Bourdet - Well testing and interpretation.pdf

WELL TESTING AND INTERPRETATION D. Bourdet

CONTENTS

Pages 1 - PRINCIPLES OF TRANSIENT TESTING..................................................................................... 1 1-1 1-2

INTRODUCTION ........................................................................................................................... 1 DEFINITIONS & TYPICAL REGIMES ................................................................................................7

2 - THE ANALYSIS METHODS ......................................................................................................... 27 2-1 2-2 2-3 2-4

LOG-LOG SCALE ........................................................................................................................ 27 PRESSURE CURVES ANALYSIS ................................................................................................... 28 PRESSURE DERIVATIVE ............................................................................................................. 37 THE ANALYSIS SCALES ...............................................................................................................44

3 - WELLBORE CONDITIONS .......................................................................................................... 47 3-1 3-2 3-3 3-4 3-5 3-6

WELL WITH WELLBORE STORAGE AND SKIN, HOMOGENEOUS RESERVOIR ................................. 47 INFINITE CONDUCTIVITY OR UNIFORM FLUX VERTICAL FRACTURE ............................................ 48 FINITE CONDUCTIVITY VERTICAL FRACTURE ............................................................................. 50 WELL IN PARTIAL PENETRATION ............................................................................................... 53 HORIZONTAL WELL ................................................................................................................... 57 SKIN FACTORS............................................................................................................................71

4 - FISSURED RESERVOIRS - DOUBLE POROSITY MODELS.................................................. 75 4-1 4-2

DEFINITIONS ............................................................................................................................. 75 DOUBLE POROSITY BEHAVIOR, RESTRICTED INTERPOROSITY FLOW (PSEUDO-STEADY STATE INTERPOROSITY FLOW).......................................................................................................................... 77 4-3 DOUBLE POROSITY BEHAVIOR, UNRESTRICTED INTERPOROSITY FLOW (TRANSIENT INTERPOROSITY FLOW) ................................................................................................................................................. 85 4-4 COMPLEX FISSURED RESERVOIRS ...............................................................................................90 5 - BOUNDARY MODELS................................................................................................................... 95 5-1 5-2 5-3

ONE SEALING FAULT ................................................................................................................. 95 TWO PARALLEL SEALING FAULTS .............................................................................................. 97 TWO INTERSECTING SEALING FAULTS...................................................................................... 101

5-4 5-5 5-6 5-7

CLOSED SYSTEM ..................................................................................................................... 104 CONSTANT PRESSURE BOUNDARY ........................................................................................... 111 COMMUNICATING FAULT......................................................................................................... 113 PREDICTING DERIVATIVE SHAPES .............................................................................................117

6 - COMPOSITE RESERVOIR MODELS....................................................................................... 119 6-1 6-2 6-3 6-4

DEFINITIONS ........................................................................................................................... 119 RADIAL COMPOSITE BEHAVIOR ............................................................................................... 120 LINEAR COMPOSITE BEHAVIOR................................................................................................ 123 MULTICOMPOSITE SYSTEMS .....................................................................................................125

7 - LAYERED RESERVOIRS - DOUBLE PERMEABILITY MODEL........................................ 127 7-1 7-2 7-3

DEFINITIONS ........................................................................................................................... 127 DOUBLE PERMEABILITY BEHAVIOR WHEN THE TWO LAYERS ARE PRODUCING INTO THE WELL 129 DOUBLE PERMEABILITY BEHAVIOR WHEN ONLY ONE OF THE TWO LAYERS IS PRODUCING INTO THE WELL ............................................................................................................................................... 131 7-4 COMMINGLED SYSTEMS: LAYERED RESERVOIRS WITHOUT CROSSFLOW ...................................133 8 - INTERFERENCE TESTS ............................................................................................................. 135 8-1 8-2 8-3 8-4 8-5

INTERFERENCE TESTS IN RESERVOIRS WITH HOMOGENEOUS BEHAVIOR .................................. 135 INTERFERENCE TESTS IN DOUBLE POROSITY RESERVOIRS ....................................................... 139 INFLUENCE OF RESERVOIR BOUNDARIES ................................................................................. 143 INTERFERENCE TESTS IN RADIAL COMPOSITE RESERVOIR ........................................................ 143 INTERFERENCE TESTS IN A TWO LAYERS RESERVOIR WITH CROSS FLOW ..................................146

9 - GAS WELLS................................................................................................................................... 149 9-1 9-2 9-3

GAS PROPERTIES ..................................................................................................................... 149 TRANSIENT ANALYSIS OF GAS WELL TESTS .............................................................................. 150 DELIVERABILITY TESTS ............................................................................................................154

10 - BOUNDARIES IN HETEROGENEOUS RESERVOIRS ........................................................ 159 10-1 10-2 10-3

BOUNDARIES IN FISSURED RESERVOIRS............................................................................... 159 BOUNDARIES IN LAYERED RESERVOIRS ............................................................................... 160 COMPOSITE CHANNEL RESERVOIRS ......................................................................................162

11 - COMBINED RESERVOIR HETEROGENEITIES ................................................................. 165 11-1 11-2 11-3

FISSURED-LAYERED RESERVOIRS ........................................................................................ 165 FISSURED RADIAL COMPOSITE RESERVOIRS......................................................................... 166 LAYERED RADIAL COMPOSITE RESERVOIRS..........................................................................167

12 - OTHER TESTING METHODS.................................................................................................. 169 12-1 12-2 12-3 12-4 12-5

DRILLSTEM TEST ................................................................................................................. 169 IMPULSE TEST ..................................................................................................................... 172 RATE DECONVOLUTION ....................................................................................................... 173 CONSTANT PRESSURE TEST (RATE DECLINE ANALYSIS) ....................................................... 174 VERTICAL INTERFERENCE TEST ............................................................................................175

13 - MULTIPHASE RESERVOIRS .................................................................................................. 179 13-1 13-2

PERRINE METHOD ............................................................................................................... 179 OTHER METHODS .................................................................................................................180

14 - TEST DESIGN ............................................................................................................................. 183 14-1 14-2 14-3

INTRODUCTION ................................................................................................................... 183 TEST SIMULATION ............................................................................................................... 183 TEST DESIGN REPORTING AND TEST SUPERVISION ................................................................184

15 - FACTORS COMPLICATING WELL TEST ANALYSIS....................................................... 185 15-1 15-2 15-3 15-4 15-5 15-6 15-7

RATE HISTORY DEFINITION .................................................................................................. 185 ERROR OF START OF THE PERIOD......................................................................................... 186 PRESSURE GAUGE DRIFT ..................................................................................................... 188 PRESSURE GAUGE NOISE ..................................................................................................... 188 CHANGING WELLBORE STORAGE ......................................................................................... 189 TWO PHASES LIQUID LEVEL ................................................................................................. 190 INPUT PARAMETERS, AND CALCULATED RESULTS OF INTERPRETATION ................................191

16 - CONCLUSION ............................................................................................................................. 193 16-1 16-2

INTERPRETATION PROCEDURE ............................................................................................ 193 REPORTING AND PRESENTATION OF RESULTS .......................................................................203

APPENDIX - ANALYTICAL SOLUTIONS..................................................................................... 205 A-1 A-2 A-3 A-4

DARCY'S LAW ......................................................................................................................... 205 STEADY STATE RADIAL FLOW OF AN INCOMPRESSIBLE FLUID .................................................. 205 DIFFUSIVITY EQUATION........................................................................................................... 206 THE "LINE SOURCE" SOLUTION ................................................................................................208

NOMENCLATURE............................................................................................................................. 209 REFERENCES..................................................................................................................................... 212

Most figures presented in this set of course notes are extracted from "Well Test Analysis: The Use of Advanced Interpretation Models", D. Bourdet, Handbook of Petroleum Exploration and Production 3, ELSEVIER SCIENCE, 2002. http://www.elsevier.com/locate/inca/628241

1 - PRINCIPLES OF TRANSIENT TESTING

1-1 Introduction 1-1.1 Purpose of well testing Description of a well test

During a well test, a transient pressure response is created by a temporary change in production rate. The well response is usually monitored during a relatively short period of time compared to the life of the reservoir, depending upon the test objectives. For well evaluation, tests are frequently achieved in less than two days. In the case of reservoir limit testing, several months of pressure data may be needed. In most cases, the flow rate is measured at surface while the pressure is recorded down-hole. Before opening, the initial pressure pi is constant and uniform in the reservoir. During flow time, the drawdown pressure response ∆p is expressed :

∆p = pi − p (t ) (psi, Bars)

( 1-1)

When the well is shut-in, the build-up pressure change ∆p is estimated from the last flowing pressure p(∆t=0) :

∆p= p(t)− p(∆t =0) (psi, Bars)

( 1-2)

Rate, q

Pressure, p

pi ∆t Dd ∆p BU

∆p Dd p(∆t=0)

drawdown

∆t BU

build-up Time, t

Figure 1-1 Drawdown and build-up test sequence.

The pressure response is analyzed versus the elapsed time ∆t since the start of the period (time of opening or shut-in).

Well test objectives

Well test analysis provides information on the reservoir and on the well. Associated to geology and geophysics, well test results are used to build a reservoir model for prediction of the field behavior and fluid recovery to different -1-

Chapter 1 - Principles of transient testing

operating scenarios. The quality of the communication between the well and the reservoir indicates the possibility to improve the well productivity. Exploration well : On initial wells, well testing is used to confirm the exploration hypothesis and to establish a first production forecast: nature and rate of produced fluids, initial pressure (RFT, MDT), reservoir properties. Appraisal well : The previous well and reservoir description can be refined (well productivity, bottom hole sampling, drainage mechanism, heterogeneities, reservoir boundaries etc.) Development well : On producing wells, periodic tests are made to adjust the reservoir description and to evaluate the need of a well treatment, such as workover, perforation strategy etc. Communication between wells (interference testing), monitoring of the average reservoir pressure are some usual objectives of development well testing.

Information obtained from well testing

Well test responses characterize the ability of the fluid to flow through the reservoir and to the well. Tests provide a description of the reservoir in dynamic conditions, as opposed to geological and log data. As the investigated reservoir volume is relatively large, the estimated parameters are average values. Reservoir description : • Permeability (horizontal k and vertical kv) • Reservoir heterogeneities (natural fractures, layering, change of characteristics) • Boundaries (distance and shape) • Pressure (initial pi and average p ) Well description : • Production potential (productivity index PI, skin factor S) • Well geometry By comparing the result of routine tests, changes of productivity and rate of decrease of the average reservoir pressure can be established.

1-1.2 Methodology The inverse problem

The objective of well test analysis is to describe an unknown system S (well + reservoir) by indirect measurements (O the pressure response to I a change of rate). This is a typical inverse problem (S=O/I).

-2-


I

S

input

system

O output

As opposed to the direct problem (O=IxS), the solution of the inverse problem is usually not unique. It implies an identification process, and the interpretation provides the model(s) whose behavior is identical to the behavior of the actual reservoir.

Interpretation models

The models used in well test interpretation can be described as a transfer function; they only define the behavior (homogeneous or heterogeneous, bounded or infinite). Well test interpretation models are often different from the geological or log models, due to the averaging of the reservoir properties. Layered reservoirs for example frequently show a homogeneous behavior during tests. Analytical solutions are used to generate pressure responses to a specific production rate history I, until the model behavior O is identical to the behavior of S.

Input data required for well test analysis

• Test data : flow rate (complete sequence of events, including any operational problem) and bottom hole pressure as a function of time. • Well data : wellbore radius rw, well geometry (inclined, horizontal etc.), depths (formation, gauges). • Reservoir and fluid parameters : formation thickness h (net), porosity φ, compressibility of oil co, water cw and formation cf, water saturation Sw, oil viscosity µ and formation volume factor B. The different compressibility's are used to define the total system compressibility ct :

ct =co(1−Sw)+cwSw+c f (psi-1, Bars-1)

( 1-3)

The reservoir and fluid parameters are used for calculation of the results. After the interpretation model has been selected, they may always be changed or adjusted if needed. Additional data can be useful in some cases : production log, gradient surveys, bubble point pressure etc. General information obtained from geologist and geophysicists are required to validate the well test interpretation results.

-3-


1-1.3 Types of tests Test procedure

• Drawdown test : the flowing bottom hole pressure is used for analysis. Ideally, the well should be producing at constant rate but in practice, drawdown data is erratic, and the analysis is frequently inaccurate. • Build-up test : the increase of bottom hole pressure after shut-in is used for analysis. Before the build-up test, the well must have been flowing long enough to reach stabilized rate. During shut-in periods, the flow rate is accurately controlled (zero). • Injection test / fall-off test : when fluid is injected into the reservoir, the bottom hole pressure increases and, after shut-in, it drops during the fall-off period. The properties of the injected fluid are in general different from that of the reservoir fluid. • Interference test and pulse test : the bottom hole pressure is monitored in a shut-in observation well some distance away from the producer. Interference tests are designed to evaluate communication between wells. With pulse tests, the active well is produced with a series of short flow / shut-in periods, the resulting pressure oscillations in the observation well are analyzed.

Rate, q

Pressure, p

• Gas well test : specific testing methods are used to evaluate the deliverability of gas wells (Absolute Open Flow Potential, AOFP) and the possibility of nonDarcy flow condition (rate dependent skin factor S'). The usual procedures are Back Pressure test (Flow after Flow), Isochronal and Modified Isochronal tests.

Initial shut-in Clean Variable up rate

Build-up Stabilized rate

Time, t

Figure 1.2 Typical test sequence. Oil well.

Well completion

• Production test : the well is completed as a production well (cased hole and permanent completion). • Drill stem test (DST) : the well is completed temporarily with a down-hole shut-in valve. Frequently the well is cased but DST can be made also in open -4-


hole. The drill stem testing procedure is used only for relatively short tests. The drill string is not used any more, and production tubing is employed.

Flowh ead B OP S tack

Casing

Tu bing Tes t tool P ack er

Figure 1.3 Onshore DST test string.

1-1.4 Well testing equipment Surface equipment

• Flow head : is equipped with several valves to allow flowing, pumping in the well, wire line operation etc. The wellhead working pressure should be greater than the well shut-in pressure. The Emergency Shut Down is a fail-safe system to close the wing valve remotely. • Choke manifold : is used to control the rate by flowing the well through a calibrated orifice. A system of twin valves allows to change the choke (positive and adjustable chokes) without shutting in the well. The downstream pressure must be less than half the upstream pressure. • Heater : Heating the effluent may be necessary to prevent hydrate formation in high-pressure gas wells (the temperature is reduced after the gas expansion through the choke). Heaters are also used in case of high viscosity oil. • Test separator : In a three phases test separator, the effluent hits several plates in order to separate the gas from the liquid phase. A mist extractor is located before the gas outlet. The oil and water phases are separated by gravity. The oil and water lines are equipped with positive displacement metering devices, the gas line with an orifice meter. Surface samples are taken at the separator oil and gas lines for further recombination in laboratory.

-5-


Flowhead

Burner

Choke maniflod Heater Gas

Rig HP pump

Gas manifold

Separator

Water

Air

pump

compressor

Water

Oil Oil manifold

Surge tank

Burner Transfer pump

Figure 1.4 Surface set up.

• Oil and gas disposal : The oil rate can be measured with a gauge tank (or a surge tank in case of H2S). Oil and gas are frequently burned. Onshore, a flare pit is installed at a safe distance from the well. Offshore, two burners are available on the rig for wind constraint. Compressed air and water are injected together with the hydrocarbon fluids to prevent black smoke production and oil drop out.

Downhole equipment

• Pressure gauges : Electronic gauges are used to measure the bottom hole pressure versus time. The gauge can be suspended down hole on a wireline, or hung off on a seating nipple. When they are not connected to the surface with a cable, the gauges are battery powered and the pressure data is stored in the gauge memory. No bottom hole pressure is available until the gauge is pulled to surface. With a cable, a surface read out system allows to monitor the test in real time, and to adjust the duration of the shut-in periods. • Down hole valve : By closing the well down hole, the pressure response is representative of the reservoir behavior earlier than in case of surface shut-in (see wellbore storage effect in Section 1-2.1). DST are generally short tests. Several types of down hole valve are available, operated by translation, rotation or annular pressure. A sample of reservoir fluid can be taken when the tester valve is closed. • Bottom hole sampler : Fluid samples can also be taken with a wire line bottom hole sampler. During sampling, the well is produced at low rate.

-6-


• RFT, MDT :The Repeat Formation Tester and the Modular Formation Dynamics Tester are open hole wire line tools. They are primary used to measure the vertical changes of reservoir pressure (pressure gradient), and to take bottom hole samples. From the pressure versus depth data, fluid contacts (oil–water OWC and gas–oil GOC) are located, communication or presence of sealing boundaries between layers can be established. RFT and MDT can also provide a first estimate of the horizontal and vertical permeability near the well by analysis of the pressure versus time response.

1-2 Definitions & typical regimes 1-2.1 Wellbore storage When a well is opened, the production at surface is first due to the expansion of the fluid in the wellbore, and the reservoir contribution is negligible. After any change of surface rate, there is a time lag between the surface production and the sand face rate. For a shut-in period, the wellbore storage effect is called afterflow. Pressure profile

Î Ï Ï

Ï

rw

r

pi

pw Figure 1-5 Wellbore storage effect. Pressure distribution.

-7-

Rate, q

Pressure, p


q surface q sand face Time, t

Figure 1-6 Wellbore storage effect. Sand face and surface rates. Wellbore storage coefficient

For a well full of a single phase fluid,

C =− ∆V =coVw (Bbl/psi, m3/Bars) ∆p

( 1-4)

where : co : liquid compressibility (psi-1, Bars -1) Vw : wellbore volume (Bbl, m3) When there is a liquid level, with ∆p = ρ g ∆h , ∆V = Vu ∆h and ρ : liquid density (lb/cu ft, kg/m3) g/gc : gravitational acceleration (lbf / lbm, kgf / kgm) Vu : wellbore volume per unit length (Bbl/ft, m3/m)

C =144

Vu (Bbl/psi) ρ (g gc) Vu (m3/Bars) ρ (g gc)

WB S

( 1-5)

m

Pressure change, ∆p

C =10197

Elapsed time, ∆t

Figure 1-7 Wellbore storage effect. Specialized analysis on a linear scale. Specialized analysis

Plot of the pressure change ∆p versus the elapsed time ∆t time on a linear scale. At early time, the response follows a straight line of slope mWBS, intercepting the origin.

-8-


∆p=

qB ∆t (psi, Bars) 24C

( 1-6)

Result : wellbore storage coefficient C.

C=

qB (Bbl/psi, m3/Bars) 24 m WBS

( 1-7)

1-2.2 Radial flow regime, skin (homogeneous behavior) When the reservoir production is established, the flow-lines converge radially towards the well. In the reservoir, the pressure is a function of the time and the distance to the well. Pressure profile

Î Ï Ï Ï Î

Ï

Î

Í

Í

Í Í

p

ri

rw

pi

r

S=0 pwf

Figure 1-8 Radial flow regime. Pressure distribution. Zero skin. p

ri

rw

r

pi

S>0

pwf(S=0) pwf(S>0)

∆p skin

Figure 1-9 Radial flow regime. Pressure distribution. Damaged well, positive skin factor.

-9-


p pi

ri

rw

pwf(S<0) pwf(S=0)

r

S<0

∆p skin

Figure 1-10 Radial flow regime. Pressure distribution. Stimulated well, negative skin factor.

Skin

The skin is a dimensionless parameter. It characterizes the well condition : for a damaged well S > 0, and for a stimulated well S < 0.

kh ∆pSkin (field units) 141.2qBµ kh S= ∆pSkin (metric units) 18.66qBµ S=

( 1-8)

• Damaged well (S > 0) : poor contact between the well and the reservoir (mudcake, insufficient perforation density, partial penetration) or invaded zone • Stimulated well (S < 0) : surface of contact between the well and the reservoir increased (fracture, horizontal well) or acid stimulated zone Steady state flow in the circular zone : k rw

ks

rs

141.2qBµ rS 141.2qBµ rS (psi, field units) ln − ln kS h rw kh rw 18.66qBµ rS 18.66qBµ rS − p w, S = 0 = − (Bars, metric units) ( 1-9) ln ln kS h rw kh rw

p w, S − p w , S = 0 =

p w, S

The skin is expressed :

 k  r S= − 1 ln S  kS  rw

( 1-10)

Equivalent wellbore radius :

rwe = rw e − S (ft, m)

( 1-11)

- 10 -


Specialized analysis


For homogeneous reservoirs, a pressure versus time semi-log straight line describes the radial flow regime. The analysis gives access to the reservoir permeability thickness product kh, and to the skin coefficient S. m

∆p(1hr)

Log ∆t

Figure 1-11 Radial flow regime. Specialized analysis on semi-log scale.

Semi-log straight line of slope m :

∆p = 162.6 ∆p = 21.5

 k qBµ  − 3.23 + 0.87 S  (psi, field units) log ∆t + log 2 kh  φ µ ct rw 

 qBµ  k 3 . 10 0 . 87 − + S  (Bars, metric units)( 1-12) log ∆t + log kh  φ µ c t rw2 

Results:

qBµ (mD.ft, field units) m qBµ (mD.m, metric units) kh = 21.5 m kh = 162.6

( 1-13)

 ∆p  k 3 23 S = 1151 .  1 hr − log + .  (field units) φµ ct rw2  m    ∆p k  (metric units) S = 1.151 1 hr − log + 3 . 10 2   m φµ c r t w  

( 1-14)

1-2.3 Examples of infinite acting radial flow behaviors In the following examples, two wells A and B are tested twice with the same rate sequence, and the four test responses are compared on linear and semi-log scales.

- 11 -


The two wells have very different characteristics. Well A is in a low permeability reservoir. During one test the skin is moderate with S=6, and during the other test the well has no skin damage (S=0). Well B is in a higher permeability reservoir (four times larger than for well A) but the skin factors are large, respectively S=25 and S=60 (this large value is relatively exceptional. It suggests a completion problem such as limited entry).

pressure, psi

6000

no skin

4000

moderate skin 2000

0 0

10

20

30

40

time, hours

Figure 1.12 Test history plot well A (low permeability).

On the test history plots Figure 1.12 and Figure 1.13, the two wells show apparently a similar behavior. For each well, the flowing pressure is low during one test (the last flowing pressure is 3200 psi before shut-in), and higher during the other test (last flowing pressure of 5500psi before shut-in).

pressure, psi

6000

high skin 4000

very high skin 2000

0 0

10

20

30

40

time, hours

Figure 1.13 Test history plot well B (higher permeability).

On semi-log scale, the pressure response is more characteristic of the well and reservoir condition than on the previous linear scale plots. In the case of well A with low permeability and low skin, the pressure drop during drawdown is mainly produced in the reservoir, and the slope of the semi-log straight line is high.

- 12 -


pressure change, psi

3000

moderate skin 2000

1000

0 0.001

∆ p skin

0.01

no skin

0.1

1

10

100

time, hours

Figure 1.14 Semi-log responses for well A.

pressure change, psi

3000

very high skin 2000

∆ p skin

1000

0 0.001

high skin

0.01

0.1

1

10

100

time, hours Figure 1.15 Semi-log responses for well B.

Conversely, with the higher permeability example of well B, most of the pressure drop is due to skin damage, and the response tends to be flat with a low semi-log straight-line slope.

1-2.4 Fractured well (infinite conductivity fracture) : linear flow regime

xf

Figure 1-16 Fractured well. Fracture geometry.

- 13 -


Linear flow regime

At early time, before the radial flow regime is established, the flow-lines are perpendicular to the fracture plane. This is called linear flow.

Figure 1-17 Infinite conductivity fracture. Geometry of the flow lines. Linear and radial flow regimes.


Plot of the pressure change ∆p versus the square root of elapsed time response follows a straight line of slope mLF, intercepting the origin.

∆p = 4.06

µ

qB hx f


∆p = 0.623

φ ct k

qB hx f

mL

∆t : the

∆t (psi, field units)

µ φ ct k

∆t (Bars, metric units)

( 1-15)

F

∆t

Figure 1-18 Infinite conductivity fracture. Specialized analysis with the pressure versus the square root of time.

Result : the half fracture length xf

x f = 4.06 x f = 0.623

µ

qB φ ct k hmLF

µ

qB φ ct k hm LF

(ft, field units)

(m, metric units)

- 14 -

( 1-16)


1-2.5 Fractured well (finite conductivity fracture) : bi-linear flow regime Bilinear flow regime

kf

wf

Figure 1-19 Finite conductivity fracture. Geometry of the flow lines during the bi-linear flow regime.

When the pressure drop in the fracture plane is not negligible, a second linear flow regime is established along the fracture extension. This configuration is called bilinear flow regime.


Plot of the pressure change ∆p versus the fourth root of elapsed time straight line of slope mBLF, intercepting the origin.

∆p = 44.11

4


4


h k f w 4 φ µ ct k qBµ

h k f wf


∆p = 6.28

qBµ

4

φµ c t k

4

∆t :

( 1-17)

m BLF

4

∆t

Figure 1-20 Finite conductivity fracture. Specialized analysis with the pressure versus the fourth root of time.

Result : the fracture conductivity kfwf

1  qBµ  k f w f = 1944.8 φµ c t k  hm BLF

  

1  qBµ  k f w f = 39.46 φµ ct k  hm BLF

2

  

- 15 -

2

(mD.ft, field units)

(mD.m, metric units)

( 1-18)


1-2.6 Well in partial penetration : spherical flow regime Spherical flow regime

Spherical flow can be observed in wells in partial penetration, before the top and bottom boundaries are reached. Later, the flow becomes radial. kV kH kH hw

h

Figure 1-21 Well in partial penetration. Geometry of the flow lines. Radial, spherical and radial flow regimes.


∆p = 70.6

qBµ φ µ ct qBµ − 2452.9 3 2 k S rS k S ∆t

(psi, field units)

∆p = 9.33

qBµ φµ c t qBµ − 279.3 3 2 k S rS k S ∆t

(Bars, metric units)

∆t . The

( 1-19)

m SP H


Plot of the pressure versus the reciprocal of the square root of time 1 response follows a straight line of slope mSPH :

1

∆t

Figure 1-22 Well in partial penetration. Specialized analysis with the pressure versus 1/ the square root of time.

Result : the spherical permeability ks

 φµ ct k S =  2452.9qBµ  mSPH 

   

23

(mD, field units)

- 16 -


 φµ c t k S =  279.3qBµ  mSPH 

   

23

( 1-20)

(mD, metric units)

The permeability anisotropy is expressed with :

kH  kH  =  kV  k s 

3

( 1-21)

1-2.7 Fissured reservoir (double porosity behavior) In fissured reservoirs, the fissure network and the matrix blocks react at a different time, and the pressure response deviates from the standard homogeneous behavior.

Pressure profile

Î Ï Ï Ï Î Î

Í Í

Ï

p pi

rw

Í Í

ri

pm

r

pf pwf Figure 1-23 Double porosity behavior. Pressure distribution. Fissure system homogeneous regime.

First, the matrix blocks production is negligible. The fissure system homogeneous behavior is seen.

- 17 -


Î Ï Ï Ï Î

Î

Ï

Î

Í Í Í Í Í Í ÍÍ

p r w

pi

ri

r

pm > pf

pwf

Figure 1-24 Double porosity behavior. Pressure distribution. Transition regime.

When the matrix blocks start to produce into the fissures, the pressure deviates from the homogeneous behavior to follow a transition regime. Î Ï Ï Ï Î

Î

Ï

Î

pi

Í Í Í

Í Í

Í Í

p r w

Í

ri

r

pm = pf pwf

Figure 1-25 Double porosity behavior. Pressure distribution. Total system homogeneous regime (fissures + matrix).

When the pressure equalizes between fissures and matrix blocks, the homogeneous behavior of the total system (fissure and matrix) is reached.

- 18 -


1-2.8 Limited reservoir (one sealing fault) When one sealing fault is present near the producing well, the pressure response deviates from the usual infinite acting behavior after some production time. Pressure profile Î Ï Ï Ï Î

Ï

Î

Í

Í

Í Í

p rw

pi

ri

L

r

pwf

Figure 1-26 One sealing fault. Pressure profile at time t1. The fault is not reached, infinite reservoir behavior. p rw

pi

L

ri

r

pwf

Figure 1-27 One sealing fault. Pressure profile at time t2. The fault is reached, but it is not seen at the well. Infinite reservoir behavior. p pi

rw

L

r

ri

pwf

Figure 1-28 One sealing fault. Pressure profile at time t3. The fault is reached, and it is seen at the well. Start of boundary effect.

- 19 -


p

rw

L

r

pi ri

pwf

Figure 1-29 One sealing fault. Pressure profile at time t4. The fault is reached, and it is seen at the well. Hemi-radial flow.

t1 : the fault is not reached, radial flow t2 : the fault is reached t3 : the fault is seen at the well, transition t4 : hemi-radial flow

Figure 1-30 One sealing fault. Drainage radius.



A second semi-log straight line with a slope double (2m). Result : the fault distance L. 2m m

Log ∆t

Figure 1-31 One sealing fault. Specialized analysis on semi-log scale.

The time intersect ∆tx between the two lines is used to estimate the fault distance L:

- 20 -


L = 0.01217

k∆t x (ft, field units) φµ ct

L = 0.0141

k∆t x (m, metric units) φµ c t

( 1-22)

1-2.9 Closed reservoir In closed reservoir, when all boundaries have been reached, the flow changes to Pseudo Steady State : the pressure decline is proportional to time. Pressure profile

As long as the reservoir is infinite acting, the pressure profile expands around the well during the production (and the well bottom hole pressure drops). Î Ï Ï ri (t1)

Ï Î

Ï

Î

Í

Í

p pi

Re

Í Í

ri (t1)

rw t1

ri (t2) = Re

t2

t3

r

t4

Infinite acting pwf

Pseudo Steady State

Figure 1-32 Circular closed reservoir. Pressure profiles. Time t1: the boundaries are not reached, infinite reservoir behavior: the pressure profile expands. Time t2: boundaries reached, end of infinite reservoir behavior. Times t3 and t4: pseudo steady state regime, the pressure profile drops.

During the pseudo steady state regime, all boundaries have been reached and the pressure profile drops (but its shape remains constant with time).

- 21 -



During drawdown, plot of the pressure versus elapsed time ∆t on a linear scale. At late time, a straight line of slope m* characterizes the Pseudo Steady State regime:

∆p = 0.234

 qB qBµ  A ∆t + 162.6 log 2 − log( C A ) + 0.351 + 0.87 S  (psi, field units) φ ct hA kh  rw 

∆p = 0.0417

 qB qBµ  A ∆t + 21.5 log 2 − log(C A ) + 0.351 + 0.87 S  (Bars, metric φ c t hA kh  rw  ( 1-23)

units)

Pressure, p

pi

ppseudo ste ady

state slope m*

Time, t

Figure 1.33 Drawdown and build-up pressure response. Linear scale. Closed system.

Result : the reservoir pore volume φ hA.

qB (cu ft, field units) ct m * qB φ hA = 0.0417 (m3, metric units) ct m *

φ hA = 0.234

( 1-24)

During shut-in, the pressure stabilizes to the average reservoir pressure p ( < pi ) .

1-2.10 Interference test Pressure profile

With interference tests, the pressure is monitored in an observation well at distance r from the producer. The pressure signal is observed with a delay, the amplitude of the response is small.

- 22 -


pi

5000

Pressure (psia)

Observation well 4500

Producing well

4000

3500 0

100

200

400

300

500

Time (hours)

Figure 1-34 Interference test. Response of a producing and an observation well. Linear scale.

Î Ï

Producing well

Observation well Ï Í Î

p pi

Í

Í

Í

ri

rw

r

pwf Figure 1-35 Interference test. Pressure distribution.

1-2.11 Well responses A limited number of flow line geometries produce a characteristic pressure behavior: radial, linear, spherical etc. For each flow regime, the pressure follows a well-defined time function: log ∆t , ∆t , 1 ∆t etc. A straight line can be drawn on a specialized pressure versus time plot, to access the corresponding well or reservoir parameter. A complete well response is defined as a sequence of regimes. By identification of the characteristic pressure behaviors present on the response, the chronology and time limits of the different flow regime are established, defining the interpretation model.

- 23 -


For a fractured well for example, the sequence of regimes is : 1. Linear (1)

2. Radial (2)

Figure 1.36 Fractured well example.

In the case of a well in a channel reservoir : 1. Radial

(1)

(2)

2. Linear Figure 1.37 Example of a well in a channel reservoir.

1-2.12 Productivity Index The Productivity Index is the ratio of the flow rate by the drawdown pressure drop, expressed from the average reservoir pressure p .

PI =

q

( p − pwf )

( 1-25)

(Bbl/D/psi, m3/D/Bars)

The Ideal Productivity Index defines the productivity if the skin of the well is zero.

PI (S=0) =

q

( p − pwf ) − ∆pskin

(Bbl/D/psi, m3/D/Bars)

( 1-26)

During the infinite acting period p ≈ pi , the Transient Productivity Index is decreasing with time.

PI =

kh

(Bbl/D/psi, field units)

  k 162.6 Bµ  log ∆t + log − 3.23 + 0.87 S  φµ ct rw2   kh (m3/D/Bars, metric units) PI =   k 21.5Bµ  log ∆t + log − 3.10 + 0.87 S  2 φµ c r t w   - 24 -

( 1-27)


The Pseudo Steady State Productivity Index is a constant

PI =

PI =

kh   A − log( C A ) + 0.351 + 0.87 S  162.6 Bµ  log rw2   kh

  A 21.5Bµ  log 2 − log(C A ) + 0.351 + 0.87 S    rw  

(Bbl/D/psi, field units)

(m3/D/Bars, metric units)

( 1-28)

1-2.13 Pressure profile and Radius of Investigation The Exponential Integral of Equation A-16 defines the pressure as a function of time and distance :

φµ ct r 2  141.2qBµ  Ei − ∆p (∆t , r ) =− 0.5  (psi, field units) kh  0.001056k ∆t  φ µ c t r 2  18.66qBµ  (Bars, metric units) ( 1-29) ∆p (∆t , r ) =− 0.5 Ei −  0.0001423k∆t  kh   For small x, Ei(− x ) =− ln (γ x ) : the Exponential Integral can be approximated by a log (with γ = 1.78, Euler's constant).

