USMS023575 Slug Test Method in Pressure-Sensitive Reservoirs

Jack Dvorkin, Stanford U.; J.D Walls, Rock PhysicsAssocs.; Amos Nur, Stanford U.

Copyright I'lV Society of Petroleum EngineersThis manuscript was provided to the Society of Petroleum Engineers for distributionand possible publication in an SPE journal. The material is subject to correctionby the author(s). Permission to copy is restricted to an abstract of not more than300 words. Write SPE Book Order Dept., Library Technician, P.O. Box 833836,Richardson, TX 75083-3836 U.S.A. Telex 730989 SPEDAL.

JUL 1 5 1991

SPETechnical Publications UNSOLICITED

Slug Test Method in PressureSensitive Reservoirs

Jack Dvorkin, Stanford D., Joel D. Walls, SPE,and Amos N ur, Stanford D.

SPE 2357'

SUMMARY

We present a methodology for the estimation of the hydrologic properties of /

highly pressure sensitive reservoirs based on the results of slug tests. These

properties include a permeability-pore pressure parameter, = t ;;, as wellas the product of a reference permeability and formation thickness koH.

We include the skin effect into consideration using the concept of effective

wellbore radius. We show that the results of the traditional interpretation of

a slug test data in a pressure sensitive reservoir may vary with varying initial

pressure in a wellbore.

Introduction

A slug test is a simple and economical means of evaluating hydrologic prop-

erties of a reservoir at a point source such as an injection or production well.

The name derives from the removal or addition of a "slug" of water and the

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resulting well level change over time. Slug tests were first used to evaluate

water wells 1,2 and were later adapted for drill stem testing of oil wells 3.

The effect of skin damage was introduced to the slug test analysis by Ramey,

et.al. 4.

Current slug test methodology 4,5 has been developed under the assump-

tion that the reservoir has a constant permeability during the test. However,

coal has been shown to exhibit strong sensitivity of fluid permeability to net

pressure changes 7. These results reveal that permeability of coal is 10 to .

100 times more sensitive to net stress than tight gas sands. Other evidence

of this effect is illustrated by well tests in coal beds of the Northeast Blanco

Unit 403 8. Permeability measured in the Basal Zone during an injection test

was 17 times higher than in the same zone during a production test .

,To address these problems, we have developed a mathematical solution to

the pressure dependent slug test problem and will show that the solution can

be applied to improve interpretation of well test data from coalbed methane

and other pressure sensitive formations. Previous examples of well test solu-

tions involving non-linear gas flow have been presented by AI-Hussainy et.al.

9, Evers and Soeiinah 10, and Ostensen 11. However, our method follows

Yilmaz et.al. 12 who introduced a new term, the permeability-pore pressure

parameter I = t ;;. Using this method, we have presented a new set of type

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curves which can be used to estimate kH, skin, and /.

Filtration in Pressure Compliant Medium

To derive the equation for single-phase liquid filtration we combine Darcy's

law with the continuity equation. The equation of mass conservation is:

8(p

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b. pore-volume compressibility

c. permeability-pressure parameter

1 dk1= kdP'

We assume that the permeability-pressure parameter is constant for the

range of pressure change under consideration 12. This implies an exponential .

form for permeability-pore pressure relation:

k = koexp[,(P - Po)],

where ko is the permeability at a certain characteristic (reference) pressure Po.

This relation is supported by experimental data 13 and 14. The exponential

form for permeability-pressure dependence was also employed by Evers and

Soeiinah 10.

These parameters, if used to express fluid density p, porosity , and

permeability k through pressure P, lead to the following form of equation

(3):

For one-dimensional plane filtration this equation is:

11(f3m + f3i )f)P _ a2P (f3 )( ap)2k at - ax2 + i + I ax .

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For radial filtration we have:

IUP(Pm + PI) 8P _ 82P ~ 8P (,8 )(8P)2. k 8t - 8r2 + r 8r + I + I 8r .

