) and pseudocritical temperature (Tpc): ∑
(5-15)
∑
(5-16)
Where, n = number of components, Xi = mole fraction of component “i”.
Fluid Viscosity Mobility ratio in a displacement process is a direct function of the viscosities of displaced and displacing fluids. For miscible displacements, assuming the relative permeability relationships of the different non aqueous fluids are the same so: ⁄ ⁄
(5-17)
Where, M = Mobility ratio, Displacing relative permeability, 280
Displaced relative permeability, Displacing fluid viscosity, Displaced fluid viscosity. So Knowledge of oil viscosity is vital to the petroleum industry, and is especially important when considering production of heavy oil and bitumen. Knowledge of oil viscosity is vital to the petroleum industry, and is especially important when considering production of heavy oil and bitumen and other EOR methods. Viscosity can determine the success or failure of a given EOR scheme and it is an important parameter for doing numerical simulation and determining the economics of a project. Viscosity is a function of pressure, temperature and composition. The viscosities of crude oils vary over a wide range. There are some correlations for prediction of liquids and gases viscosity based on the fluid composition. The following equations are the most well-known correlations for prediction of viscosity.
Lorentz-Bray-Clark method73: The procedure of this method for reservoir liquids was developed using the residual viscosity concept and theory of corresponding states. It was the first known procedure for calculating the viscosity of reservoir liquids from their compositions. Jossi et al.74, developed the following equation for viscosity of the pure liquid components: [
]
∑
(5-18)
Where, 0.023364,
0.058533,
.
Viscosity at normal pressure, Viscosity parameter: (5-19) Reduced molar density: (5-20) M = Molecular Weight, TC = Critical temperature, PC = Critical pressure Critical molar density. 73
“Calculating Viscosity of Reservoir Fluids from their Composition”, Lorentz, J., Bray, B. G. and Clark, C. R. J., Journal of Petrolleum Technology 1171, p. 231, 1964 74 “The Viscosity of Pure Substances in the Dense Gaseous and Liquid Phases”, Jossi, J. A., Stiel, L. I., Thodos G., AIChE journal. (1962) 8, 59
281
Hering and Zipper75 introduced the following equation for ∑
(
∑
)
√ ( √
of a mixture (liquid or gas): 5-21)
)
To find the viscosity of a liquid mixture Lorentz et al. used eq. (5-18) with and value for mixture. They used eq.5-21) to find the for mixture and defined of a liquid mixture as functions of the composition Xi , the component molecular weights, critical pressure and critical temperatures: [∑ [∑
(
( )]
⁄
)] [∑
⁄
(
)]
5-22)
⁄
(5-23)
∑
(
(5-24)
)
(5-25)
Where, pseudocritical density of the multi component liquid, Vpc = pseudocritical volume for the mixture, VCj = critical volume of component j, Specific gravity of C7+ fraction of the oil.
The modified Lorentz-Bray-Clark method: A modified form of the Lorentz-Bray-Clark method can also be used: (
)
(
∑
)
(
)
(5-26)
This modified equation is intended to improve the modeling of the behavior of heavy oils. In this equation the viscosity increases exponentially as the reduced molar density approaches a maximum for small values of , , the modified Lorentz-Bray-Clark methodhas the same form as eq. (5-18).
75
“Calculation of viscosity of technical gas mixtures from the viscosity of individual gases”, Herning, F. and Zipper, L., Gas u. Wasserfach (1936) 79, No. 49, 69
282
The Pederson Method76: It is a corresponding states compositional viscosity model that enables viscosity prediction for black oil systems using compositional data. The prediction of viscosity according to the oil composition, is crucial for compositional simulation of the miscible process because of composition change of the displaced and displacing fluid. There are several experimental type methods that can measure the viscosity of the fluids. From the wide range of experimental tools for viscosity measurement, NMR is explained in the following subsection. Measuring viscosity using NMR
As viscosity increases, conventional measurements become progressively less accurate and more difficult to obtain. Oil viscosities measured in the lab may also be not indicative of true in-situ viscosities. An alternate method is required for predicting oil viscosity, especially if this method can be applied insitu. Low field nuclear magnetic resonance (NMR) is an attractive alternative to conventional viscosity measurements, as its measurements are fast, non-destructive and insensitive to technician error. Bryan and Kantzas did an extensive work on the application of NMR in determination of heavy oil and bitumen viscosity as well as crude oil emulsion. Their work lead to some correlations that predict viscosities from less than 1 cP to over 3,000,000 cP over 25-80oC based on the NMR data77. Bryan et. al. [77] provided a detailed explanation viscosity that is helpful to understand the basics that make NMR as a powerful tool in viscosity determination: Newtonian fluids exhibit a linear relationship between shear stress and shear rate, given by Newton’s law of viscosity: ̇
(5-27)
Where, Shear stress (Pa), Viscosity, ̇
Shear rate (s-1).
For a Newtonian fluid at a constant temperature and pressure, viscosity is a constant value. If the fluid is non-Newtonian, however, viscosity changes with shear rate. Most crude oil emulsions that contain low to moderate dispersed water are Newtonian, but non-Newtonian effects may be observed at higher dispersed water fractions. Eyring’s theory of viscosity states that the empty space between the closely packed molecules in a liquid is not enough for the molecules to move freely by one another. In order for a molecule to move, therefore, other surrounding molecules must first give way and create a space or “hole” for this molecule to enter, eq. (5-28). Highly viscous fluids have molecules that are complex and 76
“Properties of Oils and Natural Gases”, Pederson, K. S., Fredenslund, A. and Thomassen, P., Contribution in Petroleum Geology and Engineering Vol. 5, Gulf, Houston, 1989 77 “Viscosity Predictions for Crude Oils and Crude Oil Emulsions Using Low Field NMR”, Bryan, J., Kantzas, A., Bellehumeur, C., SPE No. 77329, 2002.
283
close together, so it is difficult for their molecules to move enough to create this space. Not all molecules have enough energy to overcome the attractive forces of their neighbors enough to move, so the concentration of activated molecules that are able to move is therefore related to the viscosity of the fluid. ( )
(
)
(5-28)
Where, = Distance between molecular layers in a fluid, A = Distance between a molecule and an adjacent empty lattice site, N = Avogadro number, h = Planck constant, V = liquid molar volume, ΔG0 = free energy of act As temperature increases, more the molecules have enough energy to breach this activation energy barrier and are able to move by one another more easily. At higher temperatures the spaces between the molecules are also likely to be larger, so the free energy of the molecules required for flow will also be less. This leads to lower viscosity at elevated temperatures. The development of Eq. (5-28) allows for a qualitative understanding of what causes differences in viscosity between fluids and for viscosity to be estimated on the basis of the molecular properties of the liquid. Low-field NMR measures the response of hydrogen protons in external magnetic fields. Protons are present in oil and water and have a strong relaxation response to imposed magnetic signal pulses. In the presence of magnetic field the protons will line up either parallel or anti-parallel to the field lines. A pulse sequence is then applied to these protons to tip them 90° onto the transverse plane. As the protons give off energy, they return to their equilibrium position. A low field NMR experiment can therefore provide two pieces of information: the strength of the signal from the protons and a characteristic relaxation time. This characteristic relaxation time is either the time for the signal to reappear in the longitudinal equilibrium plane (T1), or the time for the signal to disappear in the transverse plane (T2), (for more details return to chapter 3). T2 measurements are used in viscosity measurements78. The pulse sequence applied to the protons gives them energy and tips them into the transverse plane. In order to return to their equilibrium direction, these excited protons must give off their energy. Once they do so, the signal in the transverse plane decays and relaxation is said to have occurred. In the experiments for viscosity measurements two types of relaxation occur in porous media: 1. Bulk relaxation (T2B), 2. Surface relaxation (T2S). Bulk relaxation is a fluid property, and is a measure of how easily the protons give off energy to one another. Bulk relaxation can be therefore expressed in the following form79:
78
“Oil-Viscosity Predictions from Low-Field NMR Measurements”, Bryan, J., Kantzas, A., Bellehumeur, C., SPE No. 89070, 2005
79
“In Situ Viscosity of Oil Sands Using Low Field NMR”, Bryan, J., Kantzas, A., Moon, D., Petroleum Society’s Canadian International Petroleum Conference, 2003
284
(5-29) Where, T is absolute temperature of the sample. According to eq. (5-29) the bulk relaxation rate is directly proportional to the fluid viscosity, so viscous fluids will have faster relaxation rates than fluids with lower viscosity. The reason for this is that the protons must give off their energy to other protons before they can return to their equilibrium direction. Samples with higher viscosities have molecules that cannot move past one another as easily. The protons of these fluids will therefore contact one another at a higher frequency, leading to energy being given off faster. When fluids are in constricted spaces like pores, energy can also be transferred from the proton spins to the pore walls. This is termed surface relaxation, and is given by: (5-30) where, The surface relaxivity of the pore wall, Pore Volume, Pore Surface. From eq. (5-30) it can be seen that fluid found in smaller pores (where is smaller) will tend to relax faster, and T2S will be smaller. Bryan et al. used 112 oil samples with viscosity range of around 1 to 3,000,000 cP to develop the model based on the NMR data the oil samples with known viscosity values has been tested with NMR to find the NMR parameters for each sample using linear regresion. The following equation was developed that can be used to predict the viscosity of an oil sample using the NMR parameters: (5-31) Where,
the geometric mean T2 value (ms).
285
Factors Affecting Displacement Efficiency of Miscible Displacements Displacement efficiency at microscopic (pore level) and macroscopic level in a miscible displacement process are less than 100%. The magnitudes of efficiencies depend on a number of factors, including whether a displacement is secondary or tertiary recovery process.
Microscopic displacement efficiency In a miscible displacement conducted as a secondary recovery process (assume there is not mobile water in the system), the IFT between displaced (oil) and displacing (solvent) phases is zero. According to the Capillary number (Nc) definition, Nc become infinity as IFT goes to zero: (5-32)
The residual saturation in the portion of the rock contacted by the displacing phase should be essentially zero as . Experimental studies of first contact miscibility process show that the residual saturation of the displaced phase is very small when the solvent continuously is injected. However, when the solvent is injected as a small primary slug followed by a secondary slug, recovery can be poorer as a result of dispersion and mixing of different slug materials80. Significant mixing can result in loss of miscibility at either leading or training edge of the primary slug displacement. Laboratory studies of MCM processes have shown that recoveries are somewhat poorer than for FCM processes. There are different reasons for the reduced recoveries at the microscopic level in MCM processes. One is that a finite distance of travel is required in the process before miscibility is achieved. In a vaporizing gas drive process a small amount of the liquid phase drop out can occur as a result of mixing effects when compositions are in the vicinity of the bimodal curve. Bypassing in the flow process at the microscopic level owning to smallscale heterogeneities or dead-end pores can cause mixing. The mixing process can result in overall compositions that are within the two phase region, which would lead to the trapping of a residual liquid phase, although at a small saturation. Mass transfer by dispersion in a displacement process also can lead to reduced displacement efficiency. Dispersion causes the composition path that occurs during a displacement to move into the multiphase region of the phase behavior diagram. This results in formation of a liquid phase which remains as a trapped phase because of its low saturation. However a small part of the trapped liquid could be re-vaporized to the flowing gas phase, but the recovery by a revaporization process occurs at a relatively slow rate. Gardner et al81 showed that the higher the dispersion level, the poorer the calculated recovery efficiency. They concluded that displacement efficiencies achieved in their slim tube experiments at the pressure well above the MMP were limited by the levels of dispersion in the experiments. Dispersion phenomenon will be explained separately in the following chapter. 80
“Miscible slug process”, Koch, Jr., Slobod, R.L., Trans.., AIME (1957) 210, 40-47 “The Effect of Phase Behavior on CO2 Flood Displacement Efficiency”, Garder, J.W., Orr Jr., F.M., and Patel, P.D., JPT, Nov. 1981 81
286
Macroscopic displacement efficiency Four major factors affect recovery efficiency at the macroscopic level in a miscible process: mobility ratio, viscous fingering, gravity segregation, and reservoir heterogeneity. In the present of water, water blockage effect is added to these important factors.
