Bourdet, D. - Well Testing and Interpretation

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WELL TESTING

AND

INTERPRETATION

D. Bourdet

CONTENTS Pages

1- PRINCIPLES OF TRANSIENT TESTING... 1

1-1 INTRODUCTION... 1

1-2 DEFINITIONS & TYPICAL REGIMES...7

2- THE ANALYSIS METHODS ... 27

2-1 LOG-LOG SCALE... 27

2-2 PRESSURE CURVES ANALYSIS... 28

2-3 PRESSURE DERIVATIVE... 37

2-4 THE ANALYSIS SCALES...44

3- WELLBORE CONDITIONS ... 47

3-1 WELL WITH WELLBORE STORAGE AND SKIN, HOMOGENEOUS RESERVOIR... 47

3-2 INFINITE CONDUCTIVITY OR UNIFORM FLUX VERTICAL FRACTURE... 48

3-3 FINITE CONDUCTIVITY VERTICAL FRACTURE... 50

3-4 WELL IN PARTIAL PENETRATION... 53

3-5 HORIZONTAL WELL... 57

3-6 SKIN FACTORS...71

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

4-1 DEFINITIONS... 75

4-2 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 ONE SEALING FAULT... 95

5-2 TWO PARALLEL SEALING FAULTS... 97

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5-4 CLOSED SYSTEM... 104

5-5 CONSTANT PRESSURE BOUNDARY... 111

5-6 COMMUNICATING FAULT... 113

5-7 PREDICTING DERIVATIVE SHAPES...117

6- COMPOSITE RESERVOIR MODELS... 119

6-2 RADIAL COMPOSITE BEHAVIOR... 120

6-3 LINEAR COMPOSITE BEHAVIOR... 123

6-4 MULTICOMPOSITE SYSTEMS...125

7- LAYERED RESERVOIRS - DOUBLE PERMEABILITY MODEL... 127

7-2 DOUBLE PERMEABILITY BEHAVIOR WHEN THE TWO LAYERS ARE PRODUCING INTO THE WELL129 7-3 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 INTERFERENCE TESTS IN RESERVOIRS WITH HOMOGENEOUS BEHAVIOR... 135

