Basics Bas ics of of Reser Reservoi voirr Engin Engineer eering ing – Mod Module ule I
I.1.B I.1.B – Reservoir Reservoir Fundament Fundamentals als of of Fluid Fluid Flow Flow
Porosity
Porosity of a measure of the void void spac spacee within within a rock rock Range or Typical Values: 30%, unconsolidated well-sorted sandstone 20%, clean, well-sorted consolidated sandstone 8%, low permeability reservoir rock 0.5%, natural fracture porosity
Porosity
Porosity of a measure of the void void spac spacee within within a rock rock Range or Typical Values: 30%, unconsolidated well-sorted sandstone 20%, clean, well-sorted consolidated sandstone 8%, low permeability reservoir rock 0.5%, natural fracture porosity
Reservoir Make-up
Rock matrix Pore space •
Fluids: Water, Oil and gas
Rock Matrix and Pore Space
Rock matrix
Pore space
Rock Matrix and Pore Space
Rock matrix
Water
Oil and/or gas
Porosity Definition
Porosity: The fractional void space within a rock that is available for the storage of fluids
Porosity = φ =
V p V b
=
V b − V ma V b
Effect Grain Size and Packing Cubic Packing of Spheres Porosity = 48%
Porosity Calculations – Cubic/Uniform Spheres
Calculations for Cubic Packing Bulk volume = (2r)3 = 8r 3 3
Matrix volume =
4 π r 3
Pore volume = bulk volume - matrix volume
Porosity Calculations – Cubic/Uniform Spheres
Porosity = =
Pore Volume Bulk Volume Bulk Volume − Matrix Volume Bulk Volume 3
=
8 r
3
− 4 / 3 π r 3
8 r
=1−
π
()
2 3
= 47.6%
Effect Grain Size and Packing Rhombic Packing of Spheres Porosity = 27 %
Effect Grain Size and Packing Packing of Two Sizes of Spheres Porosity = 14%
Pore-Space Classification
Total porosity, φt =
Total Pore Space Bulk Volume Effective porosity, φe =
Interconnected Pore Space Bulk Volume
Comparison of Total and Effective Porosities
Very clean sandstones : φt = φe Poorly to moderately well -cemented intergranular materials: φt ≈ φe Highly cemented materials and most carbonates: φe < φt
Factors That Affect Porosity
• Particle shape • Packing • Particle sizes • Cementing materials • Overburden stress • Vugs and fractures
Example Porosity/Overburden Pressure Relationship
50
40
Sandstones
% , 30 y ti s r o o 20 P
Shales 10
0
0
1,000
2,000
3,000
4,000
5,000
6,000
Permeability
Permeability is a measure of the rock’s ability to transmit fluids
q
p1
A
k =
qµ L A∆ p L
q
p2
Permeability Values
The quality of the reservoir, as it relates to permeability, can be classified as follows: k < 1 md 1 < k < 10 md 10 < k < 50 md 50 < k < 250 md 250 md < k
poor fair moderate good very good
Example Permeability-Porosity Relationship
Factors affecting permeability l a 1.0 n i g i r o f .8 o n o i t c .6 a r f : y t i .4 l i b a e m r .2 e P
A
Well cemented A B
Friable Unconsolidated
C
.0 0
2000
4000
6000
8000
Net overburden pressure: psi
10000
Scales of Geological Reservoir Heterogeneity
e Determined d i From Well Logs, W Seismic Lines, d Statistical l e i Modeling, F etc.
Interwell Area
Well
Well 100's m
1-10 km
l l e w r e t n I
Reservoir Sandstone
10's m
100's m
e r o B l l e W
10-100's m
10-100's mm
Hand Lens or Petrographic or Binocular Microscope Scanning Electron Microscope
1-10's m
Unaided Eye
(modified from Weber, 1986)
Scales of Investigation Used in Reservoir Characterization Relative Volume
300 m
Gigascopic
50 m
300 m
Megascopic
5m
150 m
2m 1m
Macroscopic
cm
mm - m
Microscopic (modified from Hurst, 1993)
Well Test
14
10
Reservoir Model 12 2 x 10 Grid Cell
Wireline Log Interval Core Plug Geological Thin Section
3 x 10 5 x 10 1
7 2
Permeability Exercise 1
What are the “units” of permeability?