Slug Test

(4)

A slug test involves the injection or withdrawal of a slug (certain amount)

of water into or from a well. Pressure at the wellbore bottomhole is equal to

the hydrostatic pressure of the fluid column in the wellbore:

Pr=R = pgL,

where L is the fluid level in the wellbore (Fig. 1); R is the wellbore external

radius; 9 is the gravity constant. The fluid flow rate in the wellbore Q is

related to Las:

where R1 is the wellbore internal radius. Using Darcy's law we can express

flow rate Q through pressure gradient Pr at the bottomhole:

at r = R, where H is the thickness of the formation (Fig. 1); lower case

indexes t and r denote partial derivatives. Combining these equations we

can relate Pt and Pr at the bottomhole:

P, - 2pgHRk P.t - R2 r

J.l 1

5

(5)

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at r = R.

To model a slug test we must solve equation (4) with boundary condition

at r = 00 and boundary condition (5) at r = R. Here P oo is the pressure

at infinity. An initial condition for pressure distribution in the reservoir can

be, for example, P = P00 at t = 0 for R < r < 00. An initial condition for

pressure at the bottomhole is: P = Pi at r = Rand t = 0, where Pi is the

initial hydrostatic pressure. The injection of the fluid into the formation will

occur if Pi > Poo The condition Pi < Poo means the withdrawal of a slug.

Traditional Normalization

In this section we give a solution to the problem of fluid filtration through

a pressure-compliant reservoir using the traditional approach to the normal-

ization of equations: dimensionless pressure is introduced as a ratio of actual

pressure to a certain characteristic pressure. We show that the characteristics

of pressure propagation through a pressure-compliant reservoir are strongly

affected by the pressure at the reservoir boundary. The results of a slug test

may vary significantly depending on the initial fluid level in a wellbore.

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Normalized Equation

Symbols P and f will be used below to denote the normalized pressure and

coordinate.

Normalized variables are:

P = P/Po; f = r/R; t = TT,

where T is normalized time; T = R2 /ao; ao = a(Po). Parameter a is:_ k

a(P) = /l

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Linearizing Equation

To linearize a non-linear equation (6) we introduce a function F(P):

F(P) = exp(AP), (8)

where A = ({3f + /,)Po. This approach has been used to describe the flow

of real gases through porous media 9 and to model the filtration in stress-

sensitive reservoirs 10,11.

Expressing partial derivatives of P through partial derivatives of F we

can transform equation (6) to the following one:

Fr = a(P) (Frr +Fr/f). (9)

To express normalized pressure P through function F we will use the formula:

P = (In F) / A. The boundary and initial conditions for function F can be

obtained from conditions for P using formula (8). For example, the boundary

condition at f = 1 has the following form: Fr = K Fr.

The linearization of equation (6) is important for the successful imple-

mentation of an implicit numerical scheme: the non-linear term makes this

scheme highly unstable.

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Numerical Solution - Spatial Pressure Distribution - Slug Test

To solve equation (9) we employed an implicit numerical scheme (the Crank-

Nicolson method). In our computations we assumed that Poo = 0, so that

the pressure is counted from its level at infinite distance from the wellbore.

The spatial distribution of pressure in a pressure compliant reservoir is

given in Fig. 2 for different values of parameter ,. This example shows a

plane filtration of water in an infinite half-space with constant pressure kept

at the left boundary. Different curves in every plot correspond to different

time moments. The results indicate that pressure propagation through the

reservoir becomes "piston-like" as , increases.

The numerical modeling of a pressure drawdown in a pressure-compliant

reservoir during a slug test was performed for dIfferent levels of initial pressure

in a wellbore. Properties of the reservoir as well as wellbore geometry and

fluid properties were kept unchanged. The reference pressure was Po = P00 =

1 M Pa; parameter, was 10-7 Pa-1 The initial pressure Pi was 2, 7, Ii

and 21 M Pa. Pressure at the bottomhole was related to the initial pressure

(Fig. 3).

This example shows that drawdown curves are different when initial pres-

sure changes. Thus, if the traditional method 4 is used, the hydrologic prop-

erties of the same reservoir will be found to be different in different experi-

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ments.

Proposed Method of Normalization

In this section we will introduce dimensionless variables for the slug test

problem in a different way. To model a one-dimensional radial filtration in a

pressure-compliant reservoir we use equation (4):

Wp((3m + (31) 8P _ 82P ~ 8P ((3 )(8P)2k 8t - 8r2 + r 8r + 1 +, 8r .

A slug test is described by the following boundary and initial conditions:

D _ 2pgHRk nrt - 2 rr

p.R}

at r = R; P = P00 at r = 00; P = P00 at t = 0 for R < r < 00; P = Pi at

r = Rand t = O. Permeability is related to pore pressure as:

k = koexph'(P - Po)],

where Po is a reference pressure.