8-2 INTERFERENCE TESTS IN DOUBLE POROSITY RESERVOIRS... 139

8-3 INFLUENCE OF RESERVOIR BOUNDARIES... 143

8-4 INTERFERENCE TESTS IN RADIAL COMPOSITE RESERVOIR... 143

8-5 INTERFERENCE TESTS IN A TWO LAYERS RESERVOIR WITH CROSS FLOW...146

9- GAS WELLS... 149

9-1 GAS PROPERTIES... 149

9-2 TRANSIENT ANALYSIS OF GAS WELL TESTS... 150

9-3 DELIVERABILITY TESTS...154

10- BOUNDARIES IN HETEROGENEOUS RESERVOIRS ... 159

10-1 BOUNDARIES IN FISSURED RESERVOIRS... 159

10-2 BOUNDARIES IN LAYERED RESERVOIRS... 160

10-3 COMPOSITE CHANNEL RESERVOIRS...162

11- COMBINED RESERVOIR HETEROGENEITIES ... 165

11-1 FISSURED-LAYERED RESERVOIRS... 165

11-2 FISSURED RADIAL COMPOSITE RESERVOIRS... 166

11-3 LAYERED RADIAL COMPOSITE RESERVOIRS...167

12- OTHER TESTING METHODS... 169

12-1 DRILLSTEM TEST... 169

12-2 IMPULSE TEST... 172

12-3 RATE DECONVOLUTION... 173

12-4 CONSTANT PRESSURE TEST (RATE DECLINE ANALYSIS) ... 174

12-5 VERTICAL INTERFERENCE TEST...175

13- MULTIPHASE RESERVOIRS ... 179

13-1 PERRINE METHOD... 179

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14- TEST DESIGN ... 183

14-1 INTRODUCTION... 183

14-2 TEST SIMULATION... 183

14-3 TEST DESIGN REPORTING AND TEST SUPERVISION...184

15- FACTORS COMPLICATING WELL TEST ANALYSIS... 185

15-1 RATE HISTORY DEFINITION... 185

15-2 ERROR OF START OF THE PERIOD... 186

15-3 PRESSURE GAUGE DRIFT... 188

15-4 PRESSURE GAUGE NOISE... 188

15-5 CHANGING WELLBORE STORAGE... 189

15-6 TWO PHASES LIQUID LEVEL... 190

15-7 INPUT PARAMETERS, AND CALCULATED RESULTS OF INTERPRETATION...191

16 - CONCLUSION ... 193

16-1 INTERPRETATION PROCEDURE... 193

16-2 REPORTING AND PRESENTATION OF RESULTS...203

APPENDIX - ANALYTICAL SOLUTIONS... 205

A-1 DARCY'S LAW... 205

A-2 STEADY STATE RADIAL FLOW OF AN INCOMPRESSIBLE FLUID... 205

A-3 DIFFUSIVITY EQUATION... 206

A-4 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

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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 :

)

(t

p

=

_i

−

∆

(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) :

) 0 ( ) ( − ∆ = = ∆p p t p t (psi, Bars) ( 1-2) Time, t Ra te, q P re ssur e , p ∆t_BU ∆t_Dd ∆pDd ∆p_BU pi p(∆t=0) drawdown build-up Time, t Ra te, q P re ssur e , p ∆t_BU ∆t_Dd ∆pDd ∆p_BU pi p(∆t=0) drawdown build-up

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

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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 work-over, 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).

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Chapter 1 - Principles of transient testing

I S O

input system 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 :

(

w

)

w w f o

t c S c S c

c= 1− + + (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.

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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.

• Gas well test : specific testing methods are used to evaluate the deliverability

of gas wells (Absolute Open Flow Potential, AOFP) and the possibility of non-Darcy flow condition (rate dependent skin factor S'). The usual procedures are Back Pressure test (Flow after Flow), Isochronal and Modified Isochronal tests.

Time, t R a te , q Pres sure , p Clean up Initial shut-in Variable rate Stabilized rate Build-up Time, t R a te , q Pres sure , p Clean up Initial shut-in Variable rate Stabilized rate Build-up Clean up Initial shut-in Variable rate Stabilized rate Build-up

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

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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 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.

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Burner Burner Heater Separator Surge tank Air compressor Water pump Rig HP pump Gas Oil Water Choke maniflod Flowhead Transfer pump Oil manifold Gas manifold Burner Burner Heater Separator Surge tank Air compressor Water pump Rig HP pump Gas Oil Water Choke maniflod Flowhead Transfer pump Oil manifold Gas manifold

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

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• 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 Ï Ï Ï Î r rw pi pw

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Time, t Rate, q P res s u re, p q _surface q_{sand face} Time, t Rate, q P res s u re, p q _surface q_{sand face}

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

For a well full of a single phase fluid,

w oV c p V C=−∆_∆ = (Bbl/psi, m3/Bars) ( 1-4) where :

co : liquid compressibility (psi-1, Bars -1)

V_w : wellbore volume (Bbl, m3₎

When there is a liquid level, with ∆p=

ρ

g h∆ , ∆V =V_u∆h and

ρ : liquid density (lb/cu ft, kg/m3₎

g/gc : gravitational acceleration (lbf / lbm, kgf / kgm)

V_u : wellbore volume per unit length (Bbl/ft, m3_/m)

) ( 144 c u g g V C=

_ρ

(Bbl/psi) ) ( 10197 c u g g V C=

_ρ

(m3_/Bars) _{( 1-5)} Elapsed time,∆t Pr essure change , ∆ p mW BS Elapsed time,∆t Pr essure change , ∆ p mW BS

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.

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Chapter 1 - Principles of transient testing t C B q p= ∆ ∆ ₂₄ (psi, Bars) ( 1-6)

Result : wellbore storage coefficient C.

WBS

m

qB

C 24

=

(Bbl/psi, m3/Bars) ( 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 Ï Ï Ï Î Í Í Í Í Î Î Ï pwf rw ri r p pi S = 0 pwf rw ri r p pi S = 0

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

rw r pwf(S=0) pwf(S>0) ri ∆p skin p pi S > 0 rw r pwf(S=0) pwf(S>0) ri ∆p skin p pi rw r pwf(S=0) pwf(S>0) ri ∆p skin p pi S > 0

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

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r_w r p_wf(S=0) pwf(S<0) ri ∆p skin p pi S < 0 r_w r p_wf(S=0) pwf(S<0) ri ∆p skin p pi r_w r p_wf(S=0) pwf(S<0) ri ∆p skin p pi S < 0

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.

Skin

p

qB

kh

S

=

∆

µ

2 .