Mobility Ratio [50, 82]: The mobility of a fluid (i) is defined as : (5-33) Where, Viscosity, Effective permeability of the rock to fluid i. The mobility ratio (M) of a displacement process is defined as the mobility of displacing fluid divided by the mobility of the displaced fluid. Mobility ratio is one of the most important parameters of a miscible displacement and has a profound influence on volumetric (macroscopic) sweep efficiency of the solvent and on the integrity of solvent slugs. Consider an idealized situation where solvent displaces oil at the irreducible water saturation (secondary recovery process) and where mixing of solvent with the oil is negligible. No water is flowing and the permeabilities to oil and solvent are equal. Mobility ratio in this case is simply the ratio of oil and solvent viscosities: (5-34)
In practice mixing of solvent and oil does occur during the course of the displacement, which can result in an effective viscosity ratio that is less than the ratio of pure component viscosities. The viscosities of miscible solvent are typically small (<0.1 cp) and thus the mobility ratios are greater than one. Viscosity ratio or mobility ratio, as defined by eq. (5-34), ranges from 4 to 86 in the different MCM processes. Stalkup82 reports data on a number of FCM projects. For these, the viscosity ratio ranges from 4 to 40. When the mobility ratio is smaller than one it is called favorable mobility ratio and called unfavorable when it is greater than one. Solvent/water injection is a technique to reduce solvent mobility by reducing solvent relative permeability. The most accepted mobility ratio definition at presence of mobile water is:
82
(
)
(
)
(5-35)
“Miscible Dispalcement”, Stulkup Jr., F.I., Monograph series, SPE, Richardson, TX(1983).
287
Figure 5-32 gives areal sweep efficiency (fraction of the pattern area invaded by pure solvent) at breakthrough of the displacing fluid as a function of M. the upper curve is based on photographs taken during displacement of one colored liquid by a second, miscible colored liquid in a scaled model, while the lower one is based on a material balance calculation made from the known volume of injected liquid and assuming piston like displacement. Because of mixing along the liquid-liquid interface, the area measurement yields a slightly larger areal sweep value than material balance result. As indicated in Fig. 4.4, the difference in the two curves is a measure of the mixing that occurs at the interface.
Areal Sweep Efficiency at Breakthrough, %
As seen, the areal sweep efficiency (EA) at breakthrough is a strong function of M. at M=1.0, areal sweep is about 70%. It increases slightly at smaller, favorable mobility ratios and decreases very sharply as M is increased. The poorer performance at larger M values occurs for two reasons: first viscous fingering occurs at M>1.0 and become more noticeable as M increases. Second for M>1.0 the geometry contribute to the early breakthrough. Because of the geometry, the smallest flow resistance and therefore the larger flow velocity is along the center line connecting the injection and production wells. Fluid flowing along this line breaks through first in a homogeneous reservoir. When M>1.0 the fluid with lower mobility value (displaced fluid), is replaced by the injected fluid (with higher mobility). Because the major flow is along the center line path (as the shortest pass), the resistance along that path is reduced more than along any other line other than path. The result is that the larger the value of M the earlier breakthrough occurs. 90 80 70 60 50 40
Base on Pore Volume injected
30
Based on Area Measured
20
Measure of Mixing Zone
10 0.01
0.1
1 10 Mobility Ratio, M
100
Figure 5-32 – Areal sweep efficiency as a function of mobility ratio83
Viscous Fingering: When the mobility ratio between displacing solvent and displaced oil is equal or less than one and the gravity does not influence the displacement by segregating the two fluids, the oil is displaced efficiently ahead of the solvent, and the solvent does not penetrate into the oil except dictated by dispersion. The 83
“Efficiencies of miscible displacement as a function of mobility ratio”, Trans, AIME (1960)219, 264-72
288
displacement front is stable. For mobility ratio greater than one the solvent front became unstable and a numerous fingers of solvent develop and penetrate into the oil in an irregular fashion. These viscous fingers result in earlier solvent breakthrough and poorer oil recovery. Figure 5-33 shows the viscous fingering observed in areal laboratory five spot models for displacement at various mobility ratios.
Figure 5-33 – displacement front for a miscible displacement process with unfavorable mobility ratio84.
The severity of the areal fingering increases as mobility ratio becomes more unfavorable, resulting in earlier solvent breakthroughs. The reason for displacement front instability when mobility ratio is unfavorable can be visualized with a simple illustration shown in Figure 5-34.
Solvent
L
Solvent Xf +
Xf Figure 5-34 – simplified model of frontal instability.
In this illustration, solvent displaces oil linearly from a porous medium that initially is fully saturated with. The mobility ratio is the ratio of the viscosities and dispersion is assumed negligible. In the absence of heterogeneity the front should be remain a plane surface throughout the displacement. At the time of consideration of the system, the solvent front is located at position xf along the flow path. In the flow region bounded by the dashed lines, a small perturbation or protrusion of the solvent front has occurred such that the front location is at the position xf + the focus of the analysis is the determination of conditions under which e grows in time because if does grow in time, the front will be unstable; i.e viscous finger will form along the front. At condition where does not grow or even diminishes in size,
84
“Viscous fingering in five-spot experimental porous media: new experimental results and numerical simulation”, Zhang, H.R., Sorbie, K.S., Teibuklis, N. B., Chemical Enoineerinq Science, Vol. 52, No. I, pp. 37·54, 1997.
289
the front stability or uniformity will be maintained. For the normal flat front, pressure drop across the length is given by: [
(
]
)
[
(
5-36)
)]
Where is Darcy velocity and is a negative value. The above equation can be solved for interstitial velocity, v, which is equal to the velocity of the front: (
)
[
(
)]
[
]
5-37)
Where, M is mobility ratio A similar analysis applied to the perturbed area gives: (
)
[
(
)
]
Subtracting eq. () from eq. () resulted: [
(
[
[
]
(
)
[
[ ]
[
[ For
)
] (
)
[ [
]
]
[
] (
)
] ]
] ]
5-38)
: [
5-39)
]
Therefore: [(
[
]
)
]
5-40)
Where t is time. Thus according to eq.(5-40), grows exponentially with time immediately after formation of the perturbation if M>1 but decays exponentially with time if M<1. Dispersion which was not taken into account in this simple example acts to oppose this growth by moderating the viscosity contrast. Whether or not the finger propagate after being initiated and, if so, at what rate depends on the importance of dispersion in the displacement. The initial growth rate can be estimated from the above equation but the equation is not valid for fully developed fingers.
290
Koval85 developed a model for predicting the performance of miscible displacements in the presence of viscous fingers. He defined an effective viscosity ratio, E, that characterizes the effect of viscous fingering: ⌈
(
)
⌉
5-41)
Koval determined breakthrough volumetric displacement efficiency in homogeneous linear system as a function of E (Figure 5-35 – Estimated breakthrough recovery as a function of viscosity ratio [32].). 100
Viscosity Ratio
80 60 40 20 0 0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 Fractional Recovery at breakthrough
0.9
1
Figure 5-35 – Estimated breakthrough recovery as a function of viscosity ratio [32].
Figure 5-36 shows results from two sets of experiments on a linear system. In the first set, CO2 displaces the oil by FCM process. The viscosity ratio was 16. In the second experiment CO2 displaces oil in a MCM process, while viscosity ratio is 21. There was no mobile water in these experiments. In both experiments the oil recovery was delayed because of the adverse mobility ratio, i.e., there was an early breakthrough of CO2. In FCM process, ultimate recovery approached 100% with continued injection of CO2. In the MCM process the viscous fingering not only delayed the oil recovery but also worked to reduced total recovery. The implication is that an interaction or collaboration occurs between viscous fingering and phase behavior to cause liquid-phase drop out and trapping.
85
291
Oil Recovery, Fraction of OOIP
1 MCM
0.9
FCM 0.8 0.7 0.6 0.5 0.5
0.7
0.9 1.1 1.3 1.5 Movable pore volumes CO2 injected
1.7
Figure 5-36 – Impact of viscous instability on secondary CO2 flood oil recovery efficiency86
Gravity effect [29]: From flow experiments in a vertical cross-sectional laboratory model packed with glass beads, Crane et al.87 found that four flow regimes are possible at unfavorable mobility rations, depending on the value of the dimensionless group characterizing the ratio of viscous and gravity forces (eq. (5-42)). (
⁄
)( )
(5-42)
( )
Where g is gravity acceleration. And for field units: ⁄
(
)( )
( )
(5-43)
Where, Darcy velocity, (B/Day/ft2) L = distance between injection and producing wells (length of the system) ft, H = height of the reservoir (system), ft, k = porous medium permeability, md, oil viscosity, cP, density difference between displacing and displaced fluids, g/cm3. These equations assume that vertical permeability is the same as horizontal permeability. When this assumption is not true a suggestion approximation is to substitute k with √
86
in this equation.
“An Investigation of Phase Behavior/Microscopic By-Passing Interaction in CO2 Flooding”, Gardner, J.W. and Ypma, J.G.J., SPEJ, Oct.1984.
87
292
The magnitude of viscous forces to gravity forces increases with increasing ⁄ value. At small ⁄ values, the displaced phase tends to override (Figure 5-37.a) or underride (Figure 5-38.b), depending on the magnitude of the liquids densities, which leads to early breakthrough of the displacing phase, even for M=1.
Displaced Phase Displacing Phase a)
b)
Displaced Phase
Displacing Phase
Figure 5-37– Gravity segregation in displacement processes. A) Gravity override ρo<ρs. b) Gravity Underride ρs<ρo
In this manner the displacement is characterized by a single gravity tongue (Figure 5-38, a, Regime I). The geometry of this tongue and vertical sweepout both depend on the particular ⁄ of the displacement. At higher value of ⁄ the displacement is still characterized by a single gravity tongue (Figure 5-38, a, Regime II), but vertical sweepout become independent of the particular value of ⁄ until a critical value is exceeded. Beyond this critical value a transition region is encountered (Figure 5-38, b, Regime III) where viscous fingers are formed along the primary gravity tongue. Vertical sweep is improved by the formation of viscous fingers in this regime. In this regime, sweep out for a given value of pore volumes injected increases sharply with increasing values of ⁄ . Finally, a value of ⁄ is reached where the displacement is entirely dominated by multiple fingering in the cross section and vertical sweepout again becomes independent of the particular value of the ⁄ (Figure 5-38, c, Regime IV). A gravity tongue does not form in this regime because of the strong viscous fingering. The value of ⁄ at which a transition occurs from one flow regime to another depends on the mobility ratio
(a) Region I and II
(b) Region III
(C) Region IV
Figure 5-38 – Flow regimes for miscible displacement in a vertical cross section.
Figure 5-39 further illustrate the different flow regimes in miscible displacement and shows how sweepout at solvent breakthrough in a vertical cross section is affected both by the flow regime and by mobility ratio.
293
Figure 5-39 – Flow regimes in a two-dimensional, uniform linear system (Schematic) [29].
Figure 5-40 shows a relation between mobility ratio,
⁄
and vertical sweep efficiency.
Example 5-6 A miscible displacement process will be used to displace oil from a linear reservoir having the following properties: L = 300 ft, h = 10 ft,
0.20, ko = 200 md.
Determining the effect of gravity segregation on the vertical sweep efficiency if the oil is displaced miscibly by a solvent with a density of 0.7 gr/cm3 and a viscosity of 2.3 cp at reservoir temperature. The density of the oil is 0.85 g/cm3 and the viscosity is 2.3 cp. Darcy velocity is 0.075 ft/D.
Solution Calculate viscous/gravity ratio using eq.(5-43): (
⁄
)
And ⁄
Calculate M:
From Figure 5-40 vertical displacement efficiency (EI) at breakthrough is 0.86.
294
Figure 5-40 – volumetric sweep efficiency at breakthrough as a function of the viscous/gravity force ratio88.