Assumptions and Estimates

Assuming that permeability is related to porosity by Kozeny's equation we

conclude that k is proportional to (p3, and 4> is proportional to P/3. Thus,

we can relate porosity and pore pressure:

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where

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New Function - Normalization

We introduce a dimensionless function F(P):

F(P) = expbP).

It is necessary to point out that here P is actual non-normalized pressure.

This approach (exponential normalization) differs from the traditional lin-

ear normalization of pressure. This function is related to permeability and

porosity as follows:

Coordinate r and time t are normalized in a traditional way:

J-lR2 0r = xR; t =T---"3 ko

where x and T are normalized coordinate and time.

Substituting function F(P) and normalized variables into equation (10)

we obtain:

(11)

Boundary conditions are: Fr = CIFFx at x .= 1; F = 1 at x = 00. Initial

conditions are: F = 1 at 1 < x < 00; F = C2 at x = 1 for T = O. In these

expressions we introduced two dimensionless parameters:

12

(12)

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Evaluating Hydrologic Properties

Two constants C1 and C2 define the shape of a drawdown curve F = F(T).

Using the fact that P = In F / "y, we will be able to find a match between a

given pressure drawdown curve P = P(t) and one of the curves that depend

on C1 and C2 If C1 is known, parameter "y can be found as:

where Pi is the initial pressure for the actual slug test.

Now the product H 0 can be calculated using the value of constant C1 :

3 R2 1Ho = CI -2 R~-'pg "y

Finally, if T = T m is the point of a close match corresponding to physical

time tm , we can find the ratio 0/ko:

From these equations we can easily find the product koH:

Type Curves

(13)

As we mentioned before, the hydrological properties of a pressure-compliant

reservoir can be evaluated provided that a match between an observed pres-

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sure drawdown curve and one of computed curves is found. To create a type

curve chart we calculated function In(F)1 In(C2) = PIPi versus argument

In T for different constants C1 and C2

Showing the dependence of a slug test curve on two parameters C1 and C2

requires a number of graphs or one complicated graph for computed pressure

drawdown curves. This fact introduces some difficulties and uncertainties in

a visual curve matchin~procedure. Thus, an accurate numerical matching

technique has to be developed.

In this paper we offer an approximate matching technique. Our approach

is based on the observation that at the level P/ Pi = 0.1 the product C1T is

approximately constant for different C1 and same C2 in a wide interval of C1

from 10-4 to 10-2 (Fig. 4).

This observation allows us to find a matching curve on a type curve chart

plotted for constant C1 and different C2 (Fig. 4). When C2 is found, we can

calculate the product C1Tm at P/ Pi = 0.1 and find a corresponding real time

match point tm . Then koH can be calculated from (13). This approximate

method gives an estimate for koH with accuracy within 20% range.

It is necessary to mention that if C2 < 1, an initial pressure Pi is negative

and, thus, pressure buildup in a wellbore occurs. Yet, in this case we will also

observe a PIPi drawdown because actual pressure (that is negative relative

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to the initial reservoir pressure) is related to negative Pi-

Skin Effect

To treat the skin effect we will follow 4 using the concept of "effective"

wellbore radius 15_

For a pressure compliant reservoir the relation between flow rate Q and

pressure gradient Pr is:

Here we used the following relations:

(14)

Resolving equation (14) we can find that for steady flow, .6.F between r = rl

and r = r2 is:

where F1 corresponds to rl and F2 corresponds to r2-

We introduce a skin factor s as follows:

where .6.Fs is .6.F at the wellbore interface.

15

(15)

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We define the effective wellbore radius Re as that which makes the calcu-

lated ~F in a reservoir without skin effect equal to that in an actual reservoir

with skin. Thus, from (15):

R .s = In R

e; Re = Rexp(-s). (16)

Now we can notice that equation (13) does not include parameter / and

actual external wellbore radius R. It means that the knowledge of R is not

required in this case and we can treat the problem including skin effect 4.

This approach means that the constant C1 in (12) is actually expressed

as follows:

where the effective radius Re is defined in (15). Now we can find Re :

3C1Re = R1 2 H pg / 0

If the information about the actual wellbore radius R is available, we can

find a skin factor s from (16).