[ (

]

162.6∆qBµ log 0.000264 k ∆t φµ ct r 2 + 0.809 (psi, field units) kh 21.5qBµ ∆p (∆t , r ) = log 0.000356k ∆t φµ ct r 2 + 0.809 (Bars, metric units) ( 1-30) kh ∆p( ∆t , r ) =

)

[ (

)

]

(The semi-log straight line Eq. 1-12 corresponds to Eq. 1-30 for r=rw). p

Log r

pi

t1

t2

t3

t4

pwf

Figure 1-38 Pressure profile versus the log of the distance to the well.

When presented versus log(r), the pressure profile at a given time is a straight line until the distance becomes too large for the logarithm approximation of the

- 25 -


Exponential Integral. Beyond this limit, the profile flattens, and tends asymptotically towards the initial pressure. The radius of investigation ri tentatively describes the distance that the pressure transient has moved into the formation. Several definitions have been proposed, in general ri is defined with one of the two relationships :

(0.000264k ∆t φµ c r ) = 41 or = γ1 (0.000356k ∆t φµ c r ) = 14 or = γ1 2

t i

2

2

t i

2

(field units) (metric units)

(in dimensionless terms of Equation 2.4 or 8-2, t D riD2 =

( 1-31)

1 1 2 = 2 ). or t D riD 4 γ

This gives respectively,

ri = 0.032 k∆t φµ ct (ft, field units) ri = 0.037 k∆t φµc t (m, metric units)

( 1-32)

and

ri = 0.029 k∆t φµ ct (ft, field units) ri = 0.034 k∆t φµct (m, metric units)

( 1-33)

(the radius of investigation is independent of the rate). The radius of investigation ri is sometimes viewed as the minimum distance of any event, such as a reservoir limit, that cannot be observed during the test period. With the sealing fault example of Figure 1-30, the pressure transient reaches the fault 4 times earlier the boundary can be observed on the producing well pressure behavior. In practice, for an initial flow period, the radius of investigation of Equation 1-32 or 1-33 is relatively consistent with the distance estimated by a simulation, when a boundary effect is introduced at the end of the test period. For a shut-in periods, Equations 1-32 and 1-33 are not always accurate.

- 26 -

2 - THE ANALYSIS METHODS

2-1 Log-log scale For a given period of the test, the change in pressure ∆p is plotted on log-log scale versus the elapsed time ∆t. This data plot is then compared to a set of dimensionless theoretical curves. 102 101

∆P, psi

100

10-1 10-3 (3.6 sec)

10-2 (36 sec)

10-1 (6 mn)

100

101

102

∆t, hr Figure 2-1 Log-log scale.

pD = A ∆p, t D = B ∆t ,

{ A= f ( kh,...)}

{B = g( k , C, S ...)}

( 2-1)

The shape of the response curve is characteristic : the product of one of the variables by a constant term is changed into a displacement on the logarithmic axes. If the flow rate is doubled for example, the amplitude of the response ∆p is doubled also, but the graph of log(∆p) is only be shifted by log(2) along the pressure axis. With the log-log scale, the shape of the data plot is used for the diagnosis of the interpretation model(s).

log pD = log A + log ∆p

( 2-2)

log t D = log B + log ∆t

The log-log analysis is global : it considers the full period, from very early time to the latest recorded pressure point. The scale expands the response at early time.

- 27 -

Chapter 2 - The analysis methods

2-2 Pressure curves analysis 2-2.1 Example of pressure type-curve : "Well with wellbore storage and skin, homogeneous reservoir" Dimensionless terms

Dimensionless terms are used because they illustrate pressure responses independently of the physical parameters magnitude (such as flowrate, fluid or rock properties). For example, describing the well damage with the dimensionless skin factor S is much more meaningful than using the actual pressure drop near the wellbore. Dimensionless pressure

kh ∆p (field units) 1412 . qBµ kh ∆p (metric units) pD = 18.66qBµ

pD =

( 2-3)

Dimensionless time

0.000264 k ∆t (field units) φµ ct rw2 0.000356k tD = ∆t (metric units) φµ c t rw2 tD =

( 2-4)

Dimensionless wellbore storage coefficient

CD =

CD =

0.8936C (field units) φ ct hrw2 0.1592C

φ c t hrw2

(metric units)

( 2-5)

Dimensionless time group

tD kh ∆t = 0.000295 (field units) CD µ C tD kh ∆t (metric units) = 0.00223 CD µ C - 28 -

( 2-6)


Dimensionless Pressure, pD

1 02

1060 1050 1040 1030 1020 1015 1010 8 10 106 104 103 102 10 3 1 0.3

Approximate start of semi-log straight line 10

CDe2S 1

10-1 10-1

1

102

10

103

104

Dimensionless time, tD/CD

Figure 2-2 Pressure type-curve: Well with wellbore storage and skin, homogeneous reservoir. Log-log scale. CDe(2S) = 1060 to 0.3.

Dimensionless curve group

0.8936C 2 S (field units) e φ ct hrw2 0.1592C 2S (metric units) C D e 2S = e φ c t hrw2 CD e 2 S =

( 2-7)

The curve label CD e2S defines the well condition. It ranges from CD e2S =0.3 for stimulated wells, up to 1060 for very damaged wells.

Log-log matching procedure

Pressure change, ∆p (psi)

103

102

101

1 10-3

10-2

10-1

1

101

Elapsed time, ∆t (hours)

Figure 2-3 Build-up example. Log-log plot

- 29 -

102


The log-log data plot ∆p, ∆t is superimposed on a set of dimensionless type-curves pD, tD /CD. The early time unit slope straight line is matched on the "wellbore 2S storage" asymptote but the final choice of the CD e curve is frequently not unique (Figure 2-12).

Results of log-log analysis

Pressure match PM = p D ∆p : the permeability thickness product

kh = 141.2qBµ (PM ) (mD.ft, field units) kh = 18.66qBµ (PM ) (mD.m, metric units)

( 2-8)

Time match TM = (t D C D ) ∆t : the wellbore storage coefficient

kh  1    (Bbl/psi, field units) µ  TM  kh  1  3 C = 0.00223   (m /Bars, metric units) µ  TM 

C = 0.000295

( 2-9)

Curve match : the skin

C D e 2 S Match S = 0.5 ln CD

( 2-10)

2-2.2 Shut-in periods Drawdown periods are in general not suitable for analysis because it is difficult to ascertain a constant flowrate. The response is distorted, especially with the log-log scale that expands the response at early time. Build-up periods are preferably used : the flowrate is nil, therefore well controlled.

Example of a shut-in after a single rate drawdown

Build-up responses do not show the same behavior as a first drawdown in a reservoir at initial pressure. After a drawdown of tp, the well shows a pressure drop of ∆p(tp). It takes an infinite time to reach the initial pressure during build-up, and to produce a pressure change ∆pBU of amplitude ∆p(tp). Build-up responses depend upon the previous rate history.

- 30 -


Rate, q

Pressure, p

pi ∆pBU(∆t) ∆p (tp)

∆t BU

q 0 0

tp

tp+∆t Time, t

Figure 2-4 History drawdown - shut-in.

The diffusivity equation used to generate the well test analysis solutions is linear. It is possible to add several pressure responses in order to describe the well behavior after any rate change. This is the superposition principle. For a build-up after a single drawdown at rate q, an injection period at -q is superposed to the extended flow period.

(∆p (tp+∆t) - ∆p (∆t) ) Pressure, p

pi ∆p (∆t)

∆p (tp+∆t)

Rate, q

∆p (tp) q 0 -q 0

∆t

tp Time, t

Figure 2-5 History extended drawdown + injection.

Log-log analysis : build-up type curve

[p

D

( ∆t ) D ]BU

(

= pD ( ∆t ) D − pD t p + ∆t

)

D

( )

+ pD t p

D

The pressure build-up curve is compressed on the ∆p axis when ∆t>>tp.

- 31 -

( 2-11)



10 2 CDe2S drawdown type curve

pD(tpD ) 10

build-up type curve 1 tpD 10-1 10-1

1

10 102 Dimensionless time, tD /CD

103

104

Figure 2-6 Drawdown and build-up type curves (tpD = 2).

Semi-log analysis : superposition time

[∆p(∆t )]BU [∆p(∆t )]BU

  t p ∆t k + log − 3 . 23 + 0 . 87 S log  (psi, field units) φ µ ct rw2  t p + ∆t   t p ∆t qBµ  k = 21.5 + log − 3 . 10 + 0 . 87 S log  (Bars, metric units) kh  t p + ∆t φµ ct rw2 

= 162.6

qBµ kh

( 2-12)

With the superposition time, the correction compresses the ∆t scale.


10

CDe2S drawdown type curve pD(tpD ) build-up type curve

5

tpD 0 10-1

1

10

102

103

104

Dimensionless times, tD / CD and [ tpD tD / (tpD + tD) CD ]

Figure 2-7 Drawdown and build-up type curves of Figure 2-6 on semi-log scale.

Horner method

t p + ∆t qBµ log (psi, field units) ∆t kh t p + ∆t qBµ = p i − 21.5 log (Bars, metric units) kh ∆t

pws = pi − 162.6 p ws

- 32 -

( 2-13)



10 P*

5

m

0 1

102

10

103

104

105

Horner time, [(tpD + tD) / tD ]

Figure 2-8 Horner plot of build-up type curve of Figure 2-6.

Horner analysis : • The slope m, • The pressure at ∆t =1 hour on the straight line • The extrapolated pressure to infinite shut-in time (∆t = ∞): p*. Results :

qBµ (mD.ft, field units) m qBµ kh = 21.5 (mD.m, metric units) m kh = 162.6

( 1-13)

 ∆p  tp +1 k  (field units) S = 1151 .  1 hr − log + log + 3 . 23 tp φµ ct rw2  m 

 ∆p  t p +1 k  (metric units) S = 1.151 1 hr − log + log + 3 . 10 2  m  t φµ c t rw p  

( 2-14)

In an infinite system, the straight line extrapolates to the initial pressure and p*=pi.

Multi- rate superposition

At time ∆t of flow period # n, the multi-rate type curve is :

[

pD ( ∆t ) D

]

MR

=

n −1

qi − qi −1 pD (t n − ti ) D − pD ( t n + ∆t − ti ) D + pD ( ∆t ) D ( 2-15) n −1 − qn

∑q i =1

[

]

- 33 -

Pressure, p


∆t

Rate, q

Period # 1,2,…, 5,

6,…….....10,

11

q1,…. q5=0, q6,………..q10,

q11=0

Time, t

Figure 2-9 Multi- rate history. Example with 10 periods before shut-in.

The multirate superposition time is expressed :

p ws (∆t ) = pi −162.6 p ws (∆t ) = p i −21.5

Bµ n−1 ∑ (qi − qi −1 )log(t n + ∆t − ti )+(qn − qn−1 )log(∆t ) (psi, field units) kh i =1

Bµ n −1 ∑ (qi − qi −1 ) log(t n + ∆t − t i ) + (q n − q n −1 ) log(∆t ) (Bars, metric kh i =1 ( 2-16)

units)

Limitations if the time superposition: the sealing fault example

In the following example, the well is produced 50 hours and shut-in for a pressure build-up. A sealing fault is present near the well and, at 100 hours, the flow geometry changes from infinite acting radial flow to hemi-radial flow.

5000

Pressure, psi

4500 Radial

4000

Hemi-radial

3500

Radial

0

50

Hemi-radial

100

150

Infinite reservoir Sealing fault

200

250

300

Time, hours

Figure 2-10 History drawdown – build-up. Well near a sealing fault.

During the 50 initial hours of the shut-in period (cumulative time 50 to 100 hours), both the extended drawdown and the injection periods are in radial flow regime.

- 34 -


The superposition time of Equations 2-12 or 2-13 is applicable, and the Horner method is accurate. At intermediate shut-in times, from 50 to 100 hours (cumulative time 100 to 150 hours), the extended drawdown follows a semi-log straight line of slope 2m when the injection is still in radial flow (slope m). Theoretically, the semi-log approximation of Equation 2-11 with Equation 2-12 is not correct. Ultimately, the fault influence is felt during the injection and the 2 periods follow the same semi-log straight line of slope 2m (shut-in time >> 100 hours, cumulative time >> 150 hours). The semi-log superposition time is again applicable. In practice, when the flow regime deviates from radial flow in the course of the response, the error introduced by the Horner or multirate time superposition method is negligible on pressure curve analysis results. It is more sensitive when the derivative of the pressure is considered.

Time superposition with other flow regimes

The time superposition is sometimes used with other flow regimes for straight-line analysis. When all test periods follow the same flow behavior, the Horner time can be expressed with the corresponding time function. For fractured wells, Horner time corresponding to linear (Equation 1-15) and bi-linear flow (Equation 1-17) is expressed respectively :

(t

p

+ ∆t

)

12

− ( ∆t )

12

(hr1/2)

(t p + ∆t )1 4 −(∆t )1 4 (hr

( 2-17)

1/4

( 2-18)

)

The Horner time corresponding to spherical flow of Equation 1-19 has been used for the analysis of RFT pressure data.

( ∆t )−1 2 − (t p + ∆t )

−1 2

(hr-1/2)

- 35 -

( 2-19)


2-2.3 Pressure analysis method The analysis is made on log-log and specialized plots. The purpose of the specialized analysis is to concentrate on a portion of the data that corresponds to a particular flow behavior. The analysis is carried out by the identification of a straight line on a plot whose scale is specific to the flow regime considered. The time limits of the specialized straight lines are defined by the log-log diagnosis. 4000 p*

p(1hr)

Pressure, psia

3750

slope m

slop em

3500

3250 3000 1

101

102

103

104

(tp +∆t )/ ∆t

Figure 2-11 Build-up example of Figure 2-3. Semi-log Horner analysis.


1 02

1060 1050 1040 1030 1020 1015 1010 8 10 106 104 103 102 10 3 1 0.3

10

CDe2S 1

10-1 10-1

1

102

10

103

104


Figure 2-12 Build-up example of Figure 2-3. Log-log match.

For the radial flow analysis of a build-up period, the semi-log superposition time is used. The slope m of the Horner / superposition straight line defines the final pressure match of the log-log analysis.

PM =

p D 1.151 (psi-1, Bars-1) = ∆p m

( 2-20)

2S Once the pressure match is defined, the CD e curve is known accurately. Results from log-log and specialized analyses must be consistent.

- 36 -


2-3 Pressure derivative 2-3.1 Definition The natural logarithm is used.

∆p ' =

dp dp (psi, Bars) = ∆t dt d ln ∆t

( 2-21)

The derivative is plotted on log-log coordinates versus the elapsed time ∆t since the beginning of the period.

2-3.2 Derivative type-curve : "Well with wellbore storage and skin, homogeneous reservoir" Radial flow

Log ∆p Log ∆p'

∆p' = constant

Log ∆t Figure 2-13 Pressure and derivative responses on log-log scale. Radial flow.

∆p = 162.6 ∆p = 21.5

 qBµ  k − 3.23 + 0.87 S  (psi, field units) log ∆t + log 2 φ µ ct rw kh  

 qBµ  k − 3 . 10 + 0 . 87 S log ∆t + log  (Bars, metric units)( 1-12) kh  φ µ c t rw2 

The radial flow regime does not produce a characteristic log-log shape on the pressure curve but it is characteristic with the derivative presentation : it is constant. ∆p ' = 70. 6

qB µ (psi, field units) kh

∆p ' = 9.33

qBµ (Bars, metric units) kh

In dimensionless terms,

- 37 -

( 2-22)


dp D = 0.5 d ln( t D C D )

( 2-23)

Wellbore storage

∆p =

qB ∆t 24C

(psi, Bars) ( 1-6)

qB ∆p' = ∆t (psi, Bars) 24C

( 2-24)

During wellbore storage, the pressure change ∆p and the pressure derivative ∆p' are identical. On log-log scale, the pressure and the derivative curves follow a single straight line of slope equal to unity.

Log ∆p Log ∆p'

Slope 1

Log ∆t Figure 2-14 Pressure and derivative responses on log-log scale. Wellbore storage

Derivative of Section 2-2 example

During the transition between the wellbore storage and the infinite acting radial 2S flow regime, the derivative shows a hump, function of the CD e group.

Pressure derivative, ∆p' (psi)

103

102 pe slo

101

1

0.5 line

1 10-3

10-2

10-1

1

101

102


Figure 2-15 Derivative of build-up example Figure 2-3. Log-log scale.

- 38 -


Dimensionless Pressure erivative, p'D

Derivative type-curve 1 02 CDe2S 1060

10

103 102 10 3 1 0.3

1

10-1 10-1

1

1040 1050 1030 1020 1015 1010 108 106 104

102

10

103

104


Figure 2-16 "Well with wellbore storage and skin, homogeneous reservoir" Derivative of type-curve Figure 2-2. Log-log scale. CDe(2S) = 1060 to 0.3.

Derivative match

Dimensionless Pressure Derivative, p'D

The match point is defined with the unit slope pressure and derivative straight line, and the 0.5 derivative stabilization.

1 02

10

1

10-1 10-1

1

102

10

103

104


Figure 2-17 Derivative match of example Figure 2-3. Log-log scale.

2-3.3 Other characteristic flow regimes During other characteristic flow regimes, the pressure changes with the elapsed time power 1/n : - 39 -


∆p = A (∆t )1 n + B (psi, Bars) With: • 1/n =1 • 1/n =1/2 • 1/n =1/4 • 1/n =-1/2

( 2-25)

during the pure wellbore storage and the pseudo steady state regimes, in the case of linear flow, for bi-linear flow, when spherical flow is established.

The logarithm derivative is:

∆p ' =

dp A 1n = (∆t ) (psi, Bars) d ln ∆t n

( 2-26)

The log-log pressure derivative curve (∆p', ∆t) follows a straight-line slope of 1/n.

Infinite conductivity fracture (linear flow)

On log-log scale, the pressure and derivative follow two straight lines of slope 1/2. The level of the derivative half-unit slope line is half that of the pressure.

∆p = 4.06

qB hx f

∆p = 0.623

∆p' = 2.03

qB hx f

qB hx f

∆p' = 0.311

qB hx f

µ φ ct k µ φ ct k µ φ ct k µ φ ct k

∆t (psi, field units) ∆t (Bars, metric units)

( 1-15)

∆t (psi, field units) ∆t (Bars, metric units)

( 2-27)

Slope 1/2 Log ∆p Log ∆p'

Log ∆t

Figure 2-18 Pressure and derivative responses on log-log scale. Infinite conductivity fracture.

- 40 -


Finite conductivity fracture (bi-linear flow)

A log-log straight line of slope 1/4 can be observed on pressure and derivative curves, but the derivative line is four times lower.

∆p = 44.11

∆p = 6.28

qBµ h k f w 4 φ µ ct k

4

qBµ h k f wf

∆p' = 11.03

∆p' = 1.571

4

4

φµ c t k

qBµ h k f w 4 φ µ ct k

4

qBµ h k f wf

4



( 1-17)

∆t (psi, field units) 4

φµ ct k


( 2-28)

Slope 1/4 Log ∆p Log ∆p'

Log ∆t

Figure 2-19 Pressure and derivative responses on log-log scale. Finite conductivity fracture.

Well in partial penetration (spherical flow)

∆p = 70.6

qBµ φ µ ct qBµ − 2452.9 3 2 (psi, field units) k S rS k S ∆t

∆p = 9.33

qBµ φµ c t qBµ (Bars, metric units) − 279.3 3 2 k S rS k S ∆t

∆p' = 1226.4 ∆p ' = 139.6

qBµ φ µ ct k S3 2 ∆t

qBµ φµ c t k S3 2 ∆t

( 1-19)

(psi, field units)


The shape of the log-log pressure curve is not characteristic but the derivative follows a straight line with a negative half-unit slope.

- 41 -

( 2-29)


Log ∆p Slope –1/2

Log ∆p'

Log ∆t

Figure 2-20 Pressure and derivative responses on log-log scale. Well in partial penetration.

Closed system (pseudo steady state)

The late part of the log-log pressure and derivative drawdown curves tends to a unit-slope straight line. The derivative exhibits the characteristic straight line before it is seen on the pressure response.

Log ∆p Slope 1

Log ∆p'

Log ∆t Figure 2-21 Pressure and derivative responses on log-log scale. Closed system (drawdown).

  A log 2 − log(C A ) + 0.351 + 0.87 S  (psi, field units) rw    qB qBµ  A ∆p = 0.0417 ∆t + 21.5 log 2 − log(C A ) + 0.351 + 0.87 S  (Bars, metric kh  φ c t hA rw 

∆p = 0.234

qB qBµ ∆t + 162.6 φ ct hA kh

( 1-22)

units)

qB ∆t (psi, field units) φ ct hA qB ∆p ' = 0.0417 ∆t (Bars, metric units) φ ct hA

∆p ' = 0.234

- 42 -

( 2-30)


2-3.4 Data differentiation The algorithm uses three points, one point before (left = 1) and one after (right = 2) the point i of interest. It estimates the left and right slopes, and attributes their weighted mean to the point i. On a p vs. x semi-log plot,

 ∆p   ∆p    ∆x2 +   ∆x1  ∆x  2 dp  ∆x  1 = ∆x1 + ∆x2 dx

( 2-31)

It is recommended to start by using consecutive points. If the resulting derivative curve is too noisy, smoothing is applied by increasing the distance ∆x between the point i and points 1 and 2. The smoothing is defined as a distance L, expressed on the time axis scale. The points 1 and 2 are the first at distance ∆x1,2>L. The smoothing coefficient L is increased until the derivative response is smooth enough but no more, over smoothing the data introduces distortions. With this smoothing method, L is usually no more than 0.2 or 0.3.

L Pressure change, ∆p

2 i 1 ∆x1

∆p1

∆x2

∆p2

Log (superposition)

Figure 2-22 Differentiation of a set of pressure data.

At the end of the period, point i becomes closer to last recorded point than the distance L. Smoothing is not possible any more to the right side, the end effect is reached. This effect can introduce distortions at the end of the derivative response.

2-3.5 Build-up analysis For a shut-in after a single drawdown period (the Horner method is applicable), the derivative is generated with respect to the modified Horner time given in the superposition Equation 2-12 :

- 43 -


∆p ' =

t p + ∆t dp dp ∆t = (psi, Bars) t p ∆t tp dt d ln t p + ∆t

( 2-32)

For a complex rate history, the multirate superposition time is used. In all cases, the derivative is plotted versus the usual elapsed time ∆t : the log-log derivative curve is not a raw data plot but is dependent upon the rate history introduced in the time superposition calculations.

Limitations if the time superposition: the sealing fault example

When the response deviates from the infinite acting radial flow regime, the derivative with respect to the time superposition can introduce a distortion on the response, as illustrated on the log-log derivative of the build-up example of Figure 2-10 for a well near a sealing fault.

Pressure change, ∆p and Pressure Derivative, psi

1 04

1 03

1 02

drawdown build-up

101 10-2

10-1

1

10

102

103

104

Elapsed time ∆t, hours

Figure 2-23 Log-log plot of the build-up example of Figure 2-10. Well near a sealing fault.

2-4 The analysis scales The log-log analysis is made with a simultaneous plot of the pressure and derivative curves of the interpretation period. Time and pressure match are defined with the derivative response. The CD e2S group is identified by adjusting the curve match on pressure and derivative data.

- 44 -


Dimensionless Pressure, pD and Derivative, p'D

1 02

1060 1050 1040 1030 1020 1015 1010 108 106 104 103 102 10 3 1 0.3

10

CDe2S 1

10-1 10-1

1

102

10

103

104


Figure 2-24 Pressure and derivative type-curve for a well with wellbore storage and skin, homogeneous reservoir.

The double log-log match is confirmed with a match of the pressure type-curve on semi-log scale to adjust accurately the skin factor and the initial pressure. A simulation of the complete test history is presented on linear scale in order to control the rates, any changes in the well behavior, the average pressure etc.

- 45 -

- 46 -

3 - WELLBORE CONDITIONS

3-1 Well with wellbore storage and skin, homogeneous reservoir 3-1.1 Characteristic flow regimes 1. Wellbore storage effect. Result: wellbore storage coefficient C. 2. Radial flow. Results: permeability-thickness product kh and skin S.

3-1.2 Log-log analysis


1 02 CDe2S =1030

high skin

10

1

pe slo

1

low skin

CDe2S =0.5

0.5 line 10-1 10-1

1

102

10

103

104


Figure 3-1 Responses for a well with wellbore storage and skin in an infinite homogeneous reservoir. Log-log scale. CDe(2S) = 1030 and 0.5.

3-1.3 Semi-log analysis Dimensionless Pressure, pD

50 CDe2S =1030

Slope m

40 30

∆ skin

20 10 Slope m

0 10-1

1

10

102


Figure 3-2 Semi-log plot of Figure 3-1.

- 47 -

CDe2S =0.5 103

104

Chapter 3 - Wellbore conditions

3-2 Infinite conductivity or uniform flux vertical fracture Two models are available: one considers a uniform flux distribution along the fracture length and, with the other, the fracture conductivity is infinite.

3-2.1 Characteristic flow regimes 1. Wellbore storage 2. Linear flow: 1/2 slope straight line. Results: fracture half-length xf. 3. Pseudo radial flow: derivative stabilization at 0.5. Results: permeabilitythickness product kh and the geometrical skin S.

3-2.2 Log-log analysis Dimensionless terms

t Df = t Df =

0.000264 k ∆t (field units) φµ ct x 2f 0.000356k

φµ ct x 2f

∆t (metric units)

( 3-1)


On Figure 3-3, CD = 0. The two models are slightly different during the transition between linear flow and radial flow. With the uniform flux model, the transition is shorter and the pressure curve is higher. 10

1

0.5 line

10-1

1/2 pe o l S

Uniform flux Infinite condutivity

10-2 10-4

10-3

10-2

10-1

1

10

102

103

Dimensionless time, tDf

Figure 3-3 Responses for a well intercepting a high conductivity fracture. Log-log scale. No wellbore storage effect CD = 0. Infinite conductivity and uniform flux.

Match results

The kh product is estimated from the pressure match (Eq. 2-8) and the fracture half-length xf from the time match :

- 48 -


xf =

0.000264k 1 (ft, field units) φµ ct TM

xf =

0.000264k 1 (m, metric units) φµ ct TM

( 3-2)

The fracture stimulation is seen as a negative skin during the radial flow regime. With infinite conductivity fracture, this geometrical skin effect is defined from the fracture half-length xf as :

x f = 2 rw e − S (ft, m)

( 3-3)

And, for the uniform flux solution,

x f = 2.7 rw e − S (ft, m)

( 3-4)

Figure 3-4 Flow line geometry near a fractured well.

3-2.3 Linear flow analysis


The half fracture length xf is also estimated from Equation 1-16. 1.2

m LF

0.8 0.4

Uniform flux Infinite condutivity

0 0

0.2

0.4

0.6

0.8

Square root of dimensionless time, √tDf Figure 3-5 Square root of time plot of Figure 3-3. Early time analysis.

- 49 -

1.0



3-2.4 Fractured well with wellbore storage 10

1/2 pe S lo

1

0.5 line

CD=0 10-1 104 103, 10-2 -4 -3 10 10 10-2

10-1

1

10

102

103

Dimensionless time, tDf

Figure 3-6 Responses for a fractured well with wellbore storage. Infinite conductivity fracture. Log-log scale. 3 4 CD = 0, 10 , 10 .

3-2.5 Damaged fracture with wellbore storage Dimensionless Pressure, pD and Derivative, p'D

10

1 S=1 10-1

S=0.3 S=0

10-2 10-2

10-1

1

10

102

103


Figure 3-7 Responses for a fractured well with wellbore storageand skin. Infinite conductivity fracture. Log-log scale. S = 0, 0.3, 1.

3-3 Finite conductivity vertical fracture With the finite conductivity fracture model, there is a pressure gradient along the fracture length. This happens when the permeability of the fracture is not very high compared to the permeability of the formation, especially when the fracture is long.

3-3.1 Characteristic flow regimes 1. 2. 3. 4.

Wellbore storage Bi-linear flow : 1/4 slope straight line. Results : fracture conductivity kfwf. Linear flow: 1/2 slope straight line. Results : fracture half-length xf. Pseudo radial flow : derivative stabilization at 0.5. Results : permeabilitythickness product kh and the geometrical skin S.

- 50 -


3-3.2 Log-log analysis The dimensionless fracture conductivity kfDwfD is defined as :

k fD w fD =

k f wf

( 3-5)

kx f


10

1 0.5 line 10-1

1/2 pe Slo

10-2 /4 Slope 1

10-3 10-1

1

10

102

103

104

105

Dimensionless time, tD /CD

Figure 3-8 Response for a well intercepting a finite conductivity fracture. Loglog scale. No wellbore storage effect CD = 0, kfDwfD = 100.

For large fracture conductivity kfDwfD, the bilinear flow regime is short lived and the 1/4-slope pressure and derivative straight lines are moved downwards. The behavior tends to a high conductivity fracture response (when kfDwfD is greater than 300, see Figure 3-10).


10

1 kfDwfD= 10-1

0.5 line

1 1/2 pe Sl o

10 10-2 100 10-3

/4 Slope 1

10-1

1

10

102

103

104

105


Figure 3-9 Response for a well intercepting a finite conductivity fracture. Loglog scale. No wellbore storage effect CD = 0, no fracture skin, kfDwfD = 1, 10 and 100. Match results

The kh product is estimated from the pressure match (Eq. 2-8) and the fracture half-length xf from the time match (Eq. 3-2). The fracture conductivity kfwf is estimated from the match on the bi-linear flow 1/4 slope. - 51 -


The fracture negative skin is defined by two terms: the geometrical skin of an infinite conductivity fracture (Eq. 3-3), and a correction parameter G to account for the pressure losses in the fracture.  k f wf S LKF = G  kxf 

  + ln 2rw  xf 

( 3-6)

1

rwe / xf

0.5 10-1

10-2 10-1

1

102

10

103

Dimensionless fracture conductivity, kfDwfD

Figure 3-10 Effective wellbore radius for a well with a finite conductivity fracture. Log-log scale.

3-3.3 Bi-linear and linear flow analyses The fracture conductivity kfwf is estimated with Equation 1-18, the fracture halflength form Equation 1-16.

3-3.4 Flux distribution along the fracture

Dimensionless flux, qfD

3

Uniform flux Infinite conductivity Finite conductivity

2

kfDwfD >300

1 5 0.5 0 0

.2

.4

.6

.8

1

Dimensionless distance, x /xf

Figure 3-11 Stabilized flux distribution. Uniform flux, Infinite conductivity (kfDwfD > 300) and Finite conductivity fracture (kfDwfD = 0.5 and 5) models.

- 52 -


3-4 Well in partial penetration 3-4.1 Definition

Sw

kV kH

h hw

zw

Figure 3-12 Geometry of a partially penetrating well.

hw : open interval thickness zw : distance of the center of the open interval to the lower reservoir boundary kH : horizontal permeability kV : vertical permeability

3-4.2 Characteristic flow regimes 1. Wellbore storage. 2. Radial flow over the open interval : a first derivative plateau at 0.5 h/hw. Results : permeability-thickness product for the open interval kHhw, and the skin of the well, Sw. 3. Spherical flow : -1/2 slope derivative straight line. Results : permeability anisotropy kH/kV and location of the open interval in the reservoir thickness. 4. Radial flow over the entire reservoir thickness : second derivative stabilization at 0.5. Results : permeability-thickness product for the total reservoir kHh, and the total skin ST. The total skin combines the wellbore skin Sw and an additional geometrical skin Spp due to distortion of the flow lines, as depicted on Figure 1-21: • Spp is large when the penetration ratio hw/h or the vertical permeability kV is low (high anisotropy kH/kV). • For damaged wells, the product (h/hw)Sw can be larger than 100.

ST =

h S w + S pp hw

( 3-7)

A skin above 30 or 50 is indicative of a partial penetration effect.

- 53 -




Influence of kV / kH 102

10-3 -2 10 -1 10

10 first stabilization 1

0.5 line kV/kH = 10-1

10-2

10-3

10-1 10-1

1

102

10

103

104

105

106


Figure 3-13 Responses for a well in partial penetration with wellbore storage and skin. Log-log scale. hw/h = 1/5 in center of the interval, CD = 33, Sw=0, kV / kH = 0.10, 0.01 and 0.001.

When the vertical permeability kV is low (low kV/kH), the start of the spherical flow regime is delayed (-1/2 derivative slope moved to the right).


Influence of zw/h 102

10 hem i-sp h sph eric al

1

eric al

0.5 line 10-1 10

102

103

104

105

106

107


Figure 3-14 Responses for a well in partial penetration with wellbore storage and skin. Log-log scale. hw/h = 1/10, CD = 6, Sw=0, kV/kH = 0.005, zw/h = 0.5 and 0.2. Match results

The kHh product is estimated from the pressure match (Eq. 2-8). The wellbore skin Sw and the penetration ratio hw/h are estimated from the first radial flow when present (derivative plateau at 0.5 h/hw) :

hw ∆p2nd stab. m2nd line = = ∆p1st stab. h m1st line

( 3-8)

The permeability anisotropy kV/kH and location of the open interval are estimated from the spherical flow -1/2 slope match. - 54 -



3-4.4 Semi-log analysis kV/kH =

40

10-3 10-2 10-1

Slope m

30

∆ Spp

20 10 0 10-1

1

102

10

103

104

105

106


Figure 3-15 Semi-log plot of Figure 3-13. Influence of kV / kH on Spp (Sw=0).

The final semi-log straight line defines kHh and ST. When a first semi-log straight line is seen (radial flow over the open interval), it defines the permeabilitythickness kHhw (penetration ratio hw/h with Eq. 3-8), and the wellbore skin Sw.

3-4.5 Geometrical skin Spp When the penetration ratio hw h and the dimensionless reservoir thicknessanisotropy group (h rw ) k H kV are not very small, Spp can be expressed :

S pp

 π h  h = − 1 ln   2 rw  hw

 hw kH  h  h + ln  h k V  hw  2+ w  h



(z + hw 4)(h − z + hw 4)  ( 3-9) (z − hw 4)(h − z − hw 4)  

With hw h = 0.1 and kH/kV = 1000, Spp = 68 whereas with hw h = 0.5 and kH/kV = 10, Spp = 6 only.

3-4.6 Spherical flow analysis Plot of ∆p versus 1 ∆t . The straight line is frequently not well defined and the analysis is difficult : on example kV/kH =10-3 of Figure 3-13, the spherical flow regime is established between tD/CD=104 and 106. The straight line is very compressed, it ends before 1

t D C D =0.01.