Gravity effect in dipping reservoirs [50] In some reservoirs with dip, gravity can be used to advantage to improve sweepout and oil recovery. This is achieved by injecting the solvent updip and producing the reservoir at a rate low enough for gravity to keep the less dense solvent segregated from the oil, suppressing fingers of solvent as they try to form. For example if oil were displaced from a reservoir by injecting a less dense, more mobile solvent up-dip, gravity would tend to stabilize the displacement front. That is, if the displacement velocity were sufficiently low, gravity forces would act to prevent the formation of fingers at the solvent/oil interface. Similarly, in water flood down-dip injection of water can work to stabilize the interface between the water and oil bank interface. There are some criteria to have a stable gravity displacement. With the same strategy that was used to find the criteria for grows of the viscous fingers (eq.(5-40)), in the following paragraphs we will try to find the criteria for stable displacement in a dipping reservoir. Consider a displacement in a down-dip direction in which the reservoir dip is at an angle t to the horizontal. Assume that the interface is sharp with only displacing fluid flowing ahead of the front. For miscible displacement that means there is no mixing at the interface. Assume a small perturbation, finger, or protrusion has formed at the interface (Figure 5-41). The same as viscous fingering the objective of the analysis is the determination of conditions under which the perturbation will remain stable or will grow in magnitude, leading to an unstable interface.
88
“A laboratory study of gravity segregation in frontal drives”, Craig, F.F. Jr. et al., Trans, AIME (1957) 210, 275-82.
295
Solvent Interface
s
Oil
, µs
ΔLp
Flow Direction
θ
o,
µo
Figure 5-41 - Model for determining stability criterion in a dipping reservoir
If the pressure of the displaced phase (oil) just at the interface of the finger is higher than the displacing (solvent) pressure the finger will remain stable. According to Darcy’s equation solvent and oil pressure (Ps and Po) just at the perturbation face could be find as: (see Figure 5-41) (5-44) (5-45) Where, Reference pressure at the point of the unperturbed interface. Length of perturbation. In these equation on the right hand side of equations the first term is the pressure at the unperturbed interface, second term is the gravity pressure in the flow direction caused by the oil or solvent column and the third term is the pressure difference caused by flow. For the interface to remain stable: (5-46) Substituting eqs. ( (5-44)and(5-45) ) into eq. ((5-46)): (
)
(5-47)
using eq.(5-47) critical velocity as the maximum velocity at which the interface will remain stable, defined as: 296
(
5-48)
)
In customary units, and if ks=ko and M=
(
5-49)
)
Where, is in ib/ft3, µ in cp, k in Darcies, and uc in feet per day. In derivation of eq. 5-49) we assumed that there is a sharp interface between displaced and displacing fluid, while in miscible displacement there is a mixing zone between oil and solvent that solvent concentration change from zero to pure solvent in this zone. For this condition ust is defined as the maximum flow velocity for a completely stable displacement. The velocity, ust, is called “stable rate”. Below this velocity the miscible displacement is completely stable throughout the transition zone. ust is more stringent criterion. If the actual displacement velocity, u, is greater than uc, the displacement is unstable and solvent fingers into the oil. If u
)
5-50)
⁄
Example 5-7 A miscible flood displacement is to be conducted in a laboratory experiment in which one glycerol/brine solution is displaced vertically downward by a second solution having a different concentration of glycerol. Liquid properties are as follows: Properties
Liquid # 1 (Displacing (D))
Liquid #2 (Displaced (d))
NaCl content, (gr/liter)
30.00
20.00
Glycerol content, (gr/liter)
650
700
1.1551
1.1609
7.4
1.3
3
o
Density, (gr/cm ) at 77 F o
Viscosity, (cp) at 77 F
The liquid Darcy velocity is to be 0.237 ft/D. porosity of the porous medium is 0.206 and permeability is 303 md. 297
Determine whether viscous fingering will occur? Calculate
Solution Equation 5-48) is applicable for calculation of nc:
(
)
kd
0.303 darcies
kD
0.303 darcie 72.440 72.078
=0.00123 ft/D
Because the planned velocity of 0.237 ft/D >> uc, the flow will be unstable and viscous fingering will occur. Eq. (5-50) is applicable: (
) ⁄
Fluid mixing results in an even smaller maximum velocity for stable flow. As this example showed, the critical velocity is so low and most of the time, in field application, injection velocity is higher than this critical value. Several solutions were suggested to increase the final recovery of the miscible flood. Kantzas et al.89 proposed a method to increase the final recovery of a vertical miscible performance using cyclic pressure pulsing. This method is illustrated in the next section:
Optimization of Vertical Miscible Flood Performance through Cyclic Pressure Pulsing [89]: The study of this method has been done on the Rainbow Lake reefs oil of Northern Alberta. In this field oil was displaced downward by a less viscous solvent than was injected as a slug followed by chase gas. The injection flow rate was high enough that the solvent front velocity in the formation be higher than the critical velocity for a stable front displacement so viscous fingering, which was in part triggered by local heterogeneities, was blamed for reduced sweep efficiencies and oil recoveries which were less than what was expected in the design stage. Viscous fingering leads to soon breakthrough of the solvent and chase gas. When both solvent and chase gas break through, the gas to oil ratio (GOR) increases dramatically with a comparable decrease in oil production. The standard approach in controlling excessive gas or solvent production is work overs in which the current perforations are cement squeezed and new perforations are made lower in the reef. This procedure is costly and is not always successful. Husky proposed another procedure in optimizing the displacement strategy which an improve the vertically directed miscible floods, by applying a cyclic flow interruption consisting of periodical shut-in. Experimental and numerical results showed that the 89
“Optimization of Vertical Miscible Flood Performance through Cyclic Pressure Pulsing”, Kantzas, A., Marentette, D.F., See, D., Adamache, I., Mclntyre, F.I., Sigmund, P.M., JCPT, vol. 33, July 1994
298
concept of shut-in leads to significant incremental oil recoveries. Indeed during the shut in period solvent and oil have time to settle under gravity effect and there is an extra time for mass transfer between solvent and oil. The problem with the cyclic flow interruptions is that production is discontinued. The question was raised whether a similar effect could be observed through a continuous process. It was proposed that cyclic pressure pulsing under restricted flow could increase mass transfer between solvent and oil and also force recovery from dendritic structures or other upswept areas through pressure pulsing of the solvent and gas phases. Kantzas et al. ran a series of vertical miscible flood tests on a core from the Rainbow Keg River area. All runs were performed at rates significantly higher that the critical rate for the specific system. Methane was used as the solvent and n-pentane as the oil. The choice of materials (methane and n-pentane) was made to match density and viscosity rations of the reservoir fluids while maintaining a simple composition of each phase, so that effluent analysis could be simplified. Figure 5-42 shows their laboratory apparatus. The core was always mounted in the vertical position and saturated with n-pentane. Methane was injected at various rates from the top of the core. Different pressure pulsing scenarios were also applied. Sixteen experiments were discussed in their report. For each experiment at different time steps recovery, cumulative recovery, GOR and mass fraction of each fluid in the effluent using gas chromatograph data and a Peng-Robinson type algorithm. The base case that is used to evaluate other process is continuous injection of the solvent. Injection rate was approximately ten times of the critical rate. To simulate the shut in process, the same rate was maintained, but every two hours the experiment was shut in for a day. The shut in test lasted for around 75 hours. The incremental recovery compare to the base case (continuous injection) was around 6% OOIP. So shut in can provide significant incremental oil recovery with zero incremental investment. However, since the project is extended for significantly longer periods of time, the return on the initial investment may be perceived as unattractive.
High Pressure Core Holder
nC5
Core
Effluent
C1 Water
Injection Pump
Overburden pressure GC instrument for analysis
Pump
Figure 5-42 - Schematic of the experimental apparatus.
299
Two groups of the tests were designed to study the shut in and pressure pulsing scenario. In the first group pressure pulsing and shut-in scenarios were applied with the assumption that the average rates have to be maintained at the same value, 10 times of the critical rate. The second group of experiments consisted of a combination of rate reduction tests accompanied by pressure pulsing / shut-in cycles. The pressure pulse was achieved by pressurizing the injection cylinder to approximately 2,000 kPa over the normal operating pressure while the production was shut-in. When the high limit pressure was reached, the injection was stopped and production resumed until the pressure had dropped approximately 2,000 kPa lower than the normal operating pressure (i.e. a pressure cycle of 4,000 kPa). At this point the pressure cycle was repeated. Since the pressure pulsing scenarios are evaluated for mature miscible floods, pressure pulses always started after solvent breakthrough. It was found that with respect to rate reductions reducing the rate by half does not increase the recovery of oil. However, rate reduction after breakthrough gave a recovery increase of approximately 3% OOIP. Reducing the rate and applying pressure pulsing at the same time also provides better recoveries than the base case. Up to 4% incremental recovery of OOIP can be obtained. Figure 5-43 is the conventional plotting of mole fraction of solvent in effluent as a function of pore volumes injected. As it can be seen, after every shut-in period the methane mole fraction drops, the lower the amount of pentane left in the core, the lower the drop in the methane mole fraction. This can be explained by the fact that bypassed n-pentane is allowed to mix and/or flow towards the production end, this increasing the ultimate recovery. The plots for the base case compared to a combination of pressure pulsing with rate reduction are shown in Figure 5-44. In this figure the recovery plot as a function of pore volumes of solvent injected.
Mole fraction of solvent in effluent
1.00 0.80 0.60 0.40 Shut in process 0.20
Base case
0.00 0
0.5
1
1.5
2
2.5
3
Pore volume of solvent injected Figure 5-43 - Plot of mole fraction of solvent in the effluent as a function of the pore volumes of solvent injected. Periodic shut-in case.
Each of the pressure pulses is shown in the stepwise increase of production. The horizontal part of the step is the pressure buildup, while the vertical part of the step is the pressure relief, where actually the incremental production occurs.
300
Oil recovery, fraction OOIP
1.00 0.80 0.60 0.40
pressure pulsing process Base case
0.20 0.00 0
0.5
1
1.5
2
2.5
3
Pore volume of solvent injected Figure 5-44 - Cumulative recovery for rate reduction and pressure pulsing as a function of pore volumes of solvent injected
The plot of the mole fraction of solvent in the effluent is shown in Figure 5-45. The response is similar to Figure 5-43. However, the reduction in solvent mole fraction is significantly lower in the pressure pulsing case, than in the shut-in case, indicating that the pressure pulse occurred too fast and appreciable mixing did not take place. The combination of flow reduction with pressure pulsing shown in Figure 5-44 provide the incremental recovery without any increase in GOR. From this point of view pressure pulsing accompanied by rate reduction can be considered as a tool for better reservoir management and reduction of operating costs. The combination of shut in and pressure pulsing process showed around 11% incremental recovery compare to the base case. According to the Kantzas et al. experimental results it could be concluded that the life of a mature vertical miscible flood can be extended if the miscibility is enhanced. Such an enhancement can be achieved through periodic flow interruptions (shut-in), or rate reduction combined with cyclic pressure pulsing.
Mole fraction of solvent in fluent
1.00
0.80
0.60
0.40
Base case Pressure pulsing process
0.20
0.00 0
0.5
1
1.5
2
2.5
3
Pore volume of solvent injected
301
Figure 5-45 - Mole fraction of solvent in the effluent as a function of the pore volumes of solvent injected. Rate reduction and pressure pulsing case.
Mixing of Fluid by Dispersion In all miscible displacement processes mixing occurs between the displacing and displaced fluids. That is there is dispersion between the different fluids. This dispersion dilutes the displacing fluid with the displaced fluid and thereby affects the phase behavior. Suppose a first contact miscible solvent is injected into a linear tube to displace an oil that has the same density and viscosity as the solvent. Viscosity and density difference has no effect on the displacement and flow behavior. Concentration of the solvent at the outlet face of the tube is measured with time to find a concentration profile of the solvent versus time. Figure 5-46 shows two different concentration profile of the solvent in effluent versus time. Figure 5-46.a is the concentration that we expected according to mobility ratio equal to one and no gravity effect. We expect to see the solvent in the effluent after injection of one pore volume (equal to the tube volume, here) of solvent and the concentration is expected to reach its maximum value immediately (Figure 5-46.a) . But Figure 5-46.b is the concentration profile that we get in the real word. Solvent in the effluent is detected before injection of one pore volume. S-shaped concentration profile resulted from mixing, or dispersion of solvent and oil in the tube. At first, solvent is produced at low concentration. This is followed by a period of sleepy rising concentration and finally by a period where effluent concentration gradually approaches injected concentration. The same phenomenon happens in porous media. During the solvent injection into a linear tube packed with sand there is a transition zone between pure solvent/oil because of mixing. The S-shaped curve in Figure 5-46.b is the classic concentration for miscible displacement of one fluid with a second in a linear system in which the mobility ratio is favorable and gravity effect is negligible.