141

(field units) Skin

66 .

18 qB

p

kh

S

=

∆

µ

(metric units) ( 1-8)

• Damaged well (S > 0) : poor contact between the well and the reservoir (mud-cake, 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 :

r_w r_s k_s k r_w r_s k_s k w S w S S S w S w r r kh qB r r h k qB p

p _, − _, ₌₀=141.2

µ

ln −141.2

µ

ln (psi, field units) w S w S S S w S w

r

kh

qB

r

h

k

qB

p

_,

−

_, ₌₀

=

18 .

66 µ

ln

−

18 .

66 µ

ln

(Bars, metric units) ( 1-9)

The skin is expressed :

S

k

r

S S w

=



−











1 ln

( 1-10)

Equivalent wellbore radius :

S

e

r

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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.

Log∆t P ress u re cha nge, ∆ p m ∆p_(1hr) Log∆t P ress u re cha nge, ∆ p m ∆p_(1hr)

Figure 1-11 Radial flow regime.

Specialized analysis on semi-log scale.

Semi-log straight line of slope m :

∆

p

qB

∆

kh

t

k

c r

_{t w}

S

=



+

−

+











162 6

.

µ

log

₂

3 23 0 87

.

φµ

(psi, field units)

        + − + ∆ = ∆ S r c k t kh qB p w t 87 . 0 10 . 3 log log 5 . 21 ₂

µ

φ

µ

(Bars, metric units)( 1-12)

Results: kh qB m =162 6.

µ

(mD.ft, field units) m qB kh=21.5

µ

(mD.m, metric units) ( 1-13)

S

p

m

k

c r

_{t w}

=



−

+











1151

.

∆

1 hr

log

₂

3 23

.

φµ

(field units)         + − ∆ =1.151 1hr log ₂ 3.10 w tr c k m p S

φµ

(metric units) ( 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.

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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).

0 2000 4000 6000 0 10 20 30 40 time, hours pr es s u re , ps i no skin moderate skin

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).

0 2000 4000 6000 0 10 20 30 40 time, hours pr es s u re , ps i _{high skin}

very high skin

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.

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Chapter 1 - Principles of transient testing 0 1000 2000 3000 0.001 0.01 0.1 1 10 100 time, hours pr es s u re c h a nge, ps i no skin moderate skin ∆p skin

Figure 1.14 Semi-log responses for well A.

0 1000 2000 3000 0.001 0.01 0.1 1 10 100 time, hours pr es s u re c h a nge, p s i high skin very high skin

∆p skin

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

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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

∆

t

: the

response follows a straight line of slope mLF, intercepting the origin.

∆

p

qB

∆

hx

_f

c k

_t

t

=

4 06

.

µ

φ

t

k

c

hx

qB

p

t f

∆

=

∆

φ

µ

623 .

0

Pressure cha nge, ∆ p mLF t ∆ Pressure cha nge, ∆ p mLF 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

c k

q B

hm

f t LF

=

4 06

.

µ

φ

(ft, field units) LF t f

hm

qB

k

c

x

φ

µ

623 .

0 =

(m, metric units) ( 1-16)

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

Bilinear flow regime

w k_f

w_f k_f

w k_f

w_f k_f

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 bi-linear flow regime.

Plot of the pressure change ∆p versus the fourth root of elapsed time 4

∆

t

:

straight line of slope mBLF, intercepting the origin.

4 4

11 .

44 t

k

c

w

k

h

qB

p

t f

∆

=

∆

µ

φ

µ

4 4

28 .

6 t

k

c

w

k

h

qB

p

t f f

∆

=

∆

φµ

µ

Pressure cha nge, ∆ p mBLF 4_∆_t Pressure cha nge, ∆ p mBLF 4_∆_t Pressure cha nge, ∆ p mBLF 4_∆_t

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

Result : the fracture conductivity kfwf

2

1

8 .

1944

_











=

BLF t f f

hm

qB

k

c

w

k

µ

φµ

(mD.ft, field units) 2

1

46 .

39 _











=

BLF t f f

hm

qB

k

c

w

k

µ

φµ

(mD.m, metric units) ( 1-18)

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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 k_H kH hw h kV k_H kH kV k_H kH hw h

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

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

∆

p

qB

k r

qB

c

k

t

S S t S

=

70 6

.

µ

−

2452 9

.