Use Dimensional Analysis:
k =
qµ L A∆ p
Permeability Exercise 2
Relate the permeability concept to other common fluid flow situations: laminar fluid flow through a pipe and through parallel plates (“fractures”). • What is the permeability of a circular opening (“vug”) of 0.005 inches? • What is the permeability of a fracture of 0.01 in thickness?
Saturations H2O
Fluid Saturations
Grain
Water
Gas
Oil
Definition of Fluid Saturation
Water saturation:
Oil saturation:
Gas saturation:
S w = S o =
V w V p V o V p
S g = 1.0 − S o − S w
Net Pay Thickness
Shale h1
Sand
h2
h3
h = h1 + h2 + h3
Rock Wettability
Wettability: Tendency of one fluid to spread on or adhere to a solid surface in the presence of other immiscible fluids •Wettability refers to interaction between fluid and solid phases ow ow
Water os
ws
Oil Solid
Water Oil
os
ws
Solid surface is reservoir rock (i.e., sandstone, limestone, dolomite or mixtures of each) -- Fluids are oil, water, and/or gas
Solid
Interfacial Tension and Adhesion Tension
Interfacial tension is the force per unit length required to create a new surface •
Interfacial tension is commonly expressed in Newtons/meter or dynes/cm
Adhesion tension can be expressed as the difference between two solid-fluid interfacial tensions
AT
σ os
σ ws
σ ow
cos θ
Contact Angle
ow
Oil
Water
Oil
os
ws
Solid
Oil
Wetting Phase Fluid A wetting phase preferentially wets the solid rock surface Because of attractive forces between rock and fluid, the wetting phase is drawn into smaller pore spaces of porous media Wetting phase fluid often is not very mobile Attractive forces prohibit reduction in wetting phase saturation below some irreducible value (called irreducible wetting phase saturation) Many hydrocarbon reservoirs tend to be either totally or partially water wet
Nonwetting Phase Fluid A nonwetting phase does not preferentially wet the solid rock surface Repulsive forces between rock and fluid cause nonwetting phase to occupy largest pore spaces of porous media Nonwetting phase fluid is often the most mobile fluid, especially at large nonwetting phase saturations Natural gas is never the wetting phase in hydrocarbon reservoirs
Water-Wet Reservoir Rock Reservoir rock is considered to be water-wet if water preferentially wets the rock surfaces The rock is water-wet under the following conditions: σws > σos
AT < 0 (i.e., the adhesion tension is negative) 0° < θ < 90° If θ is close to 0°, the rock is considered to be “strongly water-wet”
Force Balance - Water-Wet Rock
ow
Oil Water os
Solid
ws
Note: 0 <
< 90
Oil-Wet Reservoir Rock
Reservoir rock is considered to be oil-wet if oil preferentially wets the rock surfaces The rock is oil-wet under the following conditions: σos > σws
AT > 0 (i.e., the adhesion tension is positive) 90° < θ < 180° If θ is close to 180°, the rock is considered to be “strongly oil-wet”
Force Balance - Oil-Wet Rock
ow
Water Oil
os
Note: 90 <
ws
< 180
Solid
Implications of Wettability Wettability affects the shape of the relative permeability curves. •
Oil moves easier in water-wet rocks than oil-wet rocks.
Primary oil recovery is affected by the wettability. •
A water-wet system will exhibit greater primary oil recovery.
Oil recovery under waterflooding is affected by wettability •
A water-wet system will exhibit greater oil recovery under waterflooding.