Sol .
Oil
Oil
Effluent Solvent Concentration
Effluent Solvent Concentration
Sol.
T1pv
Time
(a). what is expected because of piston like displacement
T1pv
Tim e
(b) what happened in reality because of mixing
Figure 5-46 – profile of the solvent effluent concentration produced from a capillary tube in an equal viscosity and density.
302
As another illustration consider the injection of a small slug of fluid B into packed tube saturated with fluid A. small slug of B in followed by injection of A. the concentration profile at the inlet at the first time would be like as Figure 5-47. If the concentration of solvent is measured at different position of the capillary tube at a fixed time after injection Figure 5-47.a is what is expected in the absence of mixing, or dispersion. But in real condition the concentration profiles would tend to spreads as the fluid moved downstream, and the amplitude (maximum concentration) would decrease (Figure 5-47.b).
T = T0
Sol.
Oil
Oil
At time = T1
Distance from inlet (a). what is expected because of piston like displacement
Effluent Solvent Concentration
Effluent Solvent Concentration
Oil
Sol.
Oil
T = T0 At time = T1
Distance from Inlet (b) what happened in reality because of mixing
Figure 5-47 – Concentration profile for injection of a slug of solvent to displace oil.
This mixing in the direction of primary flow is called “Longitudinal Dispersion”. Dispersion also occurs in transvers direction – that is in a direction perpendicular to flow: Assume solvent injection to a two dimensional model. The model contains two layer of sand. One of the layers is much more permeable than the other such that solvent, which is injected across the left face of the model, mostly enters only the permeable layer. The solvent and oil have the same viscosity and density. In this experiment, the solvent not only mixes with the oil by longitudinal dispersion in the direction of flow; it also mixes with oil transverse to the direction of flow in the less permeable (Figure 5-48). If concentration were measured in situ through the section marked AA, an S-shaped concentration profile again would be found. This mixing transverse to the direction of flow is called “Transverse Dispersion”.
303
Solvent
A
Longitudinal Mixing
Solvent Concentration Oil
Solvent
A Transverse Mixing
Oil Figure 5-48 – Mixing of solvent and oil by longitudinal and transverse dispersion.
The dispersion of miscible fluids may be attributed to several phenomena: Molecular Diffusion: is present in all systems in which miscible fluids are bought into physical contact. The diffusion process is molecular in nature; i.e., it results from the random motion of molecules in solution. Diffusion is dominated dispersion mechanism if flow rates are very low in a porous medium. At rates that commonly exist in reservoir displacement processes, however, dispersion also results from bulk flow or convection phenomenon. Velocity profile (Taylor) effect: Taylor showed that dispersion that occurs during the miscible displacement process in a straight capillary tube (Figure 5-46.b), is a result of velocity profile in the capillary tube. Assume that fluid A is initially in the capillary and that the fluid B displaces fluid A. assume further that the flow rate is low so that flow is laminar. The velocity profile radially across the capillary tube would have the parabolic form (Figure 5-46.b). the profile would be such that: ̅
(5-51)
And the maximum velocity occurs at the center of tube. If the effluent from the capillary were collected and the concentration measured, breakthrough of fluid B would occur when a volume equal to one-half of the capillary volume had been injected. With continued injection concentration of B increase till there was no more fluid A in the effluent. Thus mixing occurs because of the velocity profile that develops in laminar flow in a capillary. Molecular diffusion also occurs across the fluid A/B boundaries in the capillary. Series of mixing cells: another model for the mixing of miscible fluids is based on the assumption that a porous medium is a series of mixing cells or tanks in which fluids perfectly mixed. () illustrate this concept. Fluid B (solvent) enters a pore (Tank1) from the left. Fluids in the tank are perfectly mixed; i.e. there is no concentration gradient in the tank. This assumption means that at any instance there is no concentration difference between the effluent fluid concentration of the tank and fluid in the tank. The mixed fluid from the first pore enters to the next pore, where the fluids are again perfectly mixed. So such model will lead to dispersion of fluid B into fluid A as the displacement process continuous.
304
B
A
Figure 5-49 – dispersion porous medium being viewed as a series of mixing tank.
Stagnant pockets: another contribution to dispersion can be attributed to the flow behavior around stagnant pockets or dead end pores. Assume fluid B displaces fluid A. the part of fluid A that is in the main flow channel is displaced directly by fluid B. however some part of fluid A are in in stagnant pockets or dead end pores- i.e., pore spaces that are connected to the main channels but through which there is no flow. This quantity of fluid A is not displaces directly but is initially bypassed by fluid B. displacement of this bypassed fluid does occur slowly as a result of molecular diffusion between the main flow channel and the stagnant pocket. The overall result of this process is to cause mixing of fluid A and B in the medium as the displacement progresses through the system. The consequence of finiterate mass transfer between the flowing and stagnant fractions is an increase in the length of the mixing zone. The trapping of fluid in dead-end pores is termed "capacitance." Some simple configurations of stagnant volume that have been used to model capacitance are shown in Figure 5-50.
Figure 5-50 - Stagnant volume models
The effect of the capacitance could be distinguished in effluent concentration profile. In the absence of the capacitance (stagnant volume), effluent concentration profile has a symmetric s-shaped as shown in (Figure 5-50). In the presence of stagnant pockets tracer effluent profiles from porous media are characterized by tailing or deviation from the symmetrical S-shape (Figure 5-51).
Stagnant Volume effect
305
Figure 5-51 - The effects of capacitance.
Variation in flow path: dispersion of fluid B into fluid A can result from the variation of flow paths through the media encountered by different fluid particles. As before, fluid A is being displaced by fluid B. visualize two separate particles (very small quantities) of fluid B located at point 1 in Figure 5-51. As flow progresses, the particles moves downstream to point 2 but by slightly different paths through the medium. Because the flow paths are different, the particles arrive downstream at different time. That is, dispersion of the fluid has resulted directly from the tortuosity inherent in the porous medium. Fluids become mixed because flow paths are varied as flow progresses through the system.
2
1 B
A
Figure 5-52 – Dispersion caused by variation of flow paths in a porous medium.
The Equation of Continuity Application of the principle of mass conservation of species i in a multi-component fluid mixture to an arbitrary control volume of the fluid yields the well-known equation of continuity, which, in it's most general form, can be written as follows (Bird et al., 1960): ̅
5-52)
Where, concentration of species i (mass per unit volume), ̅
the mass flux vector (mass of species i per unit area per unit time) source or sink term (mass of i per unit volume per unit time) time.
In order to obtain the concentration of species i as a function of time and space from Equation 1, a constitutive equation that expresses the relationship between the fluxes and driving forces is required. Such an equation is provided by Fick's first law of diffusion: ̅
̅
(5-53)
Where, ̅
mass average velocity vector (length per unit time), fluid (mixture) mass density (mass per unit volume), molecular diffusion coefficient (length squared per unit time), 306
mass fraction of species i (
).
Equation 5-52) states that the flux of species i relative to stationary coordinates is the resultant of the bulk motion of the fluid and molecular diffusion. Substituting Equation (5-53) into Equation 5-52) gives: ̅
(5-54)
Equation (5-54) is applicable to systems with variable and Do. When and Do are assumed constant, Equation (5-54) simplifies to: ̅
(5-55)
When Equation 4 is further simplified to represent one-dimensional flow: ̅
(5-56)
where x is distance and u is velocity, both in the direction of flow.
The Equation of Continuity in Porous Media Eqs (5-54) to (5-56) are usually derived by applying the fluid continuum approach to an element of bulk fluid, for example to the fluid continuum filling a pipe or the pore space in a porous medium. An elementary volume inside the bulk fluid is denoted "microscopic control volume." In the case of porous media, a mathematical description of the pore and flow geometry is too complex to be modeled, and it is difficult to define boundary conditions for the complex solid/fluid boundaries. A continuum approach on a coarser level is therefore applied to describe fluid flow in porous media (Bear, 1972; Rumer, 1972). The multiphase porous medium is replaced by a fictitious continuum, in which the values of the properties of the continuum can be assigned to any point in the medium (solid or fluid) and are continuous functions of space and time. This leads to the definition of the "macroscopic control volume". The macroscopic control volume in a continuum representing a porous medium must be much larger than an individual pore or grain, and much smaller than the entire flow domain, and the porosity of the macroscopic control volume must be representative of the porous medium as a whole. The properties of the macroscopic control volume are neither solid (grain) nor pore space properties, but are averages. The equation of continuity for porous media can be derived by applying a mass balance over a macroscopic control volume and replacing some of the variables with quantities more applicable to porous media. The molecular diffusion coefficient (Do), is replaced by the dispersion coefficient , a tensor, and the velocity ̅ by the apparent linear velocity ̅ , defined as: ̅
̅
(5-57)
Where, 307
̅
volumetric flow rate (volume per unit time), porous medium cross-sectional area (length squared), porosity.
The resulting equation is of the same form as that for bulk fluids: ̅
(5-58)
A more rigorous derivation of the equation of continuity for porous media uses the procedure of spacial averaging (Bear, 1972; Rumer, 1972). In such an approach, every velocity and concentration is expressed as the sum of the average value taken over the macroscopic control volume and the spacial variation at any point inside the macroscopic control volume. The expressions for the velocities and concentrations and their variations are substituted into the equation of continuity for a fluid continuum (Equation 3), and the resulting equation is averaged over the macroscopic control volume to obtain Equation 6. Carrying out the averaging procedure, as described by Bear (1972), rather than using the simpler approach of writing a mass balance over the macroscopic control volume, reveals information on the structure of the dispersion coefficient: the dispersion of a substance during flow in porous media results from molecular diffusion in the direction of flow, coupled with transverse molecular diffusion due to the velocity profiles (as in capillary tubes, see below), and mechanical mixing arising from velocity variations due to the complex nature of the pore structure. The velocity variations are not easily measured, and the dispersion coefficient has therefore frequently been correlated with more easily measurable quantities such as the apparent linear velocity and some characteristic length of the porous medium, for example particle diameter or porous medium length. The complex pore structure causes flow in the porous network to deviate from the mean direction of flow at the pore level. This effect can be described by the flow tortuosity, a tensor quantity (Bear, 1972). Mechanical mixing due to flow is different in different directions. The dispersion coefficient, a measure of mixing during flow, is therefore anisotropic, hence is a tensor quantity. In one-dimensional flow, with constant and
, Eq. (5-58) becomes:
̅
(5-59)
This equation is commonly referred to as the one-dimensional convection-dispersion equation.
Equation of Dispersion in Capillaries Dispersion in cylindrical capillaries containing flowing fluids has been described mathematically in a paper by Taylor (1953). Consider a slug of tracer that is introduced into fully developed laminar flow. The velocity profile in a capillary of circular cross-section is parabolic. Since the velocity in the tube increases from zero at the walls to a maximum value at the center, the tracer would, in the absence of other effects, eventually disperse throughout the length of the tube. However, it is observed experimentally that a symmetrical mixing zone develops around a plane that travels at the mean speed of flow. This is not unlike the stationary case above, except that the mixing zone moves along the tube. 308
The reason is molecular diffusion. Initially, the velocity profile distorts the tracer slug into parabolic shape. This sets up radial concentration gradients, which give rise to molecular diffusion in the radial direction. Given sufficient time, radial concentration gradients are eliminated by molecular diffusion. In the fully established mixing zone, the solute concentration is equalized radially, but depends on the axial distance from a plane that moves at the mean speed of flow, as illustrated in Figure 5-53. In its most general form, the tracer concentration in a capillary is given by the two-dimensional unsteady convective diffusion equation in cylindrical coordinates: [
(
)]
(5-60)
Where, point concentration (a function of radial and axial position, and time), parabolic laminar velocity profile, radial distance, axial distance.