µ φµ

_{3 2} (psi, field units)

t k c qB r k qB p S t S S ∆ − =

∆ 9.33

µ

279.3

µ

₃₂

φµ

P res s u re c han ge , ∆ p _m S P_H t ∆ 1 P res s u re c han ge , ∆ p _m S P_H t ∆ 1

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

Result : the spherical permeability ks

3 2 SPH

9 .

2452

_













=

m

c

qB

k

_S

µ

φµ

t (mD, field units)

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Chapter 1 - Principles of transient testing 3 2 SPH

3 .

279 _













=

m

c

qB

k

_S

µ

φµ

t (mD, metric units) ( 1-20)

The permeability anisotropy is expressed with :

k

H V H 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 Í Í Í Í Ï Ï Ï Ï Î Î Î Í Í Í Í Ï Ï Ï Ï Ï Ï Ï Ï Î ÎÎ Î r r_i _r r_i p_f r r_i p p_m p_wf p_i r_w rr_i_i _r_r p_f r r_i p p_m p_wf p_i r_w

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.

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Î Í Í Í Ï Ï Ï Ï Î Î Í Í ÍÍÍ Î Î Í Í Í Ï Ï Ï Ï Ï Ï Ï Ï Î Î Í Í ÍÍÍ Î rr p_m> p_f r r_i r_ii p p_wf p_i r_w _rr p_m> p_f r r_i r_ii r_i r_ii p p_wf p_i r_w

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 Í Í Í Ï Ï Ï Ï Î Î ÍÍ _ÍÍ Í Î Î Í Í Í Ï Ï Ï Ï Ï Ï Ï Ï Î Î ÍÍ _ÍÍ Í Î ririi _rr p_m= p_f r p p_wf p_i r_w ririiririiririi _rr p_m= p_f r p p_wf p_i r_w

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.

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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 Í Í Í Í Ï Ï Ï Ï Î Î Î Í Í Í Í Ï Ï Ï Ï Ï Ï Ï Ï Î Î Î Î r_i p_wf r_w r p p_i ri L p_wf r_w r p p_i L

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

r_i p_wf r_w r p p_i L ri p_wf r_w r p p_i L

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.

ri p_wf rw r p pi L _r i p_wf rw r p pi L

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r_i pwf r_w L r p p_i r_i pwf r_w L r r_i pwf r_w L r p p_i

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. Log∆t P res s u re ch ang e, ∆ p 2m m Log∆t P res s u re ch ang e, ∆ p 2m m

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

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Chapter 1 - Principles of transient testing t x

c

t

k

L

φµ

∆

=

0 .

01217

(ft, field units) t x

c

t

k

L

φµ

∆

=

0 .

0141

(m, metric units) ( 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 Í Í Í Í Ï Ï Ï Ï Î Î R_e r_i(t₁) Î Í Í Í Í Ï Ï Ï Ï Ï Ï Ï Ï Î Î Î Î R_e r_i(t₁) p_wf r_w ri (t2) = Re r p p_i t₄ t₁ t2 _t 3 Infinite acting

Pseudo Steady State r_i(t₁) p_wf r_w ri (t2) = Re r p p_i t₄ t₄ t₁ t₁ tt22 _t 3 t₃ Infinite acting

Pseudo Steady State r_i(t₁)

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).

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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

qB

∆

c hA

t

qB

kh

A

r

C

S

t w A

=

+



−

+











0 234

.

162 6

.

log

₂

log

0 351 0 87

.

φ

µ

( )

        + + − + ∆ = ∆ C S r A kh qB t hA c qB p _A w t 87 . 0 351 . 0 log log 5 . 21 0417 . 0

µ

₂

φ

(Bars, metric units) ( 1-23) Time, t Pr e s s u re , p p_i p -slope m* pseudo steady state

Time, t Pr e s s u re , p p_i p -slope m* pseudo steady state

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

Result : the reservoir pore volume φ hA.

φ

hA qB

c m_t

=0 234.

* (cu ft, field units)

* 0417 . 0 m c qB hA t =

φ

(m3, metric units) ( 1-24)

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

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

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Chapter 1 - Principles of transient testing Time (hours) P re ssu re ( p sia ) 3500 4000 4500 5000 0 100 200 300 400 500 pi Observation well Producing well Time (hours) P re ssu re ( p sia ) 3500 4000 4500 5000 3500 4000 4500 5000 0 100 200 300 400 500 pi Observation well Producing well

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

Î Î Í Í Í _Í Ï Ï Producing well Observation well Î Î Í Í Í _Í Ï Ï Producing well Observation well r p p_wf r_w r_i r p p_i r p p_wf r_w r_i r p p_i

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.