- 55 -



When the open interval is in the middle of the formation, the slope mSPH of the spherical flow straight line gives the permeability anisotropy from Equations 1-20 and 1-21. If the open interval is close to the top or bottom sealing boundary, flow is semi-spherical and the slope mSPH must be divided by two in Equation 1-20. 40 kV/kH =

35

10-3 30

slopes mSPH

10-2 10-1

15 20 0

0.02

0.04

0.06

0.08

0.1

Dimensionless time function, 1 t D CD

Figure 3-16 Spherical flow analysis of responses Figure 3-13. One over square root of time plot.

3-4.7 Influence of the number of open segments

When the open interval is distributed in several segments, the ability of vertical flow is improved compared to the single segment partially penetrating well of same hw. On the examples Figure 3-17 with 1, 2 and 4 segments, the –1/2 slope is displaced towards early time when the number of segments is increased (the global skin is respectively 17.9, 15.9 and 13.9).


102

segments 1 2 4

10

1

10-1 1

10

102

103

104

105


Figure 3-17 Responses for a well in partial penetration with wellbore storage and skin. Log-log scale. One, two or four segments. hw/h = 1/4, CD = 100, Sw=0, kV /kH = 0.10, one segment centered, two or four segments uniformly distributed in the interval.

- 56 -


3-4.8 Constant pressure upper or lower limit In the case of a bottom water / oil contact or a gas cap on top of the producing interval, no final radial flow regime develops after the spherical flow regime: the pressure stabilizes and the derivative drops.


102

10

1 oil water 10-1 1

10

102

103

104

105


Figure 3-18 Responses for a well in partial penetration with a bottom constant pressure boundary. Log-log scale. hw/h = 1/5, CD = 1000, Sw=0, kV/kH = 0.005, one segment on top. The dotted derivative curve describes the response with sealing upper and lower boundaries.

3-5 Horizontal well 3-5.1 Definition kV

kH kH h

L

L

zw

Figure 3-19 Horizontal well geometry.

L : effective half length of the horizontal well zw : distance between the drain hole and the bottom-sealing boundary kH : horizontal permeability kV : vertical permeability

- 57 -


3-5.2 Characteristic flow regimes

Vertical radial flow

Linear flow

Horizontal radial flow Figure 3-20 Horizontal well flow regimes.

1. Wellbore storage.

2. Vertical radial flow : a first derivative plateau at 0.5(h 2 L ) k H kV . Results : the permeability anisotropy kH/kV and the wellbore skin Sw (or the vertical radial flow total skin STV of Equation 3-15). 3. Linear flow between the upper and lower boundaries : 1/2 slope derivative straight line. Results : effective half-length L and well location zw of the horizontal drain. 4. Radial flow over the entire reservoir thickness : second derivative stabilization at 0.5. Results : reservoir permeability-thickness product kHh, and the total skin STH.

Dimensionless Pressure , pD and Derivative p'D

3-5.3 Log-log analysis 10

1

0.5 First stabilization

10-1

10-1

kH h

k H L2

kV k H 2 L

C

10-2 10-2

1/2 pe Slo

1

10

102

103

104

105

106


Figure 3-21 Response for a horizontal well with wellbore storage and skin in a reservoir with sealing upper and lower boundaries. Log-log scale.

With long drain holes, the 1/2 derivative slope is moved to the right and the first derivative stabilization is moved down. When the vertical permeability is increased, the first derivative stabilization is also moved down. - 58 -


Match results

The kHh product is estimated from the pressure match (Eq. 2-8). The effective half-length L and well location zw are estimated from the intermediate time 1/2 slope match. The vertical radial flow total skin STV and the permeability anisotropy kH/kV are estimated from the first radial flow in the vertical plane (permeability thickness 2 kV k H L and derivative plateau at 0.25(h L) k H kV ).

Influence of L


The examples presented Figures 3-22 to 3-41 are generated with h = 100 ft and rw = 0.25 ft. 102

10 5

1

15 L/h = 30

10-1 1

10

102

103

104

105

106


Figure 3-22 Influence of L on pressure and derivative log-log curves. CD =1000, Sw =5, kV /kH =0.004, rw =0.25ft, zw /h =0.5, L =3000, 1500 and 500ft.

When the effective well length is increased, the first derivative stabilization during the vertical radial flow is lowered and the linear flow regime is delayed.


During the linear flow, the location of the half-unit slope straight line is a function of L2. 10

1

L/h = 2.5,

5,

10

10-1 1

10

102

103

104

105

106


Figure 3-23 Influence of L on pressure and derivative log-log curves. SQRT (kV kH)*L constant, (∆p1st stab)D= 0.223. CD =100, Sw =0, kV /kH =0.2, L =250ft; kV /kH =0.05, L =500ft; kV /kH =0.0125, L =1000ft; h =100ft, rw =0.25ft, zw /h =0.5.

- 59 -


When the effective well length is short, the behavior becomes similar to that of a well in partial penetration. Dimensionless Pressure , pD and Derivative p'D

102

10

1 L/h = 2.5,

5,

10

10-1 1

10

102

103

104

105

106


Figure 3-24 Influence of L on pressure and derivative log-log curves. SQRT (kV kH)*L constant, (∆p1st stab)D =1. CD =100, Sw =0, kV /kH =0.01, L =250ft; kV /kH =0.0025, L =500ft; kV /kH =0.000625, L=1000ft; h =100ft, rw =0.25ft, zw /h =0.5.

Influence of zw


10

1

10-1 zw/h = 0.125,

0.25,

0.5

10-2 10-1

1

10

102

103

104

105


Figure 3-25 Influence of zw on pressure and derivative log-log curves. CD =1000, Sw =2, L =1500ft, kV /kH =0.02, h =100ft, rw =0.25ft, zw /h =0.5, 0.25, 0.125.

3-5.4 Dimensionless variables In the derivation of the model, the lengths are transformed in order to introduce the permeability anisotropy between vertical and horizontal directions. The apparent open interval thickness ha, the position of the horizontal drain hole with respect to the lower boundary of the zone zwa, and the apparent wellbore radius are defined as:

- 60 -


ha = h

kH kV kH kV

z wa = z w

( 3-10)

(ft, m)

( 3-11)

(ft, m)

[

1 rwa = rw 4 kV k H +4 k H kV 2

] (ft, m)

( 3-12)

Several authors use the ratio hD of the apparent thickness ha of Equation 3-10, by the well half-length L, as a leading parameter of horizontal well behavior. hD =

ha h = L L

kH kV

( 3-13)

3-5.5 Vertical radial flow semi-log analysis ∆p =

kV k H ∆t 162.6qBµ  − 3.23 log φ µ ct rw2 2 kV k H L  k  1 k + 0.87 S w − 2 log  4 V + 4 H  2  k H kV 

∆p =

kV k H ∆t 21.5qBµ  − 3.10 log φ µ ct rw2 2 kV k H L  k  1 k + 0.87 S w − 2 log  4 V + 4 H  2  k H kV 

(psi, field units)

(Bars, metric units) ( 3-14)

The skin STV measured during the vertical radial flow is expressed with the wellbore skin Sw and the anisotropy skin Sani of Equation 3-34 :

S TV = S w + S ani = S w − ln

4

kV k H + 4 k H kV 2

( 3-15)

Sometimes, the vertical radial flow skin is expressed as S'TV, defined with reference to the equivalent fully penetrating vertical well : ' STV =

h kH STV = 0.5 hD S TV 2 L kV

- 61 -

( 3-16)


3-5.6 Linear flow analysis ∆p =

8128 . qB µ ∆t 1412 . qBµ 1412 . qBµ + Sw + S z (psi, field units) φ ct k H 2 kV k H L 2L h kH h

∆p =

1.246 qB µ ∆t 18.66 qBµ 18.66 qBµ Sw + S z (Bars, metric units)( 3-17) + kH h 2Lh φ c t k H 2 kV k H L

During the linear flow regime, the flow lines are distorted vertically before reaching the horizontal well, producing a partial penetration skin Sz.

π r  kH h k   π z  log  w  1 + V  sin w   kV L k H   h    h 

S z = −1151 .

( 3-18)

3-5.7 Horizontal pseudo-radial flow semi-log analysis ∆p = 162.6 ∆p = 21.5

 k H ∆t qBµ  − 3.23 + 0.87 S TH  (psi, field units) log 2 k H h  φ µ ct rw 

 k H ∆t qBµ  − 3.10 + 0.87 S TH  (Bars, metric units) log 2 k H h  φµ c t rw 

( 3-19)

STH measured during the horizontal radial flow combines S'TV of Equation 3.16 and the geometrical skin SG of the horizontal well (function of the logarithm of the well effective length and a partial penetration skin SzT , close to the linear flow skin Sz of Equation 3.18) :

kH S w + SG kV

( 3-20)

S G = 0.81 − ln

L + S zT rw

( 3-21)

S zT = −1.151

k H h π rw log  kV L  h

S TH =

h 2L

 kV 1 +  kH  − 0.5

- 62 -

kH kV

  π z w   sin    h    h  1 z w z w2  − + L2  3 h h 2  2

( 3-22)



4

zw/h =

3 2

sm pe o Sl

F Slope m VR

0 .125 0.25 0.5

F HR

1 0 10-1

1

10

102

103

104

105



Geometrical skin, SG

2

kV/kH = 1,

0.1,

0.01,

0.001

0 -2 -4

kV/kH = ∞

-6 -8

zw/h =0.5 zw/h =0.1

- 10 102

103

104

105

Dimensionless half length, L/rw

Figure 3-27 Semi-log plot of the geometrical skin SG versus L/rw. Influence of kV/kH. h/rw =1000, zw/h=0.5, 0.1.

2 Geometrical skin, SG

1000

2000

4000

0 h/rw =

-2 -4

500

kV/kH = ∞

-6 -8

zw/h =0.5 zw/h =0.1

- 10 102

103

104

105

Dimensionless half length, L/rw

Figure 3-28 Semi-log plot of the geometrical skin SG versus L/rw. Influence of h/rw. kV/kH =0.1, zw/h=0.5, 0.1.

- 63 -


3-5.8 Discussion of the horizontal well model Several well conditions can produce a pressure gradient in the reservoir, parallel to the wellbore. The vertical radial flow regime is then distorted, and the derivative

response deviates from the usual stabilization at 0.25(h L) k H kV ). During horizontal radial flow, the geometrical skin can be larger or smaller than SG of Equation 3-21 and 3-22.

Non-uniform mechanical skin


10 Skin Swi 1

10-1

10-2 1

10

102

103

104

105

106


Figure 3-29 Influence of non-uniform skin on pressure and derivative curves. CD = 100, L =1000 ft, h =100 ft, rw =0.25 ft, zw/h =0.5, kV/kH=0.1. The well is divided in 4 segments of 500 ft with skins of Swi=4, 4, 4, 4 (uniform damage), Swi=8, 5.33, 2.66, 0 (skin decreasing along the well length), Swi=0, 8, 8, 0 (damage in the central section), Swi=8, 0, 0, 8 (damage at the two ends).

The two ends of the well are more sensitive to skin damage (the total skin STH is more negative on the curve Swi=0, 8, 8, 0).

Finite conductivity horizontal well

When the pressure gradients in the wellbore are comparable to pressure gradients in the reservoir, the flow is three-dimensional (pseudo-spherical), and the derivative is displaced upwards during the early time response. During horizontal radial flow, the total skin STH is less negative.

Partially open horizontal well

When only some sections of the well are open to flow, the response first corresponds to a horizontal well with the total length of the producing segments. Later, each segment acts like a horizontal well, and several horizontal radial flow regimes are established until interference effects between the producing sections are felt. Then, the final horizontal radial flow regime is reached for the complete - 64 -



drain hole. The more distributed the producing sections, the more negative the total skin STH. 10

1

0.5 0.25

10-1

0.125

10-2 1

10

102

103

104

105

106

107


Figure 3-30 Influence of number of open segments on pressure and derivative log-log curves. Total half-length 2000 ft, effective half-length 500 ft. CD =100, 1, 2, 4 segments with Swi =0, ΣLeff= L /4, L =2000ft, h =100ft, rw =0.25ft, zw / h =0.5, kV/kH =0.1.


When the producing segments are uniformly distributed along the drain hole, the total skin STH can be very negative even with a low penetration ratio. On the examples Figure 3-31, with penetration ratios of 100, 50, 25 and 12.5%, STH is respectively –7.9, -7.4, -6.6 and –5.1. 10

1

100% 50% 25% 12.5%

10-1

10-2 1

10

102

103

104

105

106

107


Figure 3-31 Influence of the penetration ratio on pressure and derivative loglog curves. Four segments equally spaced. CD =100, 4 segments with Swi =0, ΣLeff= L /8, L /4, L /2 and L, L =2000ft, h =100ft, rw =0.25ft, zw /h =0.5, kV /kH =0.1.

Non-rectilinear horizontal well

During the vertical radial flow, the upper and lower sealing boundaries can be reached at different times when the well is not strictly horizontal. The transition between vertical radial flow and linear flow is then distorted.

- 65 -



10

1

10-1

10-2 10-1

1

10

102 103 104 Dimensionless time, tD/CD

105

106

107

Figure 3-32 Non-rectilinear horizontal wells. Pressure and derivative curves. CD =100, L =2000ft (500+1000+500), Swi =0, h =100ft, rw =0.25ft, kV / kH =0.1, (zw / h)i=0.5 or 0.95 (average 0.725).

Anisotropic horizontal permeability

In anisotropic reservoirs, horizontal well responses are also sensitive to the well orientation.

kz

ky kx

k y L2

kz ky 2L

kx k y h

Figure 3-33 Horizontal permeability anisotropy. Effective permeability during the three characteristic flow regimes towards a horizontal well.

The final horizontal radial flow regime defines the average horizontal permeability

k H = k x k y . During the linear flow regime, only the permeability ky normal the well orientation is acting. At early time, the average permeability during the vertical radial flow is k z k y .

pD & pD'

1.0E+01

1.0E+00

k y L2 kxky h

1.0E-01

kzk y 2L 1.0E-02 1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

tD/CD

Figure 3-34 Influence of the permeability anisotropy during the three characteristic flow regimes.

- 66 -


When the isotropic horizontal permeability model is used for analysis, the apparent effective half-length is :

La = 4 k y k x L (ft, m)

( 3-23)

(the vertical permeability kz is unchanged). ky

ky

kx

kx

Figure 3-35 Horizontal well normal to the maximum permeability direction : apparent effective length increased.

ky ky kx

kx

Figure 3-36 Horizontal well in the direction of maximum permeability : apparent effective length decreased.

Horizontal wells should be drilled preferably in the minimum permeability direction.

Changes in vertical permeability

In a layered reservoir with crossflow, the horizontal radial flow regime gives the average horizontal permeability : n

k H = ∑ k Hi hi 1

n

∑ hi

( 3-24)

(mD)

1

During the vertical radial flow, the changes of permeability are acting in series. When the contrast in vertical permeability is not too large, the resulting average vertical permeability is defined (assuming the well is centered in layer j) : n j −1    ∑ hi + h j 2  ∑ hi + h j 2 j +1  + n k V = 0.5  j −1 1   ∑ hi kVi + h j 2 kVj ∑ hi kVi + h j 2 kVj   1  j +1

- 67 -

(mD)

( 3-25)


In the example Figure 3-37 with n=3 and j=2, the match with a homogeneous layer

. k H 2 and k V = 0.5 (0.082 + 0.028)k H 2 = 0.0514 k H . is defined with k H = 107 Dimensionless Pressure , pD and Derivative p'D

10

1

10-1 One equivalent layer 10-2 10-1

1

10

102

103

104

105

106

107


Figure 3-37 Horizontal well in a reservoir 3 layers with crossflow. Pressure and derivative log-log curves. CD =100, L =1000ft, Sw =0, h =100ft (30+30+40), rw =0.25ft, zw /h =0.55 (well centered in h2), kH1/kH2=1.5, kH3/kH2=0.8, (kV /kH)1=0.08, (kV /kH)2=0.05, (kV / kH)3=0.03. One layer: kH= (k1h1+ k2h2+ k3h3) / (h1+h2+h3), kV/kH=0.0514.

On Figure 3-38, a thin reduced permeability interval is introduced in the main layer. When a homogeneous layer of total thickness is used for analysis, the effective well length is too small and the vertical permeability over-estimated. Dimensionless Pressure , pD and Derivative p'D

10

1

One layer =

10-1

h1+h2+h3 h3 10-2 10-1

1

10

102

103

104

105

106

107


Figure 3-38 Horizontal well in a reservoir 3 layers with crossflow. Pressure and derivative log-log curves. CD = 100, L = 1000 ft, Sw=0, h =100 ft (h1=45ft, h2=5ft, h3=50ft), k1=k3=100k2, rw =0.25 ft, (kV/kH)i=0.1, zw/h = 0.25 (well centered in h3). • One layer (h1+h2+h3) : k= (k1h1+ k2h2+ k3h3) / (h1+h2+h3), L = 550 ft, Sw=-0.2, kV/kH=0.4, zw/h = 0. 5 (well centered in h1+h2+h3). • One layer (h3) : k= k3, L = 1000 ft, Sw=0, kV/kH=0.1, zw/h = 0. 5 (well centered in h3).

Presence of a gas cap or bottom water drive

When the constant pressure boundary is reached at the end of the vertical radial flow regime (or hemi radial in the examples Figure 3-39), the pressure stabilizes and the derivative drops. It the thickness of the gas zone is not large enough, the derivative stabilizes at late time to describe the total oil + gas mobility thickness. - 68 -



10

1 No gas cap 10-1 hgas = 20 ft 10-2

100 ft

hgas hoil

10-3 10-1

1

10

102

500 ft 103

104

105

106


Figure 3-39 Horizontal well in a reservoir with gas cap and sealing bottom boundary. Pressure and derivative log-log curves. CD = 100, L = 1000 ft, Sw=2, h =100 ft, rw =0.25 ft, (kV/kH)=0.1, zw/h = 0.2 (well close to the bottom boundary). Gas cap : hgas= 0.20, 1.0, 5.0 h, µgas=0.01 µoil, ct gas=10 ct oil.

3-5.9 Other horizontal well models Multilateral horizontal well

As for partially penetrating horizontal wells, the different branches of multilateral wells start to produce independently until interference effects between the branches distort the response. At later time, pseudo radial flow towards the multilateral horizontal well develops.


In the case of intersecting multilateral horizontal wells in reservoir with isotropic horizontal permeability, increasing the number of branches does not improve the productivity. With the examples of Figure 3-40, the total skin STH of the horizontal well is STH =-6.8 (one branch) and respectively –6.6 and –6.2 with two and four branches. 10

1

10-1

10-2 10-1

1

10

102

103

104

105

106

107


Figure 3-40 Multilateral horizontal wells. Pressure and derivative curves. CD = 100, L = 1000 ft (500+500 or 250+250+250+250), Swi=0, h =100 ft, rw=0.25 ft, kV/kH=0.1, zw/h = 0.5.

- 69 -



When the distance between the two producing segments is large enough, the response becomes independent of the orientation of the branches. The responses Figure 3-41 tend to be equivalent to the example with two segments of Figure 330. The total skin STH is more negative when the distance between the branches is increased. For the two multilateral horizontal wells of Figure 3-41, STH =-7.1 (and STH =-6.8 with one branch). 10

1

10-1

10-2 10-1

1

102

10

103

104

105

106

107


Figure 3-41 Multilateral horizontal wells. Pressure and derivative curves. CD = 100, L = 1000 ft (500+500), Swi=0, h =100 ft, rw=0.25 ft, kV/kH=0.1, zw/h = 0.5. The distance between the 2 parallel branches is 2000ft, on the second example the intersection point is at 1000ft from the start of the 2 segments.

Fractured horizontal well

Two configurations are considered : longitudinal and transverse fractures. At early time, the different fractures produce independently until interference effects are felt. With longitudinal fractures, bi-linear and linear flow regimes can be observed, possibly followed by horizontal radial flow around the different fractures. For a single fracture of half-length xf, the slope mBLF and mLF are expressed :

m BLF = 44.11 m BLF = 6.28

m LF = 4.06

qBµ xf kf

φ µ ct k H

qBµ x f k f w f 4 φµ ct k H

qB hx f

m LF = 0.623

w4

qB hx f

µ k H φ ct µ φ ct k H

(psi.hr-1/4, field units)

(Bars.hr-1/4, metric units)

( 3-26)



( 3-27)

With transverse fractures, the flow is first linear in the formation and radial in the fracture, it changes into linear flow, and later into the horizontal radial flow regime around the fracture segments. The radial linear flow regime yields a semi-log straight line whose slope is function of the fracture conductivity. For a single transverse fracture of radius rf, the slope mRLF and mLF are:

- 70 -


m RLF = 81.3

qBµ kf w

m RLF = 10.75

m LF = 5.17

qBµ k f wf

qB hr f

m LF = 0.793

(psi, field units)

µ φ ct k H

qB hr f

( 3-28)


µ φ ct k H



( 3-29)

Once the interference effect between the different fractures is fully developed, the final pseudo radial flow regime towards the fractured horizontal well establishes. As for partially open horizontal wells, the time of start of the final regime is a function of the distance between the outermost fractures.

3-6 Skin factors 3-6.1 Anisotropy pseudo-skin An equivalent transformed isotropic reservoir model of average radial permeability is used, by a transformation of variables in the two main directions of permeability kmax and kmin. With

k = k max k min (mD) x' = x

y' = y

k k max k k min

( 3-30)

=x 4

k min k max

(ft, m)

( 3-31)

= y4

k max k min

(ft, m)

( 3-32)

The wellbore is changed into an ellipse whose area is the same as in the original system, but the perimeter is increased. The elliptical well behaves like a cylindrical hole whose apparent radius is the average of the major and minor axes, and produces an apparent negative skin :

rwa =

1 rw 2

[

4

k min k max + 4 k max k min

- 71 -

]

(ft, m)

( 3-33)


Sani = − ln

= − ln

4

k min k max + 4 k max k min 2 k min + k max

( 3-34)

2 k

Sani is in general low but, for horizontal wells, when kV/kH <<1, Sani =-1 may be observed.

3-6.2 Geometrical skin A

B

C


Figure 3-42 Configuration of wells A, B and C. A = fully penetrating vertical well, B = well in partial penetration, C = horizontal well.

102 SG>0 10 SG<0

1 10-1 10-2 10-2

A : vertical well B : partial penetration C : horizontal well

10-1

1

10

102

103

104

105

106


Figure 3-43 Pressure and derivative response of wells A, B and C. Log-log scale.

- 72 -

Dimensionless Pressure , pD


30 A : vertical well B : partial penetration C : horizontal well

20

SG>0

10 SG<0 0 10-2

10-1

1

10

102

103

104

105

106

Dimensionless time, tD/CD Figure 3-44 Semi-log plot of Figure 3-43 examples.

3-6.3 The different skin factors

Name Sw SG

Sani SRC

S2φ D.q

Description

Type

Infinitesimal skin at the wellbore.

Positive or negative

Geometrical skin due to the streamline curvature (fractured, partial penetration, slanted or horizontal wells). Skin factor due to the anisotropy of the reservoir permeability. Skin factor due to a change of reservoir mobility near the wellbore (permeability or fluid property, radial composite behavior). Skin factor due to the fissures in a double porosity reservoir. Turbulent or inertial effects on gas wells.

Positive or negative

- 73 -

Negative Positive or negative Negative Positive

- 74 -

4 - FISSURED RESERVOIRS - DOUBLE POROSITY MODELS 4-1 Definitions 4-1.1 Permeability The fluid flows to the well through the fissure system only and the radial permeability of the matrix system does not contribute to the mobility (km = 0). The permeability thickness product kh estimated by the interpretation is used to define an equivalent bulk permeability of the fissure network, over the complete thickness h:

kh = k f h f (mD.ft, mD.m)

( 4-1)

Matrix Fissure Vug

Figure 4-1 Example of double porosity reservoir, fissured and multiple-layer formations.

4-1.2 Porosity φf and φm : ratio of pore volume in the fissures (or in the matrix), to the total volume of the fissures (of the matrix).

Vf and Vm : ratio of the total volume of the fissures (or matrix) to the reservoir volume (Vf + Vm = 1).

φ = φ f V f + φ mVm

( 4-2)

In practice, φf and Vm are close to 1. The average porosity of Equation 4.2 can be simplified as :

φ = Vf + φm

( 4-3)

4-1.3 Storativity ratio ω

ω=

(φ Vct ) f (φ Vct ) f = (φ Vct ) f + (φ Vct )m (φ Vct ) f +m - 75 -

( 4-4)

Chapter 4 - Fissured reservoirs

4-1.4 Interporosity flow parameter λ

λ = α rw2

km kf

( 4-5)

α is related to the geometry of the fissure network, defined with the number n of families of fissure planes. For n = 3, the matrix blocks are cubes (or spheres) and, for n = 1, they are slab.

α=

n(n + 2) -2 -2 (ft , m ) rm2

( 4-6)

rm is the characteristic size of the matrix blocks. It is defined as the ratio of the volume V of the matrix blocks, to the surface area A of the blocks :

rm = nV A (ft, m)

( 4-7)

When a skin effect (Sm in dimensionless term) is present at the surface of the matrix blocks, the matrix to fissure flow is called restricted interporosity flow.

Sm =

k m hd rm k d

( 4-8)

km rm

hd kd n=3, cubes

n=1, slabs

Figure 4-2 Matrix skin. Slab and sphere matrix blocks.

The analysis with the restricted interporosity flow model (pseudo-steady state interporosity flow) provides the effective interporosity flow parameter λeff :

λ

eff

=n

rw2 k d rm hd k f

( 4-9)

λeff is independent of the matrix block permeability km.

- 76 -


4-1.5 Dimensionless variables

kh ∆p (field units) 1412 . qBµ kh ∆p (metric units) pD = 18.66qBµ

pD =

tD kh ∆t = 0.000295 (field units) CD µ C tD kh ∆t (metric units) = 0.00223 CD µ C

C Df = C Df =

( 4-10)

( 4-11)

0.8936C (field units) (φ Vct ) f hrw2 0.1592C

(φVct ) f hrw2

(metric units)

C Df + m =

0.8936C (field units) (φ Vct ) f +m hrw2

C Df + m =

0.1592C (metric units) (φ Vct ) f + m hrw2

( 4-12)

( 4-13)

The storativity ratio ω correlates the two definitions of dimensionless wellbore storage :

C Df + m = ω C Df

( 4-14)

4-2 Double porosity behavior, restricted interporosity flow (pseudo-steady state interporosity flow) 4-2.1 Log-log analysis Pressure type curves

Three component curves : 1. - (CDe2S)f at early time, during fissure flow. 2. - λeff e-2S during transition regime, between the two homogeneous behaviors. 3. - (CDe2S)f+m at late time, when total system behavior is reached. - 77 -


A double porosity response goes from a high value (CDe2S)f when the storativity corresponds to fissures, to a lower value (CDe2S)f+m when total system is acting.


102

CDe2S = 1030 λe-2S = 10-30 1010 10-10 103 10-6 5 0.1 10-2 5x10-3

Start of semi-log radial flow

10

1

0.5

10-1 10-1

1

102

10

103

104


Figure 4-3 Pressure type-curve for a well with wellbore storage and skin in a double porosity reservoir, pseudo steady state interporosity flow.

Typical responses

The limit "approximate start of the semi-log straight line" shows that the wellbore storage stops during the fissure regime with example A. With example B, wellbore storage lasts until the transition regime and, during the fissure regime, the fissure (CDe2S)f curve does not reach the semi-log straight-line approximation.


102

CDe2S


λe-2S = 10-30

= 1030 1010 105 104 1 0.1 5x10-3

B

10

10-7 3x10-4 10-2

1

10-1 10-1

A

1

10

102

103

104

105

Dimensionless time, tD/CD Figure 4-4 Pressure examples for a well with wellbore storage and skin in a double porosity reservoir, pseudo steady state interporosity flow. 2S 2S -2S o = A : (CDe )f = 1, (CDe )f+m = 0.1, ω = 0.1, λeffe = 3.10-4. 2S 2S -2S ■ = B : (CDe )f = 105, (CDe )f+m = 104, ω = 0.1, λeffe = 10-7.

On semi-log scale, two parallel straight lines are present with example A. With example B, only the total system straight line is seen.

- 78 -



10 em slop

8

B

6 em sl o p

4 A slop

2

em

0 10-1

1

102

10

103

104

105


Figure 4-5 Semi-log plot of Figure 4-4 examples.


102

CDe2S λe-2S = 10-30

10

B

1030

10-7

A

1010

3x10-4

105

= 1030 1010 1054 10 1 0.1 5x10-3

10-2

1

B 1

A 3x10-5

10-3

0.1 λCD/ω(1-ω) = 10-2

10-1 10-1

1

102

10

λCD/(1-ω)

3x10-4

103

104

105


Figure 4-6 Pressure and derivative examples of Figure 4-4 for a well with wellbore storage and skin in a double porosity reservoir, pseudo steady state interporosity flow. λeffCDf+m/ω(1-ω) =10-2, 3x10-4. λeffCDf+m/(1-ω) = 10-3, 3x10-5.

With the derivative, example A shows two stabilizations on 0.5. The derivative of example B stabilizes on 0.5 only during the total system homogeneous regime. On the derivative type-curve, the transition is described with two curves, labeled

(λ

eff

CD f +m

) [ω (1 − ω )] (decreasing derivative) and (λ

eff

CD f +m

) (1 − ω ) .

Match results

kh = 141.2qBµ (PM ) (mD.ft, field units) kh = 18.66qBµ (PM ) (mD.m, metric units) C = 0.000295

kh  1    (Bbl/psi, field units) µ  TM 

- 79 -

( 2-8)


C = 0.00223

S = 0.5 ln

(C

kh  1  3  (m /Bars, metric units)  µ  TM 

De

2S

)

( 2-9)

f +m

( 4-15)

C Df + m

(C e ) ω= (C e ) 2S

D

f +m

( 4-16)

2S

D

(

f

)

λ eff = λ eff e −2 S e 2 S

( 4-17)

Pressure and derivative response


When the three characteristic regimes of the restricted interporosity flow model are developed, the derivative exhibits a valley shaped transition between the two stabilizations on 0.5. 10-2

10 0.5 line

1

10-1 10-1

1

10

102

103

104

105


Figure 4-7 Pressure and derivative response for a well with wellbore storage in double porosity reservoir, pseudo-steady state interporosity flow. CDf+m = 103, S = 0, ω = 0.1, λeff= 6.10-8 (CDe2Sf =104, λeffe-2S= 6.10-8 and CDe2Sf+m = 103)

4-2.2 Influence of the heterogeneous parameters ω and λeff Influence of ω

With small ω values, the transition regime from CDe2Sf to CDe2Sf+m is long. On the derivative responses, the transition valley drops when ω is reduced. On semilog scale, the first straight line is displaced upwards and the horizontal transition between the two parallel lines is longer.

- 80 -



102 ω = 10-3

10

10-1

0.5

1 10-1

10-1

10-2 ω = 10-3

10-2 10-3 10-1

1

102

10

103

104

105

106

107

108


Figure 4-8 Double porosity reservoir, pseudo-steady state interporosity flow. Influence of ω. Log-log scale. CDf+m =1, S =0, λeff=10-7 and ω =10-1, 10-2 and 10-3 Dimensionless Pressure , pD

10 ω = 10-3

8

10-2

m pe slo

10-1

6 slo

4

m pe

2 0 10-1

1

10

102

103

104

105

106

107

108


Figure 4-9 Semi-log plot of Figure 4-8. Influence of λeff

The interporosity flow parameter defines the time of end of the transition regime. The smaller is λeff, the later the start of total system flow. On the pressure curves, the transition regime occurs at a higher amplitude and, on the derivative responses, the transition valley is displaced towards late times. Dimensionless Pressure , pD and Derivative p'D

102 λ = 10-8

10

10-6 1 10-1 λ =

10-6 ,

10-7 ,

10-8

104

105

10-2 10-1

1

10

102

103

106

107


Figure 4-10 Double porosity reservoir, pseudo-steady state interporosity flow. Influence of λeff. Log-log scale. CDf+m =100, S =0, ω =0.02 and λeff=10-6, 10-7 and 10-8

- 81 -



12 λ

10-8

= 10-7

8

10-6 em slop

slop

em

4

0 10-1

1

102

10

103

104

105

106

107



4-2.3 Analysis of the semi-log straight lines


10 8

Double porosity

6 4 2

em slop

0 10-1

1

Homogeneous

10

102

103

104

105


Figure 4-12 Semi-log plot of homogeneous and double porosity responses. CD = CDf+m = 100, S = 0, ω = 0.01 and λeff= 10-6

During fissure flow, when the first semi-log line is present,

  k log ∆t + log  (psi, field units) 3 . 23 0 . 87 − + S (φVct ) f µ rw2    qBµ  k log ∆t + log  (Bars, metric units)(4-18) 3 . 10 0 . 87 ∆p = 21.5 − + S 2 kh  ( ) φ µ V c r  t w f  ∆p = 162.6

qBµ kh

The second line, for the total system regime is :

∆p = 162.6

 qBµ  k log ∆t + log  (psi, field units) 3 . 23 0 . 87 S − + 2 kh  ( ) φ µ Vc r  t f +m w 

- 82 -


∆p = 21.5

 qBµ  k log ∆t + log  (Bars, metric units)( 4-19) 3 . 10 0 . 87 S − + 2 kh  ( ) φ µ V c r  t f +m w 

The vertical distance δp between the two lines gives ω :

ω = 10 −δp m

( 4-20)

When only the first semi-log straight line for fissure regime is present, if the total storativity is used instead of that of the fissure system, the calculation of the skin gives an over estimated value Sf :

S f = S + 0.5 ln

1

( 4-21)

ω

4-2.4 Build-up analysis Log-log pressure build-up analysis

When the production time tp is small, the three characteristic regimes of a double porosity response are not always fully developed on build-up pressure curves. Whatever long are the three build-up examples of Figure 4-13, only example A3 exhibits a clear double porosity response. The build-up curve A1 does not show a double porosity behavior, but only the build-up response of the fissures. For example A2, the build-up curve flattens at the same ∆p level as the λeffe-2S transition, there is no evidence of total system flow regime.