309
(a)
(b)
(c)
(d)
Figure 5-53 Development of the mixing zone as a function of time during laminar flow in a capillary (after Nunge and Gill, 1970).
For “long” times, i.e. when the mixing zone in Figure 5-53.d has been fully established, which is equivalent to saying that equalization of concentrations by molecular diffusion in the radial direction is fast compared to mixing by convection, Eq. (5-60) can be shown to simplify into one-dimensional form (Taylor, 1953): (5-61) Where, Tracer concentration at distance z and time t, mean speed of flow, hydrodynamic dispersion coefficient in the capillary. Note that cm is no longer a function of radius, since concentrations are equalized radially. Also note that Eq. (5-61) is of the same form as the one-dimensional equations of continuity discussed above (Eqs (5-56) and (5-59)), without the source/sink term. Thus, for “long” times, flow in the capillary can be described by the one-dimensional convection-dispersion equation. The dispersion coefficient for porous media or capillaries is a parameter in Eqs (5-58), (5-59), and (5-61). Hydrodynamic dispersion is the macroscopic outcome of the mixing of one fluid in a second, miscible fluid of different composition during flow through capillary spaces or porous media. The coefficient of hydrodynamic dispersion is a measure at which material will spread out in the system, hence of the length of the mixing zone. The mixing zone can be defined as that region of a capillary or porous medium where the concentration of an injected species changes from zero to the injected concentration, or from some imposed lower to an upper limit. The dispersion coefficient has the same units as the molecular diffusion coefficient (length squared per unit time).
310
For “long” times, it has been shown that the effective dispersion coefficient for flow in capillaries can be expressed as follows (Perkins and Johnston, 1963; Nunge and Gill, 1970; Fried and Combarnous, 1971; Bear, 1972): (5-62) Where, effective dispersion coefficient for flow in capillaries, constant that depends on capillary radius, characteristic dimension of the capillary cross-section. For a capillary of circular cross-section, a is radius, and =1/48. The right hand side of Eq. (5-62) contains a diffusive and a convective term, and reflects the fact that dispersion in capillaries is the result of molecular diffusion and mechanical mixing due to flow. represents axial (longitudinal) dispersion, since concentrations are equalized radially. The diffusion term, the first term on the right hand side, is a measure of dispersion by axial molecular diffusion, and the second term represents mixing due to flow and radial diffusion. At high velocities, the molecular diffusion term becomes negligible and dispersion is dominated by the convective term. At low velocities or high rates of diffusion, axial molecular diffusion contributes to dispersion. Molecular diffusion in the axial direction increases dispersion. Molecular diffusion in the radial direction decreases dispersion. Dispersion increases at higher velocities and with increasing tube radius. “Long” time, as referred to above, refers to times long enough to eliminate radial concentration gradients by molecular diffusion. “Long” in this context is a relative term, depending on the relative rates of diffusion and convection. The minimum time required for the convection-dispersion model (or dispersion model, for short) to hold has been discussed in detail by Nunge and Gill (1970). It depends on a number of factors, such as capillary geometry, inlet boundary conditions, the type of concentration change (step change versus slug stimulus), laminar or turbulent flow, developing or fully developed flow fields, and fluid properties (density and viscosity differences). The minimum time is frequently expressed as dimensionless time () and correlated with the Peclet number (Pe), 5-63) (5-64) As an example, for fully developed laminar flow in capillary tubes with a step change in input concentration, the dispersion model holds for > 0.8. A summary of the regions of applicability of different models, taken from Dullien (1992) and Nunge and Gill (1970), is shown in Figure 5-54. The Taylor-Aris theory in Figure 5-54 is represented by Eq. (5-62), while the Taylor theory is Eq. (5-62) without the diffusion term (the first term on the right hand side). When < min, the dispersion model (Eq. (5-61)) does not apply and more complex models (Eq. (5-60)) have to be used.
311
10,000
Pure Convection
Taylor Theory
Peclet number
1,000
100
Taylor -Aris Theory 10
Pure Diffusion 1
0.01
0.1
1
10
100
Dimensionless Time Figure 5-54 Summary of the regions of applicability of various analytical solutions for dispersion in capillary tubes with step change in inlet concentration as a function of dimensionless time and Peclet number (from Dullien, 1992).
Equation of Dispersion in Porous Media Flow in porous media is different from flow in simple capillaries in that a major contribution to mechanical dispersion arises from the complex geometry of the flow paths. Dispersion in porous media results from the same mechanisms that are active in capillaries (molecular diffusion and velocity profiles across the flow channels). However, as mentioned before, additional mechanisms arise from the fact that flow takes place in capillaries that have non-uniform, variable cross-sections and different conductivities. The capillaries run at various angles to the net direction of flow, and mixing and resplitting of streams takes place at pore junctions. A component of dispersion thus arises from the existence of the porous medium itself. In general, the dispersion coefficient in porous media is a second-order tensor that depends on local variations of the velocity field and on porous medium characteristics (Bear, 1972). The dependence of dispersion on these quantities cannot be measured easily, and the dispersion coefficient is therefore commonly reduced to a longitudinal component parallel to the net direction of flow, and a transverse component perpendicular to the net direction of flow. Experimentally, it is observed that longitudinal dispersion is larger than transverse dispersion (Blackwell, 1962; Fried and Combarnous, 1971). Dispersion is thus anisotropic. Theories attempting to model dispersion in porous media have included capillary tube and mixing cell models, networks of fixed and random capillaries, and models that use special averaging and the macroscopic equation of continuity (Fried and Combarnous, 1971; Bear, 1972). Exact mathematical models have been developed only for very simple geometries, such as single capillaries or bundles of capillaries, and for arrays of mixing cells. While capillary models cannot adequately represent porous media, they are useful from a conceptual point of view. The theory of random walk has often been applied to dispersion modeling. The basic postulate in this approach is that, although it is impossible to predict the exact path of an individual tracer particle, the 312
rules of statistics may be applied to predict the overall behavior of a cloud of tracer particles. The statistical approach has been applied to fixed and random capillary patterns representing simple porous media. A criticism of these geometrical models has been that they are restrictive through the choice of a specific geometrical pattern. The macroscopic convection-dispersion equation (Equations 6 and 7), which treats the porous medium as a continuum and uses a "lumped" dispersion parameter, has been used most commonly to model dispersion in practice. However, even this, perhaps most general, approach is still closely linked to experiment for estimation of the model parameters. The dispersion coefficient in porous media is a function of the fluid velocity. Mathematical models for flow in porous media have yielded velocity exponents of either 1 or 2 (Collins, 1961; Fried and Combarnous, 1971; Bear, 1972). Bear (1972) has pointed out that models which consider the combined effects of a velocity distribution across the flow channels and molecular diffusion predict a dispersion coefficient proportional to v2, as in the case of capillary tubes (see Equation 10). Models that take into account only the mean motion of the fluid in the flow channels and assume that complete mixing takes place at their junctions (perfect mixers) lead to a linear relationship between D and v. It has been observed experimentally that the velocity exponent of the dispersion coefficient in porous media is often close to 1.2 (Collins, 1961; Brigham et al., 1961; Bear, 1972). A general equation that expresses the dependence of dispersion in porous media on velocity can be written as follows (Perkins and Johnston, 1963; Fried and Combarnous, 1971): [
]
(5-65)
Where, porous medium dispersion coefficient, apparent molecular diffusion coefficient in the porous medium, a constant, a measure of the inhomogeneity of the porous medium, apparent linear velocity, average particle diameter, exponent that depends on the flow regime. The dimensionless quantity
is the particle Peclet number of molecular diffusion. It expresses the
ratio of convective to diffusive mixing, with the particle size as characteristic length. Porous media are generally too complex for exact mathematical description, necessitating the parameters and . Eq. (5-65) shows that dispersion in porous media can be modeled as a sum of diffusive and convective terms, as in the case of capillaries (Eq. (5-62)). Eq. (5-65) is often represented graphically in the form of a log-log plot of dimensionless dispersion (D/D0) versus particle Peclet number (
), or modified particle 313
Peclet number (
). A typical example is shown in Figure 3, from which the following flow regimes can
be identified (Perkins and Johnston, 1963).
Figure 5-55 Dependence of dispersion coefficient on Peclet number in different flow regimes. The scales on the axes depend on porous medium and other factors. The curve shown approximates longitudinal dispersion in unconsolidated random packs (after Perkins and Johnston, 1963).
Regime a: At low velocities or high molecular diffusion coefficients, molecular diffusion dominates dispersion. The second term in Eq. (5-65) can be neglected, and the dispersion coefficient is constant and equal to the apparent molecular diffusion coefficient (Dapp), which is lower than the true molecular diffusion coefficient because of porous medium tortuosity. Regime b: The effect of mechanical dispersion becomes significant, and both molecular diffusion and convection contribute to dispersion. Regime c: In this regime, the porous medium can be modeled as a perfect mixer, and m=1. This flow regime applies to longitudinal dispersion in random packs when ( dispersion in similar packs when
), and to transverse
(Perkins and Johnston, 1963).
Regime d: When the Peclet number becomes very high, the concentration in each mixing cell cannot be equalized, and m becomes greater than unity. The exponent m has been found to increase from 1 to approximately 1.25 in this flow regime. In regimes c and d, molecular diffusion is often neglected, and for practical purposes an equation of the form D=v, being a constant, is used (Fried and Combarnous, 1971; Bear, 1972; van Genuchten and Wierenga, 1977; Satter et al., 1980).
314
The flow regimes described by different authors are not always consistent. The above description follows that of Perkins and Johnston (1963). Fried and Combarnous (1971) and Bear (1972) have used m=1 to 1.2 in Regime c and m=1 in Regime d, and have identified a fifth flow regime for longitudinal dispersion at very high Peclet numbers. In this fifth regime the effects of turbulence become appreciable and Darcy's law no longer applies. Turbulence causes increased transverse mixing, which reduces longitudinal dispersion; the slope of the curve thus tends to become less than unity. In a porous medium, there is no sharp transition from laminar to turbulent flow. Defining the particle Reynolds number (Re) in packs of spheres as 5-66) Where, fluid density, fluid viscosity. Laminar flow has been observed at Reynolds numbers lower than 10, while turbulence is fully developed at Reynolds numbers greater than 1000 (Perkins and Johnston, 1963). In the turbulent region, dispersion is often expressed as a particle Peclet number (Pe) of dispersion: (5-67) In fully developed turbulent flow, the longitudinal Peclet number approaches 2, and the transverse Peclet number approaches 11 with increasing Reynolds number, Figure 4 (Aris and Amundson, 1957; Perkins and Johnston, 1963). Turbulent flow is not likely to be encountered in a petroleum reservoir. In a typical reservoir rock we may have: ,
, Then:
315
Figure 5-56 Dependence of Peclet number on Reynolds number for an aqueous system (from Perkins and Johnston, 1963).
The general form of eq. (5-65) and the shape of the curve in Figure 5-55 apply to longitudinal and transverse dispersion and to different types of porous media, but the transition between flow regimes and the magnitude of the dispersion coefficient are system dependent. Figure 5-56 is a typical example for longitudinal dispersion in unconsolidated random packs. Blackwell (1962) has given a set of empirical curves that allow estimation of transverse and longitudinal dispersion coefficients in sandpacks (Figure 5-57).
Figure 5-57 Range of dispersion coefficients for various sandpacks. The lower curves are for coarse sand packed to a porosity of about 34%. The upper curves are for finer sand or looser packings (porosity > 34%). Fine sand 200-270 mesh, medium sand 40-200 mesh, coarse sand 20-30 mesh (after Blackwell, 1962).