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For a fractured well for example, the sequence of regimes is : (1) (2) (1) (2) 1. Linear 2. Radial

Figure 1.36 Fractured well example.

In the case of a well in a channel reservoir :

(2) (1)

(2)

(1) 1. Radial

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

_wf (Bbl/D/psi, m3/D/Bars) ( 1-25)

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

(

)

₍

₎

PI

_S=0

=

−

q

p

_wf

∆

p

_skin (Bbl/D/psi, m3/D/Bars) ( 1-26)

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

PI= + − +    _ kh B t k c r S t w 162 6 3 23 0 87 2 .

µ

log log . .

φµ

∆

(Bbl/D/psi, field units)

      + − + ∆ = S r c k t B kh w t 87 . 0 10 . 3 log log 5 . 21 PI 2

φµ

µ

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The Pseudo Steady State Productivity Index is a constant

( )

PI= − + +    _ kh B A r C S w A 162 6.

µ

log ₂ log 0 351 0 87. .

(Bbl/D/psi, field units)

( )

_













+

−

=

S

C

r

A

B

kh

A w

87 .

0

351 .

0 log

log

5 .

21 PI

2

µ

(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 :

(

)

_       ∆ − − = ∆ ∆ t k r c kh qB r t p t 001056 . 0 Ei 2 . 141 5 . 0 , 2

φµ

µ

(

)

_       ∆ − − = ∆ ∆ t k r c kh qB r t p t 0001423 . 0 Ei 66 . 18 5 . 0 , 2

µ

φ

µ

For small x,

Ei

( )

−

x

=

−

ln

( )

γ

x

: the Exponential Integral can be approximated by a log (with γ = 1.78, Euler's constant).

(

)

[

(

)

]

∆ ∆p t r ∆qB ∆

kh k t c rt

, =162 6.

µ

log .0 000264

φµ

2 +0 809. (psi, field units)

(

∆ ,

)

= 21.5

[

log

(

0.000356 ∆ 2

)

+0.809

]

∆ k t c r kh qB r t

p

µ

φµ

_t (Bars, metric units) ( 1-30)

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

pwf Log r p pi t4 t₁ _t 2 t3 pwf Log r p pi t4 t4 t₁ t₁ _t 2 t₂ tt₃₃

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

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Exponential Integral. Beyond this limit, the profile flattens, and tends asymptotically towards the initial pressure.

The radius of investigation r_i tentatively describes the distance that the pressure transient has moved into the formation. Several definitions have been proposed, in general r_i is defined with one of the two relationships :

(

0 000264

)

1 4 2 . k t∆

φµ

c r_{t i} = or = 1₂

γ

(field units)

(

)

₄1 000356 . 0 k∆t

φµ

ctri2 = or = 1 2

γ

(metric units) ( 1-31)

(in dimensionless terms of Equation 2.4 or 8-2, t_D r_iD2 1

4

= or t_D r_iD2 = 1₂

γ

).

This gives respectively,

r

_i

=

0 032

.

k t

∆

φµ

c

_t (ft, field units) t i

k

t

c

r

=

0 .

037 ∆

φµ

(m, metric units) ( 1-32) and

r

_i

=

0 029

.

k t

∆

φµ

c

_t (ft, field units) t i

k

t

c

r

=

0 .

034 ∆

φµ

(m, metric units) ( 1-33)

(the radius of investigation is independent of the rate).

The radius of investigation r_i 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.

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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)

∆

t, hr

100 ₁₀1₁₀2

Figure 2-1 Log-log scale.

(

)

{

}

(

)

{

}

p

A p

A f kh

t

B t

B g k C S

D D

=

∆

,

,...

,

, , ...

( 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

p

A

p

t

B

t

D D

=

+

=

+

∆

( 2-2)

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.

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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

p

kh

qB

p

D

=

1412

.

µ

∆

(field units) p qB kh p_D = ∆

µ

66 . 18 (metric units) ( 2-3) Dimensionless time t k c r t D t w =0 000264. ₂

φµ

∆ (field units)

t

r

c

k

t

w t D

=

₂

∆

000356

.

0 φµ

(metric units) ( 2-4)

Dimensionless wellbore storage coefficient

C C c hr D t w =0 8936. ₂

φ

(field units) 2

1592

.