Homogeneous behaviour, ( fissures CDe2Sf= 1 and total system CDe2Sf+m= 0.1) Double porosity, ( drawdown and build-up) 10 A3 A2 A1

1 tp1 = 102

tp2 = 9x103

tp3 = 3x105

10-1 10-1

1

10

102

103

104

105

106


Figure 4-13 Drawdown and build-up pressure responses for a well with wellbore storage and skin in double porosity reservoir, pseudo-steady state interporosity flow. Log-log scale. CDf+m = 0.1, S = 0, ω = 0.1, λeff= 3.10-4 (CDe2Sf =1, λeffe-2S= 3.10-4 and CDe2Sf+m = 0.1). tpD/CD = 100 (A1), 9.103 (A2), 3.105 (A3).

- 83 -



8 tp3 = 3x105

drawdown build-up 6

m pe slo A3

tp2 = 9x103

4 pe slo

2

tp1 = 102

A2

m

A1

0 10-1

1

102

10

103

104

105

106


Figure 4-14 Semi-log plot of drawdown and build-up pressure responses of Figure 4-13.

Horner & superposition analysis

Dimensionless Pressure Difference, (p - pi)D

In example A3, the initial pressure pi is obtained by extrapolation of the second straight line, the first one extrapolates to pi + m ln (1/ω). If the drawdown stops during the transition (example A2), only the first semi-log straight is seen and its extrapolated pressure p* is between pi and pi + m ln (1/ω), depending upon tp.

0 slo pe

m

-2 A1 -4

p* > pi

slo pe m

A2 p* = pi

A3

-6 10-1

1

10-2

10-3

10-4

10-5

10-6

Horner time, (tpD+ tD)/ tD

Figure 4-15 Horner plot of the three Build-ups of Figure 4-13. A1 (tpD/CD = 100), A2 (tpD/CD = 9.103) and A3 (tpD/CD = 3.105). Derivative build-up analysis

Dimensionless Pressure Derivative p'D

1 A3

0.5 10-1

A2

A1

Drawdown Build-up 10-2 10-1

1

10

102

103

104

105

106


Figure 4-16 Drawdown and build-up derivative responses of Figure 4-13.

- 84 -


4-3 Double porosity behavior, unrestricted interporosity flow (transient interporosity flow) 4-3.1 Log-log analysis Pressure type-curve

Two pressure curves : 1. - β ' at early time, during transition regime before the homogeneous behavior of the total system 2. - (CDe2S)f+m later, when the homogeneous total system flow is reached The two families of curves have the same shape: the β ' transition curves are equivalent to CDe2S curves whose pressure and time are divided by a factor of two.

β ' is defined as :

β '= δ '

(C

De

2S

λe

)

f +m

( 4-22)

−2 S

The constant δ' is related to the geometry of the matrix system. For slab matrix blocks δ '=1.89, and for sphere matrix blocks δ ' = 1.05. 102 Dimensionless Pressure, pD


CDe2S = 1030 β ' = 1030 1010

10

103 5 0.1

1010 103 5 0.1 5x10-3

1

10-1 10-1

1

102

10

103

104


Figure 4-17 Pressure type-curve for a well with wellbore storage and skin in a double porosity reservoir, transient interporosity flow.

- 85 -


Typical responses

A long transition on a β ' curve is seen on example A. With example B, the wellbore storage is large, and the transition is shorter on the tD/CD time scale.


102

CDe2S


β' = 1030

B

10

1010 106 5

A

= 1030 1010 6x103 10 0.1

1

10-1 10-1

1

10

102

103

104

105

Dimensionless time, tD/CD Figure 4-18 Pressure examples for a well with wellbore storage and skin in a double porosity reservoir, transient interporosity flow, and slab matrix blocks. 2S -2S o = A : (CDe )f+m = 10, ω = 0.001, β' = 106, λe = 1.8914*10-5. 2S -2S ■ = B : (CDe )f+m = 6.103, ω = 0.001, β' = 1010, λe = 1.1348*10-6.

On semi-log scale, example A shows a first straight line of slope m/2 during transition, before the total system straight line of slope m. With example B, only the total system straight line is present.


10 em slop

8 B 6 4

slope m

em

A

2 0 10-1

slop

/2

1


104

105

Figure 4-19 Semi-log plot of Figure 4-18 examples.

- 86 -



102

CDe2S β' = 1030

B

10

1010 106 5

A 6x106

1

10

4

1

1010 6x103 10 0.1

B

5 λCD/(1-ω)2 = 3x10-2

10-1 10-1

1030

= 1030

A 3x10-3

102

10

3x10-4

103

3x10-5

104

105


Figure 4-20 Pressure and derivative examples of Figure 4-18. 2 λCDf+m (1-ω) = 3.10-2, 3.10-3, 3.10-4, 3.10-5.

With the derivative, example A shows a first stabilization on 0.25 before the final stabilization on 0.5 for the total system homogeneous regime. The derivative of example B exhibits only a small valley before the stabilization on 0.5. The end of transition, and the start of the total system homogeneous regime, is described by a (λ C D )

(1 − ω )2

derivative curve.

Match results

On a double porosity response with unrestricted interporosity flow, after the wellbore storage hump the derivative exhibits a first stabilization on 0.25 before the final stabilization on 0.5.

λ =δ'

(C

De

2S

β 'e

)

f +m −2 S

( 4-23)

ω is difficult to access with the transient interporosity flow model. Slab and sphere matrix blocks

With the two types matrix geometry, the pressure curves look identical but the derivatives are slightly different. At late transition time, the change from 0.25 to the 0.5 level is steeper on the curve generated for slab matrix blocks.

- 87 -



10

1

0.5 sphere 0.25

10-1 10-1

1

slab

102

10

103

104

105


Figure 4-21 Double porosity reservoir, transient interporosity flow, slab and sphere matrix blocks. Log-log scale. CDe2Sf+m=1, β'=104 and ω=10-2. Slab: λe-2S = 1.89 10-4, Sphere: λe-2S = 1.05 10-4.

4-3.2 Influence of the heterogeneous parameters ω and λ


Influence of ω 102 10

ω = 10-3 ω = 10-1

1

0.5

ω = 10-1 ω = 10-3

10-1 10-2 10-1

1

10

102

103

0.25

104

105

106

107

108


Figure 4-22 Double porosity reservoir, transient interporosity flow, slab matrix blocks. Influence of ω on pressure and derivative curves. CDf+m =1, S =0, λ =10-7 and ω =10-1, 10-2 and 10-3 Dimensionless Pressure , pD

10 8 6

ω = 10-3 m /2 slope

m pe sl o

10-2

4

10-1

2 0 10-1

1

10

102

103

104

105

106



- 88 -

107

108



Influence of λ 102 λ = 10-8

10

λ = 10-6 λ = 10-6, 10-7,

1

10-8 0.5

10-1

0.25

10-2 10-1

1

10

102

103

104

105

106

107

108


Figure 4-24 Double porosity reservoir, transient interporosity flow, slab matrix blocks. Influence of λ on pressure and derivative curves. CDf+m =100, S =0, ω =0.02 and λ =10-6, 10-7 and 10-8


10

m pe sl o λ = 10-8

8

10-7 10-6

6 m/2 slope

4 2 0 10-1

1

102

10

103

104

105

106

107

108




4-3.3 Build-up analysis 1

A3 0.5 A2

10-1

A1

Drawdown Build-up -2

10

10-1

1

10

102

103

104

105

106


Figure 4-26 Drawdown and build-up derivative responses, double porosity reservoir, unrestricted interporosity flow, slab matrix blocks. CDf+m = 0.1, S = 0, ω = 0.1, λ = 3.10-4. tpD/CD = 100 (A1), 9.103 (A2), 3.105 (A3).

- 89 -


4-4 Complex fissured reservoirs


4-4.1 Matrix skin 10

0.5

1 0.25 Sm= 0

10-1

0.1 1 10

10-2 1

10

102

103

104

100

105

106

107



Figure 4-27 Double porosity reservoir, transient interporosity flow, slab matrix blocks with interporosity skin. CDf+m = 1, S = 0, ω = 0.01, λ = 10-5. Sm = 0, 0.1, 1, 10, 100. 1

10-1

10-2 10

Sm= 1 102

103

10

104

105

100 106

107



Figure 4-28 Comparison of Figure 4-27 derivative responses with the restricted interporosity flow model. λ eff = 2.500x10-6 (Sm = 1), λ eff = 3.323x10-7 (Sm = 10), λ eff = 3.333x10-8 (Sm = 100).

10

0.5

1 0.25 10-1

Sm= 0 0.1 1

10

10-2 1

10

102

103

104

105

100 106

107


Figure 4-29 Double porosity reservoir, transient interporosity flow, sphere matrix blocks with interporosity skin. CDf+m = 1, S = 0, ω = 0.01, λ = 10-5. Sm = 0, 0.1, 1, 10, 100.

- 90 -



1

10-1

10-2 10

Sm= 1 102

103

104

10

100

105

106

107


Figure 4-30 Comparison of Figure 4-29 derivative responses with the restricted interporosity flow model. λ eff = 1.66x10-6 (Sm = 1), λ eff = 1.96x10-7 (Sm = 10), λ eff = 2.00x10-8 (Sm = 100).


10 unrestricted slab unrestricted sphere

1

0.5

0.25 10-1 restricted 10-2 10-1

1

10

102

103

104

105


Figure 4-31 Log-log plot of pressure and derivative responses for a well with wellbore storage and skin in double porosity reservoir, restricted and unrestricted interporosity flow, slab and sphere matrix blocks. 2S -2S CDf+m = 1, S = 3, ω = 0.02, λ = 10 -4. CDe f+m=403, λe = 2.48*10-7. Slab: β' = 3.07*10 9, Sphere: β' = 1.71*10 9

4-4.2 Triple porosity solution The model considers two sizes of matrix blocks. The blocks are uniformly distributed in the reservoir. Alternatively, the matrix blocks can be fissured.

fissure, block 1, block 2

Two block sizes

fissure, microfissure, block

Fissured matrix blocks

Figure 4-32 Multiple matrix blocks.

- 91 -


When the blocks are uniformly distributed, δi defines the contribution of the group i to the total matrix storage (δ1 + δ2 =1):


δ i=

(φVct = )mi (φVct )mi = (φVct )m1 + (φVct )m2 (φVct )m

( 4-24)

10

0.5

1

fissure

10-1

10-2

1

fissure + group 1

102

10

103

104

total system

105

106

107


Figure 4-33 Triple porosity reservoir, pseudo steady state interporosity flow, two sizes of matrix blocks uniformly distributed, different λeff. CDf+m = 1, S = 0, ω = 0.01, λeff1 =10-5, δ1 =0.1, λeff2 =5x10-7, δ2 =0.9.


10 m pe (slo

8 6 1 up gro + ure fiss

ur e fiss

4 2

)

m ste l sy a t o t

0 10-1

1

10

102

103

104

105

106

107



Figure 4-34 Semi-log plot of Figure 4-33 example.

10

0.5

1 group 1 fissure

10-1

total system group 2

10-2 1

10

102

103

104

105

106

107


Figure 4-35 Triple porosity reservoir, pseudo steady state interporosity flow, two sizes of matrix blocks uniformly distributed, same λeff. CDf+m = 1, S = 0, ω = 0.01, λeff1 = λeff2 =10-6 , δ1 =0.1, δ2 =0.9. The dashed curves describe the double porosity responses for only blocks 1 (small valley) and only blocks 2.

- 92 -



10 group 1 8

group 2

6 4 2

e lop e (s r u fiss

m)

ls tota

m pe slo ( em yst

)

0 10-1

1

10


105

106

107

Figure 4-36 Semi-log plot of Figure 4-35 example. The thin curves describe the double porosity responses for only blocks 1 (final semi-log straight line for fissures + blocks 1) and only blocks 2 (final semi-log straight line for fissures + blocks 2).

- 93 -

- 94 -

5 - BOUNDARY MODELS 5-1 One sealing fault 5-1.1 Definition L

L

Image (q)

Well (q)

LD =

L rw

( 5-1)

5-1.2 Characteristic flow regimes 1. Radial flow 2. Hemi-radial flow

Dimensionless Pressure pD and Derivative p'D


101 1 1

10-1 10-1

0.5

1

101

102

103

104

105


Figure 5-1 Pressure and derivative response for a well with wellbore storage and skin near one sealing fault in a homogeneous reservoir. Log-log scale. CD = 104, S = 0, LD = 5000.

- 95 -

Chapter 5 - Boundary models


102

101 LD=100 1 300

1000

3000

10-1 101

1

102

103

104

105


Figure 5-2 Responses for a well with wellbore storage and skin in a homogeneous reservoir limited by one sealing fault. Several distances. CD = 100, S = 5, LD = 100, 300, 1000, 3000.

5-1.4 Semi-log analysis The time of intercept ∆tx between the two semi-log straight lines can be used to estimate the distance between the well and the sealing fault :

k∆t x (ft, field units) φµ ct

L = 0.01217 L = 0.0141

k∆t x (m, metric units) φµ ct

( 1-22)

Dimensionless Pressure pD

20 LD=100 300 1000 3000

m e2 slop

15 10

slope m

5 0 1

101

102

103



- 96 -

104

105


5-2 Two parallel sealing faults 5-2.1 Definition

L2 Well L1

5-2.2 Characteristic flow regimes

1. Radial flow 2. Linear flow

5-2.3 Log-log analysis Dimensionless Pressure pD and Derivative p'D

102 ºA

ºB

101

1 0.5 10-1 10-1

1 pe slo

B

1

101

102

/2

A 103

104

105


Figure 5-4 Responses for a well with wellbore storage in a homogeneous reservoir limited by two parallel sealing faults. Log-log scale. One channel width, two well locations. CD = 3000, S = 0, L1D = L2D = 3000 (curve A) and L1D = 1000, L2D = 5000 (curve B).

- 97 -



102 L1D= L2D= 500 1000 2500 5000

101

1

10-1 10-1

101

1

102

103

104

105


Figure 5-5 Responses for a well with wellbore storage and skin near two parallel sealing faults. Homogeneous reservoir. The well is located midway between the two boundaries, several distances between the two faults are considered. CD = 300, S = 0 L1D = L2D = 500, 1000, 2500 and 5000.

5-2.4 Semi-log analysis On semi-log scale, only one straight line is present. During the late time linear flow, the responses deviate in a curve above the radial flow line. The time of end of the semi-log straight line is function of the channel width and the well location.


40 L1D= L2D=

500

30 1000 20 2500 5000

10 slope m

0 10-1

1

101

102

103



- 98 -

104

105



20

15

B

10

A slope

5

m

0 10-1

1

101

103

102

105

104



5-2.5 Linear flow analysis


40 L1D= L2D= 500 slope mch

30

1000 20

2500 5000

10 0 0

50

100

150

200

250

300

350

(tD /CD)1/2

Figure 5-8 Square root of time plot of Figure 5-5.

The pressure change ∆p is plotted versus the square root of the elapsed time ∆t . The slope mch and the intercept ∆pchint of the linear flow straight line are used to estimate the channel width and the well location.

mch = 8.133

µ qB (psi.hr-1/2, field units) h(L1 + L2 ) kφ ct

mch = 1.246

µ qB (Bars.hr-1/2, metric units) h(L1 + L2 ) kφ ct

L1 + L2 = 8.133

qB hmch

µ kφ ct

(ft, field units)

L1 + L2 = 1.246

qB hmch

µ kφ ct

(m, metric units)

- 99 -

( 5-2)

( 5-3)


kh ∆pchint − S (field units) 141.2qBµ kh = ∆p ch int − S (metric units) 18.66 qBµ

S ch = S ch

 L + L2 −Sch L1 1 = arcsin 1 e L1 + L2 π  2π rw

( 5-4)

  

( 5-5)


5-2.6 Build-up analysis 102 ºD

ºC 101

D

1

pe slo

0.5 10-1

10-1

1

101

102

1 /2

C 103

104

105

106


Figure 5-9 Build-up responses for a well with wellbore storage in a homogeneous reservoir limited by two parallel sealing faults. One channel width, two well locations. The dotted curves describe the drawdown responses. CD = 3000, S = 0, L1D = L2D = 5000 (curve C) and L1D = 2000, L2D = 8000 (curve D). Production time: tpD/CD = 2000.

9 Dimensionless Pressure pD

D 8 C 7

slop em

6 5 4 3 1

101

102

103

(tpD +tD )/ tD

Figure 5-10 Horner plot of Figure 5-9.

The extrapolation p* of the Horner straight line does not correspond to the infinite shut-in time pressure.

- 100 -



9 D

slope mch

8 C 7 6 5 4 3 0

20

10

30

[(tpD +tD )/CD]1/2 -

40

50

[tD /CD]1/2

Figure 5-11 Square root of time plot of Figure 5-9. pD versus [(tpD+tD)/CD]1/2 - [tD/CD]1/2.

For an infinite channel, when both the drawdown and the shut-in periods are in linear flow regime, the superposition function is expressed as

t p + ∆t − ∆t .

The extrapolation of the linear flow straight line to infinite shut-in time, at

t p + ∆t − ∆t = 0 , is used to estimate the initial reservoir pressure.

5-3 Two intersecting sealing faults 5-3.1 Definition

L2

θ

Well

L1

θw

The angle of intersection θ between the faults is smaller than 180°, the wedge is otherwise of infinite extension. LD is the dimensionless distance between the well and the faults intercept. The well location in the wedge is defined with θw. The distances L1 and L2 between the well and the sealing faults are expressed as :

L1 = LD rw sin θ w (ft, m)

( 5-6)

- 101 -


L2 = LD rw sin(θ − θ w ) (ft, m)

( 5-7)

5-3.2 Characteristic flow regimes 1. Radial flow 2. Linear flow 3. Fraction of radial flow



If for example the angle between the faults is 60° (π/3), the wedge is 1/6 of the infinite plane (2π), and the derivative stabilizes at 3.

102 ºB ºA

101

180°/ θ = 3 B

1 0.5

A

102

103

10-1 10-1

1

101

104

105


Figure 5-12 Responses for a well with wellbore storage in a homogeneous reservoir limited by two intersecting sealing faults. Log-log scale. CD = 3000, S = 0, LD = 5000, θ = 60°, θw = 30° (curve A) and θw = 10° (curve B).

θ = 360°

∆p1st stab. ∆p2nd stab.

( 5-8)

Between the two stabilizations, the derivative follows a half unit slope straight line.

- 102 -



102 θ= 10° 20° 45° 90° 135°

10° 101 180° 1 180° 10-1 10-1

101

1

102

103

104

105

106


Figure 5-13 Responses for a well with wellbore storage in a homogeneous reservoir limited by two intersecting sealing faults. Log-log scale. Several angles of intersection θ, the well is on the bisector θw = 0.5 θ, the distance to the two faults is constant L1D = L2D = 1000, the distance LD to the fault intercept changes. CD = 1000, S = 0, θ = 10°, LD = 11473; θ = 20°, LD = 5759; θ = 45°, LD = 2613; θ = 90°, LD = 1414; θ = 135°, LD = 1082; θ = 180°, LD = 1000.

5-3.4 Semi-log analysis On a complete response, two semi-log straight lines can be identified. The first, of slope m, describes the infinite acting regime. The second, with a slope of (360/θ)m, defines the fraction of radial flow limited by the wedge.


60

θ

= 10° 20°

40

slope (360°/θ) m 45° 90° 135°

20 slope m

0 10-1

1

180° 101

102

103

104

105

106



θ = 360°

m1st line m2nd line

( 5-9)

The end of the first semi-log straight line, and the level of the second straight line, is a function of the well location in the wedge.

- 103 -


6m

B

15

sl op e


20

A

10 m slope

5 0 10-1

1

101

102

103

104

105



5-4 Closed system 5-4.1 Definition A rectangular reservoir shape is considered. The well is at dimensionless distances L1D, L2D, L3D, and L4D from the four sealing boundaries, the dimensionless area of the closed reservoir is expressed as:

A = ( L1D + L3 D )( L2 D + L4 D ) rw2

( 5-10)

5-4.2 The pseudo steady state regime

Pressure, p

pi

ppseudo ste ady

state slope m*

Time, t

Figure 5-16 Drawdown and build-up pressure response. Linear scale. Closed system.

The well, at initial reservoir pressure pi, is produced at constant rate until all reservoir boundaries are reached. At the end of the drawdown, the pseudo steady state regime is shown by a linear pressure trend. The well is then closed for a shutin period, the pressure builds up until the average reservoir pressure p is reached, - 104 -


and the curve flattens. The difference pi − p , between the initial pressure and the final stabilized pressure defines the depletion.

5-4.3 Log-log behavior


On log-log scale, a straight line of slope unity on the late time drawdown pressure and derivative curves characterizes the pseudo steady state flow regime. During build-up, the pressure curves flattens to ∆ p and the derivative drops.

102 slope 1

ºB ºA

101

A&B B 1 0.5 A 10-1 10-1

1

101

102

103

104

105

106


Figure 5-17 Drawdown and build-up responses for a well with wellbore storage in a closed square homogeneous reservoir. Log-log scale. The dotted curves describe the drawdown responses. CD = 25000, S = 0. Curve A: L1D = L2D = L3D = L4D = 30000. Curve B: L1D = L2D = 6000, L3D = L4D = 54000. (tp/C)D = 1000. (tp/C)D = 1000.

5-4.4 Drawdown analysis


Log-log analysis 102

101 A/rw2 = 106

107

108

1 0.5 10-1 10-1

1

101

102 103 104 Dimensionless time, tD /CD

105

106

Figure 5-18 Drawdown responses for a well with wellbore storage in a closed square homogeneous reservoir. Three reservoir sizes, the well is centered or near one of the boundaries. CD = 100, S = 0, A/rw2 = 106, 107, 108 (L1D = 200).

- 105 -



102 ºD

slope 1

ºC

101 D

C

1

pe slo

0.5

1/2

10-1 10-1

101

1

102

103

104

105

106


Figure 5-19 Pressure and derivative drawdown responses for a well with wellbore storage in a closed channel homogeneous reservoir. CD = 1000, S = 0. Curve C: L1D = L3D = 20000, L2D = L4D = 2000. Curve D: L1D = L2D = L3D = 2000, L4D = 38000.

Analysis of semi-log straight lines


20 A/rw2 = 106

15

107

108

2m

10

e slop

5

m slope

0 10-1

1

101

102

103

104

105

106




30 B 20 pe slo

10

0 10-1

4m

slope m

1

A

101

102

103

104

105


Figure 5-21 Semi-log plot of Figure 5.17 drawdown examples.

- 106 -


Linear and semi-linear flow analysis


50

D

40 30 2 pe slo

20

C

mh slope c

10 0

m ch

0

20

40 (tD /CD

60

80

)1/2

Figure 5-22 Linear flow analysis plot of Figure 5-19.

The slope for the infinite channel behavior (curve C of Figure 5-19) is expressed in Equation 5.2. For the limited channel (curve D) the slope of the linear flow straight line is double :

mhch = 16.27

qB µ (psi.hr-1/2, field units) h(L2 + L4 ) kφ ct

m hch = 2.494

qB µ (Bars.hr-1/2, metric units) h(L1 + L2 ) kφ c t

( 5-11)


Pseudo-steady state analysis 50 A/rw2= 106

40

107

30

slope m*

20

108

10 0 0

200 000

400 000

600 000

800 000


Figure 5-23 Pseudo steady state flow analysis plot of Figure 5-18.

During pseudo-steady state regime, the drawdown dimensionless pressure is expressed as :

p D = 2π t DA + 0.5 ln

A 2.2458 + 0.5 ln +S 2 CA rw

The dimensionless time tDA is defined with respect to the drainage area : - 107 -

( 5-12)


0.000264 k ∆t (field units) φµ ct A 0.000356k = ∆t (metric units) φµ c t A

t DA = t DA

( 5-13)

The "shape factor" CA characterizes the geometry of the reservoir and the well location. With real data, the pressure during pseudo steady state flow regime is expressed :

∆p = 0.234

 qBµ  A qB ∆t + 162.6 log 2 − log(C A ) + 0.351 + 0.87 S  (psi, field units) kh  rw φ ct hA 

∆p = 0.0417

 qB qBµ  A ∆t + 21.5 log 2 − log(C A ) + 0.351 + 0.87 S  (Bars, metric kh  φ c t hA rw  (1-22)

units)

the slope m* of the pseudo-steady state straight line provides the reservoir connected pore volume :

qB (cu ft, field units) ct m * qB φ hA = 0.0417 (m3, metric units) ct m *

φ hA = 0.234

( 1-23)

When kh and S are known from semi-log analysis, the shape factor CA is estimated from the intercept ∆pint of the pseudo-steady state straight line :

C A = 2.2458e or

2.303 pi − p*int  m−log A rw2  −0.87 S  



[ (



) ]

* m − 2.303 p i − pint m C A = 5.456 e m*

- 108 -

( 5-14)

( 5-15)


5-4.5 Build-up analysis


Log-log analysis of build-up 102 º 101 tpDA=0.6

1 0.5

tpDA=10, 2

10-1 101

1

102

103

104

105

106


Figure 5-24 Build-up responses for a well with wellbore storage and skin in a closed rectangle homogeneous reservoir. The well is close to one boundary. Three production times are considered. CD = 292, S = 0, L1D = 500, L2D = 1000, L3D = 3500, L4D = 1000 tpD/CD (tpDA) = 16400 (0.6), 54600 (2), 273000 (10).

The rectangular reservoir configuration used for the build-up examples of Figure 5-24 is described in the Shape Factors Tables with CA = 0.5813 and the start of pseudo steady state is defined at tDA = 2 (Eq. 5-13 or, with Eq. 2-6, tD/CD = 54600). The well is closed for build-up before (tpDA = 0.6) or during the pure pseudo steady state flow regime (tpDA = 2 and 10). When all reservoir boundaries have been reached during drawdown, the shape of the subsequent build-up is independent of tp on log-log scale. At late times, the stabilized dimensionless pressure p D is expressed as :

pD

  A rw2 = 1151 .  log + 0.35 + S CA  

( 5-16)

Semi-log analysis of build-up

When tp>>∆t, the Horner time can be simplified with tp+∆t ≅ tp :

log

t p + ∆t

∆t

= log t p − log ∆t

( 5-17)

For different production time tp in a depleted reservoir, the Horner straight lines of slope m are parallel.

- 109 -



10

p-D

8

slop em

6 4 tpDA = 0.6,

2

2,

10

0 101

1

102

103 (tpD +tD )/ tD

104

105

106

Figure 5-25 Horner plot of Figure 5-24.

The Horner plot Figure 5-25 is presented in dimensionless terms. The straight line extrapolated pressure p * D changes with tp and, later, the curves flatten to reach

p D = 8.62 of Equation 5.16. For examples tpDA = 2 and 10, p*D > p D , but not for the example with tpDA = 0.6. With real pressure, the average pressure p decreases when tp increases. When the same production time is used for Horner analysis of the three build-up periods (tpDA = 2 on Figure 5-26), the difference between the straight line

extrapolated pressure p * and the average shut-in pressure p becomes a constant.


9 tpDA=2, 10

p-D p*D= 8.1

tpDA=0.6 7

slo pe

5

m

3 1

101

102 (tpD +tD )/ tD

103

104

Figure 5-26 Horner plot of Figure 5-24 with same tp. For the three examples, the Horner time is tpD/CD = 16400 (tpDA =0.6).

- 110 -


5-5 Constant pressure boundary 5-5.1 Definition gas

water

L

L Well

Image

(q)

(-q)

5-5.2 Log-log analysis The dimensionless stabilized pressure is defined as :

p D = ln(2 LD ) + S

( 5-18)


The derivative follows a negative unit slope straight line. 102

101

1 LD=100

300

1000

3000

10-1 1

101

102

103

104

105


Figure 5-27 Responses for a well with wellbore storage and skin near one constant pressure linear boundary in a homogeneous reservoir. Several distances. CD = 100, S = 5, LD = 100, 300, 1000, 3000.

- 111 -



102

101

sealing fault : 1

1 0.5

constant pressure

10-1 10-1

101

1

102

103

104

105


Figure 5-28 Pressure and derivative responses for a well with wellbore storage near two perpendicular boundaries in a homogeneous reservoir. The closest boundary is sealing, the second at constant pressure. CD = 100, S = 0, θ= 90°, θw = 20°, LD = 1000.


5-5.3 Semi-log analysis LD= 3000 1000 300 100

15

10

slope m

5

0

1

101

102

103

104

105



The time of intercept ∆tx between the semi-log straight line and the constant pressure is used, as for a sealing fault, to estimate the distance of the boundary :

L = 0.01217 L = 0.0141

k∆t x (ft, field units) φµ ct k∆t x (m, metric units) φµ c t

( 1-22)

The difference of pressure between the start of the period and the final stabilized pressure, [ p − p( ∆t = 0) ], can also be used to estimate L :

L = 0.5rw e

[1.151 (p − p(∆t = 0)) m − S ] (ft, m)

- 112 -

( 5-19)


5-6 Communicating fault In the case of communicating fault, two different configurations are considered. With the semi-permeable boundary model, also called leaky fault, the vertical plane fault is not sealing but acting as a flow restriction. Conversely, a finite conductivity fault improves the drainage because the fault permeability is larger than the surrounding permeability of the reservoir.

5-6.1 Semi permeable boundary Definition

The partially communicating fault, at distance L from the well, has a thickness wf and a permeability kf. The dimensionless fault transmissibility ratio α is expressed as :

α=

k f wf

( 5-20)

k L

Characteristic flow regimes

wf

1. Radial flow 2. Hemi-radial flow 3. Leak 4. Radial flow

kf

Log-log analysis


102

101

1 0.5 10

0.5

-1

10-1

1

101

102

103

104

105

106


Figure 5-30 Pressure and derivative response for a well with wellbore storage near a semi-permeable linear boundary. Homogeneous reservoir. Log-log scale. CD = 104, S = 0, LD = 5000, α = 0.05.

- 113 -



102

101 α = 0.001 1 1 0.5 α=1,

10-1 10-1

101

1

102

103

0.1, 104

0.01 105

106


Figure 5-31 Responses for a well with wellbore storage and skin near a semipermeable linear boundary. Several transmissibility ratios. CD = 100, S = 5, LD = 300, α = 1, 0.1, 0.01, 0.001.

Semi-log analysis


20 15

e slop

α=1

m slope

10

2m

0.1 0.01 0.001

5 0 10-1

1

101

102

103

104

105

106



5-6.2 Finite conductivity fault Definition

With the finite conductivity fault model, flow is possible along the fault plane, depending upon the fault dimensionless conductivity FcD (a zero fault conductivity FcD corresponds to the semi-permeable fault solution).

FcD =

k f wf

( 5-21)

kL

The resistance to flow across the fault plane is described with the skin factor Sf. The definition of the dimensionless skin Sf includes the possibility of a region of altered permeability ka with an extension wa around the fault:

- 114 -


Sf =

2π k  wa w f + L  k a 2k f

   

( 5-22)

The skin factor Sf is related to the transmissibility ratio a of Eq. 5-20:

α=

π

( 5-23)

Sf

Characteristic flow regimes

1. Radial flow 2. Constant pressure boundary effect 3. Bi-linear flow 4. Radial flow

kf L wf


Log-log analysis 102 101 0.5

1

10-1

1

101

102

0.5

103

104

105

106

107

108


Figure 5-33 Pressure and derivative responses for a well with wellbore storage near a finite conductivity fault. No fault skin. Log-log scale. 3 CD = 10 , S = 0, LD = 1000, FcD= 100, Sf = 0.

- 115 -



102

101

FcD = 1

1

10

100

1000

10000

0.5 10-1

10-2 101

102

103

104

105

106

107

108

109


Figure 5-34 Responses for a well with wellbore storage and skin near a finite conductivity fault. No fault skin and several conductivity. Log-log scale. CD = 100, S = 5, LD = 300, Sf = 0, FcD = 1, 10, 100, 1000, 10000.


102

101 1 1

Sf=1000

0.5 Sf=10

10-1 101

102

103

0.5

Sf=100 104

105

106

107

108

109


Figure 5-35 Responses for a well with wellbore storage and skin near a finite conductivity fault. Several fault skin and conductivity. Log-log scale. CD = 100, S = 5, LD = 300, FcD = 10, 1000, Sf = 10, 100, 1000.

Semi-log analysis


15 Sf = 100

em slop

2m pe sl o

10

e slop

5

Sf = 0

m

0 1

101

102

103

104

105

106

107

108


Figure 5-36 Semi-log plot for a well with wellbore storage near a finite conductivity fault. 3 CD = 10 , S = 0, LD = 1000, FcD = 100, Sf = 0 or 100.

- 116 -


5-7 Predicting derivative shapes

Figure 5-37 Closed reservoir example.

Example of a drawdown in a closed system, the shape of the reservoir is a trapezoid. After wellbore storage, the response shows : 1 - the infinite radial flow regime (derivative on 0.5), 2 - one sealing fault (derivative on 1), 3 - the wedge response (derivative on π /θ), 4 - linear flow (derivative straight line of slope 1/2),


5 - pseudo steady state (straight line of slope 1). 103 102 101

180/θ

1

0.5

1

pe slo e slop

1

1/ 2

10-1 1

101

102

103

104

105

106

107


Figure 5-38 Derivative response for a well in a closed trapezoid.

- 117 -

108

- 118 -

6 - COMPOSITE RESERVOIR MODELS 6-1 Definitions With the radial composite model, the well is at the center of a circular zone of radius r. With the linear composite model, the interface is at a distance L. The well is located in the region "1". The parameters of the second region are defined with a subscript "2".

(k/µ)2, (φct)2

(k/µ)2, (φct)2

(k/µ)1, (φct)1

(k/µ)1, (φct)1

R

L

Radial composite

Linear composite

Figure 6-1 Models for composite reservoirs.

6-1.1 Mobility & storativity ratios

(k µ )1 (k µ )2

M=

F=

( 6-1)

(φ ct )1 (φ ct )2

( 6-2)

6-1.2 Dimensionless variables The dimensionless variables (including the wellbore skin Sw) are expressed with reference to the region "1" parameters.

pD =

k 1h ∆p (field units) 1412 . qBµ 1

pD =

k1 h ∆p (metric units) 18.66qBµ 1

tD k h ∆t = 0.000295 1 (field units) µ1 C CD

- 119 -

( 6-3)

Chapter 6 - Composite reservoir models

tD k h ∆t (metric units) = 0.00223 1 CD µ1 C CD =

0.8936C (field units) (φ ct )1 hrw2

CD =

0.1592C (metric units) (φ ct )1 hrw2

( 6-4)

( 6-5)

k1h ∆pskin (field units) 141.2qBµ1 k1h ∆pskin (metric units) Sw = 15.66qBµ1 Sw =

rD =

LD =

( 6-6)

r rw

( 6-7)

L rw

( 6-8)

6-2 Radial composite behavior 6-2.1 Influence of heterogeneous parameters M and F Dimensionless Pressure , pD and Derivative p'D

102 10 M = 10 M=2

1

M = 0.5

0.5 10-1 10-2 10-1

M = 0.1

0.5 M 1

10

102

103

104

105

106


Figure 6-2 Radial composite responses, well with wellbore storage and skin, changing mobility and constant storativity. Log-log scale. The two dotted curves correspond to the closed and the constant pressure circle solutions. CD = 100, Sw = 3, rD = 700, M = 10, 2, 0.5, 0.1, F =1.