316
Some values for the parameters in eq. (5-65) are as follows (Perkins and Johnston, 1963; Fried and Combarnous, 1971): 0.5
longitudinal dispersion
0.016
longitudinal dispersion
3.5
unconsolidated random bead pack
1.0
unconsolidated medium of identical spherical beads
0.25-0.46 cm
Berea sandstone
0.36 cm
average for some consolidated sandstones
In consolidated porous media, the particle diameter dp loses it's meaning, and and dp are lumped together. It must be kept in mind that these parameters are strongly system dependent and cannot be universally applied (Brigham et al., 1961; Blackwell, 1962; Fried and Combarnous, 1971; Batycky et al., 1982). The majority of studies have been carried out with unconsolidated media, but some values for consolidated rocks are also available. Brigham et al. (1961) list parameters for glass bead packs of different bead diameters, Torpedo sandstone, and Berea sandstone. Spence and Watkins (1980) have determined dp for a large number of Berea and Boise sandstone and San Andres carbonate cores. Some of the factors that determine dispersion in porous media have been described by Perkins and Johnston (1963), and include particle size distribution, particle shape, packing or permeability heterogeneities, density and viscosity effects, turbulence, the presence of an immobile phase, capacitance, and wall effects. Patel and Greaves (1987) present experimental data to illustrate the effects of permeability, porous medium type (unconsolidated sand and consolidated sandstone), phase saturation, phase type, surfactant concentration, and interfacial tension. As has been mentioned, the longitudinal dispersion coefficient is usually found to depend on velocity raised to a power between 1 and 1.25 at experimental conditions typical for laboratory core floods. This dependency has been found to hold for different types of porous media in single and two-phase flow. Some examples are bead packs (Brigham et al., 1961; de Smedt and Wierenga, 1979b), consolidated sandstone (Brigham et al., 1961; Baker, 1977; Ramirez et al., 1980; Spence and Watkins, 1980; Salter and Mohanty, 1982; Sorbie et al., 1985), consolidated sandstones with two-phase flow (Ramirez et al., 1980), and carbonate rocks, which usually have a more heterogeneous pore structure than sandstones and may exhibit dual porosity (Baker, 1977; Spence and Watkins, 1980). Although the velocity dependence of the dispersion coefficient is similar in different porous media, the magnitude of dispersion is usually found to be higher in consolidated than in unconsolidated media (Brigham et al., 1961; Perkins and Johnston, 1963; Fried and Combarnous, 1971), and higher in carbonates than in sandstones (Baker, 1977; Batycky et al., 1980; Spence and Watkins, 1980; Bretz et al., 1986). It increases when the pore size distribution becomes wider (Spence and Watkins, 1980; Bretz et al., 1986) or the phase saturation decreases (Stalkup, 1970; Ramirez et al., 1980; Patel and Greaves, 1987; Sorbie et al., 1985), and is higher in the non-wetting than in the wetting phase (Patel and Greaves, 1987; Salter and Mohanty, 1982; Maini et al., 1986). The effect of phase saturation and wettability on 317
dispersion can be attributed to the connectivity and tortuosity of the phase in question. The presence of a second, immiscible phase changes the flow geometry, and wettability determines liquid distributions. Dispersion increases when the viscosity ratio of displaced to displacing fluid increases towards unity. When the viscosity ratio exceeds unity, the one-dimensional convection-dispersion equation (Equation 7) no longer applies.
Solutions to the One-Dimensional Convection-Dispersion Model Infinite System For a single solute and no source/sink, Eq. (5-56) becomes: (5-68) The following assumptions are implicit in eq. (5-68): homogeneous porous medium of constant cross-section; bulk flow in the axial direction at constant interstitial velocity; constant fluid density; constant dispersion coefficient; incompressible porous medium; uniform concentration distribution in the direction perpendicular to flow (i.e., time is “long” enough for the convection-dispersion model to hold); - no solute sources or sinks. Eq. (5-68) is a second order partial differential equation, requiring two boundary and an initial condition for its solution. -
When the space variable x is translated such that the transformed distance x' becomes the distance from the flood front rather than the distance from the inlet of the porous medium, eq. (5-68) reduces to the one-dimensional unsteady diffusion or heat conduction equation, also known as Fick’s second law (Taylor, 1953; Nunge and Gill, 1970): (5-69) Where,
And, distance of the flood front from the porous medium entrance. This equation has been solved analytically for a variety of initial and boundary conditions (see, for example, Carslaw and Jaeger, 1959; Özişik, 1980). Consider an infinite porous medium at zero initial concentration, with a step change in inlet concentration (c0): t=0
(5-70) 318
t>0 The solution to eq. (5-69) with initial and boundary conditions (5-70) is (Danckwerts, 1953; Brigham et al., 1961): [
(
√
)]
(5-71)
Where, normalized concentration, error function of a variable r: √
∫
(5-72)
Some properties of the error function are
Then, from eq. (5-71),
Typical concentration profiles, obtained from Equation 23, are shown in Figure 6. The curves are symmetrical about the point (z’=0, c/c0=1/2). The profiles become more dispersed when D increases or as time progresses. In non-dimensional form, eq. (5-68) becomes: (5-73) Where,
(5-74) V = volume injected Vp = pore volume
319
5-75) Initial and boundary conditions can similarly be put into non-dimensional form.
Figure 5-58 Concentration as a function of transformed distance for different values of dispersion coefficient or time, calculated from eq. (5-72), infinite system.
The convection-dispersion model thus contains a single parameter: the dispersion coefficient, or, in dimensionless form, the Peclet number. The Peclet number represents the ratio of characteristic times for mass transfer by flow and by dispersion, and is a measure of the length of the mixing zone relative to the length of the porous medium. The Peclet number, as defined in eq.5-75), contains the porous medium length as characteristic length, because L is a parameter easily measured in laboratory experiments. Other characteristic lengths have been used in the definition of the Peclet number, as is evident from the discussions above. Also, depending on the process under investigation, the denominator may contain the diffusion coefficient or the dispersion coefficient. Regions of flow where different models apply in any given system are often characterized in terms of the Peclet number, as, for example, in Figure 5-56.
Determination of the Dispersion Coefficient Because of its simplicity, eq. (5-71), although representing infinite systems, is often used to approximate concentration distributions in finite systems. In a typical laboratory core flood experiment, a porous medium is fully saturated with a fluid. A miscible fluid or tracer, such as a salt solution or dye or radioactive material, is then injected at a known, constant flow rate. Effluent samples are collected at the core exit and analyzed for tracer concentration. In such an experiment, eq. (5-71) represents the effluent concentration, i.e., the concentration at a fixed distance (z=L, or z’=L-vt). In an experiment such as this one, the dispersion coefficient can be determined using a simple procedure described by Brigham et al. (1961). When eq. (5-71) is used to calculate effluent 320
concentrations from a finite porous medium (z=L or z'=L-vt), it is convenient to transform variables such that time is replaced by throughput in pore volumes (Brigham et al., 1961; Brigham, 1974). Change variables such that: (5-76) (5-77) Eq. (5-71) then becomes: [
(
√
⁄
)]
[
(
√ ⁄
)]
(5-78)
Where, ⁄ (5-79)
√ ⁄
When is plotted against the percentage of displacing fluid in the effluent on arithmetic probability coordinates (Figure 5-59), and provided the convection-dispersion model holds, a straight line results from which the longitudinal dispersion coefficient can be obtained using the following equation (Brigham et al., 1961; Perkins and Johnston, 1963): [
]
(5-80)
Figure 5-59 Typical probability plot for determination of longitudinal dispersion coefficient.
10 and 90 are the values of when the effluent concentration is 10% and 90% of the injected value, respectively, and v and L are known experimental parameters. The constant 3.625 arises from the error 321
function and the choice of 10% and 90% concentrations. When 20% and 80%, or 5% and 95%, concentrations are used instead, this constant becomes 2.380 or 4.653, respectively. Alternatively, the dispersion coefficient can be determined by plotting experimental effluent concentration profiles against volume injected and adjusting D until Equation 30 fits the experimental data. This is commonly done with effluent concentrations, since they are easily measured through chemical analysis. In-situ concentrations may also be used, but generally require more sophisticated experimental methods, for example x-ray or nuclear magnetic resonance techniques. In-situ concentrations can be modeled as a function of time and distance using eq. (5-71). When experimental effluent concentrations are plotted on probability coordinates, it is assumed that they are described by the error function (eq. (5-78)), i.e. that the porous medium can be approximated by an infinite system. Analytical solutions to eq. (5-68) for semi-infinite or finite systems contain an error function in addition to other terms. Theoretically, they should not result in a linear probability plot, particularly at low Peclet numbers (Pe < 30). However, in practice it has been found that a probability plot is often linear for finite systems and, in addition, yields the correct dispersion coefficient even at Peclet numbers as low as 14 (Brigham, 1974). Shuler (1978) found that dispersion coefficients determined by fitting a finite system convection-dispersion model to tracer data were very similar to coefficients obtained from probability plots for the range of Peclet numbers investigated (Pe between 140 and 420). The probability plot is therefore commonly used to determine dispersion coefficients for systems other than infinite. In fact, a probability plot is often used to determine whether the convection-dispersion model is valid for a given set of experimental data (Brigham, 1974). Deviation of a probability plot from linearity indicates that a more complex model may be required to describe mass transport.
Normalized Effluent Concentration
1.2 1 0.8
Pe = 8
0.6
Pe = 80 0.4
Pe = 320 Pe = Infinite
0.2 0 0
0.5
1
1.5
2
2.5
Produced Pore Volume Figure 5-60 Effluent concentration profiles for a range of Peclet numbers, calculated from eq. (5-78), infinite system.
Figure 5-60 shows effluent concentrations plotted as a function of pore volumes injected (eq. (5-78)). As in Figure 5-58, the mixing zone becomes wider as the dispersion coefficient increases, or as Pe decreases. According to eq. (5-78), c/c0=1/2 at one pore volume injected. When D0 (Pe∞), plug 322
flow is obtained and the effluent profile is a step function. When D∞ (Pe0), the porous medium becomes a continuous stirred tank, and the dispersion model no longer holds. It is of interest to note that the curves in Figure 5-58, which represent concentrations at a fixed value of time plotted as a function of distance, are always symmetrical. By contrast, the concentrations in Figure 5-60, which are concentrations at a fixed distance (z=L) plotted as a function of time or throughput, are not symmetrical, particularly at low Peclet numbers. This may seem surprising, since both sets of curves are derived from solutions to the same equation. The reason for the difference in symmetry is evident from eqs. (5-71) and (5-78). At a fixed value of time, concentrations calculated from equation 23 are always symmetrical about the flood front (z=vt or z'=0). Concentrations calculated as a function of time or pore volumes injected at z=L from eq. (5-78) are not symmetrical about a throughput of one pore volume because of the term ( V / Vp ) that appears in the denominator of the error function argument. The asymmetry becomes more pronounced at low Peclet numbers. Brigham (1974) has previously noted this, and has discussed the implications in detail.