Implications of Wettability
Core Percent no silicone Wettability
i o
S , t n e c r e p , y c n e i c i f f e y r e v o c e R
1 2 3 4 5
80 1
2
60
3
0.00 0.649 0.0200 0.176 0.200 - 0.222 2.00 - 0.250 - 0.333 1.00
Curves cut off at Fwd •100
4 40
5
20 0
1
2
3
4
5
6
7
8
Water injected, pore volumes
9
10
11
12
Implications of Wettability
Squirr Squirrel el oil - 0.10 0.10 N NaCl NaCl - Torpe Torpedo do core core ( • 33 O W • 663, K • 0945, Swi • 21.2 21.20% 0%))
i p
S t n e c r e p , y80 c n e i c60 i f f e y r 40 e v o c e20 R 0
Squirrel Squirrel oil - 0.10 N NaCl • Torpedo Torpedo Sands Sandstone tone core, core, after remaining in oil for 84 days ( • 33.0 W • 663, K • 0.925, Swi • 23.2 23.28% 8%))
1
2
3
4
5
6
7
Water injection, pore volumes
8
9
10
Capillary Pressure Definition Capillary Pressure is the pressure difference existing across the interface separating two immiscible fluids. It is usually calculated as: P c = pnwt - pwt ow
Po, Oil Po, Oil
ow
Pw, Water
Pw, Water os
ws
os
ws
Capillary Example 1
Define capillary pressure in the following systems: •
Water-gas system
•
Water-wet water-oil system
•
Oil-gas system
Capillary Tube Model. Air-Water System
Air h
Water
Capillary Tube Model. Air-Water System The height of water is a function of •
The adhesion tension between the air and water
•
The radius of the tube
•
The density difference between fluids
∆h =
2 σ aw cos θ
r g ∆ ρ aw
Capillary Tube Model. Air-Water System
pa1 h
pw1
Air
pa2 pw2 Water
Capillary Pressure. Air-Water System
Combining the two relations results in the following expression :
P c =
2 σ aw cos θ
r
Capillary Pressure. Oil-Water System
From a similar derivation, the equation for capillary pressure for an oil/water system is
P c =
2 σ ow cos θ
r
Imbibition and Drainage & Capillary Pressure
Imbibition: Fluid flow process in which the saturation of the wetting phase increases and the nonwetting phase saturation decreases Mobility of wetting phase increases as wetting phase saturation increases Drainage: Fluid flow process in which the saturation of the nonwetting phase increases Mobility of nonwetting fluid phase increases as nonwetting phase saturation increases
Typical Drainage and Imbibition Capillary Pressure Curves
The pc-drainage curve is always higher than the pcimbibition curve.
Drainage (1) Imbibition (2)
P c
P d
S i
0
S m
0.5 S
1.0
Capillarity Exercise 2 50 45 40 35 30
(1)
P c , psi 25
(2)
20 15 10 5 0
0
0.25
0.5 S w
0.75
1.0
Converting Laboratory Capillary Pressure Data to Reservoir Conditions Based on our previous derivation, we use the following basic equations:
P cL = P cR =
2 σ L cos θ L
r L 2 σ R cos θ R
r R
Setting r L = r R and combining equations yields r L = r R →
2 σ L cos θ L
P cL
=
2 σ R cos θ R
P cR
Capillary pressure at reservoir conditions
P cR =
σ R
cos θ R
σ L
cos θ L
P cL
Capillarity Example 3 Converting Laboratory Capillary Pressure Data to Reservoir Conditions 2000
a i s 1600 p , e r u 1200 s s e r P n 800 o i t c e 400 j n I 0 80
60
40
20
Mercury Saturation, percent pore space
0
Example 3 Solution
Mercury Saturation (S Hg ) % 70 60 50 40 30 20 10
P cL psia
P cR psia
1,320 820 560 410 310 240 200
80.5 50.0 34.2 25.0 18.9 14.6 12.2
Effects of Reservoir Properties on Capillary Pressure
Capillary pressure characteristics in reservoir are affected by •
Variations in permeability
•
Grain size distribution
•
Saturation history
•
Contact angle
•
Interfacial tension
•
Density difference between fluids
Effect of Permeability 20 Decreasing Permeability 16
e r u s s e r P y r a l l i p a C
12 C B
A
8
4
0 0
0.2
0.4
0.6
Water Saturation
0.8
1.0
Effect of Grain Size Distribution
a i s p , e r u s s e r p y r a l l i p a C
Well-sorted
Poorly sorted
Effect of Saturation History
a i s p , e r u s s e r p y r a l l i p a C
Drainage
Imbibition
Water saturation, %
Effect of Contact Angle
20
16
e r u s s 12 e r P y r a 8 l l i p a C
Decreasing
R R
=0
= 30 R
4
R
= 60
R
= 80
0 0
0.2
0.4
0.6
Water Saturation
0.8
1.0
Effect of Interfacial Tension
l e v e L r e t a W e e r F e v o b A t h g i e H
High Tension
Low Tension
0
1.0
Water Saturation
Effect of Density Difference
l e v e L r e t a W e e r F e v o b A t h g i e H
Small Density Difference
Large Density Difference 0
1.0
Water Saturation
Uses of Capillary Pressure Data
• Determine initial water saturation in the reservoir • Determine fluid distribution in reservoir • Determine residual oil saturation for water flooding applications • Determine pore size distribution index • May help in identifying zones or rock types • Input for reservoir simulation calculations.