Application of the Method

During the slug test of combined Mary Lee and Blue Creek seam (coal thick-

ness 2.06 m) water was slugged to surface from its initial level 54 m. This

means that Pi = -O.54MPa. The normalized pressure drawdown curve is

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given in Fig. 6. Applying this curve to the type curve chart in Fig. 5 we can

find a match at C2 :::::: 0.3. Thus, 1'=2.23 . 10-6 Real time for the matching

point at PjPi =O.l is approximately 2.1 .105s; dimensionless time T :::::: 2100.

Thus, from (13) koH :::::: 5Ri . 10-12 ; ko :::::: 2.43Ri 10-12

The interpretation of observed pressure change curve in the example gave

realistic estimates of parameters I' and koH. The value I' can be used to

predict reservoir performance with pressure depletion.

It is important to mention that the pressure change curve presented in

Fig. 6 could be also interpreted by means of the traditional method 4. Our

method must be used if there is a difference in permeability values obtained

with different initial pressures in the same reservoir. We believe this effect

may be practically important in coalbed methane wells tested by alternately

injecting and withdrawing fluid.

Conclusions

A new methodology for the estimation of the hydrologic properties of highly

pressure sensitive reservoirs based on the results of slug tests is developed.

These properties include a permeability-pore pressure parameter I' = t;; aswell as the product of a reference permeability and formation thickness koH.

We show that a slug test dynamics depends on two dimensionless parameters

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C1 and C2 An approximate technique of matching an observed pressure

drawdown curve with computed type curves is offered. The interpretation

of observed pressure change curve in the example gave realistic estimates of

parameters, and koH. We include the skin effect into consideration using

the concept of effective wellbore radius. We show that the results of the

traditional interpretation of a slug test data in a pressure sensitive reservoir

may vary with varying initial pressure in a wellbore.

Nomenclature

Co- sound velocity in fluid;

C1 , C2 - dimensionless parameters, eq. (12);

F(P) - "pseudopressure", eq..(8);

9 - gravity acceleration;

H - formation thickness;

k - formation permeability;

ko - reference permeability;

L - fluid level in a wellbore;

P, P - pore pressure;

Po - reference pore pressure;

P00 - initial reservoir pressure;

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Pi - initial pressure at the bottomhole;

Q - flow rate in a wellbore;

qx, %" qz, - filtration flow rates;

R - wellbore external radius;

R1 - wellbore internal radius;

Re - wellbore effective radius;

s - skin factor, eq. (15);

t - time;

tm - real time at a matching point;

x, y, z - spatial coordinates;

(31 - fluid compressibility;

(3m - pore-volume compressibility;

I - permeability-pressure parameter;

J.l - fluid viscosity;

p - fluid density;

T - normalized time;

Tm - dimensionless time at a matching point;

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Acknowledgments

The authors wish to express their appreciation to GRI and Taurus Ex-

ploration for providing the slug test data used here. We also thank Dr.

H.J.Ramey for his review and helpful suggestions with regard to slug test

data interpretation.

References.

1. Ferris, J .G. and Knowles, D.B.: "The slug test for estimating transmissi-

bility," USGS Ground Water Note 26, (1954) 1-7.

2. Maier, L.F.: "DST interpretation calculations for water reservoi~s." Hal-

liburton Services Ltd. Report, Calgary, Alberta, Canada (1970).

3. Kohlhaas, C.A.: "A method for analyzing pressures measured during

drillstem test flow periods," J. Petro Tech., v.24 (October), (1972) 1278-

1282.

4. Ramey, H.J., Jr., Agarwal, R.G., and Martin, I.: "Analysis of 'Slug Test'

or DST flow period data," J. Can. Petr. Tech., v.13, no 3 (July-Sept.),

(1975) 37-47.

5. Koenig, R.A. and Schraufnagel, R.A.: "Application of the slug test in

coalbed methane testing," Coalbed Methane Symp., Tuscaloosa, AL (1987).

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6. Rushing, J.A., Blasingame, T.A., Poe, B.D., Jr., Brimhall, R.M., and

Lee, W.J.: "Analysis of slug test data from hydraulically fractured coalbed

methane wells," SPE 21492, Gas Technology Symp., Jan. 22-24, (1991).