Alternative Boundary Conditions Alternative boundary conditions have been used to solve eq. (5-68), specifying concentration (Dirichlet boundary condition), flux (Neumann boundary condition), or a combination of both (Robin's boundary condition). The following initial and boundary conditions have been commonly used to represent finite and semi-infinite systems: 5-81) 5-82) Inlet:
or
5-83) When
5-84) 5-85)
Outlet: t>0
or
5-86)
or
5-87)
where co can be a function of time, but is often taken as a constant, and ca is the initial concentration in the porous medium and can be zero, or a constant, or a function of distance. These boundary conditions have been discussed quite widely, for example by Danckwerts (1953), Kramers and Alberda (1953), Aris and Amundson (1957), Bischoff and Levenspiel (1962), Coats and Smith (1964), and Brigham (1974). Inlet condition 5-83) states that the solute concentration at the entrance to the porous medium is determined by bulk flow and diffusion. Outlet condition 5-86) is somewhat intuitive, and is necessary to prevent the solute concentration in a porous medium of finite 323
length from passing through a maximum or minimum (Danckwerts, 1953). Conditions 5-83) and 5-86) are often referred to as "Danckwerts' boundary conditions." Outlet condition 5-87) has been used by Batycky et al. (1982), who argue that it is less restrictive than the Danckwerts condition 5-86), and is computationally more convenient to use than the semi-infinite bed condition. Analytical solutions to eq. (5-71) with different boundary conditions are readily available in the literature. References to some of these papers have been tabulated, with boundary conditions, by Mannhardt and Nasr-El-Din (1994), and include Danckwerts (1953), Gershon and Nir (1969), Bear (1972), Brighham (1974), Coats and Smith (1964), Fried and Combarnous (1971), Brenner (1962), Sorbie et al. (1985). Van Genuchten and co-workers have published extensive compilations of analytical solutions (Parker and van Genuchten, 1984; Toride et al., 1993; van Genuchten, 1981; van Genuchten and Alves, 1982). Many solutions to the convection-dispersion equation for systems other than infinite contain the error function in addition to other terms, and c/co is therefore different from 0.5 at V/Vp=1. Different concentration profiles are obtained when different boundary conditions are used to solve eq. (5-61), particularly when the mixing zone is large compared to the length of the porous medium, i.e., when Pe is small (Brigham, 1974). An example of effluent profiles with boundary conditions 5-83) and 5-86), and zero initial concentration, is shown in Figure 5-61 (Brenner, 1962). In the absence of dispersion (Pe, D=0), the effluent concentrations are a step function. As dispersion increases (Pe decreases), the effluent concentrations take the symmetrical S-shape often observed in experimental data. c/c0 is greater than ½ at one pore volume injected at small Pe, but approaches ½ with increasing Pe. When Pe=0 (D), the porous medium becomes a continuous stirred tank. The convection-dispersion model no longer applies, and the effluent concentrations become an exponential function of the volume injected. Normalized effluent concentration
1.2 1 0.8 0.6
Pe = 0
0.4
Pe = 8
0.2
Pe = 80
0 0
0.5
1
1.5
2
Injected pore volume Figure 5-61 Effluent concentration profiles for a range of Peclet numbers, finite system with Danckwerts’ boundary conditions (Brenner, 1962).
The solution to eq. (5-71) is thus sensitive to the boundary conditions. Examples of effluent curves for an infinite system with a step change in input concentration (boundary conditions (5-70)) and for a finite system with boundary conditions 5-83) and 5-86) are shown in Figure 5-62. At a Peclet number of 24, 324
differences in effluent concentrations from the two systems become quite small. A detailed comparison of solutions to the convection-dispersion model, obtained with different boundary conditions, is provided by Gershon and Nir (1969) and van Genuchten and Alves (1982). The latter authors suggest that differences between solutions become small compared to experimental errors when the Peclet number exceeds 30. 1.2
Normalized effluent concentration
Finite system with Danckwerts boundary condition 1
Infinite system
0.8 0.6 0.4 0.2 0 0
0.5
1
Injected pore volume
1.5
2
1.2
Normalized effluent concentration
Finite system with Danckwerts boundary condition 1
Infinite system
0.8 0.6 0.4
Pe = 24
0.2 0 0
0.5
1
Injected pore volume
1.5
2
Normalized effluent concentration
1.2
Finite system with Danckwerts boundary condition 1
Infinite system
0.8 0.6 0.4
Pe = 80
0.2 0 0
0.5
1
Injected pore volume
1.5
2
Figure 5-62 Effect of boundary conditions on solutions to the convection-dispersion equation at different Peclet numbers.
The proper choice of inlet boundary condition in a semi-infinite system with outlet condition 5-84), with ca=0, has been discussed by Brigham (1974), and later by Parker and van Genuchten (1984) and Toride 325
et al. (1993), who argue that one must distinguish between "flowing" and "in-situ" concentrations, related through the following equation: (5-88) Where, Flowing concentration, In-situ concentration. Eq. (5-88) states that the concentration flowing from a differential volume element of porous medium is different from the in-situ concentration inside the volume element because mass transfer of solute is due to the combined effects of convection and dispersion. Depending on the concentration gradient, the flowing concentration can be larger or smaller than the in-situ concentration. Eq. (5-71) represents both flowing and in-situ concentrations. However, which concentration is represented by any specific solution to Eq. (5-71) is determined by the boundary conditions. Boundary condition 5-82) represents flowing concentration (the concentration being injected into the porous medium), and the solution to equation (5-68) at z=L using this boundary condition thus represents effluent concentrations ("effluent" implying "flowing"). When the convection-dispersion model is applied to concentrations that are measured in-situ, boundary condition (5-83), representing in-situ concentrations, should be used. No matter which concentrations are used, one can interconvert between in-situ and flowing concentrations using equation 40. Brigham (1974) concludes that it makes little difference which boundary conditions one uses, provided the solution to Equation (5-68) is interpreted properly in terms of in-situ and flowing concentrations. This is particularly important when Pe is small. At large Pe, all solutions approach one another, and the choice of boundary conditions becomes less critical (Coats and Smith, 1964; Gershon and Nir, 1969; van Genuchten and Alves, 1982; Brigham, 1974). As noted above, a Peclet number larger than approximately 30 to 50 can be considered large in this context. Eq. (5-78) predicts that the concentration at z=L in an infinite system is always one-half the injected value at V/Vp = 1. As pointed out by Brigham (1974), the concentration given by eq. (5-78) is the in-situ concentration. The flowing concentration at x=L is larger than 0.5 by (4Pe)-1/2 at a throughput of one pore volume. The same is true for semi-infinite systems (Coats and Smith, 1964). Similarly, the finite system boundary conditions given by eqs 5-83) and 5-86) yield a value of c/co larger than 0.5 by 5/(16Pe)1/2 when one pore volume has been injected (Coats and Smith, 1964). Depending on the boundary conditions used with the dispersion model, the effluent concentration from a porous medium may thus be larger than 0.5 at a throughput of one pore volume, but approaches 0.5 for large values of Pe. This is clearly evident from the curves in Figure 5-61. Eq. (5-71) is not applicable to miscible displacements with unfavorable viscosity ratios or density differences (Brigham et al., 1961; Nunge and Gill, 1970). In these situations, the displacement is dominated by viscous fingering or gravity override, the length of the mixing zone is greatly increased, and the probability plot is no longer linear. A one-dimensional equation is not adequate to describe viscous fingering. Whether instabilities of the flood front are treated as fingering or dispersion depends 326
on the ratio of the scale of the instability to the scale of the porous medium. If this ratio is small, the instability can be treated as a dispersion phenomenon.
The Capacitance Model Tracer effluent profiles from porous media are often characterized by tailing or deviation from the symmetrical S-shape given by solutions to the convection-dispersion equation at high Peclet number. As a result, a probability plot of these tracer concentrations is concave upward at the high concentration end (Figure 5-63). As mentioned before this behavior has been attributed to dead-end pores, stagnant regions of pore space not contributing to flow, or partially saturated porous media. It has frequently been modeled by dividing the pore space into a flowing and a stagnant fraction, with some resistance to mass transfer between them. The concept of capacitance is obviously an oversimplification, but provides a convenient tool to model regions of large velocity contrast. The consequence of finite-rate mass transfer between the flowing and stagnant fractions is an increase in the length of the mixing zone.
Figure 5-63 The effects of capacitance. (a) Illustration of symmetrical and skewed effluent profiles. (b) Deviation of probability plot from linearity due to asymmetry in effluent profile.
In the presence of capacitance, the one-dimensional convection-dispersion equation becomes (5-89) where f is the flowing fraction of the pore volume, and c* is the average concentration in the stagnant fraction. An additional equation specifying the mode of mass transfer between the flowing and stagnant fractions is required. Various models have been used, one of the most common ones being the first order mass transfer model of Coats and Smith (1964): (5-90) where c* is assumed to be uniformly mixed, and K is the mass transfer coefficient. Coats and Smith's model extends a mixing cell capacitance model developed by Deans (1963). Using the dimensionless variables in eqs (5-74) and 5-75), and
327
(5-91) Eqs (5-89) and (5-90) can be put into non-dimensional form:
c c * 1 2c D c D f D (1 f ) D 2 Pe z D z D t D t D (1 f )
c D * St (c D c D *) t D
5-92
5-93
Where, St is the Stanton or Damköhler number, defined by:
St
KL v
5-94)
Any of the initial and boundary conditions previously discussed can be used with the capacitance model. An initial condition for c* is also required. As well, the discussion of flowing and in-situ concentrations in a previous section applies to the capacitance model (Brigham, 1974; Baker, 1977). The capacitance model has frequently been solved by numerical techniques, but analytical solutions are also available. When mass transfer is slow, solutions of the capacitance model revert to those for the convection-dispersion equation, with tD replaced by tD/f. Some analytical solutions to the capacitance model and discussions of boundary conditions can be found in papers by Coats and Smith (1964), Bennett and Goodridge (1970), Brigham (1974), de Smedt and Wierenga (1979a,b), and Patel and Greaves (1987).
Factors Influencing Capacitance Model Parameters Mass transfer into the stagnant volume is often assumed to take place by molecular diffusion, and, in some models, has been described by the unsteady diffusion equation (Turner, 1958; Gottschlich, 1963; Shearn and Wakeman, 1978). Jasti et al. (1987) have argued that mass transfer should be proportional to molecular diffusivity in the absence of eddies, and Stalkup (1970) has shown experimentally that mass transfer is strongly dependent on molecular diffusivity. However, a large amount of experimental work from different investigators has shown that the mass transfer coefficient, obtained by fitting the Coats and Smith model to experimental data, increases with increasing velocity (Baker, 1977; van Genuchten and Wierenga, 1977; Ramirez et al., 1980; Spence and Watkins, 1980; Batycky et al., 1982; Bretz and Orr, 1987; Patel and Greaves, 1987). The velocity dependence of the mass transfer coefficient indicates that mixing takes place by mechanisms other than pure molecular diffusion, and possibly indicates the inadequacy of a model that divides the pore space into a fraction that contributes to flow and one that is completely stagnant. Some flow which is likely to be present in the "stagnant" pore space or the presence of eddies may contribute to mixing. In the presence of a second, immiscible phase, the morphology of each phase depends on its saturation, and the mass transfer coefficient thus also depends on phase saturation (Ramirez et al., 1980; Batycky et al., 1982; Salter and Mohanty, 1982) and on interphase mass transfer (Shearn and Wakeman, 1978).
328
The flowing fraction depends on pore morphology and, in two-phase systems, on the morphology of the phase in question, this in turn being determined by wettability and phase saturation. The flowing fraction in sandstones containing a single fluid phase is frequently between 0.9 and 1.0 (Coats and Smith, 1964; Baker, 1977; Ramirez et al., 1980; Spence and Watkins, 1980; Batycky et al., 1982; Sorbie et al., 1985; Bretz et al., 1986; Patel and Greaves, 1987). The flowing fraction in carbonates can be much lower (as low as 0.38), reflecting the dual pore structure often encountered in carbonate rocks (Baker, 1977; Spence and Watkins, 1980; Batycky et al., 1982; Bretz et al., 1986; Bretz and Orr, 1987). Conflicting results regarding the dependence on velocity have been reported: some investigators have found f to be independent of velocity (Coats and Smith, 1964; Baker, 1977; Ramirez et al., 1980; Jasti et al., 1987; Patel and Greaves, 1987), while others have reported a decrease in f with increasing velocity (Spence and Watkins, 1980; Batycky et al., 1982; Bretz and Orr, 1987). Van Genuchten and Wierenga (1977) and Batycky et al. (1982) report an increase in f with increasing velocity for certain systems. The velocity dependence of f likely depends on porous medium structure. In a heterogeneous, poorly connected medium, such as a vuggy limestone, velocity contrasts in different regions of the pore space might be expected to increase with increasing velocity, resulting in a decrease in the flowing fraction. On the other extreme, in a relatively homogeneous porous medium an increase in velocity may lead to better mixing and a lower stagnant fraction. In two-phase systems, the flowing fraction of a phase is expressed as a fraction of the phase saturation. Several investigators have found the flowing portion of one phase to decrease when the saturation of that phase decreases (Stalkup, 1970; Shuler, 1978; Batycky et al., 1982), while others found no correlation between flowing fraction and saturation (Ramirez et al., 1980). The most comprehensive study on this aspect of the capacitance model has been conducted by Salter and Mohanty (1982), who found that the flowing fraction of a non-wetting phase increases as its saturation increases, this phase becoming increasingly more connected. By contrast, the flowing fraction in the wetting phase goes through a minimum when the wetting phase saturation increases. The reason for this may be a transition of wetting phase flow through fine pores and films surrounding the grains, to simultaneous flow of wetting and non-wetting phases, and finally to flow of the wetting phase through the larger flow channels. These transitions between flow regimes parallel changes from irreducible wetting phase saturation at one extreme to residual non-wetting phase saturation at the other.