- 120 -



25 M=10

20

M=2

15

M=0.5

slope m

10

M=0.1

5

slopes m M

0 10-1

1

102

10

103

104

105

106




102 F = 10

10

F = 0.1

1

F = 10

0.5

0.5

F = 0.1

10-1 10-1

1

10

102

103

104

105

106


Figure 6-4 Radial composite responses, well with wellbore storage and skin, constant mobility and changing storativity. Log-log scale. CD = 100, Sw = 3, rD = 700, M = 1, and F =10, 2, 0.5, 0.1.


15 F=10

10

m slope

F=0.1

sm slope

104

105

5

0 10-1

1

10

102

103

106



6-2.2 Log-log analysis The permeability thickness product k1h of the inner region is estimated from the pressure match, and C from the time match :

k1h = 141.2qBµ1 (PM ) (mD.ft, field units) k1 h = 18.66 qBµ1 (PM ) (mD.m, metric units) - 121 -

( 6-9)


k1h  1   (Bbl/psi, field units)  µ1  TM  k h 1  3 C = 0.00223 1   (m /Bars, metric units) µ1  TM 

C = 0.000295

( 6-10)

At early time, the homogeneous (CD e2S)1 curve defines the wellbore skin factor Sw. The mobility ratio M is estimated from the two derivative stabilizations.

M=

∆p2nd stab. ∆p1st stab.

( 6-11)

6-2.3 Semi-log analysis The first semi-log straight line defines the mobility of the inner zone, and the wellbore skin factor Sw.

∆p = 162.6

 qBµ 1  k1  log ∆t + log  − 3 . 23 + 0 . 87 S w  (psi, field units) k1h  (φµ ct )1 rw2 

∆p = 21.54

 k1 qBµ1   (Bars, metric units) ( 6-12) − + S log ∆t + log 3 . 10 0 . 87 w  k1h  (φµ ct )1 rw2 

The second line, for the outer zone, defines M and the total skin ST.

∆p = 162.6

∆p = 21.5

qBµ 2 k2 h

qBµ 2 k2h

  k2  log ∆t + log  − 3 . 23 + 0 . 87 S T  (psi, field units)  (φµ ct )2 rw2  

 k2  log ∆t + log − 3.10 + 0.87 ST  (φµ ct )2 rw2 

  (Bars, metric units) ( 6-13)  

The total skin ST includes two components : the wellbore skin factor Sw and a radial composite geometrical skin effect SRC of Equation 1-10, function of the mobility ratio M and the radius rD of the circular interface :

ST =

1  1  S w +  − 1 ln rD M M 

When the mobility near the wellbore is higher than in the outer zone (M>1), the geometrical skin is negative.

- 122 -

( 6-14)


6-2.4 Build-up analysis Dimensionless Pressure , pD and Derivative p'D

102 Drawdown Build-up 10 1.5 1

0.5

10-1 10-1

1

102

10

103

104

105


Figure 6-6 Drawdown and build-up responses for a well with wellbore storage and skin in a radial composite reservoir, changing mobility and constant storativity. Log-log scale. The dotted curves describe the drawdown response. CD = 11500, Sw = 5, rD = 2000, M = 3, F=1.


With a strong reduction of mobility (M>>10), drawdown and build-up responses can show the behavior of a closed depleted system, before the influence of the outer region is seen. 10-2 Drawdown 50

Build-up 10

1

0.5 tp

10-1 1

10

102

103

104

105

106

107


Figure 6-7 Drawdown and build-up responses for a well with wellbore storage and skin in a radial composite reservoir. The dotted pressure and derivative curves correspond to the drawdown solution. CD = 1000, Sw = 0, rD = 10000, M =100, F =1 and tp/CD=3200.

6-3 Linear composite behavior 6-3.1 Influence of heterogeneous parameters M and F The second homogeneous behavior is defined with the average properties of the two regions :

1   k  = 0.5 1 +   k µ  (mD/cp) 1  µ  APPARENT  M

- 123 -

( 6-15)



102 10 M = 10

1 0.5

M = 0.5

10-1

M = 0.1

10-2 10-1

1

102

10

103

104

105

106


Figure 6-8 Linear composite responses, well with wellbore storage and skin, changing mobility and constant storativity. Log-log scale. The two dotted curves correspond to the sealing fault and the constant pressure boundary solutions. CD = 100, Sw = 3, LD = 700, M = 10, 2, 0.5, 0.1, F=1.


15

M=10 M=2 M=0.5 M=0.1

m slope

10

5

0 10-1

1

102

10

103

104

105

106





Radial

10

Linear Radial

1

Linear

0.5 10-1 10-1

1

10

102

103

104

105


Figure 6-10 Comparison of radial and linear interfaces. Well with wellbore storage and skin in composite reservoirs. Log-log scale. CD = 200, Sw = 0, F=1, rD = LD = 300. Linear composite : M = 5. Radial composite : M =1.667.

The two derivative stabilizations are used to estimate the mobility ratio M :

- 124 -


M=

∆p2nd stab. 2 ∆p1st stab. − ∆p2nd stab.

( 6-16)

6-4 Multicomposite systems


6-4.1 Three inner regions with abrupt change of mobility 10

1

RD=1000, M=0.1 RD=2500, M=0.15 RD=50000, M=0.5

0.5 0.33

10-1

0.1 0.05

10-2 1

10

102

103

104

105

106


Figure 6-11 Pressure and derivative responses for a well with wellbore storage and skin in a 4 regions radial composite reservoir. CD = 5440, Sw = 0, F =1. r1D = 1000, k/µ2 = 1.5 k/µ1, r2D = 2500, k/µ3 = 5 k/µ1, r3D = 50,000, k/µ4 = 10 k/µ1. The dashed curves correspond to radial composite responses with only one zone (RD = 1000, M = 0.1, RD = 2500, M = 0.15, RD = 50,000, M = 0.5).


6-4.2 Two inner regions with a linear change of mobility 10

1

0.5 RD=10000 RD=1000

10-1

0.05

10-2 1

10

102

103

104

105

106

107


Figure 6-12 Pressure and derivative responses for a well with wellbore storage and skin in a radial composite reservoir, linear change of transmissivity. CD = 1000, Sw = 0, F =1. From R1D = 1000 to R2D = 10,000, M decreases linearly from 1 to 0.1. The dashed curves correspond to radial composite responses (M=0.1, RD = 1000, RD = 10,000).

- 125 -

- 126 -

7 - LAYERED RESERVOIRS - DOUBLE PERMEABILITY MODEL

7-1 Definitions The layer "1" is assumed to be the high permeability layer. The two-layers model can be used for multi-layers systems. Layer "1" describes the sum of the high permeability zones, and layer "2" the lower permeability intervals.

S1

h1, k1, kZ1 h',

S2

k'Z

h2, k2, kZ2

Figure 7-1 Model for double permeability reservoir.

7-1.1 Permeability and porosity khTOTAL = k1h1 + k 2 h2

(mD.ft, mD.m)

(φ ct h)TOTAL = (φ ct h)1 + (φ ct h)2

(ft/psi, m/Bars)

( 7-1) ( 7-2)

7-1.2 Mobility ratio κ

κ=

k1h1 k1h1 = k1h1 + k 2 h2 khTOTAL

( 7-3)

When κ=1, the response is double porosity.

7-1.3 Storativity ratio ω

ω=

(φ ct h)1 (φ ct h)1 = (φ ct h)1 + (φ ct h)2 (φ ct h)TOTAL

- 127 -

( 7-4)

Chapter 7 - Layered reservoirs

7-1.4 Interlayer cross flow coefficient λ

rw2 λ= k1h1 + k 2 h2

2 h h h' + 1 + 2 2 k ' Z k Z1 k Z 2

( 7-5)

λ is a function of the vertical permeability k z' in the low permeability "wall" of

thickness h' between the layers, and of vertical permeabilities in the two layers kz1 and kz2. If the vertical resistance is mostly due to the "wall", a simplified λ can be used to characterize this interlayer skin :

λ=

rw2 k 'Z k1h1 + k 2 h2 h'

( 7-6)

When there is no skin at the interface and the vertical pressure gradients are negligible in the high permeability layer 1, λ is expressed:

rw2 kZ2 λ= k1h1 + k 2 h2 h2 2

( 7-7)

When λ=0, there is no reservoir crossflow.

7-1.5 Dimensionless variables k1h1 + k 2 h2 ∆p (field units) 1412 . qBµ 1 k h + k 2 h2 pD = 1 1 ∆p (metric units) 18.66qBµ

pD =

tD k h + k 2 h2 ∆t = 0.000295 1 1 (field units) C µ CD tD k h + k 2 h2 ∆t (metric units) = 0.00223 1 1 CD µ C

CD =

0.8936C

[(φ c h) t

CD =

]

+ (φ ct h) 2 rw2 1

0.1592C

[(φ ct h )1 + (φ ct h)2 ]rw2

( 7-8)

( 7-9)

(field units)

(metric units)

- 128 -

( 7-10)


7-2 Double permeability behavior when the two layers are producing into the well 7-2.1 Log-log pressure and derivative responses Three characteristic flow regimes : 1. First, the behavior corresponds to two layers without cross flow. 2. At intermediate times, when the fluid transfer between the layers starts, the response follows a transition regime. 3. Later, the pressure equalizes in the two layers and the behavior describes the equivalent homogeneous total system. The derivative stabilizes at 0.5. Dimensionless Pressure , pD and Derivative p'D

102

10

0.5

1

10-1 10-1

1

10

102

103

104

105


Figure 7-2 Response of a well with wellbore storage and skins in a double permeability reservoir. The two layers are producing into the well. CD = 1000, S1 =S2 = 0, ω = 0.02, κ = 0.8, λ = 6.10-8

k1h1 + k 2 h2 = 141.2qBµ (PM ) (mD.ft, field units) k1 h1 + k 2 h2 = 18.66qBµ (PM ) (mD.m, metric units) k1h1 + k 2 h2  1    (Bbl/psi, field units) µ  TM  k h + k 2 h2  1  3 C = 0.00223 1 1  (m /Bars, metric units)  µ  TM 

( 7-11)

C = 0.000295

( 7-12)

The heterogeneous parameters κ, ω and λ are adjusted preferably with the derivative curve. When the two skins S1 and S2 are different, the well condition influences the shape of the derivative transition, and it is difficult to conclude the match uniquely.

λ provides an estimate of the vertical permeabilities. From Equations 7-6 and 7-7 :

k ' Z = ( k1h1 + k 2 h2 )

λ rw2

h' (mD)

- 129 -

( 7-13)


k Z 2 = ( k1h1 + k 2 h2 )

λ h2

( 7-14)

(mD)

rw2 2

7-2.2 Influence of the heterogeneous parameters κ and ω It is assumed in that the two skin coefficients are equal: S1 = S2 ( = 0). Dimensionless Pressure , pD and Derivative p'D

10 κ = 0.999 0.6

1

0.5 0.6 0.9

10-1

0.99 0.999 κ= 1

10-2 10-1

1

102

10

103

104


Figure 7-3 Double permeability responses when the two layers are producing into the well. Well with wellbore storage and skins, high storativity contrast. The two dotted curves describe the homogeneous reservoir response (CDe2S = 1) and the double porosity response (κ = 1). CD = 1, S1 = S2 = 0, ω = 10-3, λ = 4.10-4. Four mobility ratios : κ = 0.6, 0.9, 0.99 and 0.999.

κ = 0.99


6 κ = 1,

4

κ = 0.6

0.999

em slop

0.99 0.9 0.6

2 Two layers no crossflow Double permeability

0 10-1

1

10

102

103

104


Figure 7-4 Semi-log plot of Figure 7-3. The thick dotted curves correspond to the homogeneous reservoir response (CD e2S = 1) and the double porosity response (κ = 1).The thin dotted curves correspond to the two layers responses with no reservoir crossflow (for κ = 0.6 and 0.99, λ = 0).

- 130 -



10 κ = 0.999 0.6

1

0.5 0.6 0.9 κ = 1

10-1 10-1

1

10 102 Dimensionless time, tD/CD

0.99 0.999

103

104

Figure 7-5 Double permeability responses when the two layers are producing into the well. Well with wellbore storage and skins, low storativity contrast. Log-log scale. The two dotted curves describe the homogeneous reservoir response (CDe2S = 1) and the double porosity response (κ = 1). CD = 1, S1 = S2 = 0, ω = 10-1, λ = 4.10-4. Four mobility ratios : κ = 0.6, 0.9, 0.99 and 0.999.


6 Two layers no crossflow Double permeability

κ = 0.99 κ = 0.6

m pe slo

4 κ=

2

0 10-1

1

1 0.999 0.99 0.9 0.6

10 102 Dimensionless time, tD/CD

103

104

Figure 7-6 Semi-log plot of Figure 7-5. The thick dotted curves correspond to the homogeneous reservoir response (CD e2S = 1) and the double porosity response (κ = 1).The thin dotted curves correspond to the two layers responses with no reservoir crossflow (for κ = 0.6 and 0.99, λ = 0).

7-3 Double permeability behavior when only one of the two layers is producing into the well 7-3.1 Log-log pressure and derivative responses Three characteristic flow regimes : 1. First, the perforated layer response is seen alone, and the behavior is homogeneous. 2. When the second layer starts to produce by reservoir cross flow, the response deviates in a transition regime. The derivative drops. 3. Later, the pressure equalizes in the two layers, and the equivalent homogeneous behavior of the total system is seen. The derivative stabilizes at 0.5.

- 131 -



102 layer 2 produces 10 0.5/(1-κ) 1

0.5

10-1 10-1

1

102

10

103

104

105

106


Figure 7-7 Response for a well with wellbore storage and skin in double permeability reservoir, only layer 2 produces into the well. Log-log scale. CD =1000, S1 = 100, S2 = 0, ω = 0.1, κ = 0.9, λ = 6.10-8.

7-3.2 Discussion of double permeability parameters When only the low permeability layer is producing, the derivative tends to stabilize at 0.5/(1-κ) during the first homogeneous regime. The response is then similar to the behavior of a well in partial penetration. Dimensionless Pressure , pD and Derivative p'D

102 the two layers produce

layer 2

10

layer 1 layer 2 produces

1 layer 1 produces

0.5

10-1 10-1

1

10

102

103

104

105


Figure 7-8 Response for a well with wellbore storage and skin in double permeability reservoir, only one layer is producing into the well. The dotted curve describes the double permeability response when the two layers are producing into the well (no skin). CD = 1, ω = 0.2,κ = 0.9, λ = 10-4, S1 = 100, S2 = 0 and S1 = 0, S2 = 100.

When only the high permeability layer produces into the well, the two derivative stabilizations are almost at the same level: 0.5/κ for the first (0.55 in the example of Figure 7-8) and 0.5 for the second. The response tends to be equivalent to the double porosity solution with restricted interporosity flow.

7-3.3 Analysis of semi-log straight lines The response can follow two semi-log straight lines. When one of the two layers (called layer i) starts to produce alone, the first line is expressed :

- 132 -



30 the two layers produce

slope m

layer 2 produces

20

10 layer 1 produces slope m

0 10-1

1

10

102

103

104

105


Figure 7-9 Semi-log plot of Figure 7-8. The dotted curve corresponds to the homogeneous reservoir response, no skin (CD e2S = 1).

∆p = 162.6

 ki qBµ  − 3 . 23 + 0 . 87 S log ∆t + log i  (psi, field units) k i hi  (φ ct )i µ rw2 

∆p = 21.54

 ki qBµ  − 3 . 10 + 0 . 87 S log ∆t + log  (Bars, metric units)( 7-15) i k i hi  (φ ct )i µ rw2 

The second line, for the total system regime, gives the total mobility :

∆p = 162.6

 kTOTAL qBµ  − 3 . 23 + 0 . 87 S  log ∆t + log  (psi, field units) khTOTAL  (φ ct )TOTAL µ rw2 

∆p = 21.54

qBµ khTOTAL

  khTOTAL − 3.10 + 0.87 S  (Bars, metric log ∆t + log 2 (φ ct h)TOTAL µrw   ( 7-16)

units)

The global skin S measured on the total system semi-log straight line is not only a function of the two layers skins S1 and S2, but also of κ, ω and λ.

7-4 Commingled systems: layered reservoirs without crossflow 7-4.1 Same initial pressure When there is no reservoir crossflow, the amplitude of the response is larger than that of the equivalent homogenous system (thin dashed curves on Figure 7-4 and Figure 7-6). The semi-log slope decreases slowly with time, to reach the equivalent total system slope of Equation 7-16. In a n layers system, the pseudo-skin factor SL due to layering is defined as : - 133 -


SL =

(kh φ ct h) j 1 n k jhj ln ∑ 2 j =1 khTOTAL ( kh φ ct h) TOTAL

( 7-17)

On the example κ=0.999 and ω=0.001 of Figure 7-4, the pseudo-skin is estimated at SL=3.5. For the curve κ=0.9 and ω=0.1 of Figure 7-6, SL is only 0.9. When the layers have different mechanical skin factors Si, the response is also a function of the skin contrast between the different layers. The global skin can be defined with two components : SL of Equation 7-17, and an average mechanical skin S . The average mechanical skin S is approximated with : n

S=∑

k jhj

j =1 khTOTAL

n

S j = ∑κ j S j

( 7-18)

i =1

7-4.2 Different initial pressure When the layers have a different initial pressure, the bottom hole pressure tends asymptotically towards the average initial pressure if the well is not opened to surface production. For an infinite system, p i is defined as : n

pi = ∑

k jhj

j =1 khTOTAL

pi j (psi, Bars)

( 7-19)

If the non-producing commingled reservoir is closed, the final average reservoir pressure is p : n

p=∑

V j ct j

j =1 Vct TOTAL

pi j (psi, Bars)

( 7-20)

where Vj is the pore volume of layer j. The final average reservoir pressure p can be greater or smaller than the "infinite" average initial pressure pi of Equation 719.

- 134 -

8 - INTERFERENCE TESTS

8-1 Interference tests in reservoirs with homogeneous behavior 8-1.1 Responses of producing and observation wells 4930 pi

5000

Pressure (psia)

Observation well

Observation well

4500

Producing well

pwf

4920

4000

4910

3500 0

100

200

300

400

500

180

Time (hours)

200

220

Time (hours)

Figure 8-1 Response of a producing and an observation well. Linear scale. On the second graph, the observation well pressure is presented on enlarged scale at time of shut-in.

103

Pressure Change, ∆p and Derivative (psi)

Producing well 102

101 Observation well 1 10-2

1

10-1

101

102

103


Figure 8-2 Build-up response of the producing and observation wells. Loglog scale.

8-1.2 Log-log analysis with line-source solution Dimensionless parameters

The line source solution, also called the exponential integral (Ei), or Theis solution, is expressed as :

- 135 -

Chapter 8 - Interference tests

(

p D = − 1 2 Ei − rD2 4t D

)

( 8-1)

pD is defined in Equation 2-3 and the time group tD/rD2 is :

t D 0.000263k = ∆t (field units) 2 φµ ct r 2 rD t D 0.000356k = ∆t (metric units) rD2 φ µ ct r 2 Dimensionless Pressure pD and Derivative p'D

101 1

( 8-2)

PRESSURE Intersection DERIVATIVE

10-1 Approximate start of radial flow

10-2 10-3 10-2

10-1

1

102

101

Dimensionless time,

103

104

tD /rD2

Figure 8-3 The Theis solution (exponential integral). Log-log scale, pressure and derivative responses.

With the line source response, the pressure and derivative curves intersect at tD/rD2 = 0.57 and pD = p'D = 0.32. The 0.5 derivative stabilization starts 10 times later, approximately at tD/rD2 = 5.

Match results

The permeability thickness product kh is estimated from the pressure match with

(

)

Equation 2-8. The time match t D r 2D ∆t gives the effective porosity compressibility product φ ct :

0.000263 k  1    (psi-1, field units) µ r 2  TM  0.000356 k  1  φ ct =   (Bars-1, metric units) 2 µr  TM 

φ ct =

- 136 -

( 8-3)


8-1.3 Influence of wellbore storage and skin effects at both wells 101


Line source well 1

10-1

10-2

C:

rD = 300,

CD = 3000,

S = 30

B:

rD = 1000,

CD = 10000,

S = 10

A:

rD = 1000,

CD = 3000,

S=0

10-3 10-4 10-2

1

10-1

101

102

103

Dimensionless time, tD /rD2

Figure 8-4 Influence of wellbore storage and skin effects on interference pressure responses. Log-log scale. The dotted curve corresponds to the Theis solution. Two distances: rD = 1000 : CD = 3000, S = 0 (curve A) and CD = 10000, S = 10 (curve B). rD = 300 : CD = 3000, S = 30 (curve C).


101 Line source well 1

10-1

10-2

C:

rD = 300,

CD = 3000,

S = 30

B:

rD = 1000,

CD = 10000,

S = 10

A:

rD = 1000,

CD = 3000,

S=0

10-3

10-4 10-2

10-1

1

101

102

103

Dimensionless time, tD /rD2

Figure 8-5 Derivative curves of Figure 8-4. Log-log scale. The dotted derivative curve corresponds to the Theis solution.

- 137 -



1 Intersections

Line source well

A

10-1

B 10-2 10-1

10-2

101

1

Dimensionless time,

tD /rD2

Figure 8-6 Pressure an derivative curves of Figure 8-4 and Figure 8-5, examples A and B. Log-log scale. The dotted derivative curve corresponds to the Theis solution.

8-1.4 Semi-log analysis of interference responses When tD/rD2 > 5, the infinite acting radial flow regime is reached.

pi − p wf =

 k 162.6 qBµ   log ∆t + log − 3.2275 (psi, field units) 2 kh φ µ ct r  

p i − p wf =

 21.5 qBµ  k  (Bars, metric units) log ∆t + log − 3 . 10  kh  φ µ ct r 2 

( 1-30)

8-1.5 Anisotropic reservoirs y Observation well at (x, y)

kmax

kmin θ

x

Active well

Figure 8-7 Interference test in an anisotropic reservoir. Location of the active well and the observation well.

With a coordinate system centered on the active well, the observation well location is defined as (x,y) and kx, ky, kxy are the components of the permeability tensor.

- 138 -


When several observation well responses are matched against the exponential integral type curve of Figure 8-3, the pressure match is the same for all responses and only the time match changes. The apparent permeability is : 2 k = k max k min = k x k y − k xy

( 8-4)

(mD)

The apparent distance rD,x,y of the observation well is function of the well location with respect to the main permeability directions. The dimensionless time corresponding to well (x,y) is defined as :

  tD  k max k min 0.000263∆t   (field units)  2 = φ µ ct  k x y 2 + k y x 2 − 2 k xy xy   rD  x , y  tD  r 2  D

   k max k min  (metric units)  = 0.000356∆t  2 2    φ µ k y + k x − 2 k xy c t y xy  x, y  x 

( 8-5)

With three observation well responses, kx, ky and kxy can be estimated. The major and minor reservoir permeability kmax and kmin are be defined with

 k max = 0.5k x + k y +  k x − k y  

)

 k min = 0.5k x + k y −  k x − k y  

)

(

(

2

2  + 4 k xy 

2

2  + 4 k xy 

1/ 2

1/ 2

  (mD) 

( 8-6)

  (mD) 

( 8-7)

The angle between the major permeability axis and the x-axis of the coordinate system is expressed with :

 k max − k x    k xy 

θ = arctan

( 8-8)

When only one observation well response is available for interpretation, the reservoir anisotropy is not accessible. The pressure match gives the average permeability k max k min but the porosity compressibility product φ ct estimated from the time match with Equation 8-3 can be wrong.

8-2 Interference tests in double porosity reservoirs The responses are expressed with the dimensionless pressure pD versus the dimensionless time group tfD/rD2 defined with reference to the fissure system storativity (φ V ct)f : - 139 -


t Df r 2D t Df rD2

= =

0.000263k∆t (field units) (φV ct ) f µ r 2 0.000356k

(φVct ) f µ r 2

∆t (metric units)

( 8-9)

8-2.1 Double porosity reservoirs with restricted interporosity flow Pressure type curves

Three curves are needed to define to a double porosity interference response : 1. During the fissure flow regime, the interference response follows the exponential integral solution. 2. When the transition starts, the response deviates from the fissure curve and follows a λ rD2 transition curve. 3. Later, the total system equivalent homogeneous regime is reached and a second exponential integral curve is seen at late time. The distance between the two homogeneous regime curves is a function of the storativity ratio ω. The level of the pressure change ∆p during the transition is defined by λ rD2. When the distance rD between the active and the observation wells is large, the λ rD2 transition stabilizes at a low ∆p value and, beyond a certain distance riD, ∆p becomes less than the pressure gauge resolution. This distance riD represents the radius of influence of the fissures around the active well.


101

0.01 0.1

1

1 λ rD2 = 5

10-1

10-2 10-1

ω =0.1 1

ω =0.01

ω =0.001

102 101 Dimensionless time, tD f /rD2

103

104

Figure 8-8 Interference pressure type-curve for a double porosity reservoir, restricted (pseudo-steady state) interporosity flow. 2 λrD = 5, 1, 0.1, 0.01.

- 140 -



When the observation well is located inside the radius of influence riD, the fissure flow regime is seen first. The interference response is observed faster than for the equivalent homogeneous reservoir. The tDf time scale of Figure 8-9 shows that the transition is observed at the same time in the active well and in the observation wells. With the tDf/rD2 time scale of Figure 8-10, the time of transition is a function of the λ rD2 group.


101 Active well 1

10-1 A 10-2

B rD=5000

rD=1000

104

105

106

107

108

109

Dimensionless time, tD f

Figure 8-9 Interference responses in double porosity reservoirs with restricted interporosity flow (tDf time scale). ω = 0.1, λ = 5 X 10-8, two distances : rD = 1000 (curve A) and rD = 5000 (B). The dotted curve describes the derivative response at the active well.


101 A 1

B

10-1

A

B rD=5000 10-2 10-2

10-1

1

rD=1000

101

102

103

Dimensionless time, tD f /rD2 2

Figure 8-10 Interference responses of Figure 8-9, tDf /rD time scale.

- 141 -


8-2.2 Double porosity reservoirs with unrestricted interporosity flow Pressure type-curve

Two pressure curves : 1. - The interference response starts on a β rD2 transition curve. 2. - When the total system equivalent homogeneous regime is reached, the response follows the exponential integral curve.


101 6 60 600

1

β rD2 = 6000 10-1

ω =0.1

10-2 10-1

ω =0.001

ω =0.01 101

1

102

Dimensionless time,

103

104

tD f /rD2

Figure 8-11 Interference pressure type-curve for a double porosity reservoir, unrestricted (transient) interporosity flow β rD2= 6*103, 6*102, 60 and 6.

For slab matrix blocks, β = 3λ 5ω and, for sphere matrix blocks β = λ 3ω .



101 A 1

B

A 10-1

B

rD=1000

rD=5000

10-2 10-2

10-1

1

101

102

103

104

Dimensionless time, tD f /rD2

Figure 8-12 Interference responses in double porosity reservoirs with unrestricted interporosity flow. Log-log scale. Two wells, with same parameters as on Figure 8-10

- 142 -


8-3 Influence of reservoir boundaries

Period #2 Period #3

Period #3 O1

A

Linear sealing fault

O2

Active well

Figure 8-13 Interference in a reservoir with a sealing fault. Location of the active well A and the two observation wells O1 and O2.

In case of one sealing fault, the derivative stabilizes at p'D=1 at late time. The time of transition from 0.5 to 1 can be earlier, or later, than in the active well.

102


O1

101

Active well

O2

1 10-1

1

101

102

103


Figure 8-14 Interference in a reservoir with a sealing fault. Pressure and derivative curves of the two observation wells. Log-log scale.

8-4 Interference tests in radial composite reservoir When the mobility around the active well is higher than the mobility of the reservoir (Figure 8-16), the interference signal travels faster. When the active well is located in a low mobility region (Figure 8-17), the interference signal is delayed.

- 143 -


(k/µ)1

(k/µ)2

O2

O1

A

R

Active well

2R

R/2

Figure 8-15 Interference in a radial composite reservoir. Location of the active well A and the observation wells O1 and O1.

103


Active well 102 O1

101 O2 Line source region 2

1 10-1

101 102 Elapsed time, ∆t (hours)

1

103

Figure 8-16 Interference responses in a radial composite reservoir. The mobility of the inner zone is 4 times larger (M=4, F=1). The dotted derivative curves correspond to the active well A and to the Theis solution for region 2 parameters.


103

102

Active well

101 O2

O1

Line source region 2

1 10-1

1

101

102

103


Figure 8-17 Interference responses in a radial composite reservoir. The mobility of the inner zone is 4 times smaller (M=1/4, F=1). The dotted derivative curves correspond to the active well A and to theTheis solution for region 2 parameters.

- 144 -


Pressure Change, ∆p (psi)

103

102

O1 101

O2

M=4 M=1/4

M=4 M=1/4 Line source region 2

1 10-1

101

1

102

103


Figure 8-18 Interference responses in a radial composite reservoir. Pressure curves of examples Figure 8-16 and Figure 8-17. The mobility of the inner zone is 4 times smaller or 4 times larger. The dotted pressure curve corresponds to the Theis solution for region 2 parameters.

When there is a reduction of storativity φct around the active well, the interference signal reaches the observation well faster (Figure 8-19).


103

Active well

102

101

O2 Line source region 2

1 10-1

1

101

102

103


Figure 8-19 Interference responses in a radial composite reservoir. Well O2. The storativity of the inner zone is 4 times smaller (M=1, F=1/4). The dotted derivative curves correspond to the active well A and to the Theis solution for region 2 parameters.

- 145 -



103

Active well

102


101

O2

1 10-1

101

1

103

102


Figure 8-20 Interference responses in a radial composite reservoir. Well O2. The storativity of the inner zone is 4 times larger (M=1, F=4). The dotted derivative curves correspond to the active well A and to the Theis solution for region 2 parameters.


When both the active well and the observation well are located in the inner reservoir region, the interference response can show the 2 usual derivative stabilizations of the radial composite model (Figure 8-21).

102

101


Active well

O1

1 10-2

10-1

1

101

102

103

104


Figure 8-21 Interference responses in a radial composite reservoir. Well O1. The mobility and the storativity of the inner zone are 10 times larger (M=F=10). The dotted derivative curves correspond to the active well A and to the Theis solution for region 2 parameters.

8-5 Interference tests in a two layers reservoir with cross flow The dimensionless pressure p1+2D and the dimensionless time group t1+2D/rD2 are defined with the parameters of the total system. For the example used in the following, the contrast between the layers is not high (ω =0.4 and κ =0.7), and the active well is expected to show the equivalent homogeneous behavior. - 146 -


On Figure 8-22, only one layer is perforated at the observation well. When only the high permeability layer 1 is communicating with the observation well, the response is seen before the equivalent homogeneous solution for the total system. When the interference is monitored through the low permeability layer 2, the early time response is delayed compared to the Theis solution for the total system. After the double permeability transition, the two responses merge on the equivalent homogeneous total system curve.


1 Layer 1

10-1 Layer 2

Line source total system

10-2 10-2

10-1

1

Dimensionless time, tD 1+2 /rD

101 2

Figure 8-22 Interference responses in a double permeability reservoir, one layer is perforated in the observation well. Log-log scale. The dotted pressure and derivative curves correspond to the Theis solution for the total system equivalent homogeneous reservoir. ω=0.4, κ=0.7 and λ=10-6.

When two layers are perforated, a cross flow is present in the well at the start of the interference response, and the observation well becomes active (even though it is not producing at surface). The resulting response (Figure 8-23) is close to the response of layer 1 alone : when several layers are perforated, the high permeability layer dominates the observation well behavior.


1

10-1 Line source total system 10-2 10-2

10-1 1 Dimensionless time, tD 1+2 /rD2

101

Figure 8-23 Interference responses in a double permeability reservoir, the two layers are perforated in the observation well. Same parameters as on Figure 8-22, the dotted curves correspond to the total system equivalent homogeneous Theis solution.

- 147 -

- 148 -

9 - GAS WELLS Two different types of test are used with gas wells. Transient analysis provides a description of the producing system, as for oil wells. With deliverability testing, the theoretical rate at which the well would flow if the sandface was at atmospheric pressure, "the Absolute Open Flow Potential" AOFP, is estimated.

9-1 Gas properties 9-1.1 Gas compressibility and viscosity The viscosity µ and the compressibility of gas cg change with the pressure.

cg =

1 1 ∂Z − p Z ∂p

(psi-1, Bars-1)

( 9-1)

Z is the real gas deviation factor. For an ideal gas Z=1, and the compressibility is cg=1/p.

9-1.2 Pseudo-pressure The pseudo-pressure m(p), also called "real gas potential", is defined :

m( p) = 2

p

p

∫ µ ( p)Z ( p) dp

(psia2/cp, Bars2/cp)

( 9-2)

p0

The pressure p is expressed in absolute unit, m(p) has the unit of (pressure)2 / viscosity , (psia2 / cp with the usual system of units). The reference pressure p0 is an arbitrary constant, smaller than the lower test pressure. The complete pressure data is converted into pseudo-pressure m(p) before analysis. The change of pseudo-pressure, expressed as m(p)-m(p[∆t=0]), is independent of the reference pressure p0.