7. Koenig, R.A.: "Application of hydrology to evaluation of coalbed methane

reservoirs," Gas Research Institute, Topical Report on Contract 5087-214-

1489, (1989) 114.

8. Mavor, M.J., Britton, R.N., Close, T.M., Erwin, T.M., Logan, T.L., and

Marshall, R.B.: "Evaluation of the Cooperative Research Area, Northeast

Blanco Unit," GRI Topical Report GRI-90/0041 (1989).

9. AI-Hussainy, R., Ramey, H.J, Jr., and Crawford, P.B.: "The flow of real

gases through porous media," J. Petro Tech., v.18 (May), (1966) 624-636.

10. Evers, J.F., and Soeiinah, E.: "Transient tests and long-range perfor-

mancein stress-sensitive gas reservoirs," J. Petro Tech., v.29 (August), (1977)

1025-1030.

11. Ostensen, R.W.: "The effect of stress-dependent permeability on gas

production and well testing," SPE Formation Evaluation, v.1, no 3 (June),

(1986) 227-235.

12. Yilmaz, 0., Nur, A., and Nolen-Hoeksema, R.: "Pore pressure fronts in

fractured and compliant rocks," submitted to "Geophysics" (1990).

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SPE 23575

13. Thomas, R.D., and Ward, D.S., "Effect of overburden pressure and water

saturation on gas permeability of tight sandstone cores," J. Petro Tech., v.24

(February), (1972) 120-124.

14. Walls, J., Nur, A., and Bourbie, T.: "Effect of pressure and partial water

saturation on gas permeability in tight sands: experimental results," J. Petro

Tech., v.34 (April), (1982) 930-936.

15. Matthews, C.S., and Russel, D.G.: "Pressure Buildup and Flow Tests in

Wells," New York, Dallas, SPE of AIME, (1967) 167.

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FIGURE CAPTIONS

FIGURE 1. Slug test in a wellbore.

FIGURE 2. Spatial pressure distribution in a pressure-compliant reservoir:

plane filtration in a half-space with constant pressure kept at the left bound-

ary; different curves at each plot correspond to different time moments.

FIGURE 3. Pressure drawdown curves in the same reservoir for different

initial pressure.

FIGURE 4. PjPi drawdown curves for C2 = 4 and C1 = 0.01,0.001 and

0.0001 (from left to right).

FIGURE 5. Type curve chart for C1 = 0.01 and C2 increasing from

0.000625 to 32 (C2=32, 16, 8, 4, 2, 1.7, 1.5, 1.3, 1.1, 0.9, 0.7, 0.5, 0.25,

0.125, 0.0025, 0.00125 and 0.000625 from left to right).

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FIGURE 6. P / Pi dropdown curve during the slug test of combined Mary

Lee and Blue Creek seam.

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L

Fluid level

R

...

H

Formation

FIGURE 1. Slug test in a wellbore.

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Distance

..,=110-6 l/Pa

Distance

..,=710-1 I/Pa

Distance

..,=21O-6 1/Pa

Distance

FIGURE 2. Spatial pressure distribution in a pressure-compliant reservoir:plane filtration in a half-space with constant pressure kept at the left bound-ary; different curves at each plot correspond to different time moments.

SPE 2357;

I. I

8

Lnr

l'i, MPa

12

FIGURE 3. Pressure drawdown curves in the same reservoir for differentinitial pressure.

0.14

0.12

0.1

0.08

0.06

0.04

0.02

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I

n 1\

I

1\I

,1\

rT

"1\

"

1

I""

,,",

'"'" l't.7 8 9

Lnr10 11 12 13

FIGURE 4. P/Pi drawdown curves for O2 = 4 and 0 1 = 0.01,0.001 and0.0001 (from left to right).

0.8

0.6

0.4

0.2

2 4Lnr

. 6 8

FIGURE 5. Type curve chart for C1 = 0.01 and C2 increasing from0.000625 to 32 (C2=32, 16, 8, 4, 2, 1.7, 1.5, 1.3, 1.1, 0.9, 0.7, 0.5, 0.25,

. 0.125, 0.0025, 0.00125 and 0.000625 from left ~o right).

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864Lnt2o

~~

\,

~'\

I

P/Pi r\\

1\

\\

\

1\

\\~-

o -2

004

FIGURE 6. P/ Pi dropdown curve during the slug test of combined MaryLee and Blue Creek seam.

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