Scaling Capacitance Model Parameters to the Field In improved oil recovery by miscible displacement, a large dispersion coefficient and the presence of capacitance increase the length of the mixing zone, thereby contributing to slug degradation and low oil recovery efficiency. While the capacitance model has been used to gain a better understanding of mass transport processes in miscible displacements, it is not altogether clear how parameters determined in the laboratory should be scaled to the field. It has been suggested that the capacitance model simply provides a convenient mathematical tool which allows improved estimates of dispersion coefficients from laboratory experiments by correcting for laboratory restrictions. Scaling of the model parameters to field conditions is not straightforward (Coats and Smith, 1964; Stalkup, 1970; Brigham, 1974; Baker, 1977; Shuler, 1978; Bretz et al., 1986; Bretz and Orr, 1987).
329
Mixing due to capacitance depends on the Stanton number, St=KL/v. On the field scale, L is very much larger and v is frequently smaller than in the laboratory. Mass transfer is thus instantaneous on the field scale, and the effects of capacitance observed in the laboratory are reduced or absent in the field. However, it is still necessary to take into account capacitance when modeling a laboratory displacement in order to determine the proper dispersion coefficient to use in field calculations. The length of a mixing zone in a laboratory core with capacitance depends both on dispersion and on mass transfer into the stagnant volume. If both mixing effects are lumped into the dispersion coefficient, the magnitude of the dispersion coefficient, and therefore the length of the mixing zone and the slug size required in the field, can be grossly overestimated. Capacitance may thus not be important in the field, but is important in obtaining meaningful parameters from laboratory experiments. The choice of boundary conditions for the capacitance model is also important if parameters are to be scaled to the field (Brigham, 1974).
The Convection-Dispersion Model with Adsorption The convection-dispersion equation has frequently been used with a sink or source term to account for solute adsorption and desorption at the solid/liquid interface. Adsorption during flow through porous media is of interest in a number of disciplines, such as adsorptive separation processes, chromatography, soil science, and improved oil recovery. A large body of literature on the subject exists. The term “adsorption” refers to physical (not chemical) bonding. If a solute has a higher affinity for the solid in a porous medium than for the bulk fluid, the concentration of solute near the solid surface will be higher than in the bulk fluid. Adsorption is an equilibrium process. Solute molecules are constantly exchanged between bulk and adsorbed phases, but, at equilibrium, the average concentration of solute in the bulk and adsorbed phases is constant. The most thermodynamically consistent approach to describing adsorption is the Gibbs surface excess (Schay, 1969; Sircar and Myers, 1970, 1971; Sircar et al., 1972; Larionov and Myers, 1971; Everett, 1973; Chattoraj and Birdi, 1984; Sircar, 1985). The surface excess is defined as the excess of solute near an interface over the amount that would be present if the interface had no preference for the solute. The surface excess concept can be applied to liquids and gases, and to different kinds of interfaces (solid/liquid, solid/gas, liquid/liquid, liquid/gas). The surface excess concept has been incorporated into the convection-dispersion model by Huang and Novosad (1986) and Mannhardt and Novosad (1988, 1990). Adsorption depends on fluid phase composition (or gas pressure) and on temperature. The dependence of adsorption on concentration (or pressure), at constant temperature, is given by an adsorption isotherm. A large number of adsorption isotherms for gases and liquids on solid surfaces is available in the literature. A discussion of some of these can be found in Adamson's (1990) text on the physical chemistry of surfaces. The choice of adsorption isotherm and adsorption kinetics is obviously systemspecific and should reflect the interfacial chemistry and physics of the system of interest. However, the choice has often been empirical and based on considerations of simplicity or fitting of experimental data. Only one simple example of a commonly used adsorption model, the Langmuir model, will be discussed here. A large number of adsorption models that have appeared in the petroleum literature are summarized in a review by Mannhardt and Nasr-El-Din (1994).
330
A second order reversible rate expression that has been widely used is given by Langmuir ratecontrolled adsorption, also sometimes referred to as "bilinear adsorption kinetics:" 5-95) where q is amount adsorbed, qmax is the maximum adsorptive capacity of the adsorbent, and k1 and k2 are rate constants of adsorption and desorption, respectively. At equilibrium, q / t 0 , and Equation 5-95) becomes: ⁄ ⁄
(5-96)
Where, ⁄
,
⁄
Amount absorbed
and a and b are the Langmuir constants. Equation (5-96) is the Langmuir adsorption isotherm. Its typical shape is shown in Figure 5-64. At low concentration, qac, and the isotherm becomes nearly linear. At high concentration, the isotherm asymptotically approaches qmax.
Concentration Figure 5-64 Typical shape of a Langmuir adsorption isotherm.
The Langmuir equilibrium isotherm has been widely used for adsorption at gas/solid and liquid/solid interfaces because it describes the general shape of many adsorption isotherms adequately. While it contains too many simplifying assumptions to represent many real adsorption systems, it has retained great utility because of its simplicity. Many adsorption processes from gases and liquids can be modeled by the Langmuir equation despite the fact that the simplifying assumptions made in its derivation are rarely met. With adsorption, the one-dimensional convection-dispersion equation becomes: (5-97) Where, amount adsorbed (mass of solute per unit mass of solid), 331
porosity, solid (grain) density (mass of solid per unit volume). In addition to initial and boundary conditions, a condition specifying the initial value of adsorption is required; often q=0 at t=0. An adsorption model, such as the Langmuir model, is also required to describe q in eq. (5-97). When adsorption kinetics are important, adsorption is modelled as a dynamic process by substituting eq. 5-95) (or alternative models) for q / t in eq. (5-97). When adsorption is fast compared to convective mass transfer, adsorption equilibrium can be assumed. Then, (5-98) Substituting Equation (5-98) into Equation (5-97) gives: [
]
(5-99)
where dq/dc is the slope of the adsorption isotherm. The factor multiplying the time derivative is frequently referred to as "retardation factor," since it is a measure of the rate of propagation of an adsorbing versus a non-adsorbing solute. Any adsorption isotherm can be substituted for dq/dc. Using again the Langmuir adsorption isotherm (eq. (5-96)) as an example, eq. (5-99) becomes: [
]
(5-100)
Adsorption models with Langmuir rate-controlled or equilibrium adsorption constitute non-linear partial differential equations. No analytical solutions are available, and numerical solutions have to be resorted to. However, at low concentration, the Langmuir adsorption isotherm becomes nearly linear. When q=ac, the adsorption model can be solved analytically with boundary conditions for infinite, semi-infinite and finite domains (Gershon and Nir, 1969; Satter et al.; 1980, van Genuchten and Alves, 1982). Transport of surfactants in petroleum reservoir rocks or of pollutants in ground water has frequently been modelled using Langmuir adsorption, for example by Gupta and Greenkorn (1973), Trogus et al. (1977), Shuler (1978), Satter et al. (1980), Ramirez et al. (1980), Ziegler and Handy (1981), Foulser et al. (1989), Novosad et al. (1986), and Huang and Novosad (1986). The shape of a concentration profile for an adsorbing system is determined by the relative magnitudes of the rates of dispersion, convection, adsorption, and desorption, and by the adsorptive capacity of the rock. The effect of these groups on effluent profiles has been discussed in detail by Satter et al. (1980) using Langmuir adsorption. Examples of concentration profiles for a slug of an adsorbing solute injected into a porous medium that is fully saturated with solvent, obtained using the model of Mannhardt and Novosad (1988), are shown in Figure 5-65.
332
Figure 5-65 Effect of adsorption model parameters on adsorbate effluent concentrations. (a) Effect of adsorptive capacity. (b) Effect of rate of adsorption. (c) Effect of rate of desorption.
Increasing the adsorptive capacity of the solid delays breakthrough of the adsorbing solute at the porous medium outlet. When the rate of adsorption is high compared to convective mass transfer, equilibrium is reached at the solid surface. The leading edge of the concentration profile is similar in shape to the zero adsorption (tracer) case, but is displaced towards higher injection volumes, as some of the solute is held back on the solid surfaces. When the rate of adsorption is decreased, solute breakthrough occurs earlier and the profile becomes skewed. At very low rates of adsorption, solute breakthrough almost
333
coincides with tracer breakthrough, and the profile again becomes steep, but the solute concentration becomes nearly constant at a value lower than the injected concentration. The kinetics of desorption mirror the kinetics of adsorption, at the trailing edge of the solute slug. The concentrations now reflect injection of solvent into a porous medium that is fully saturated with solute. When desorption is very slow, tracer and solute are produced almost simultaneously. As the rate of desorption increases, solute is eluted for longer periods of time than tracer, as the adsorbed solute is released from the solid surfaces. At high rates of desorption, solute is eluted quickly until, at very high desorption rates, all the adsorbed solute is recovered in the effluent over a short injection volume. The validity of the adsorption model with Langmuir rate-controlled or equilibrium adsorption (or any other adsorption term) can be tested by comparing predicted to experimental effluent concentrations of adsorbing chemicals from porous media, as has frequently been done with surfactant solutions used in improved oil recovery. The model parameters, which include the dispersion coefficient, Langmuir constants (a and b), and rate constants of adsorption and desorption, can be determined by fitting experimental effluent data. Alternatively, some of the parameters may be measured independently. For example, the dispersion coefficient may be found from tracer data (Foulser et al., 1989), or by measuring surfactant dispersion in a porous medium whose surfaces have been pre-saturated with surfactant (Friedman, 1976; Shuler, 1978). The Langmuir constants may be obtained from adsorption isotherms measured statically in test tubes, or by performing material balance calculations in dynamic test runs at different input concentrations (Trogus et al., 1977; Ramirez et al., 1980; Ziegler and Handy, 1981). This would leave only the rate constants as adjustable parameters for fitting of adsorbate effluent profiles. An adsorption isotherm such as eq. (5-96) describes the dependence of adsorption on concentration at equilibrium. The dependence of adsorption on time at a fixed concentration can be obtained by solving the appropriate rate equation. With initial condition q=q0, solving eq. 5-95) gives the time dependence of adsorption assuming Langmuir kinetics, as follows:
q
k1q max c k q c q 0 1 max e ( k1c k 2 ) t k1c k 2 k1c k 2
(5-101)
Whether adsorption equilibrium can be assumed (eqs (5-99) or (5-100)) or the kinetics of adsorption need to be taken into account (eq. (5-97)) depends on the relative rates of adsorption and convective mass transport.
Some Other Forms of the Convection-Dispersion Model These notes have discussed some examples of models used to describe various mass transfer processes during miscible displacement in porous media. Other mechanisms, or combinations of mechanisms, can be encountered and described by similar models. Some of these mechanisms include -
chemical reaction, precipitation, ion exchange, inaccessible pore volume (capacitance with no mass transfer into stagnant pore space), and 334
-
partitioning of species between immiscible phases.
Laboratory core floods are commonly used to provide information about these mechanisms. The effluent profiles from core floods can provide a “finger print” of the porous medium and its interaction with the fluids flowing through it. Thus, a core flood can be designed to yield information on -
the porous medium (dispersion, heterogeneity, capacitance, ion exchange, adsorption, inaccessible pore volume), the fluids, solutes, or chemical species flowing through the porous medium (dispersion, adsorption, reaction, partitioning, precipitation, ion exchange), and the distribution of immiscible phases in the porous medium (wetting, connectivity, phase saturation, trapping).
335