9-1.3 Pseudo-time The pseudo-time tps is sometimes used as a complement of m(p). t

t ps = ∫ 0

1 dt µ ( p)ct ( p)

(hr.psi/cp, hr.Bars/cp)

In order to estimate µ and ct before calculation of the superposition with the pseudo time tps, the pressure must be known during the complete flow rate sequence

- 149 -

( 9-3)

Chapter 9 - Gas wells

9-2 Transient analysis of gas well tests 9-2.1 Simplified pseudo-pressure for manual analysis On Figure 9-1, µZ is plotted versus p for a typical natural gas at constant temperature : - When the pressure is less than 2000 psia, the product µZ is almost constant and m(p) simplifies into :

m( p) =

2 µZ

p

∫

pdp =

p0

p 2 − p02 µ i Zi

(psia2/cp, Bars2/cp)

( 9-4)

On low-pressure gas wells, it is possible to analyze the test in terms of pressuresquared p2. - When the pressure is higher than 3000 psia, the product µZ tends to be proportional to p and p/µZ can be considered as a constant. The pseudo-pressure m(p) becomes :

2p m( p) = µZ

p

2 pi (psia2/cp, Bars2/cp) Z i i

∫ dp = ( p − p0 ) µ

p0

( 9-5)

On high-pressure wells, the gas behaves like a slightly compressible fluid, and the pressure data can be used directly for analysis. - Between 2000 psia and 3000 psia, no simplification is available and m(p) must be used.

0.04

µ Z (cp)

0.03 µ

o al t ion t r o rop Zp

p

0.02 µ Z constant 0.01

0.00 0

2000

4000

6000

8000

Pressure (psia)

Figure 9-1 Isothermal variation of µZ with pressure.

- 150 -


9-2.2 Dimensionless parameters Nomenclature

In field units, the standard pressure is psc =14.7psia and the temperature is Tsc = 520°R (60°F, all temperatures are expressed in absolute units). The gas rate is expressed in standard condition as qsc in Mscf/D (103scft/D ). With the metric system, psc =1 Bar, Tsc = 288.15°K (15°C) and cubic meters are used for gas rates (m3/D.). When the pseudo-pressure is used, the dimensionless terms are defined with respect to the gas properties at initial condition (subscript i). With the pressure and pressure-squared approaches, the properties are defined at the arithmetic average pressure of the test (symbol ).

Dimensionless pressure

m(p):

p D = 1.987 ∗ 10 −5

kh Tsc [m( pi ) − m( p )] Tq sc p sc

kh = 7.03 ∗ 10 [m( pi ) − m( p )] Tq sc T sc kh [m( p i ) − m( p)] pD = 37.33T q sc p sc

(field units)

−4

kh [m( p i ) − m( p)] = 0.1296T q sc p2:

p D = 1.987∗10 −5

(

(metric units)

kh Tsc 2 pi − p 2 µ ZTq sc psc

( (

) )

kh = 7.03∗10 pi2 − p 2 µ ZTq sc kh Tsc 2 pi − p 2 pD = 37.33µ Z Tqsc psc −4

(

kh pi2 − p 2 = 0.1296 µ zTqsc

)

) (field units)

(metric units)

p:

p D = 3.976∗10 −5 = 1406 . ∗10

−3

kh p Tsc ( pi − p) µ ZTq sc psc kh p ( pi − p) µ ZTq sc

- 151 -

( 9-6)

(field units)

( 9-7)


pD =

kh p 18.66 µ Z Tq sc kh p

=

0.0648µ Z Tq sc

Tsc ( pi − p) p sc

(metric units)

( 9-8)

( pi − p)

Dimensionless time

m(p):

tD = tD =

0.000263k

φµ i cti rw 2 0.000356k

φ µ i cti rw 2

∆t (field units) ∆t (metric units)

( 9-9)

p2 and p:

tD = tD =

0.000263k

φ µ ct rw 2 0.000356k

φ µ c t rw 2

∆t (field units) ∆t (metric units)

( 9-10)

Dimensionless wellbore storage

As for oil wells, the wellbore storage coefficient is expressed in Bbl/psi (or m3/Bars). m(p):

CD = CD =

0.8936C (field units) φ cti hrw2 0.1592C

φ c ti hrw2

(metric units)

( 9-11)

p2 and p:

CD = CD =

0.8936C (field units) φ ct hrw2 0.1592C

φ c t hrw2

(metric units)

Dimensionless time group tD/CD

m(p):

tD kh ∆t = 0.000295 (field units) CD µi C - 152 -

( 9-12)


tD kh ∆t (metric units) = 0.00223 CD µi C

( 9-13)

p2 and p:

tD kh ∆t = 0.000295 (field units) CD µ C tD kh ∆t (metric units) = 0.00223 CD µ C

( 9-14)

Skin

On gas wells, the skin coefficient S' is expressed with a rate dependent term, also called turbulent effect or non-Darcy skin.

S ' = S + Dq sc

( 9-15)

In a multirate sequence, the analysis is made with respect to the rate change (qn qn-1), and the skin is estimated from the change of ∆pskin between the flow periods n and n-1. S' is expressed :

S' =

q n ( S + Dq n ) − q n −1 ( S + Dq n −1 ) q n − q n −1

= S + D(q n + q n −1 )

( 9-16)

During shut-in periods (qn =0) and during a period immediately after shut-in (qn-1 = 0), the actual flow rate is used in Equation 9-16.

S'=S+D(qn+qn-1)

12

10

lope D=s

8

S = intercept

6 0

2000

4000 qn+qn-1 (Mscf/D)

6000

8000

Figure 9-2 Variation of the pseudo-skin with the rate qn + qn-1.

- 153 -


9-3 Deliverability tests 9-3.1 Deliverability equations Empirical approach (Fetkovich, or "C & n")

(

2 q sc = C pi2 − pwf

)

n

(Mscf/D, m3/D)

( 9-17)

The initial pressure pi and the stabilized flowing pressures pwf are expressed in absolute units. The coefficients C and n are two constant terms. n can vary from 1 in the case of laminar flow, to 0.5 when the flow is fully turbulent. 109

108 1/n =s lop e

pi2- pwf2 (psia2)

pwf=14.7 psia

107 AOF=9000 Mscft/D

106 103

104 Rate, qsc (Mscf/D)

105

Figure 9-3 Deliverability plot for a backpressure test. Log-log scale, pressure-squared method.

The Absolute Open Flow Potential (AOF) is the theoretical rate for a bottom hole flowing pressure pwf = 14.7 psia (pwf =1 Bar). Theoretical approach (LIT, or Houpeurt's, or Jone's, or "a & b")

In a closed system, the difference between the pseudo-steady state flowing pressure pwf and the following shut-in average pressure p is expressed from Equation 5-16 as :

() ( )

m p − m p wf = 1637

 A rw2 T T  2 (psia2/cp,  log + 0.35 + 0.87 S  q sc + 1422 Dq sc kh kh  CA 

field units)

- 154 -


() ( )

m p − m p wf

 A rw2 T T  2 log = 0.1491 + 0.351 + 0.87 S q sc + 0.1296 D q sc   kh kh  CA 

(Bars2/cp, metric units)

( 9-18)

With a circular reservoir of radius re, CA = 31.62 and ∆m(p) is simplified :

() ( )

m p − m pwf = 1637

 0.472re T  T 2 + 0.87 S  q sc + 1422 Dq sc (psia2/cp, field units)  2 log kh  rw kh 

() ( )

m p − m p wf = 0.1491

 0.472re T  T 2  2 log + 0.87 S q sc + 0.1296 D q sc kh  rw kh 

(Bars2/cp, ( 9-19)

metric units)

∆m(p)/q (psia2D/cpMscf)

40,000 ed biliz st a

35,000

30,000 a = intercept

tra

,b ent ns i

e l op =s

25,000

20,000 0

2000

4000 Rate, qsc (Mscf/D)

6000

8000

Figure 9-4 Deliverability plot for an isochronal or a modified isochronal test. Linear scale, pseudo-pressure method.

Before the pseudo-steady state regime, the response follows the semi-log approximation and ∆m(p) is :

() ( )

m p − m p wf = 1637

 T T  k∆t 2 S  log + 3 . 23 + 0 . 87  q sc + 1422 Dq sc 2 kh kh  φµ i cti rw 

(psia2/cp, field units)

() ( )

m p − m p wf = 0.1491

 T  k∆t 2 q sc + 0.1296 T D q sc log 3 . 10 0 . 87 S + + 2   kh  kh φ µ i c ti rw 

(Bars2/cp, metric units)

( 9-20)

The two ∆m(p) deliverability relationships can be expressed as a(t) qsc + b q2sc. During the infinite acting regime, a(t) is an increasing function of the time whereas "a" is constant when pseudo-steady state is reached. The coefficient "b" is the same in the two equations. The Absolute Open Flow Potential is :

- 155 -


q sc , AOF =

(

− a + a 2 + 4b m( p) − m( psc )

)

(Mscf/D, m3/D)

2b

( 9-21)

9-3.2 Back pressure test (Flow after flow test) The well is produced to stabilized pressure at three or four increasing rates and the different flow periods have the same duration. pi pwf1

pwf2

30,000

pwf3

pwf4 20,000

6900 10,000

6800 0

Rate, qsc (Mscf/D)

Pressure (psia)

7000

0 200

400

600

800

1000

Time (hours)

Figure 9-5 Pressure and rate history for a backpressure test.

∆m(p)/q (psia2D/cpMscf)

3500

3000

b=

pe slo

2500 a = intercept 2000 0

2000

4000

6000

8000

Rate, qsc (Mscf/D)

Figure 9-6 Deliverability plot for a backpressure test. Linear scale, pseudo-pressure method.

9-3.3 Isochronal test The well is produced at three or four increasing rates and a shut-in period is introduced between each flow. The drawdown periods, of same duration tp, are stopped during the infinite acting regime. The intermediate build-ups last until the initial pressure pi is reached. A final flow period is extended to reach stabilized flowing pressure.

- 156 -


Pressure (psia)

30,000 pwf1 pwf2

pwf, stab

6900

pwf3 pwf4

6800

20,000

0

200

400 Time, hours

10,000

Rate, qsc (Mscf/D)

pi 7000

0 800

600

Figure 9-7 Pressure and rate history for an isochronal test.

sta bil tra ize ns d 1/ ien n= t, slo pe

pi2 (or pws2 )- pwf2 (psia2)

108

107

pwf=14.7 psia

106 AOF=8000 Mscft/D

105 103

104

105

Rate, qsc (Mscf/D)

Figure 9-8 Deliverability plot for an isochronal or a modified isochronal test. Log-log scale, pressure-squared method.

9-3.4 Modified isochronal test The intermediate shut-in periods have the same duration tp as the drawdown periods, and the last flow is extended until the stabilized pressure is reached.

6900

pws2

pws3

pws4

pi 30,000

pwf1 pwf2

6700

pwf, stab

pwf3 pwf4

6500

10,000

6300 0

100

200

20,000

300 400 Time (hours)

500

Rate, qsc (Mscf/D)

Pressure (psia)

7100 pws1

0 600

Figure 9-9 Pressure and rate history for a modified isochronal test.

- 157 -

- 158 -

10 - BOUNDARIES IN HETEROGENEOUS RESERVOIRS

10-1 Boundaries in fissured reservoirs


A sealing fault can be reached during the fissure flow regime (Figure 10-1). The double porosity transition is observed during the semi-radial flow regime, after a first derivative stabilization at 1. 102

101 start of the sealing fault 1

1

1

0.5 fissure regime

10-1 10-1

101

1

102

transition 103

104

total system

105

106


Figure 10-1 Well with wellbore storage near a sealing fault, double porosity reservoir, pseudo-steady state interporosity flow. CD = 104, S = 0, LD = 5000, ω = 0.2, λeff = 10-9.


In a channel double porosity reservoir with unrestricted interporosity flow, a 1/4 slope derivative straight line can be observed at transition time (Figure 10-2). 102 º slo

101

slope

1

10-1 10-1

1/4

pe

1/ 2

0.5

0.25 1

101

102 103 104 105 Dimensionless time, tD /CD

106

107

108

Figure 10-2 Well with wellbore storage in a double porosity channel reservoir, unrestricted interporosity flow, slab matrix blocks. The thin curves correspond to the infinite double porosity reservoir response. CD = 10, S = 0, L1D = L2D = 300, ω = 10-3, λ = 10-6.

When the four sealing boundaries of a closed system are reached during the fissure flow, the double porosity transition is superimposed to the start of the pseudosteady state regime (Figure 10-3). With mixed boundaries, derivative responses can exhibit several consecutive humps (Figure 10-4).

- 159 -


Chapter 10 - Boundaries in heterogeneous reservoirs

102 º

101

0.5

1

10-1 10-1

1

101

102

103

104

105



Figure 10-3 Drawdown response for a well with wellbore storage at the center of closed square double porosity reservoir, pseudo steady state interporosity flow. The thin dotted curves correspond to the equivalent homogeneous closed square reservoir. The infinite reservoir double porosity derivative response is presented by the thick dotted curve. CD = 100, S = 0, LiD = 1000, ω = 0.1, λeff = 10-6.

102 º 101

2 1 0.5 10-1 10-1

1


104

105

Figure 10-4 Well with wellbore storage in a square double porosity reservoir with composite boundaries, pseudo steady state interporosity flow. The dotted curve corresponds to the equivalent infinite double porosity reservoir. CD = 100, S = 0, ω = 0.1, λeff = 10-6, L1D = L2D = 500 (sealing), L3D = 1500 (constant pressure) and L4D = 1500 (sealing).

10-2 Boundaries in layered reservoirs On Figure 10-5, the reservoir cross flow is not established when the fault is seen. The boundary is reached first in Layer 1, and the derivative deviates earlier than on the equivalent homogeneous response. In layered channel reservoirs, the channel width can appear smaller (Figure 10-6).

- 160 -



102

101 1 1 0.5 10-1 10-1


1

104

105


Figure 10-5 Well with wellbore storage in a double permeability reservoir with a sealing fault. The dotted curves describe the sealing fault response in the equivalent homogeneous reservoir. CD = 100, S1 = S2 = 0, LD = 500, ω = 0.15, κ = 0.7, λ = 10-10. 102

101 /2 e1 slop

1 0.5 10-1 10-1

1

101

102

103

104

105


Figure 10-6 Well with wellbore storage in a double permeability reservoir with two parallel sealing faults. The dotted curves describe to the channel response of the equivalent homogeneous reservoir. CD = 100, S1 = S2 = 0, L1D = L2D = 1000, -10 ω = 0.15, κ = 0.7, λ = 10 .

In a closed double permeability reservoir, a derivative hump can be observed at intermediate time, as on the composite example of Figure 10-4. On Figure 10-7, the closed circular boundary is reached during the early time commingled response. After the wellbore storage effect and the early time infinite behavior, a second unit slope straight line, followed by a hump is seen. Later, the derivative stabilizes at 0.5 / (1 - κ) until the final unit slope line for the pseudo steady state regime becomes evident. The first unit slope straight line describes the wellbore storage, the second is a function of layer 1 storage ω A/rw2 and the final corresponds to the reservoir storage (A/rw2 in dimensionless terms).

- 161 -

102 1

º

pe

101

slo



pe slo

1

1

0.5/(1-κ) 0.5

10-1 10-1

1

101


105

106

107

Figure 10-7 Drawdown response for a well with wellbore storage in a closed circle double permeability reservoir. The dotted curves correspond to the closed equivalent homogeneous reservoir. CD = 100, S1 = S2 = 0, rD = 5000, ω = 0.002, κ = 0.7, λ = 10-10.

10-3 Composite channel reservoirs In channel reservoirs, when the mobility changes near the edges of the channel banks (Figure 10-8), or along the channel length (Figure 10-9), the responses tend to be equivalent to that of a homogeneous channel with a different width. When the mobility contrast is large, drawdown responses can show at intermediate time a closed system behavior, or channel with constant pressure boundary response (Figure 10-10). Build-up responses can be severely distorted (Figure 1011).


102 M= 5 0.2

º 101 1/2 pe s lo

1 0.5

M=0.2, 1,

5

10-1 1

101

102

103

104

105

106


Figure 10-8 Well with wellbore storage in a composite channel. The interfaces are parallel to the boundaries. CD = 100, S = 0, L1D = L2D =1000, d1D = d2D =500, M1 = M2 = 0.2, 1 and 5.

- 162 -



102

M =0.2 5

º 101 pe slo

1/2

1 0.5

M=0.2, 1,

5

10-1 1

101


105

106

Figure 10-9 Well with wellbore storage in a composite channel. The interfaces are perpendicular to the boundaries. CD = 100, S = 0, L1D = L2D =1000, d1D = d2D =2000, M1 = M2 = 0.2, 1 and 5.

Dimensionless Derivative p'D

103 closed channel

º

102

slo

101

pe

1

M= 50

e1 slop

/2

M=0.02

1

channel with constant pressure

0.5 10-1 101

102


106

107

Figure 10-10 Drawdown responses for a well with wellbore storage in composite channel. The interfaces are perpendicular to the boundaries. On the dotted curves, the interfaces are changed into sealing and constant pressure boundaries. CD = 100, S = 0, L1D = L2D =500, d1D = d2D =1500, M1 = M2 = 0.02, 1 and 50.


102 º 101

M=5, 1, 0.2 M = 50

1 0.5 10-1 101

102

103

104

105


Figure 10-11 Pressure and derivative drawdown and build-up responses of curve M=50 of Figure 10-10. The two dotted derivative curves are drawdown, the build-up response (thick line) is generated for (tp/C)D = 650.

- 163 -

- 164 -

11 - COMBINED RESERVOIR HETEROGENEITIES

11-1 Fissured-layered reservoirs On Figure 11-1, a double permeability response where the two layers are fissured is presented. For each layer, restricted interporosity flow is assumed. The parameters correspond to the triple porosity example of Figure 4.33. When the vertical communication is good in a fissured layered reservoir, grouping of matrix size by layers has no effect on the response.


When reservoir cross flow between layers is not allowed (λ =0), the response is different. 10 double permeability

1

0.5

10-1

no crossflow crossflow oooo

triple porosity

10-2 1

10

102

103

104

105

106

107


Figure 11-1 Fissured layered reservoir, pseudo steady state interporosity flow, different λ in each layer. CDf+m = 1, S1 = S2 = 0, ω = 0.1, κ= 0.7, λ =10-3 or λ =0. ω1 =0.01, λeff1=10-5, ω2 =0.01, λeff2 =5x10-7. The (o) dotted curve corresponds to the triple porosity response of Figure 4.33.

Fissured layered responses depend upon which transition, the double porosity or the double permeability transition, is seen first. On Figure 11-2, the high permeability layer 1 is fissured and not layer 2. When the interporosity flow parameter is small (λeff1 =10-8), layer 1 is in fissure regime when the double permeability transition starts. The reservoir cross flow is established between the layer 2 and the fissure network of layer 1 and the response becomes equivalent to the double permeability response κ = 0.99 of Figure 7-3 (for a storativity ratio ω =10-3). If layer 1 is in total system flow (λeff1 =10-3) at start of the double permeability transition, the double porosity transition in layer 1 is first seen during the two layers no cross flow regime. The double permeability transition tends to be similar to that of the double permeability response κ = 0.99 of Figure 7-5 (ω =10-1).

- 165 -

Chapter 11 - Combined heterogeneities


10

double permeability ω=10-1

1

λ 1= 10-3

0.5

10-1 λ 1= 10-8

double permeability ω=10-3

10-2 10-1

1

102

10

103

104


Figure 11-2 Fissured layered reservoir, pseudo steady state interporosity flow, only layer 1 is fissured. CDf+m = 1, S1 = S2 = 0, ω = 0.1, κ = 0.99, λ =4.10-4, ω1 =0.01, λeff1 =10-3 or λeff1 =10-8. The (o) dotted curve corresponds to the double permeability response of Figure 7-3 with ω = 10-3, κ = 0.99 and λ =4.10-4 and the ( ) to the double permeability response of Figure 7-5 with ω = 10-1, κ = 0.99 and λ =4.10-4.

11-2 Fissured radial composite reservoirs On Figure 11-3, the inner region of a radial composite reservoir is fissured. The radial composite model corresponds to the curve M=10 of Figure 6-2.


When λeff1 =10-4, the response shows first a characteristic double porosity valley transition. After, it is equivalent to the radial composite with a homogeneous inner region. When λeff1 =10-6, the radial composite interface is seen during the fissure regime. The two transitions are combined at the same time. 102 double porosity λ1=10-6 radial composite

10

1

0.5

λ1=10-6

λ1=10

-4

10-1 10-1

1

10

102

103

104

105

106


Figure 11-3 Radial composite reservoir, the inner region is fissured, pseudo steady state interporosity flow. CD = 100, S = 3, M=10, F =1 rD = 700. ω1 =0.01, λeff1=10-4 or λeff1=10-6. The (o) dotted curve corresponds to the radial composite response of Figure 6-2 with M=10, the dashed curve describes the double porosity response with ω1 =0.01 and λeff1=10-6.

- 166 -


11-3 Layered radial composite reservoirs


On Figure 11-4, the reservoir is two-layer without cross flow, but layer 2 is radial composite with a strong reduction of mobility at r2D = 100. The derivative tends to follow a unit slope straight line at intermediate time (examples M2 =100 or 1000). After the derivative hump, the two layers commingled infinite reservoir response is seen, and the derivative tends to stabilize. 102

M2=1000

M2=1000

100

10

M2=10

10

1 0.5 10-1 10-1

1

10

102

103

104

105

106


Figure 11-4 Layered reservoir, no cross flow, layer 1 homogeneous, layer 2 radial composite. CD = 30, S1 = S2 =0, ω=0.1, κ=0.5, λ=0. r2D = 100, M2 = 10, 100, 1000, F2 = 1.


The radial composite double permeability model can be used to describe the presence of a flow barrier between the layers. When no cross flow is allowed in the inner region of radius rD, the valley shaped derivative transition is delayed, and it tends to be steeper than the double permeability infinite reservoir response (Figure 11-5). When the reservoir cross flow is only possible in the inner region, the responses change to the two layers without cross flow at late time (Figure 116). Before, the derivative deviates above the 0.5 stabilization and produces a smooth hump. 10

1

10-1 10-1

0.5

rD=30

1

10

102

rD=100

103

300

104

105

106


Figure 11-5 Layered reservoir, no cross flow in the inner region. -4 CD = 1, S1 = S2 =0, ω=0.1, κ=0.9, M=F =1. λ1=0, λ2=4 10 , rD=30, 100, 300. The dotted curves correspond to the double permeability response of Figure 7-5 with κ=0.9.

- 167 -



10

1

rD=30

rD=100

300

0.5 10-1 10-1

1

10

102

103

104

105

106


Figure 11-6 Layered reservoir, no cross flow in the outer region. -4 CD = 1, S1 = S2 =0, ω=0.1, κ=0.9, M=F =1. λ1=4 10 , λ2=0, rD=30, 100, 300. The dotted curves correspond to the double permeability response of Figure 7-5 with κ=0.9 and the dashed curves to the commingled reservoir (λ=0).

- 168 -

12 - OTHER TESTING METHODS

12-1 Drillstem test 12-1.1 Test description During a drillstem test, a down hole shut-in valve controls the well. Before opening, the well is partially filled with a liquid cushion designed to apply a pressure p0 above the valve, smaller than the formation pressure pi. When the tester valve is opened, an instantaneous drop of pressure (pi - po) is applied to the sandface. The formation starts to produce into the well, the level rises in the drill string and the bottom hole flowing pressure increases. If the liquid level reaches the surface, the rate tends to stabilize and the DST procedure becomes similar to that of a standard production test. When no flow to surface is desired, the down hole valve is closed before the liquid has reached the surface (Figure 12-1). This flow period is called a "slug test".

5100 pi Pressure (psia)

5000 4900

shut-in

4800 p0

4700 4600 0

1

2

3

4

5

6

Time (hours)

Figure 12-1 Example of DST pressure response. The rate is less than critical. Linear scale. The sequence is initial flow, initial shut-in, flow period and final shut-in.

12-1.2 Slug test analysis During a slug test period, the pressure increases and the flow rate declines. In some cases, the rate is not controlled by the downstream pressure but by the well condition. It becomes constant and the pressure increases linearly with time. With this flow condition, called critical flow, the flowing pressure is not suitable for interpretation. When rate is less than critical, slug test analysis methods use a dimensionless pressure ratio prD, defined as the drop of pressure (pi-pwf ) normalized by (pi - po).

- 169 -

Chapter 12 - Other testing methods

prD =

pi − p wf (t )

( 12-1)

pi − p 0

The ratio prD is very sensitive to the accuracy of the initial pressure pi, especially after some production time, when (pi - pwf ) becomes small.

Slug test pressure type curve

On the type curve Figure 12-2, the dimensionless pressure ratio prD is presented versus the dimensionless time tD/CD. The CDe2S curves describe the well condition.

Dimensionless pressure ratio, prD =[pi- pwf(t)]/[pi- p0]

1

CDe2S=1060

10-1 CDe2S=10-1

10-2 -1

10-3 10-1

1

slo

101

pe

102

103


Figure 12-2 Slug test type curves on log-log scale.

When the well is opened, prD = 1 and, when the liquid level rises in the well, the ratio drops. The same pressure ratio is used for the data and the dimensionless curves, the pressure match is PM =1. Knowing the wellbore storage coefficient from the changing liquid level relationship of Equation 1-5, the time match gives the permeability thickness product:

µC

 tD CD  (mD.ft, field units)   0.000295  ∆t  MATCH µC  t D CD  (mD.m, metric units) kh =   0.00223  ∆t  MATCH kh =

the skin is estimated from the CDe2S curve match with Equation 2-10.

- 170 -

( 12-2)


Analysis of slug test with the derivative type curve

The product of the slug test pressure change (pi-pwf ) by the elapsed time ∆t can be matched directly against a derivative type-curve, without having to differentiate the data.

(

)

(

)

dp D 0.000295kh = ∆t pi − p wf (t ) (field units) d ln t D C µ ( pi − p0 ) dp D 0.00223kh = ∆t pi − p wf (t ) (metric units) d ln t D Cµ ( pi − p0 )

( 12-3)

The permeability thickness product is estimated either from the time match (Equation 12.2) or from the pressure match :

kh =

µ C ( pi − p0 )  dp D d ln t D 

(mD.ft, field units)

kh =

µ C ( p i − p 0 )  dp D d ln t D 

(mD.m, metric units)

(

)

0.000295  ∆t pi − p wf (t )  MATCH 0.00223

(

)

 ∆t p i − p wf (t )    MATCH

( 12-4)

12-1.3 Build-up analysis When the well is closed down hole before the liquid level has reached the surface, the decreasing rate has to be estimated as a function of time in order to analyze the subsequent build-up. 5000 p6

400 p6

4800

p2

p1

4700

300

p1

p0

200

q1

4600

100 q5

q6

4500 1

1.2

Rate (BOPD)

Pressure (psia)

4900

1.4

1.6

1.8

2

0 2.2

Time (hours)

Figure 12-3 Example of rate estimation during a DST flow period.

The increasing pressure curve of the flow period is discretized into constant pressure steps (Figure 12-3). Knowing the liquid gravity, the pressure difference is converted into the corresponding height of fluid. From the capacity of the drill pipe, the height is converted into volume.

- 171 -


12-2 Impulse test 12-2.1 Test description With impulse tests, the well is produced only a few minutes and then closed. 5100

Pressure (psia)

pi

4900

4700 ∆t

tp 4500 0

0.5

1

1.5

2

Time (hours)

Figure 4 Example of impulse pressure response. Linear scale.

12-2.2 Impulse test analysis The complete well pressure response is analyzed on a single analysis plot. During the short flow, the impulse response is expressed as pi − pwf t p and, during the

(

(

)

)

shut-in, as ( pi − pws ) t p + ∆t . The pressure and derivative type curves are used to analyze the pressure response: during the flowing time, the impulse response is matched against a pressure type curve and, during the shut-in period, the response deviates from the usual pressure response to reach the derivative curve with same CDe2S. The pressure match is defined, as in Equation 12-3 :

(

)

dp D 0.000295kh t p + ∆t ( pi − p ws ) (field units) = d ln t D Qt µ dp D 0.00223kh = t p + ∆t ( pi − p ws ) (metric units) d ln t D Qt µ

(

)

where Qt is the amount of fluid produced during the short flow tp.

- 172 -

( 12-5)


Pressure change, ∆p= (pi-pwf)tp or (pi-p)(tp+∆t) (psi)

102

well flowing

well shut-in

101

1 10-3

10-2

10-1

101

1


Figure 12-5 Impulse match.

As for slug test analysis, the result of impulse test interpretation is very sensitive to the accuracy of the initial pressure pi used for the data plot. The results can be controlled with a conventional analysis of the shut-in period after the few minutes flow period (Figure 12-6). The derivative analysis is not affected by a possible error in initial pressure, and the pressure curve can be used to estimate the skin accurately.

Pressure change, ∆p and Derivative (psi)

103

102

101 10-2

10-1

1

101


Figure 12-6 Pressure and derivative analysis of the impulse shut-in period. Log-log scale, ∆p and ∆p' versus ∆t.

12-3 Rate deconvolution In the multi rate superposition method presented in Section 2-2.2 (Eq. 2-15), the rate history is described by several step-rate changes occurring at different flow times ti. In the case of a variable production, the rate increments are infinitesimal and the multi rate superposition is changed into the convolution integral. The pressure response due to a variable rate q(t) can be expressed with the time derivative of the rate history: - 173 -


141.2 Bµ ∆p(t ) = kh ∆p(t ) =

18.66 Bµ kh

t

∫ q' (τ) p

D (t

− τ)dτ (psi, field units)

∫ q' (τ) p

D (t

− τ)dτ (bars, metric units)

τ=0 t

( 12-6)

τ=0

The objective of the deconvolution is to transform the measured pressure response ∆p(t), after any variable rate sequence q(t), into an equivalent constant flow rate test that can be analyzed with the usual methods. Several algorithms have been proposed for deconvolution of well test measurements, using real data of Laplace transformed data. Results are very dependent upon the quality of the rate curve. The technique has also been envisaged for interpretation of build-up tests affected by wellbore storage effect. With accurate sandface flow rate measurement at early shut-in time, the effect of afterflow can theoretically be eliminated from the pressure build-up response.

12-4 Constant pressure test (rate decline analysis) When a well is producing at constant wellbore pressure, the declining rate can be analyzed versus time.

Dimensionless rate, qD

1 Infinite reservoir 10-1 5000 10-2

2500 re/rwe

10-3 103

104

105

106

107

= 1000 108

Effective dimensionless time, tDe

Figure 12-7 Decline curves on log-log scale. Closed reservoir. qD versus tDe.

With log-log rate type curves, the dimensionless flow rate qD is expressed as :

qD =

1412 . Bµ

(

kh pi − pwf

)

q (t ) (field units)

- 174 -


qD =

18.66 Bµ q (t ) (metric units) kh p i − p wf

(

)

( 12-7)

For semi-log analysis, the reciprocal of the rate 1/q is graphed vs. log ∆t.

  1 Bµ k = 162.6 − 3.23 + 0.87 S  (D/Bbl, field units) log ∆t + log 2 q kh pi − p wf  φ µ ct rw 

(

)

  Bµ k 1 = 21.5 − 3.10 + 0.87 S  (D/m3, metric units)( 12-8) log ∆t + log 2 q kh ( pi − pwf )  φ µ ct rw  Results: the permeability is estimated from the slope mq of the 1/q straight line and the skin from the intercept at 1 hour.

kh = 162.6 kh = 21.5

Bµ (mD.ft, field units) m q ( p i − p wf )

Bµ (mD.m, metric units) m q ( p i − p wf )

( 12-9)

1 q (1hr )  k S = 1.151 − log + 3 . 23  φ µ ct rw2  mq  1 q (1hr )  k S = 1.151 − log + 3 . 10  φ µ c t rw2  m q 

( 12-10)

12-5 Vertical interference test Vertical interference tests are used to estimate vertical permeability in a single layer, or quantify the presence of a sealing interval. An example of usual application is the characterization of low permeability in feasibility studies related to underground storage projects. Different types of equipment can be used in order to isolate several intervals in the same well.

- 175 -


kH1, kV1

hw-obs

kH2, kV2

hw zw

kV

kH3, kV3

zw-obs

kH

Homogeneous reservoir

Three layers reservoir


Figure 12-8 Well and reservoir configurations.

102

101

1

0.5 line Zw-obs/h = 0.6

0.7 0.8

10-1 10

102

103

104

105

106

107



Figure 12-9 Vertical interference responses from a well in partial penetration with wellbore storage. Log-log scale. Several distances. CD = 6, Sw=0, kV/kH = 0.005. Producing segment: hw/h = 1/10, zw/h = 0.5; observation segment: hw-obs/h = 1/100, zw-obs /h = 0.6, 0.7, 0.8.

102

101

1

kV/kH =

0.5 0.05

10-1

10

102

103

0.5 line

0.005 104

105

106

107


Figure 12-10 Vertical interference responses from a well in partial penetration with wellbore storage. Log-log scale. Several vertical permeability. CD = 6, Sw=0. Producing segment: hw/h = 1/10, zw/h = 0.5; observation segment: hw-obs/h = 1/100, zw-obs /h = 0.6. Vertical permeability: kV/kH = 0.5, 0.05, 0.005.

- 176 -


With the double-stage testing method, two tests are performed on the same layer: the first, on a thick interval, is used to define the horizontal permeability. By inflating internal packer in the thick interval, three discrete intervals are isolated to provide vertical interference responses.

Observation interval

Flowing interval

Observation interval Test 1 : radial flow

Test 2 : spherical flow


Figure 12-11 Double-stage test.

102 Partial penetration 101 Observation

Test 1 1

0.5 line 10-1

1

10

102

103

104

105

106


Figure 12-12 Double-stage test log-log responses. CD = 7, Sw=0. Producing segment: hw/h = 1/10, zw/h = 0.5; observation segment: h.w-obs/h = 1/20, zw-obs /h = 0.35. Vertical permeability: kV/kH = 0.3.