Capillary Pressure Data Using the Leverett J-Function A universal capillary pressure curve is impossible to generate because of the variation of properties affecting capillary pressures in reservoir The Leverett J-function was developed in an attempt to convert all capillary pressure data to a universal curve
J (S w ) =
0.22 P c
k
σ cos θ φ
Example J-Function for West Texas Carbonate 10.00
9.00 Jc Jmatch 8.00
Jn1 Jn2
7.00
Jn3
6.00 n o i t c n u f J
5.00
4.00
3.00
2.00
1.00
0.00 0.00
0.10
0.20
0.30
0.40
0.50
0.60
Water saturation, fraction
0.70
0.80
0.90
1.00
Use of Leverett J-Function J-function is useful for averaging capillary pressure data from a given rock type from a given reservoir. J-function can sometimes be extended to different reservoirs having same lithologies. J-function usually does not predict an accurate correlation for different lithologies. If J-functions are not successful in reducing the scatter in a given set of data, then this suggests that we are dealing with different rock types.
Capillarity Example 4 Calculation of J-function
A v e r a g e d A ir /B r in e C a p illa r y P re s s u re D a ta P c S w psia % 1 9 8 .3 2 9 8 .3 4 9 6 .8 8 5 9 .0 15 3 6 .3 35 2 5 .4 500 1 5 .3
Capillarity Example 4 Solution
Calculated J-Functions 0.22* S w J(S w ) % 98.3 0.22 98.3 0.43 96.8 0.86 59.0 1.73 36.3 3.24 25.4 7.57 15.3 108.1
Capillarity Example 4 Solution 120 100 ) w 80 S ( J * 60 2 2 . 0 40 20 0 0
20
60 40 Sw , %
80
100
Capillarity Example 5 Estimating P c from J-function
Estimate capillary pressures from Leverett J-function calculated in Example 4 for a different core sample. Properties of core sample: k = 100 md φ = 10 %
Capillarity Example 5 Solution
Estimated Capillary Pressures for the 100-md Permeability Core Sample Sw , P c, % psia 98.3 2.27 98.3 4.45 96.8 8.91 59.0 17.91 36.3 36.90 25.4 78.18 15.3 1118.63
Intro to the Relative Permeability Concept
Permeability is a property of the porous medium and is a measure of the capacity of the medium to transmit fluids When the medium is completely saturated with one fluid, then the permeability measurement is often referred to as absolute permeability
Calculating Absolute Permeability
Absolute permeability is often calculated from the flow equation:
q=
k A ∆ p µ L
Effective Permeability When the rock pore spaces contain more than one fluid, then the permeability to a particular fluid is called the effective permeability Effective permeability is a measure of the fluid conductance capacity of a porous medium to a particular fluid when the medium is saturated with more than one fluid
Calculating Effective Permeability
Oil
Water
Gas
qo =
qw =
q g =
k eo A ∆ po µ o L k ew A ∆ pw µ w L k eg A ∆ p g µ g L
Relative Permeability
Relative permeability is defined as the ratio of the effective permeability to a fluid at a given saturation to a base permeability The base permeability is commonly taken as the effective permeability to the fluid at 100% saturation (absolute permeability) or the effective non-wetting phase permeability at irreducible wetting phase saturation
Calculating Relative Permeabilities
Oil
Water
Gas
k ro =
k rw =
k rg =
k eo k k ew k
k eg k
Fundamental Concepts Water phase •
Water is located in smaller pore spaces and along sand grains
•
Therefore, relative permeability to water is a function of water saturation only (i.e., it does not matter what the relative amounts of oil and gas are)
•
Thus, we can plot relative permeability to water against water saturation on Cartesian coordinate paper
Fundamental Concepts Oil phase •
Oil is located between water and gas in the pore spaces, and to a certain extent, in the smaller pores
•
Thus, relative permeability to oil is a function of oil, water, and gas saturations
•
If the water saturation can be considered constant (i.e., the minimum interstitial water saturation), then k ro can be plotted against So on Cartesian coordinate paper