0

w t D

hr

c

C

φ

=

(metric units) ( 2-5)

Dimensionless time group t C kh t C D D =0 000295.

µ

∆ (field units)

C

t

kh

C

t

D D

₌

∆

µ

00223

.

0

(metric units) ( 2-6)

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Chapter 2 - The analysis methods Di m ensi onl ess P re ssure, pD Dimensionless time, tD/CD 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 1 02 10 1 10-1 Approximate start of semi-log straight line

C_De2S 1060 1050 1040 1030 1020 1015 1010 108 ₁₀6 104 ₁₀3 102 ₁₀ 3 1 0.3 Di m ensi onl ess P re ssure, pD Dimensionless time, tD/CD 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 1 02 10 1 10-1 Di m ensi onl ess P re ssure, pD Dimensionless time, tD/CD 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 1 02 10 1 10-1 Di m ensi onl ess P re ssure, pD Dimensionless time, tD/CD 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 1 02 10 1 10-1 Dimensionless time, tD/CD 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 1 02 10 1 10-1 Approximate start of semi-log straight line

C_De2S 1060 1050 1040 1030 1020 1015 1010 108 ₁₀6 104 ₁₀3 102 ₁₀ 3 1 0.3

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

C e S C c hr e S D t w 2 0 8936 2 2 = .

φ

(field units)

S

e

hr

c

C

S

e

C

w t D

2 1592

.

0

2

φ

=

(metric units) ( 2-7)

The curve label CDe2S defines the well condition. It ranges from CDe2S =0.3 for stimulated wells, up to 1060_{for very damaged wells.}

Log-log matching procedure

Elapsed time,∆t (hours)

P res sur e c hang e, ∆ p( p s i) 1 101 102 103 10-3 ₁₀-2 ₁₀-1 ₁ ₁₀1 ₁₀2

Elapsed time,∆t (hours)

P res sur e c hang e, ∆ p( p s i) 1 101 102 103 1 101 102 103 10-3 ₁₀-2 ₁₀-1 ₁ ₁₀1 ₁₀2 10-3 ₁₀-2 ₁₀-1 ₁ ₁₀1 ₁₀2

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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 storage" asymptote but the final choice of the CD e2S curve is frequently not unique (Figure 2-12).

Results of log-log analysis

Pressure match PM

=

p

_D

∆

p

: the permeability thickness product

( )

PM

2 .

141 qB

µ

kh

=

(mD.ft, field units)

( )

PM

66 .

18 qB

µ

kh

=

(mD.m, metric units) ( 2-8)

Time match TM

=

(

t

_D

C

_D

)

∆

t

: the wellbore storage coefficient       = TM 1 000295 . 0

µ

kh

C (Bbl/psi, field units)













=

TM

1 00223

.

0 µ

kh

C

(m3_{/Bars, metric units) (}_2-9)

Curve match : the skin

D Match S D C e C S 2 ln 5 . 0 = ( 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

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Chapter 2 - The analysis methods Rate, q Pressure, p Time, t pi 0 t_p t_p+∆t q 0 ∆t_BU ∆pBU(∆t) ∆p (tp) Rate, q Pressure, p Time, t pi 0 t_p t_p+∆t q 0 ∆t_BU ∆pBU(∆t) ∆p (tp)

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.

Ra te , q Pr es s u re , p Time, t q 0 -q p_i 0 t_p ∆t ∆p₍_∆_t) ∆p_(tp+_∆_t) ∆p_(tp) (∆p_(tp+_∆_t)- ∆p₍_∆_t)) Ra te , q Pr es s u re , p Time, t q 0 -q p_i 0 t_p ∆t Ra te , q Pr es s u re , p Time, t q 0 -q p_i 0 t_p ∆t ∆p₍_∆_t) ∆p_(tp+_∆_t) ∆p_(tp) (∆p_(tp+_∆_t)- ∆p₍_∆_t))

Figure 2-5 History extended drawdown + injection.