- 177 -

- 178 -

13 - MULTIPHASE RESERVOIRS

13-1 Perrine method 13-1.1 Hypothesis and definitions An equivalent monophasic liquid of constant properties is defined as the sum of the three phases: oil, water and gas. The three phases are assumed to be uniformly distributed in the reservoir, and the saturations are constant during the test period. The equivalent rate is expressed:

(q B ) t

= q o Bo + q w Bw + q g B g

(

)

= q o Bo + q w Bw + q sg − q o Rs B g

(Bbl/D, m3/D)

( 13-1)

where qsg is the gas rate measured at surface, and qo Rs the dissolved gas at bottom hole conditions. It is assumed that the total mobility (k/µ)t of the equivalent monophasic fluid can be expressed as the sum of the effective phase mobilities :

(k µ )t

= k o µ o + k w µ w + k g µ g (mD/cp)

( 13-2)

The effective total compressibility ct includes the effect of free gas liberated (or dissolved) in the oil and the water phases :

(

ct = c f + S o co + S w cw + S g c g + S o B g Bo (psi-1, Bars−1)

) ∂∂ Rp

s

(

+ S w B g Bw

) ∂∂Rp

sw

( 13-3)

13-1.2 Analysis In the usual equations for oil reservoirs, the mobility k/µ and the rate q are changed into the total mobility (k/µ)t and the equivalent rate (qB)t. For log-log analysis, dimensionless pressure and time are respectively :

(k µ )t h ∆p (field units) 1412 . (qB) t (k µ )t h pD = ∆p (metric units) 18.66 (qB )t

pD =

- 179 -

( 13-4)

Chapter 13 - Multiphase reservoirs

(k tD = 0.000295 CD (k tD = 0.000223 CD

µ )t h C

µ )t h

C

∆t (field units)

∆t (metric units)

( 13-5)

The slope m of the semi-log straight line is expressed

(qB )t (k µ )t h (qB )t m = 21.5 (k µ )t h m = 162.6

(psi, field units)

( 13-6)


The analysis yields the effective mobility of this equivalent fluid. When the relative permeabilities kr"o,w,g" of the different phases are known, the absolute permeability can be estimated :

(k µ )t

(

)

= k k ro µ o + k rw µ w + k rg µ g (mD/cp)

( 13-7)

13-2 Other methods 13-2.1 Multiphase pseudo-pressure For solution gas drive reservoir, the pseudo pressure is expressed : p

k ro ( S o ) dp (psi/cp, Bars/cp) µ o Bo 0

m( p) = ∫

( 13-8)

For gas condensate reservoir, the molar density of the oil and gas phases ρo,g are used:

 k ro k rg   ρ + ρ ∫  o µ o g µ g  dp (psi/cp, Bars/cp) p0 p

m( p) =

( 13-9)

The relative permeability curves are needed to calculate the multiphase pseudopressure functions. As the saturation profile depends upon the rate history, m(p) depends upon the test sequence.

- 180 -

Chapter 13 - Multiphase reservoirs

13-2.2 Pressure squared method For log-log analysis, dimensionless pressure is expressed with respect to the oil rate:

( )

ah ∆ p 2 (field units) 282.4 q o ah pD = ∆ p 2 (metric units) 37.33 q o

pD =

( )

( 13-10)

where a is assumed to be a constant, defined as :

ko = ap µ o Bo

( 13-11)

- 181 -

- 182 -

14- TEST DESIGN

14-1 Introduction Once the objectives of the test have been defined, the program is established taking into account the different operational constraints. Test simulations are generated to ensure the objectives can be achieved, and to define the optimum testing sequence. Test programming and conduct, as well as the definition of the responsibilities during testing, are presented in a different section. In the following, only test simulation is discussed.

14-2 Test simulation 14-2.1 Simulation procedure • Before generating the simulations, all parameters must have been defined: static parameters, reservoir parameters and the anticipated flow rate. • In order to evaluate the expected reservoir model, a first simulation can be generated for a long constant rate drawdown. • By examination of this ideal response, the minimum duration of the flow and shut-in periods can be estimated. • A multirate simulation is generated for prediction of the actual test response. Taking into account possible pressure gauge noise or drift, the test program is adjusted to ensure a complete and significant pressure response for the lowest test duration. • The simulation can be converted into data in order to control the quality of the future analysis.

14-2.2 Test design tips Test design is a compromise between cost and reliability. The final test program is defined from not only technical considerations, but also taking into account the desired degree of confidence in the results. Test sequences are sometimes designed with two or several buildup periods after different flow rates, some relatively short, since wellbore problems frequently distort early time data. For gas wells for example, the Modified Isochronal test sequence, possibly followed by a long buildup period, is well adapted to transient analysis purpose.

- 183 -

Chapter 14 - Test design

In multirate testing, an increasing flowrate sequence is preferred to a decreasing rate history. With decreasing rates, the multirate correction with the time superposition function can be very sensitive to inaccurate rate data.

14-3 Test design reporting and test supervision Test design is not limited to the definition of the different flow periods. From examination of the pressure change observed on the test simulation, the requirements for the pressure gauge characteristics are defined. Guidelines for clean up (gas wells) and initial shut-in can be established. If the reservoir pressure is decreasing, it may be necessary to evaluate the pressure trend accurately before the test (interference test design). In such a case, the duration of the reservoir pressure survey before the start of the operation is part of the design program. Experience of tests in neighboring wells can be used to establish specifications such as gauge depths, use of a down hole shut-in tool, etc. In the ideal case, the same person is in charge of the design and of the test supervision. The experience gained from the design study can be used to adjust in real time the program to any unexpected event (well shut-in for operational or safety reason), or to a different pressure behavior. During the test supervision, any action that can affect the pressure data must be recorded (such as leak, operation on the well or change of annular pressure during shut-in, etc.)

- 184 -

15 - FACTORS COMPLICATING WELL TEST ANALYSIS

15-1 Rate history definition Two approaches can be used in order to simplify the rate history: 1. An equivalent production time is defined as the ratio of the cumulative production divided by the last rate (called equivalent Horner time). On the test example of Figure 15-1, tp=120. 2. When there is a shut-in period in the rate history, if the bottom hole pressure has almost reached the initial pressure pi, it is assumed that the rate history prior this shut-in is negligible. On the test example, tp=20.

Pressure, p

4000 3900 3800

Rate, q

3700 3600

tp=120 tp=20

3500 0

50

100

150

200

250

300

350

400

450

500

Time, t

Figure 15-1 Example of a two drawdowns test sequence. Linear scale.

Pressure change ∆p and pressure derivative ∆p’ (psi)

103

102 tp=20

101 tp=120

1 10-2

10-1

1 101 Elapsed time, ∆t (hours)

102

103

Figure 15-2 Log-log plot of the final build-up. The derivative is generated with three different rate histories.

In practice, if the duration of the analyzed period is ∆t, it is possible to simplify the rate history for any rate changes that occurred at more than 2∆t before the start of the period. All rate variations immediately before the analyzed test period must be introduced in the superposition time.

- 185 -

Chapter 15 - Factors complicating well test analysis

15-2 Error of start of the period 3830

e

Pressure, p

3810

d a

3790

b 3770

c 3750 169.7

169.8

169.9

170.0 Time, t

170.1

170.2

170.3

Figure 15-3 Example of Figure 15-1 at time of shut-in. Time and pressure errors. - Shut-in time error: curve a = 0.1 hr before and curve b = 0.1 hr after the actual shut-in time. - Shut-in pressure error: curve c = 10 psi below and curve d = 10 psi above the last flowing pressure. - Error in time and pressure: curve e.


103

102

101

a

1 10-2

10-1

1

101

102

103


Figure 15-4 Case a: shut-in time too early.


103

102

101

b

1

10-2

10-1

1

101

102


Figure 15-5 Case b: shut-in time too late.

- 186 -

103



103

102

101

c

1 10-2

10-1

1

101

102

103


Figure 15-6 Case c: last flowing pressure too low.

Pressure change ∆p and pressure derivative ∆ p’ (psi)

103

102

101

d

1 10-2

10-1

1

101

102

103


Figure 15-7 Case d: last flowing pressure too high.


103

102

101

e

1 10-2

10-1

1

101

102

103


Figure 15-8 Case e: shut-in time too late, last flowing pressure is taken in the build-up data, during the wellbore storage regime.

A good log-log match can be obtained in case e but the resulting skin is under estimated. Pressure errors are clearly shown on the linear scale test simulation plot.

- 187 -


15-3 Pressure gauge drift Pressure change ∆p (psi)

300

Drift + 200

Drift 100

0 0

100

200

300


Figure 15-9 Final build-up of Figure 15-1. Drift of ± 0.05 psi/hr. Linear scale.


103

102

Drift + 101

Drift 1 10-2

10-1

1

101

102

103


Figure 15-10 Log-log plot of the build-up example. Drift of ± 0.05 psi/hr.

The effect of a constant drift is inverse during flow and shut-in periods.

15-4 Pressure gauge noise Pressure change ∆p (psi)

250 200 150 100 50 0 0

100

200

300


Figure 15-11 Final build-up of Figure 15-1. Noise of +1 psi every 2 points. Linear scale.

- 188 -



103

102

101

1 10-2

10-1

1

101

102

103


Figure 15-12 Log-log plot of the build-up example. Noise of +1 psi every 2 points. Three points derivative algorithm. No smoothing.

15-5 Changing wellbore storage Changing wellbore storage happens when the compressibility of the fluid in the wellbore is not constant. It is observed for example when, in a damaged oil well, free gas is liberated in the production string.


103

102

101

C oil C gas

1 10-2

10-1

1

101

102

103


Figure 15-13 Log-log plot of a drawdown example of changing wellbore storage.

During drawdown, the response describes first the compressibility of the oil but, when the pressure drops below bubble point, the gas compressibility dominates. The wellbore storage coefficient of Equation 1-4 is then increased.

- 189 -



103

102

101

C oil

C gas 1 10-2

10-1

1

101

102

103


Figure 15-14 Log-log plot of a build-up example of changing wellbore storage

During build-up periods, the response corresponds to the gas wellbore storage coefficient immediately after shut-in, and changes to the lower oil wellbore storage later. This produces a steep increase of derivative and, in some cases; the derivative follows a slope greater than unity at the end of the gas dominated early time response. Due to the variable compressibility of gas, changing wellbore storage is also frequently evident on gas wells with a large drawdown.

15-6 Two phases liquid level In diphasic wells (oil + water, or gas + condensate), a phase redistribution in the wellbore can produce a characteristic humping effect.

diphasic flow

changing liquid level

end of phase segregation effect Figure 15-15 Changing liquid level after phase segregation.

When, after shut-in, water falls at the bottom of the well for example, the weight of the column between the pressure gauge and the formation is not constant as long as the water level rises and the gauge pressure is not parallel to the formation pressure. In some cases, the build-up pressure can show a temporary decreasing trend after some shut-in time. During this time interval, the derivative becomes negative.

- 190 -


Rate, q

Pressure, p

4000 3500

humping

Pressure difference after phase segregation

3000 Pressure difference before phase segregation

2500

2000 18

28 Time, t

Figure 15-16 Example of build-up response distorted by phase segregation. Humping effect.

If the interface between the two phases stabilizes, or reaches the depth of the pressure gauge, the pressure difference between gauge and formation returns to a constant, and the remaining build-up data can be properly analyzed.


104

103

102

101 10-3

10-2

10-1

1

101

102


Figure 15-17 Log-log plot of the build-up example of phase segregation.

When phase redistribution is expected, the pressure gauge should be as close as possible to the perforated interval (or even below).

15-7 Input parameters, and calculated results of interpretation Errors in the static parameters influence the calculated interpretation results, but the choice of the interpretation model is in general not affected. Frequently, the analysis is initialized with approximate values, and refined with adjusted parameters later, without significantly changing the interpretation model. The net thickness h and the oil viscosity µ are for example frequently not accurately defined during exploration testing. Well test interpretation provides the kh/µ group from the log-log pressure match or the semi-log slope m. Any error on h or µ directly influences the permeability estimate k. The skin Equation 1-14 - 191 -


shows that, for a given kh/µ group, S is hardly dependent upon h (with a logarithm relationship), and not upon the viscosity µ. (present in the k/µ group). From the equations used to calculate the different interpretation results, the influence of any error in the static parameters can be evaluated. The radius of investigation for example, and the distance to a possible boundary, are dependent upon h (with the square root relationship of Equation 1-32 or 1-22), but independent of µ. Before comparing results of interpretation to geological or geophysical data, the significance of the model parameters must be clearly understood. This can be illustrated with the different averaging methods used for the permeability: • The apparent vertical permeability kV is a harmonic average as shown in Eq. 325 • The horizontal permeability kH, is the arithmetic average of each layer permeability (Eq. 3-24 for example). • In the case of permeability anisotropy, the horizontal permeability is defined as the geometric average of Eq. 8-4. Boundary distances are frequently estimated by assuming strictly radial flow in a single homogeneous layer. In the case of a permeability anisotropy or heterogeneous reservoir properties such as layering (see Section 10-2) the distance to a reservoir boundary can be different from that indicated by the simple interpretation model used for analysis.

- 192 -

16 - CONCLUSION 16-1 Interpretation procedure 16-1.1 Methodology Well test analysis is a three steps process: 1. Identification of the interpretation model. The derivative plot is the primary identification tool. 2. Calculation of the interpretation model. The log-log pressure and derivative plot is used to make the first estimates. 3. Verification of the interpretation model. The simulation is adjusted on the three usual plots: log-log, test history and superposition.

Log-log analysis

Model selection (derivative)

1

Estimate parameters : kh, C, heterogeneities , boundaries (derivative) and S (pressure)

Simul

Test history simulation

#1 . . . . . . #n

•Adjust initial pressure pi •Check the data (variable skin, consistent rate history) •Check the model response on a larger time interval

Superposition simulation

2

3

Adjust parameters (pi, S, C...)

Next model End The consistency of the interpretation model is finally checked against non-testing information.

- 193 -

Chapter 16 - Conclusion

16-1.2 The diagnosis: typical pressure and derivative shapes Flow regime identification

GEOMETRY

LOG-LOG shape

TIME RANGE

slope

Early

Intermediate

Late

Radial

No 0

Double porosity restricted

Homogeneous Semi infinite behavior reservoir

Linear

1/2 1/2

Infinite conductivity fracture

Horizontal well

Two sealing boundaries

Bi-linear

1/4 1/4

Finite conductivity fracture

Finite conductivity fault

Double porosity unrestricted with linear flow

Spherical

No -1/2

Well in partial penetration

Wellbore storage

Pseudo Steady State

1 1

Steady State

0 -1 (−∞) Pressure curve Derivative curve

- 194 -

Layered no crossflow with boundaries

Closed reservoir (drawdown)

Conductive fault

Constant pressure boundary


Changes of properties during radial flow


Pressure derivative, log (∆p’)

Mobility decreases : Sealing boundaries, composite reservoirs, horizontal well with a long drain hole.

>

m1

m2 m1

Elapsed time, log (∆t)


Figure 16-1 The mobility decreases (kh ↓). Log-log and semi-log scales.



Mobility increases : Composite reservoirs, constant pressure boundaries, layered systems, wells in partial penetration.

m2 < m1

m1



Figure 16-2 The mobility increases (kh ↑). Log-log and semi-log scales.



Storativity increases : Double porosity reservoirs, layered and composite reservoirs.


m2

m1


Figure 16-3 The storativity increases (φ φ ct h ↑). Log-log and semi-log scales.

Storativity decreases : Composite systems.

- 195 -

=m

1




m2

= m1

m1



Figure 16-4 The storativity decreases (φ φ ct h ↓). Log-log and semi-log scales.

16-1.3 Summary of usual log-log responses Well models 1

1 2

Wellbore storage, C Radial, kh and S

∆p' & ∆p

Wellbore storage and Skin (3.1) C

S kh ∆t

1 2

Linear, xf Radial, kh and ST

∆p' & ∆p

Infinite conductivity fracture (3.2) 1/2

kh, S xf ∆t

Finite conductivity fracture (3.3) Bi-linear, kf wf Linear, xf Radial, kh and ST

∆p' & ∆p

1 2 3

xf kh, ST 1/2

kfwf

1/4 ∆t

1 2 3

Radial, hw and Sw Spherical (mobility ↑), kV Radial, kh and ST

∆p' & ∆p

Partial penetration (3.4) -1/2 kV hw , Sw ∆t

- 196 -

kh, ST


1 2 3

Radial vertical, kV and Sw Linear (mobility ↓), L Radial, kh and ST

∆p' & ∆p

Horizontal well (3.5) 1/2 kh, ST

L

kV, Sw ∆t

1 2 3

Radial fissures, k Transition (storativity ↑), ω and λ Radial fissures + matrix, kh and S

Double porosity, unrestricted interporosity flow (4.3) 1 2

Transition, λ Radial fissures + matrix, kh and S

ω

kh, S

λ ∆t

∆p' & ∆p

Double porosity, restricted interporosity flow (4.2)

∆p' & ∆p

Reservoir models

kh, S λ ∆t

1 2 3

Radial inner, k1h and Sw Transition (mobility ↑ or ↓), r Radial outer, k2h and ST

∆p' & ∆p

Radial composite (6.2)

k2h, ST

k1h, Sw r

k1h > k2h; or

k1h < k2h

∆t

1 2 3

Radial inner, k1h and Sw Transition (mobility ↑or ↓), L Radial total, (k1h+k2h)/2 and ST k1h > k2h; or k1h < k2h

∆p' & ∆p

Linear composite (6.3) (k1+k2)h/2, ST k1h, Sw

L ∆t

- 197 -

1 2 3

No crossflow Transition (storativity ↑), ω, κ and λ (kV) Radial, kh1+kh2 and ST

Double permeability, partial penetration S1= ∞ (7.3) 1 2 3

Radial, k2h2 and S2 Transition (mobility ↑), λ (kV) Radial, kh1+kh2 and ST

ω, κ

kh, ST λ

∆t

∆p' & ∆p

Double permeability, same skin S1=S2 (7.2)

∆p' & ∆p


k2h2, Sw kh, ST

λ ∆t

Sealing fault (5.1) 1 2 3

Radial, kh and S Transition (mobility ↓), L Hemi-radial

∆p' & ∆p

Boundary models

kh, S L

Channel closed at one end (5.4) Centered : 1 Radial, kh and S 2 Linear, L1+L2 3 Transition (mobility ↓), L3 4 Hemi-linear

1/2 L1 L1+L2 kh, S ∆t

1/2

∆p' & ∆p

Channel (5.2) Centered : 1 Radial, kh and S 2 Linear, L1+L2 Off-centered : 1 Radial, kh and S 2 Hemi-radial, L1 3 Linear, L1+L2

∆p' & ∆p

∆t

1/2 L3 L1+L2

kh, S ∆t

- 198 -

Closed system centered (5.4) Drawdown : 1 Radial, kh and S 2 Pseudo steady state, A Build-up : 1 Radial, kh and S 2

θ

L1 1/2 L1+L2 kh, S ∆t

P

∆p' & ∆p

Intersecting faults (5.3) Centered : 1 Radial, kh and S 2 Linear, L1+L2 3 Fraction of radial, θ Off-centered : 1 Radial, kh and S 2 Hemi-radial, L1 3 Linear, L1+L2 4 Fraction of radial, θ

∆p' & ∆p


1 A kh, S

Average pressure, p and A

∆t

Closed with intersecting faults (5.4) Drawdown : 1 Radial, kh and S 2 Linear, L1+L2 3 Fraction of radial, θ 4 Pseudo steady state, A Build-up : 1 Radial, kh and S 2 Linear, L1+L2 3 Fraction of radial, θ

∆p' & ∆p

A P

∆t

θ

P A

1/2 kh, S

L1+L2 ∆t

Average pressure, p and A

Constant pressure boundaries (5.5) 1 2

L1+L2

1/2

kh, S

Radial, kh and S Transition (mobility ↑), L One boundary Multiple boundaries

∆p' & ∆p

4

1

∆p' & ∆p

Closed channel (5.4) Drawdown : 1 Radial, kh and S 2 Linear, L1+L2 3 Pseudo steady state, A Build-up : 1 Radial, kh and S 2 Linear, L1+L2 3 Average pressure, p and A

L

kh, S -1 ∆t

- 199 -


16-1.4 Consistency check with the test history simulation In the following examples, the initial pressure is 5000 psi. The interpretation model, defined from log-log analysis of the short shut-in period, may be inconsistent when applied to the complete rate history.

Increase of derivative response after the last build-up point (second sealing boundary)


The log-log derivative plot suggests the presence of a sealing fault. 103

102

101

1 10-3

10-2

10-1

101

1

102

103

104


Figure 16-5 Log-log plot of the final build-up. Homogeneous reservoir with a sealing fault.

Rate, q

Pressure, p

The sealing fault model is not applicable on the extended production history. 5000

pi=4914 psia

4800

4600 4400 0

200

400

600

800

1000

1200

Time, t

Figure 16-6 Test history simulation. Linear scale. Homogeneous reservoir with a sealing fault.

When a second sealing fault, parallel to the first, is introduced farther away in the reservoir, the extended production history match is correct.

- 200 -



103

102

101

1 10-3

10-2

10-1

101

1

102

103

104


Rate, q

Pressure, p

Figure 16-7 Log-log plot of the final build-up. Homogeneous reservoir with two parallel sealing faults.

5000

pi=5000 psia

4800

4600 4400 0

200

400

600

800

1000

1200

Time, t

Figure 16-8 Test history simulation. Linear scale. Homogeneous reservoir with two parallel sealing faults.

Decrease of derivative response after the last build-up point (Layered semi infinite reservoir)


The log-log derivative plot suggests the presence of two parallel sealing faults. 103

102

101 10-3

10-2

10-1

1

101

102

103


Figure 16-9 Log-log plot of the final build-up. Homogeneous reservoir with two parallel sealing faults.

With the parallel sealing faults model, the initial pressure before the production history is too high.

- 201 -

Pressure, p


5000

pi=5443 psia

4500

Rate, q

4000 3500 3000 0

200

400

600

800

1000

Time, t

Figure 16-10 Test history simulation. Linear scale. Homogeneous reservoir with two parallel sealing faults.


The reservoir is a two layer no crossflow, one layer is closed. At late time, the derivative stabilizes to describe the radial flow regime in the infinite layer. The hump at intermediate time corresponds to the storage of the limited zone.

103

102

101 10-3

10-2

10-1

101

1

102

103

104


Figure 16-11 Log-log plot of the final build-up. Two layers reservoir, one infinite and one closed layer.

Rate, q

Pressure, p

5000 pi=5000 psia 4500 4000 3500 3000 0

200

400

600

800

1000

Time, t

Figure 16-12 Test history simulation. Linear scale. Two layers reservoir, one infinite and one closed layer.

- 202 -


16-2 Reporting and presentation of results 16-2.1 Objectives A well test interpretation report should present not only the different matches, but also all information necessary to re-do the analysis. The analysis work may be checked several years after completion. When all rates and parameters used to generate the interpretation solution are not clearly defined, it is may be impossible to re-evaluate the test.

16-2.2 Example of interpretation report contents Summary conclusion

• Main results, • Hypothesis used (if any), • Problems and inconsistencies not solved (if any). Test data

• Rate history (sequence of events for the test), • Static parameters, • Comparison of the gauge responses and choice of the pressure gauge used for analysis (when several gauges have been used). Analysis procedure

• Diagnosis (comparison of different periods, discussion of the pressure response). • Choice of the interpretation model(s) and justification. • Discussion of the results, sensitivity to the hypothesis etc. Match with the different models

• Log-log, • Semi-log, • Test simulation.

- 203 -

- 204 -

Appendix - ANALYTICAL SOLUTIONS A-1 Darcy's law Darcy's law expresses the rate through a sample of porous medium as a function of the pressure drop between the two ends of the sample. q

Figure A-1 Rate through a sample.

A dp / dl

q k dp =V = A µ dl With:

q A V k

µ

(A-1)

: volumetric rate : cross sectional area of the sample : flow velocity : permeability of the porous medium : viscosity of the fluid

The flow velocity V is proportional to the conductivity k/µ and to the pressure gradient dp/dl.

A-2 Steady state radial flow of an incompressible fluid

re

q

q rw

Figure A-2 Radial flow.

In case of radial flow, the Darcy's law is expressed, in the SI system of units:

q k dp =V = 2πrh µ dr

(A-2)

For steady state flow condition, the pressure difference between the external and the internal cylinders is:

pe − p w =

r qµ ln e 2π kh rw

(A-3)

This relationship is used in the definition of the dimensionless pressure Equation 2-3.

- 205 -

Appendix - Analytical solutions

A-3 Diffusivity equation A-3.1 Hypotheses • Constant properties: k, µ, φ and the system compressibility. • Pressure gradients are low. • The formation is not compressible and saturated with fluid.

A-3.2 Darcy's law →

V=

k

µ

→

grad p

(A-4)

A-3.3 Principle of conservation of mass (continuity equation) The difference between the mass flow rate in, and the mass flow rate out the element, defines the amount of mass change in the element during the time dt. →

div ρ V = −φ The density ρ =

∂ρ ∂t

(A-5)

m is used. v

A-3.4 Equation of state of a constant compressibility fluid The compressibility, defined as the relative change of fluid volume, is expressed with the density ρ:

c=−

1 ∂v 1 ∂ ρ = v∂p ρ∂p

(A-6)

With a constant compressibility, the fluid equation of state is:

ρ = ρ0 e

ct ( p − p 0 )

(A-7)

For a liquid flow in a porous medium, the total system compressibility ct is attributed to an equivalent fluid:

ct = c o S o + c w S w + c f

(1-3)

- 206 -


A-3.5 Diffusivity equation Combining Equations 4 and 5, then 7:

 k →  ∂ρ ∂p = φ ρ ct div  ρ grad p  = φ ∂t ∂t  µ 

(A-8)

With radial coordinates,



∂  rρ

∂ p  ∂ r 

1  r ∂r

∂p ∂ p ∂ ρ  φ ρ µ ct ∂ p 1 ∂ 2 p =  rρ +ρ +r = 2 r ∂r ∂r ∂ r ∂ r  k ∂t

(A-9)

And with Equation 7,

∂ρ ∂p = ρ ct ∂r ∂r

(A-10)

( )

∂p ∂p 1  ∂ 2 p rρ +ρ + r ρ ct 2  ∂r r ∂r ∂r

2

 = φ ρ µ ct ∂ p  k ∂t 

With the condition of low-pressure gradients, the approximation

(A-11)

( ) ∂p ∂r

2

≅ 0 is

used to linearize.

 ∂ p  φµ ct ∂ p  1  ∂r   div  grad p  = = ∇2 p = k ∂t  r ∂r  →

The ratio

∂  r

(A-12)

k is called hydraulic diffusivity. φµ ct

A-3.6 Diffusivity equation in dimensionless terms (customary oil field system of units and metric system of units)

kh ∆p (field units) 141.2qBµ kh pD = ∆p (metric units) 18.66qBµ pD =

- 207 -

(2-3)


0.000264k ∆t (field units) φµ ct rw2 0.000356k = ∆t (metric units) φµ c t rw2

tD = tD

rD =

r rw

(2-4)

(6-7)

The diffusivity equation is :



1 rD

∂  rD 

∂ pD ∂ rD

  

∂ rD

= ∇ 2 pD =

∂ pD ∂ tD

(A-13)

A-4 The "line source" solution • Initial condition : the reservoir is at initial pressure. pD = 0 at tD < 0 • Well condition : the rate is constant, the well is a "line source".

 ∂ pD Lim  rD r → 0  ∂ rD

  = −1 

(A-14)

• Outer condition : the reservoir is infinite.

Lim p D = 0 r→∞

(A-15)

The solution is called Exponential Integral.

p D (t D ,rD ) =−

1  rD2 Ei − 2  4t D

   

(8-1)

∞

e −u du u x

Ei(− x ) =− ∫

(A-16)

- 208 -

NOMENCLATURE Customary Units and Metric System of Units

A B cg co ct ct−

= = = = = =

C CA D e Ei F k kd kf kH km ks kV h hd hw L m m(p) m* M n p pf PI pi PM pm psc pw p* p− q

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Quantity and customary unit (Conversion to Metric unit) Surface, sq ft (*9.290 304*10-2 = m2) Formation volume factor, RB/STB (m3/m3) -1 1 Gas compressibility, psi (*1.450 377*10 = Bars-1) Oil compressibility, psi-1 (*1.450 377*101 = Bars-1) -1 Total compressibility, psi (*1.450 377*101 = Bars-1) Total compressibility at the average pressure of the test, psi-1 (*1.450 377*101 = Bars-1) Wellbore storage coefficient, Bbl/psi (*2.305 916 = m3/Bars) Shape factor Turbulent flow coefficient Exponential (2.7182 . . .) Exponential integral Storativity ratio (inner zone / outer zone) Permeability, mD (mD) Matrix skin permeability, mD (mD) Fracture or fissures permeability, mD (mD) Horizontal permeability, mD (mD) Matrix blocks permeability, mD (mD) Spherical permeability, mD (mD) Vertical permeability, mD (mD) Thickness, ft (*3.048*10-1 = m) Matrix skin thickness, ft (*3.048*10-1 = m) Perforated thickness, ft (*3.048*10-1 = m) Distance, or half length of an horizontal well, ft (*3.048*10-1 = m) Straight line slope (semi-log or other) Pseudo pressure or gas potential, psia2/cp (*4.753767*10-3 = Bars2/cp) Slope of the pseudo steady state straight line, psi/hr (*6.894757*10-2 = Bars/hr) Mobility ratio (inner zone / outer zone) Number of fissure plane directions, or turbulent flow coefficient Pressure, psi (*6.894757*10-2 = Bars) Fissure pressure, psi (*6.894757*10-2 = Bars) Productivity index, Bbl/D/psi (*2.305 916 = m3/D/Bars) Initial pressure, psi (*6.894757*10-2 = Bars) -1 Pressure match, psi (*1.450 377*101 = Bars-1) Matrix blocks pressure, psi (*6.894757*10-2 = Bars) Standard absolute pressure, 14.7 psia (1 Bara) Well pressure, psi (*6.894757*10-2 = Bars) Extrapolated pressure, psi (*6.894757*10-2 = Bars) Reservoir average pressure, or during the test, psi (*6.894757*10-2 = Bars) Flow rate, bbl/D (*1.589 873*10-1 = m3/D) 3 or Mscf/D (= 10 scft/D) (*2.831 685*101 = m3/D)

- 209 -

Nomenclature - Systems of units

r rf ri rm Rs rw S Sm Spp ST Sw t tp T TM Tsc v V xf wa wf zw Z Z−

= = = = = = = = = = = = = = = = = = = = = = = =

Radius, ft (*3.048*10-1 = m) Fracture radius in a horizontal well, ft (*3.048*10-1 = m) Radius of investigation or influence of the fissures, ft (*3.048*10-1 = m) Matrix blocks size, ft (*3.048*10-1 = m) Dissolved Gas Oil ratio, cf/bbl (*1.7810*10-1 = m3/m3) Wellbore radius, ft (*3.048*10-1 = m) Skin coefficient, or saturation Matrix skin Geometrical skin of partial penetration Total skin Skin over the perforated thickness Time, hr (hr) Horner production time, hr (hr) Temperature absolute, °R (*5/9 = °K) Time match, hr-1 (hr-1) Standard absolute temperature, 520°R (15°C = 288.15°K) Volume, cu ft (*2.831 685*10-2 = m3) Volume ratio (fissures or matrix), or flow velocity Half fracture length, ft (*3.048*10-1 = m) Width of altered permeability region near a conductive fault, ft (*3.048*10-1 = m) Fracture width, ft (*3.048*10-1 = m) Distance to the lower reservoir limit, ft (*3.048*10-1 = m) Real gas deviation factor Real gas deviation factor at the average pressure of the test

α β δ

= = = = = = = = = = = = = = = = = =

Geometric coefficient in λ , or transmissibility ratio of a semi-permeable fault Transition curve of a double porosity transient interporosity flow Constant of a β curve Difference Euler's constant (1.78 . . . ) Porosity, fraction Fissures porosity, fraction Matrix blocks porosity, fraction Mobility ratio Interporosity (or layer) flow coefficient Effective interporosity flow coefficient Viscosity, cp (cp) Viscosity at the average pressure of the test, cp (cp) Angle between two intersecting faults Well location between two intersecting faults Geometrical coefficient of the location of a well in a channel Storativity ratio Density, lb/cu ft (*1.601 646*101 = kg/m3)

∆

γ φ φf φm κ λ λeff µ µ− θ θw σ ω ρ

- 210 -

Nomenclature - Systems of units

Subscripts a AOF BLF BU ch cp d D e eff f G H hch i int L LF m max min o p pp ps PSS q r RC RF RLF S sc SLF SPH t, T V w wf ws WBS z 1 2

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Apparent or altered permeability region near a conductive fault Absolute Open Flow Potential Bi-linear flow (slope m) Build-up Channel (slope m) Constant pressure (slope m) Damage (matrix skin) Dimensionless Equivalent, External Effective Fracture, fissures, fault or formation Geometrical Horizontal Channel closed at one end (slope m) Initial or investigation Intersection of straight line Layer Linear flow (slope m) Matrix Maximum permeability direction Minimum permeability direction Oil Production (time) Partial penetration Pseudo (time) Pseudo steady state Rate decline (slope m) Ratio, or relative Radial-Composite Radial flow (slope m) Radial-linear flow (slope m) Skin, or spherical Standard conditions Semi linear flow (slope m) Spherical flow (slope m) Total Vertical Well, or water Flowing well Shut-in well Wellbore storage regime (slope m) Partial penetration Inner zone, or high permeability layer(s) Outer zone, or low permeability layer(s)

- 211 -

REFERENCES

Chapter 1 1-1. Matthews, C. S. and Russell, D.G.: "Pressure Build-up and Flow Tests in Wells", Monograph Series no 1, Society of Petroleum Engineers of AIME, Dallas (1967). 1-2. Earlougher, R. C., Jr.: "Advances in Well Test Analysis", Monograph Series no 5, Society of Petroleum Engineers of AIME, Dallas (1977). 1-3. Lee, J.: "Well Testing", Textbook Series, Vol. 1, Society of Petroleum Engineers of AIME, Dallas (1982). 1-4. Bourdarot, G.: " Well Testing : Interpretation Methods," Editions Technip, Institut Français du Pétrole. 1-5. van Everdingen, A. F. and Hurst, W.: "The Application of the Laplace Transformation to Flow Problems in Reservoirs," Trans., AIME ( 1949) 186, 305-324. 1-6. van Everdingen, A. F.: "The Skin Effect and its Influence on the Productive Capacity of a Well." Trans., AIME ( 1953) 198, 171-176. 1-7. Miller, C. C., Dyes, A. B., and Hutchinson, C. A.: "Estimation of Permeability and Reservoir Pressure from Bottom-Hole Pressure Build-up Characteristics," Trans., AIME ( 1950) 189, 91-104. 1-8. Russell, D. G. and Truitt, N. E.:"Transient Pressure Behavior in Vertically Fractured Reservoirs,"J. Pet. Tech. ( Oct., 1964) 1159-1170. 1-9. Clark, K. K.:"Transient Pressure Testing of Fractured Water Injection Wells," J. Pet. Tech. ( June, 1968) 1639-643; Trans., AIME ( 1968) 243. 1-10. Gringarten, A. C., Ramey, H. J., Jr. and Raghavan, R.: "Applied Pressure Analysis for Fractured Wells,"J. Pet. Tech. ( July, 1975) 887-892. 1-11. Gringarten, A. C., Ramey, H. J., Jr. and Raghavan, R.: "Unsteady-State Pressure Distribution Created by a Well with a Single Infinite Conductivity Fracture," Soc. Pet. Eng. J. ( Aug., 1974) 347-360. 1-12. Cinco-Ley, H., Samaniego-V, F. and Dominguez, N.: "Transient Pressure Behavior for a Well with a Finite Conductivity Vertical Fracture," Soc. Pet. Eng. J. ( Aug., 1978) 253-264. 1-13. Agarwal, R.G., Carter, R. D. and Pollock, C. B.: "Evaluation and Performance Prediction of Low-Permeability Gas Wells Stimulated by Massive Hydraulic Fracturing,"J. Pet. Tech. ( March, 1979) 362-372.