Fundamental Concepts Gas phase •
Gas is located in the center of the larger pores
•
Therefore, the relative permeability to gas is a function of gas saturation only (i.e., it does not matter what the relative amounts of oil and water are)
•
Thus ,we can plot k rg against Sg (or Sw + So) on Cartesian coordinate paper
Common Multi-Phase Flow Systems
Water-oil systems Oil-gas systems Water-gas systems Three phase systems (water, oil, and gas)
Relative Permeability Exercise 1
What are the relative permeability data sets we need to use for the following situations? •
Water flooding an oil reservoir above the bubble point
•
Production from an oil reservoir with a gas-cap and water aquifer
Relative Permeability Exercise 1 Solution For water flooding an oil reservoir above the bubble point : •
Water-oil relative permeability
For three phase flow : •
Water-oil relative permeability
•
Gas-oil (or gas-liquid) relative permeability
•
3 phase relative permeability
Oil-Water Relative Permeability
100
k ro @ S wi
) % ( 80 Two-Phase Flow y Residual Oil Region t i l Saturation i b a 60 e m r Oil e P 40 Irreducible e v Water i t a Saturation l 20 e R k rw @ S or Water
0
0
20
40
60
80
Water Saturation (%)
100
Oil-Gas Relative Permeability 100
) % ( 80 y t i l i b a 60 e m r e P 40 e v i t a l e 20 R 0
Oil
k rg k ro
Gas
0
20
40
60
80
100
Total Liquid Saturation - % of Pore Volume S L = S o + S wi
Relative Permeability Exercise 2
1.0
n o i t 0.8 c a r F , y t i l i 0.6 b a e m r 0.4 e P e v i t a 0.2 l e R
(1)
(2)
0 0
20
40
60
80
Water Saturation (% PV)
100
Importance of Relative Permeability Data
Relative permeability data affect the flow characteristics of reservoir fluids. Relative permeability data affect the recovery of oil and/or gas.
Relative Permeability Example 3 Effect of Relative Permeability 100
) % ( y 80 t i l i b a 60 e m r e 40 P e v i t 20 a l e R
Rock Type 2 Rock Type 1
0 0
20
40
60
Water Saturation (%)
80
100
Relative Permeability Example 3 Effect of Relative Permeability l i100 O e l b 80 a r e v 60 o c e R40 f o t n e 20 c r e P 0
Rock Type 1 Rock Type 2 0
2
4
6
8
Pore Volumes Injected
10
Factors Affecting Effective and Relative Permeabilities Fluid saturations Geometry of the rock pore spaces and grain size distribution Rock wettability Fluid saturation history (i.e., imbibition or drainage)
Effect of Wettability
1.0
1.0
n o i t c a r 0.8 F , y t i l i 0.6 b a e m r 0.4 e P e v i t 0.2 a l e R 0
n o i t c a r 0.8 F , y t i l i 0.6 b a e m r 0.4 e P e v i t 0.2 a l e R
Oil
Water 0
20
40
60
80
100
0
Oil
0
20
40
Water
60
80
Water Saturation (% PV)
Water Saturation (% PV)
Strongly Water-Wet Rock
Strongly Oil-Wet Rock
100
Effect of Saturation History
Types of relative permeability curves •
Drainage curve Wetting phase is displaced by the nonwetting phase, i.e., the wetting phase saturation is decreasing
•
Imbibition Curve Non-wetting phase is displaced by wetting phase, i.e., the wetting phase saturation is increasing
Effect of Saturation History
100
% 80 , y t i l i b 60 a e m r e P 40 e v i t a l e 20 R
Imbibition Drainage
Interstitial wetting phase saturation
Residual non-wetting phase saturation
0 0
20
40
60
80
Wetting Phase Saturation, % PV
100
Choosing the Right Curve
When simulating the waterflood of a water-wet reservoir rock, imbibition relative permeability curves should be used. When modeling gas injection into an oil reservoir, drainage relative permeability curves should be used.
Three-Phase Relative Permeabilities
100% Gas
100% Water
100% Oil
Relative Permeability to Water in a Three-Phase System 100% Gas
0% 10% 20% 40% 60% k rw = 80% 100% Water
100% Oil
Relative Permeability to Oil in a Three-Phase System
100% Gas
5% 10% 20% 30% 40% k ro = 50%
100% Oil
Uses of Effective and Relative Permeability
Reservoir simulation Flow calculations that involve multi-phase flow in reservoirs Estimation of residual oil (and/or gas) saturation