Log-log analysis : build-up type curve

( )

[

p

_D

t

]

p

( )

t

p t

(

t

)

p t

( )

D BU D D D p _D D _{p D}

∆

=

∆

−

+

∆

+

( 2-11)

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Dimensionless time, t_D/C_D 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 Dim ensionl ess P re s s ure, pD 10 2 10 1 10-1

build-up type curve

t_pD p_D(t_pD) C_De2S_drawdown type curve Dimensionless time, t_D/C_D 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 Dim ensionl ess P re s s ure, pD 10 2 10 1 10-1 Dimensionless time, t_D/C_D 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 Dim ensionl ess P re s s ure, pD 10 2 10 1 10-1

build-up type curve

t_pD p_D(t_pD)

C_De2S_drawdown

type curve

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

Semi-log analysis : superposition time

( )

[

]













+

−

+

∆

+

∆

=

∆

S

r

c

k

t

kh

qB

t

p

w t p p BU

162 .

6 log

log

_φ

_µ

2

3 .

23

0 .

87 µ

(psi, field units)

( )

[

]













+

−

+

∆

+

∆

=

∆

S

r

c

k

t

kh

qB

t

p

w t p p

87 .

0

10 .

3 log

log

5 .

21

₂ BU

φµ

µ

(Bars, metric units)

( 2-12)

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

Dimensionless times, tD/ CDand [ tpD tD/ (tpD+ tD) CD] 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 D im ens io nle s s P re s s ur e , pD 10 5 0

build-up type curve

tpD pD(tpD)

CDe2Sdrawdown type curve

Dimensionless times, tD/ CDand [ tpD tD/ (tpD+ tD) CD] 10-1 ₁ ₁₀ ₁₀2 ₁₀3 ₁₀4 D im ens io nle s s P re s s ur e , pD 10 5 0

build-up type curve

tpD pD(tpD)

CDe2Sdrawdown type curve

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

p

qB

kh

t

ws i p

= −

162 6

.

µ

log

+

∆

t t t kh qB p p_ws _i p ∆ ∆ + −

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Chapter 2 - The analysis methods Horner time, [(t_pD+ t_D) / t_D] 1 10 102 ₁₀3 ₁₀4 ₁₀5 D im e nsi o n less P re s s ur e, pD 10 5 0 m P* Horner time, [(t_pD+ t_D) / t_D] 1 10 102 ₁₀3 ₁₀4 ₁₀5 D im e nsi o n less P re s s ur e, pD 10 5 0 m P*

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 : kh qB m =162 6.

µ

(mD.ft, field units) m qB kh=21.5

µ

(mD.m, metric units) ( 1-13) S p m k c r t t t w p p =  − + + +   _ 1151. ∆ 1 hr log ₂ log 1 3 23.

φµ

(field units)













+

−

∆

=

1 .

151

1hr

log

₂

log

1

3 .

10

p p w t

t

r

c

k

m

p

S

φµ

(metric units) ( 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 :

( )

[

p

t

]

q

[

(

)

(

)

]

( )

q

p

t

p

t

p

t

D _{D MR} i i n n i n D n _{i D} D n _{i D} D _D

∆

=

−

∆

−

+

−

+

− = −

∑

1 1 1 1 ( 2-15)

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Time, t Ra te , q P re s su re , p ∆t Period # 1,2,…, 5, 6,……...10, 11 q1,…. q5=0, q6,………..q10, q11=0 Time, t Ra te , q P re s su re , p ∆t Period # 1,2,…, 5, 6,……...10, 11 q1,…. q5=0, q6,………..q10, q11=0

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

The multirate superposition time is expressed :

(

q

) (

t

) (

q

) ( )

t

kh

B

p

t

p

_n _n n i i n i i i ws

∆

=

−

+

∆

−

+

−

∆

− = −

∑

log

6 .

162 )

(

1 ₁ 1 1

µ

(

) (

log

) (

)

log

(

)

5 .

21 )

(

1 ₁ 1 1

t

q

t

q

kh

B

p

t

p

_n _n n i i n i i i ws

∆

=

−

+

∆

−

+

−

∆

− = −

∑

µ

(Bars, metric units) ( 2-16)

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.

Pr e s s u re , ps i Time, hours 0 50 100 150 200 250 300 5000 4500 4000 3500 Radial Hemi-radial Radial Hemi-radial Infinite reservoir Sealing fault Pr e s s u re , ps i Time, hours 0 50 100 150 200 250 300 5000 4500 4000 3500 Radial Hemi-radial Radial Hemi-radial Infinite reservoir Sealing fault

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.

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Chapter 2 - The analysis methods

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

)

1 2 −

( )

∆t 1 2 (hr1/2) ( 2-17)

(

)

14

( )

14 t t t_p +∆ −∆ (hr1/4) ( 2-18)

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