- 212 -

References

1-14. Cinco-Ley, H. and Samaniego-V, F:"Transient Pressure Analysis for Fractured Wells,"J. Pet. Tech.( Sept., 1981) 1749-1766. 1-15. Brons, F. and Marting, V. E.: "The Effect of Restricted FluidEntry on Well Productivity,"J. Pet. Tech. ( Feb., 1961) 172-174; Trans., AIME ( 1961) 222. 1-16. Moran, J. H. and Finklea, E. E.:"Theoretical Analysis of Pressure Phenomena Associated with the Wireline Formation Tester," J. Pet. Tech.( Aug., 1962) 899-908. Trans., AIME ( 1962), 225. 1-17. Culham, W. E.:"Pressure Build-up Equations for Spherical-Flow Problems," Soc. Pet. Eng. J. ( Dec., 1974) 545-555. 1-18. Warren , J. E. and Root, P. J.:"Behavior of Naturally Fractured Reservoirs" Soc. Pet. Eng. J. (Sept., 1963) 245; Trans., AIME ( 1963) 228. 1-19. Brons, F. and Miller, W. C.:"A Simple Method for Correcting Spot Pressure Readings," J. Pet. Tech.( Aug., 1961) 803-805. 1-20. Jones, P.: "Reservoir Limit Tests," Oil and Gas J. ( June 18, 1956) 54, no 59, 184.

Chapter 2 2-1. Ramey, H. J., Jr.: "Short-Time Well Test Data Interpretation in The Presence of Skin Effect and Wellbore Storage," J. Pet. Tech. ( Jan., 1970) 97. 2-2. Agarwal, R.G., Al-Hussainy, R. and Ramey, H. J., Jr.: "An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow. I: Analytical Treatment," Soc. Pet. Eng. J. ( Sept., 1970) 279. 2-3. McKinley, R. M.: "Wellbore Transmissibility from Afterflow Dominated Pressure Build-up Data," J. Pet. Tech. ( July, 1971) 863. 2-4. Earlougher, R. C., Jr., Kersh, K. M. and Ramey, H. J., Jr.:"Wellbore Effects in Injection well Testing," J. Pet. Tech.( Nov., 1973) 1244-1250. 2-5. Gringarten, A. C., Bourdet D. P., Landel, P. A. and Kniazeff, V. J.: "A Comparison between Different Skin and Wellbore Storage Type-Curves for Early-Time Transient Analysis," paper SPE 8205, presented at the 54th Annual Technical Conference and Exhibition of SPE, Las Vegas, Nev., Sept. 23-26, 1979. 2-6. Ramey, H.J., Jr. and Cobb, W.M.:"A General Pressure Build-up Theory for a Well in a Closed Drainage Area," J. Pet. Tech.( Dec., 1971) 1493-1505; Trans., AIME ( 1971), 252. 2-7. Horner, D. R.: "Pressure Build-ups in Wells", Proc., Third World Pet. Cong., E. J. Brill, Leiden (1951) II, 503-521. Also, Reprint Series, No. 9 —

- 213 -

References

Pressure Analysis Methods, Society of Petroleum Engineers of AIME, Dallas ( 1967) 25-43. 2-8. Agarwal, R. G.:"A New Method to Account for Production Time Effects When Drawdown Type Curves Are Used to Analyze Buildup and Other Test Data," paper SPE 9289, presented at the 55th Annual Technical Conference and Exhibition of SPE, Dallas, Tx., Sept. 21-24, 1980. 2-9. Raghavan, R.:"The Effect of Producing Time on Type Curve Analysis," J. Pet. Tech.( June, 1980) 1053-1064. 2-10. Bourdet, D. Ayoub, J. A. and Pirard, Y. M.: "Use of Pressure Derivative in Well-Test Interpretation", SPEFE (June 1989) 293-302 2-11. Balsingame, T.A., Johnston, J.L. and Lee, W.;J.: "Type-Curves Analysis Using the Pressure Integral Method," paper SPE 18799 presented at the 1989 SPE California Regional Meeting, Bakersfield, April 5-7. 2-12. Balsingame, T.A., Johnston, J.L. Rushing, J.A., Thrasher, T.S. Lee, W.;J. and Raghavan, R. : " Pressure Integral Type-Curves Analysis-II: Applications and Field Cases," paper SPE 20535 presented at the 1990 SPE Annual Technical Conference and Exhibition, New Orleans, Sept. 23-26. 2-13. Onur, M. and Reynolds, A.C.: "A New Approach for Constructing Derivative Type Curves for Well Test Analysis," SPEFE (March 1988) 197206. 2-14. Duong, A.N.: "A New Set of Type Curves for Well Test Interpretation Using the Pressure Derivative Ratio," paper SPE 16812 presented at the 1987 SPE Annual Technical Conference and Exhibition, Dallas, Sept. 27-30.

Chapter 3 3-1. Bourdet, D. P., Whittle, T. M., Douglas, A. A. and Pirard, Y. M.: "A New Set of Type Curves Simplifies Well Test Analysis," World Oil ( May, 1983) 95106. 3-2. Tiab, D. and Puthigai, S. K.:”Pressure-Derivative Type Curves for Vertically Fractured Wells,” SPEFE ( March, 1988) 156-158. 3-3. Alagoa, A., Bourdet, D. and Ayoub, J.A.:”How to Simplify The Analysis of Fractured Well Tests,” World Oil ( Oct. 1985) 3-4. Wong, D.W., Harrington, A.G. and Cinco-Ley, H.:”Application of the Pressure-Derivative Function in the Pressure-Transient Testing of Fractured Wells,"SPEFE.( Oct., 1985) 470-480. 3-5. Gringarten, A. C.and Ramey, H. J. Jr.: "An Approximate Infinite Conductivity Solution for a Partially Penetrating Line-Source Well", Soc.Pet.Eng. J. (Apr.1975) 347-360. - 214 -

References

3-6. Kuchuk, F.J. and Kirwan, P.A.: "New Skin and Wellbore Storage Type Curves for Partially Penetrated Wells". SPEFE, Dec. 1987, 546-554. 3-7. Papatzacos, P. : "Approximate Partial-Penetration Pseudoskin for InfiniteConductivity Wells", SPE-R.E. (May 1987) 227-234. 3-8. Daviau, F., Mouronval, G., Bourdarot, G and Curutchet P.: "Pressure Analysis for Horizontal Wells",. paper S.P.E. 14251, presented at the SPE 60th Annual Fall Meeting, Las Vegas, Nev., Sept. 22-25, 1985. 3-9. Clonts, M. D. and Ramey, H. J. Jr.: "Pressure Transient Analysis for Wells with Horizontal Drainholes",. paper S.P.E. 15116, presented at the 56th California Regional Meeting, Oakland, CA., April 2-4, 1986. 3-10. Goode, P. A. and Thambynayagam, R. K. M.: "Pressure Drawdown and Buildup Analysis of Horizontal Wells in Anisotropic Media", SPEFE (Dec. 1987) 683-697. 3-11. Kuchuk, F. J., Goode, P.A., Wilkinson, D.J. and Thambynayagam, R. K. M.: "Pressure-Transient Behavior of Horizontal Wells With and Without Gas Cap or Aquifer", SPEFE (March 1991) 86-94. 3-12. Kuchuk, F.: "Well Testing and Interpretation for Horizontal Wells", JPT (Jan. 1995) 36-41. 3-13. Ozkan, E., Sarica, C., Haciislamoglu, M. and Raghavan, R.: "Effect of Conductivity on Horizontal Well Pressure Behavior", SPE Advanced Technology Series, Vol. 3, March 1995, 85-94. 3-14. Ozkan , E. and Raghavan, R.: "Estimation of Formation Damage in Horizontal Wells", paper S.P.E. 37511, presented at the 1997 Production Operations Symposium, Oklahoma City, Oklahoma, 9-11 March 1997. 3-15. Yildiz, T. and Ozkan, E.: "Transient Pressure Behavior of Selectively Completed Horizontal Wells", paper S.P.E. 28388, presented at the SPE 69th Annual Fall Meeting, New Orleans, LA, Sept. 25-28, 1994. 3-16. Larsen, L. and Hegre, T.M.: "Pressure Transient Analysis of Multifractured Horizontal Wells", paper S.P.E. 28389, presented at the SPE 69th Annual Fall Meeting, New Orleans, LA, Sept. 25-28, 1994. 3-17. Larsen, L.: "Productivity Computations for Multilateral, Branched and Other Generalized and Extended Well Concepts", paper S.P.E. 36754, presented at the SPE Annual Fall Meeting, Denvers, Colorado, Oct. 6-9, 1996. 3-18. Kuchuk, F.J. and Habashy, T.: "Pressure Bahavior of Horizontal Wells in Multilayer Reservoirs With Crossflow", SPEFE (March 1996) 55-64. 3-19. Brigham, W. E. :"Discussion of Productivity of a Horizontal Well", SPERE (May. 1990) 254-255.

- 215 -

References

Chapter 4 4-1. Barenblatt , G. E., Zheltov, I.P. and Kochina, I.N.: "Basic Concepts in the Theory of Homogeneous Liquids in Fissured Rocks" J. Appl.. Math. Mech..(USSR) 24 (5) (1960)1286-1303). 4-2. Warren , J. E. and Root, P. J.:"Behavior of Naturally Fractured Reservoirs" Soc. Pet. Eng. J. (Sept., 1963) 245-255; Trans., AIME, 228. 4-3. Odeh, A.S.: "Unsteady-State Behavior of Naturally Fractured Reservoirs" Soc. Pet. Eng. J. (Mar., 1965) 60-64; Trans., AIME, 234. 4-4. Kazemi, H.: "Pressure Transient Analysis of Naturally Fractured Reservoirs with Uniform Fracture Distribution" Soc. Pet. Eng. J. (Dec., 1969) 451-462; Trans., AIME, 246. 4-5. de Swaan, O. A.: "Analytic Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing", Soc. Pet. Eng. J. (June, 1976) 117-122; Trans., AIME, 261. 4-6. Najurieta, H.L.: "A Theory for Pressure Transient Analysis in Naturally Fractured Reservoirs" J. Pet. Tech. (July 1980), 1241. 4-7. Streltsova, T.D.: "Well Pressure Behavior of a Naturally Fractured Reservoir", Soc. Pet. Eng. J. (Oct., 1983) 769. 4-8. Moench, A. F.: "Double-Porosity Models for a Fissured Groundwater Reservoir With Fracture Skin", Water Resources Res., Vol. 20, NO. 7 (July 1984) 831-846. 4-9. Mavor, M. J. and Cinco, H.: "Transient Pressure Behavior of Naturally Fractured Reservoirs", paper SPE 7977, presented at the 1979 California Regional Meeting of the SPE of AIME, Ventura, California, April 18-20, 1979. 4-10. Bourdet, D. and Gringarten, A. C.: "Determination of Fissure Volume and Block Size in Fractured Reservoirs by Type-Curve Analysis", paper S.P.E. 9293, presented at the SPE-AIME 55th Annual Fall Meeting, Dallas, TX.., Sept. 21-24, 1980. 4-11. Bourdet, D. Ayoub, J. A, Whittle, T. M., Pirard, Y. M. and Kniazeff V.: "Interpreting Well Test in Fractured Reservoirs", World Oil (Oct., 1983) 77-87. 4-12. Gringarten, A. C.: "Interpretation of Tests in Fissured and Multilayered Reservoirs with Double-Porosity Behavior: Theory and Practice", J. Pet. Tech. (April 1984), 549-564. 4-13. Bourdet, D. Ayoub, J. A. and Pirard, Y. M.: "Use of Pressure Derivative in Well-Test Interpretation", SPEFE (June 1989) 293-302. 4-14. Bourdet, D., Alagoa A., Ayoub J. A. and, Pirard, Y. M. : "New Type Curves Aid Analysis of Fissured Zone Well Tests", World Oil (April, 1984) 111-124. - 216 -

References

4-15. Cinco-Ley, H., Samaniego, F. and Kuchuk, F.: "The Pressure Transient Behavior for Naturally Fractured Reservoirs With Multiple Block Size", paper SPE 14168, presented at the 60th Annual Fall Meeting, Las Vegas, NV, Sept. 22-25, 1985. 4-16. Abdassah, D. and Ershaghi, I.: "Triple-Porosity Systems for Representing Naturally Fractured Reservoirs", SPEFE, April 1986, 113-127. 4-17. Belani, A.K. and Yazdi, Y.J.: "Estimation of Matrix Block Size Distribution in Naturally Fractured Reservoirs", paper SPE 18171, presented at the 63rd Annual Fall Meeting, Houston, Tex., Oct.; 2-5, 1988. 4-18. Stewart, G. and Ascharsobbi, F.: "Well Test Interpretation for Naturally Fractured Reservoirs", paper SPE 18173, presented at the 63rd Annual Fall Meeting, Houston, Tex., Oct.; 2-5, 1988.

Chapter 5 5-1. Clark, D. G. and Van Golf-Racht, T. D.: "Pressure Derivative Approach to Transient Test Analysis: A High-Permeability North Sea Reservoir Example," J. Pet. Tech. ( Nov., 1985) 2023-2039. 5-2. Wong, D.W., Mothersele, C.D., Harrington, A.G. and Cinco-Ley, H.: "Pressure Transient Analysis in Finite Linear Reservoirs Using Derivative and Conventional Techniques: Field Examples", paper S.P.E. 15421, presented at the 61st Annual Fall Meeting, New Orleans, La., Oct. 5-8, 1986. 5-3. Larsen, L., and Hovdan, M.: "Analysis of Well Test Data from Linear Reservoirs by Conventional Methods", paper SPE 16777, presented at the 62d Annual Fall Meeting, Dallas, Tex., Sept. 27-30, 1987. 5-4. Tiab, D. and Kumar, A.:”Detection and Location of Two Parallel Sealing Faults around a Well,” J. Pet. Tech. (Oct., 1980), 1701-1708. 5-5. van Poollen, H. K.:"Drawdown Curves give Angle between Intersecting Faults", The Oil and Gas J. (Dec.20, 1965), 71-75. 5-6. Prasad, Raj K.: "Pressure Transient Analysis in the Presence of Two Intersecting Boundaries" J. Pet. Tech. ( Jan., 1975) 89-96. 5-7. Tiab, D. and Crichlow, H.B..:”Pressure Analysis of Multiple-Sealing-Fault Systems and Bounded Reservoirs by Type Curve Matching,” SPEJ ( Dec., 1979) 378-392. 5-8. Brons F. and Miller, W.C.: "A Simple Method for Correcting Spot Pressure Readings", J. Pet. Tech. (Aug. 1961), 803-805; Trans. AIME, 222. 5-9. Dietz D.N.: "Determination of Average Reservoir Pressure From Build-Up Surveys", J. Pet. Tech. (Aug. 1965), 955-959 - 217 -

References

5-10. Earlougher, R.C. Jr.:"Estimating Drainage Shapes From Reservoir Limit Tests", J. Pet. Tech. (Oct. 1971), 1266-1268; Trans. AIME, 251 5-11. Matthews, C.S., Brons, F. and Hazebroek, P.: "A Method for Determination of Average Pressure in a Bounded Reservoir", Trans., AIME (1954) 201, 182191. 5-12. Yaxley, L.M.: "The Effect of a Partially Communicating Fault on Transient Pressure Behavior," paper S.P.E. 14311, presented at the 60th Annual Fall Meeting, Las Vegas, NV, Sept. 22-25, 1985. 5-13. Cinco, L.H., Samaniego, V.F. and Dominguez, A.N.: "Unsteady-State Flow Behavior for a Well Near a Natural Fracture", paper S.P.E. 6019, presented at the 51st Annual Fall Meeting, New Orleans, LA., Oct. 3-6, 1976. 5-14. Abbaszadeh, M.D. and Cinco-Ley, H. :"Pressure Transient Behavior in a Reservoir With a Finite-Conductivity Fault", SPEFE, (March 1995) 26-32.

Chapter 6 6-1. Carter R.D.: "Pressure Behavior of a Limited Circular Composite Reservoir," Soc. Pet. Eng. J., Dec. 1966, 328-334; Trans., AIME, 237. 6-2. Satman, A.: "An Analytical Study of Transient Flow in Systems With Radial Discontinuities," paper S.P.E. 9399, presented at the 55th Annual Fall Meeting, Dallas, Tex., Sept. 21-24, 1980 6-3. Olarewaju, J.S. and Lee, W.J.: "A Comprehensive Application of a Composite Reservoir Model to Pressure-Transient Analysis", SPE-RE, Aug. 1989, 325-231. 6-4. Abbaszadeh, M. and Kamal, M.M. :"Pressure-Transient Testing of WaterInjection Wells", SPE-RE, Feb. 1989, 115-124. 6-5. Ambastha, A.K., McLeroy, P.G. and Sageev, A.: " Effects of a Partially Communicating Fault in a Composite Reservoir on Transient Pressure Testing," paper S.P.E. 16764, presented at the 62nd Annual Fall Meeting, Dallas, Tex., Sept. 27-30, 1987. 6-6. Kuchuk, F.J. and Habashy, T.M. :"Pressure Behavior of Laterally Composite Reservoir", SPEFE, (March 1997) 47-564. 6-7. Levitan, M.M. and Crawford, G.E. : "General Heterogeneous Radial and Linear Models for Well Test Analysis," paper S.P.E. 30554, presented at the 70th Annual Fall Meeting, Dallas, TX, Oct. 22-25, 1995. 6-8. Oliver, D.S.: "The Averaging Process in Permeability Estimation From Well-Test Data," SPEFE, (Sept. 1990) 319-324.

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References

Chapter 7 7-1. Tariq, S. M. and Ramey, H. J., Jr.: "Drawdown Behavior of a Well with Storage and Skin Effect Communicating with Layers of Different Radii and Other Characteristics," paper S.P.E. 7453, presented at the 53rd Annual Fall Meeting, Houston, Tex., Oct. 1-3, 1978. 7-2. Gao, C-T.: "Single-Phase Fluid Flow in a Stratified Porous Medium With Crossflow, SPEJ, Feb. 1984, 97-106. 7-3. Wijesinghe, A.M. and Culham, W.E.: "Single-Well Pressure Testing Solutions for Naturally Fractured Reservoirs With Arbitrary Fracture Connectivity", paper S.P.E. 13055, presented at the 59th Annual Fall Meeting, Houston, Tex., Sept. 16-19, 1984. 7-4. Bourdet, D.: "Pressure Behavior of Layered Reservoirs with Crossflow", paper S.P.E. 13628, presented at the SPE California Regional Meeting, Bakersfield, CA, March. 27-29, 1985. 7-5. Prijambodo, R., Raghavan, R. and Reynolds, A.C.: "Well Test Analysis for Wells Producing Layered Reservoirs With Crossflow", SPEJ, June 1985, 380396. 7-6. Ehlig-Economides, C.A. and Joseph, J.A. : "A New Test for Determination of Individual Layer Properties in a Multilayered Reservoir", paper S.P.E. 14167, presented at the 60th Annual Fall Meeting, Las Vegas, NV, Sept. 22-25, 1985. 7-7. Larsen, L.: "Similarities and Differences in Methods Currently Used to Analyze Pressure-Transient Data From Layered Reservoirs", paper S.P.E. 18122, presented at the 63rd Annual Fall Meeting, Houston, TX, Oct. 2-5, 1988. 7-8. Larsen, L. : "Boundary Effects in Pressure-Transient Data From Layered Reservoirs", paper S.P.E. 19797, presented at the 64th Annual Fall Meeting, San Antonio, TX, Oct. 8-11, 1989. 7-9. Park, H. and Horne, R.N.: "Well Test Analysis of a Multilayered Reservoir With Crossflow", paper S.P.E. 19800, presented at the 64th Annual Fall Meeting, San Antonio, TX, Oct. 8-11, 1989. 7-10. Chen, H-Y, Poston, S.W. and Raghavan, R. : "The Well Response in a Naturally Fractured Reservoir: Arbitrary Fracture Connectivity and Unsteady Fluid Transfer", paper S.P.E. 20566, presented at the 65th Annual Fall Meeting, New Orleans, LA, Sept. 23-26, 1990. 7-11. Liu, C-q. and Wang, X-D.: "Transient 2D Flow in Layered Reservoirs With Crossflow", SPE-FE, Dec. 1993, 287-291.

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References

7-12. Larsen, L.: "Experiences With Combined Analyses of PLT and PressureTransient Data From Layered Reservoirs", paper SPE 27973 presented at University of Tulsa Centennial Symposium, Tulsa, OK, Aug. 29-31, 1994. 7-13. Boutaud de la Combe, J.-L., Deboaisne, R.M. and Thibeau, S.: "Heterogeneous Formation: Assessment of Vertical Permeability Through Pressure Transient Analysis - Field Example", paper SPE 36530, presented at the 1996 Annual Fall Meeting, Denvers, CO, Oct. 6-9, 1996. 7-14. Larsen L.: "Wells Producing Commingled Zones with Unequal Initial Pressures and Reservoir Properties", paper SPE 10325, presented at the 56th Annual Fall Meeting, San Antonio, TX, Oct. 5-7, 1981. 7-15. Agarwal, B., Chen, H-Y. and Raghavan, R.: "Buildup Behaviors in Commingled Reservoirs Systems With Unequal Initial Pressure Distributions: Interpretation", paper SPE 24680, presented at the 67th Annual Fall Meeting, Washington, DC, Oct. 4-7, 1992. 7-16. Aly, A., Chen, H.Y. and Lee, W.J.: "A New Technique for Analysis of Wellbore Pressure From Multi-Layered Reservoirs With Unequal Initial Pressures To Determine Individual Layer Properties", paper SPE 29176, presented at the Eastern Regional Conference, Charleston, WV, Nov. 8-10, 1994. 7-17. Gao, C., Jones, J.R., Raghavan, R. and Lee, W.J.: "Responses of Commingled Systems With Mixed Inner and Outer Boundary Conditions Using Derivatives," SPEFE (Dec. 94) 264-271. 7-18. Chen, H-Y., Raghavan, R. and Poston, S.W.: "Average Reservoir Pressure Estimation of a Layered Commingled Reservoir," paper SPE 26460 presented at the 68th Annual Fall Meeting, Houston, Tex., Oct. 3-6, 1993.

Chapter 8 8-1. Theis, C.V.: "The Relation Between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well Using Ground-Water Storage," Trans., AGU (1935), 519-524. 8-2. Tiab, D. and Kumar, A.:”Application of the p’D Function to Interference Analysis,” J. Pet. Tech. (Aug., 1980), 1465-1470. 8-3. Jargon, J.R.:" Effect of Wellbore storage and Wellbore Damage at the Active Well on Interference Test Analysis," J. Pet. Tech. (Aug. 1976) 851-858. 8-4. Ogbe, D.O. and Brigham, W.E.:" A Model for Interference Testing with Wellbore Storage and Skin Effects at Both Wells," paper S.P.E. 13253, presented at the 59th Annual Fall Meeting, Houston, TX, Sept. 16-19, 1984. 8-5. Papadopulos, I.S.: "Nonsteady Flow to a Well in an Infinite Anisotropic Aquifer," Proc. 1965 Dubrovnik Symposium on Hydrology of Fractured Rocks - 220 -

References

8-6. Ramey, H.J. Jr.: "Interference Analysis for Anisotropic Formations-A Case History," J. Pet. Tech. (Oct. 1975) 1290-98; Trans., AIME, 259. 8-7. Deruyck, B.G., Bourdet, D.P., DaPrat G. and Ramey, H.J. Jr.: "Interpretation of Interference Tests in Reservoirs with Double Porosity Behavior - Theory and Field Examples", paper S.P.E. 11025, presented at the 57th Annual Fall Meeting, New Orleans, La., Sept. 22-25, 1982. 8-8. Ma, Q. and Tiab, D: "Interference Test Analysis in Naturally Fractured Reservoirs," paper SPE 29514, presented at the SPE Production Operations Symposium, Oklahoma City, OK, April 2-4, 1995. 8-9. Satman, A. et Al.: "An Analytical Study of Interference in Composite Reservoirs," Soc. Pet. Eng. J., Apr. 1985, 281-290. 8-10. Chu, L. and Grader, A.S.: "Transient Pressure Analysis of Three Wells in a Three-Composite Reservoir," paper SPE 22716, presented at the 66th Annual Fall Meeting, Dallas, TX., Oct. 6-9, 1991. 8-11. Chu, W.C. and Raghavan, R.: "The Effect of Noncommunicating Layers on Interference Test Data," J. Pet. Tech. (Feb. 1981) 370-382. 8-12. Onur, M. and Reynolds, A.C.: "Interference Testing of a Two-Layers Commingled Reservoir," SPEFE. (Dec. 1989) 595-603. 8-13. Brigham, W.E.: "Planning and Analysis of Pulse-Tests," J. Pet. Tech. (May 1970) 618-624; Trans., AIME, 249 8-14. Kamal, M. and Brigham, W.E.: "Pulse-Testing Response for Unequal Pulse and Shut-In Periods," Soc. Pet. Eng. J. (Oct. 1975) 399-410; Trans., AIME, 259 8-15. Kamal, M.: "Interference and Pulse Testing - A Review," J. Pet. Tech. (Dec. 1983) 2257-70

Chapter 9 9-1. Al-Hussainy, R., Ramey, H.J. Jr. and Crawford. P. B.:"The Flow of Real Gases Through Porous Media", J. Pet. Tech. (May 1966), 624-636; Trans. AIME, 237 9-2. Al-Hussainy, R. and Ramey, H.J. Jr.:"Application of Real Gas Flow Theory to Well Testing and Deliverability Forecasting", J. Pet. Tech. (May 1966), 637642; Trans. AIME, 237 9-3. Agarwal, R.G.:"Real Gas Pseudo-Time - A New Function for Pressure Build-up Analysis of MHF Gas Wells", paper S.P.E. 8279, presented at the 54th Annual Fall Meeting, Las Vegas, NV, Sept. 23-26, 1979.

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References

9-4. Houpeurt A.:"On the Flow of Gas in Porous Medias", Revue de l'Institut Français du Pétrole, 1959, XIV (11), 1468-1684. 9-5. Wattenbarger, R.A. and Ramey, H.J. Jr.:"Gas Well Testing with Turbulence, Damage and Wellbore Storage", J. Pet. Tech. (Aug. 1968), 877-887. 9-6. "Theory and Practice of the Testing of Gas Wells", Energy Resources Conservation Board, Calgary, Alta., Canada (1975). 9-7. Bourdarot, G.: " Well Testing : Interpretation Methods," Editions Technip, Institut Français du Pétrole, p. 258. 9-8. Rawlins, E.L. and Schellardt, M.A.:"Back-Pressure Data on Natural-Gas Wells and Their Application to Production Practices," Monograph 7, USBM (1936). 9-9. Katz, D.L., Cornell, D., Kobayashi, R., Poettmann, F.H., Vary, J.A., Elenbaas, J.R. and Weinaug, C.F.:"Handbook of Natural Gas Engineering," McGraw-Hill Book Co.,Inc., New York (1959). 9-10. Bourgeois, M.J. and Wilson, M.R. :"Additional Use of Well Test Analytical Solutions for Production Prediction," paper S.P.E. 36820, presented at the 1996 SPE EUROPEC, Milan, Italy, Oct. 22-24, 1996.

Chapter 10 10-1. Stewart, G.: "Future Developments In Well Test Analysis: Introduction of Geology", Hart's Petroleum Engineer International (Sept. 1997), 73-76. 10-2. Larsen, L.: "Boundary Effects in Pressure-Transient Data From Layered Reservoirs,". paper S.P.E. 19797, presented at the 64th Annual Fall Meeting, San Antonio, Tex., Oct. 8-11, 1989. 10-3. Joseph, J., Bocock, A., Nai-Fu, F. and Gui, L.T.: "A Study of Pressure Transient Behavior in Bounded Two-Layered Reservoirs: Shengli Field, China", paper SPE 15418, presented at the 61st Annual Fall Meeting, New Orleans, LA, Oct. 5-8, 1986. 10-4. Bourgeois, M.J., Daviau, F.H. and Boutaud de la Combe, J-L. : "Pressure Behavior in Finite Channel-Levee Complexes", SPEFE, (Sept. 1996) 177-183.

Chapter 11 11-1. Al-Ghamdi, A. and Ershaghi, I.: "Pressure Transient Analysis of Dually Fractured Reservoirs", paper SPE 26959, presented at the III Latin American Conference, Buenos Aires, Argentine, April 27-29, 1994.

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References

11-2. Larsen, L.: "Similarities and Differences in Methods Currently Used to Analyze Pressure-Transient Data From Layered Reservoirs", paper S.P.E. 18122, presented at the 63rd Annual Fall Meeting, Houston, TX, Oct. 2-5, 1988. 11-3. Poon, D.C.C. :"Pressure Transient Analysis of a Composite Reservoir With Uniform Fracture Distribution," paper SPE 13384 available at SPE, Richardson, TX. 11-4. Satman, A.: "Pressure-Transient Analysis of a Composite Naturally Fractured Reservoir," SPE-FE, June 1991, 169-175. 11-5. Kikani, J. and Walkup, G.W.: "Analysis of Pressure-Transient Tests for Composite Naturally Fractured Reservoirs," SPE-FE, June 1991, 176-182. 11-6. Hatzignatiou, D.G., Ogbe, D.O., Dehghani, K. and Economides, M.J.: "Interference Pressure Behavior in Multilayered Composite Reservoirs," paper S.P.E. 16766, presented at the 62nd Annual Fall Meeting, Dallas, Tex., Sept. 27-30, 1987.

Chapter 12 12-1. Ramey, H.J. Jr., Agarwal, R.G. and Martin, I.: "Analysis of 'Slug Test' or DST Flow Period Data," J. Cdn. Pet; Tech. (July-Sept.. 1975) 14, 37. 12-2. de Franca Correa A.C. and Ramey, H.J. Jr. "A Method for Pressure Buildup Analysis of Drillstem Tests," paper S.P.E. 16808, presented at the 62nd Annual Fall Meeting, Dallas, TX, Sept. 27-30, 1987. 12-3. Peres, A.M.M., Onur, M. and Reynolds, A.C.: "A New General PressureAnalysis Procedure for Slug Tests," SPEFE. (Dec. 1993) 292-98. 12-4. Ayoub, J.A., Bourdet, D.P. and Chauvel, Y.L.: "Impulse Testing," SPEFE. (Sept. 1988) 534-46; Trans., AIME, 285 12-5. Cinco-Ley, H. et al.: "Analysis of Pressure Tests Through the Use of Instantaneous Source Response Concepts," paper S.P.E. 15476, presented at the 61st Annual Fall Meeting, New Orleans, LA, Oct. 5-8, 1986. 12-6. Kucuk, F, and Ayestaran, L,: "Analysis of Simultaneously Measured Pressure and Sandface Flow Rate in Transient Well Testing," paper S.P.E. 112177, presented at the 58th Annual Fall Meeting, San Francisco, CA, Oct. 58, 1983. 12-7. Bourdet D. and Alagoa A.: "New Method Enhances Well Test Interpretation," World Oil ( Sept, 1984). 12-8. Jacob, C.E. and Lohman, S.W.: "Nonsteady Flow to a Well of Constant Drawdown in an Extensive Aquifer," Trans., AGU (Aug. 1952) 559-569.

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References

12-9. Uraiet, A.A. and Raghavan, R.: "Unsteady Flow to a Well Producing at a Constant Pressure". J. Pet. Tech., Oct. 1980, 1803-1812. 12-10.Ehlig-Economides, C.A. and Ramey, H.J. Jr.: "Pressure Buildup for Wells Produced at Constant Pressure". SPEJ, Feb. 1981, 105-114.

Chapter 13 13-1. Perrine, R.L.:"Analysis of Pressure Build-up Curves", Drill. and Prod. Prac., API (1956), 482-509. 13-2. Martin, J.C.:"Simplified Equations of Flow in Gas Drive Reservoirs and the Theoretical Foundation of Multiphase Pressure Buildup Analyses," Trans., AIME (1959) 216, 309-311. 13-3. Fetkovich, M.J.:"The Isochronal Testing of Oil Wells," paper S.P.E. 4529, presented at the 48th Annual Fall Meeting, Las Vegas, Nev., Sept. 30- Oct.3, 1973. 13-4. Raghavan, R.: "Well Test Analysis: Wells Producing by Solution Gas Drive Wells," SPEJ, (Aug. 1976) 196-208; trans., AIME, 261. 13-5. Al-Khalifah, A.A., Aziz, K. and Horne, R.N.:"A New Approach to Multiphase Well Test Analysis", paper S.P.E. 16473 presented at the 62nd Annual Fall Meeting, Dallas, TX, Sept. 27-30, 1987. 13-6. Weller, W.T.:"Reservoir Performance During Two-Phase Flow," J. Pet. tech. (Feb. 1966) 240-246; Trans., AIME, Vol 240. 13-7. Raghavan, R.: "Well Test Analysis for Multiphase Flow" SPEFE, (Dec.1989) 585-594 13-8. Jones, J.R. and Raghavan, R.: "Interpretation of Flowing Well Responses in Gas-Condensate Wells" SPEFE, (Sep.1988) 578-594. 13-9. Jones, J.R., Vo, D.T. and Raghavan, R.: "Interpretation of Pressure Build-up Responses in Gas-Condensate Wells" SPEFE, (March 1989) 93-104.

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