Dake L.P. Fundamentals of reservoir engineering

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OILWELL TESTING 231

Since F is a constant, dependent upon the well design and fluid compressibility, then

by simple proportionality

()

T/F

=×

()

T/F

=×

Apart from the two methods mentioned in this text, afterflow analysis techniques relying

on the use of type curves have also been presented by Ramey

and Earlougher and

Kersch

. Which, of all these methods, is the most reliable is a question which is still

unanswered. One point on which all the authors of papers on the subject agree is that

afterflow analysis techniques should not be used for pressure buildup analysis when

there is a clearly defined linear portion of the conventional Horner plot which can be

subjected to the analysis techniques described in sec. 7.7. This is because the physical

and mathematical concepts involved in the description of afterflow are vastly more

complex than for the simple pressure buildup theory and therefore, the analysis results

may be less reliable.

Nevertheless, it is recommended that engineers should experiment with one or all of

these methods in their own fields to determine which, if any, is suitable. To do this,

pressure buildup tests which provide both a linear trend on the conventional Horner

plot and also a significant deviation from this trend due to afterflow, for small values of

∆t, should be analysed using both the normal and afterflow analysis techniques, and

the results compared. If the comparison is favourable and statistical confidence is built

up in one of the afterflow methods, then the engineer can use this method for test

analysis on such occasions as when buildup surveys are recovered which are

dominated by the effects of afterflow.

EXERCISE 7.9 AFTERFLOW ANALYSIS TECHNIQUES

A twelve hour pressure buildup test was conducted in a flowing oil well from which the

pressure-time record was recovered, as listed in table 7.17.

The production data and reservoir and fluid properties are as follows:

= 30,655 stb

=0.2

q = 231 stb/d

= 0.6 cp

h = 10 ft B

= 1.3 rb/stb

= 0.3 ft c = 20 × 10

-6

/psi

OILWELL TESTING 232

∆t (mins) p

(psi) ∆t (mins) p

(psi)

0 1600 (p

)

20 1920 270 2879

40 2160 300 2900

60 2350 360 2935

90 2525 420 2960

120 2650 480 2980

150 2726 540 2998

180 2779 600 3011

210 2822 660 3022

240 2852 720 3035

TABLE 7.17

The conventional Horner plot of p

versus log

+∆

∆

does not become linear, even for

the largest values of ∆t, and therefore the afterflow analysis techniques presented in

this section must be used to analyse this test.

1) Evaluate k and S using the Russell method.

2) Determine the permeability in the vicinity of the wellbore and of the undamaged

formation using the McKinley method.

EXERCISE 7.9 SOLUTION

1) Russell Analysis

As suggested by Russell, the analysis should only be applied for pressures measured

after ∆t = 1 hour. In table 7.18 several values of the parameter C have been selected in

an attempt to linearize the plot of equ. (7.77).

OILWELL TESTING 233

p/(1 1/C t)∆−∆

∆t (hrs) log ∆t ∆p (psi) C=1.7 C=2.1 C=2.5

1.5 .176 925 1522 1355 1261

2.0 .301 1050 1487 1378 1314

2.5 .398 1126 1472 1391 1340

3.0 .477 1179 1467 1401 1360

3.5 .544 1222 1469 1414 1380

4.0 .602 1252 1468 1421 1391

4.5 .653 1279 1471 1430 1404

5.0 .699 1300 1473 1437 1413

6.0 .778 1335 1480 1450 1430

7.0 .845 1360 1485 1459 1442

8.0 .903 1380 1490 1467 1453

9.0 .954 1398 1496 1476 1463

10.0 1.000 1411 1499 1482 1470

11.0 1.041 1422 1502 1486 1476

12.0 1.079 1435 1509 1494 1484

TABLE 7.18

As shown in fig. 7.43, the correct value of the parameter C to obtain a linear Russell

plot is C = 2.1.

Since the slope of this line is 151 psi/log cycle, then the kh product can be evaluated

using equ. (7.78) as

162.6 231 0.3 1.3

kh 194 mD.ft; and k 19.4 mD

151

×××

===

OILWELL TESTING 234

1500

1400

1300

1.0

C= 1.7

C= 2.1

C= 2.5

∆

1 -

∆

(psi)

log t

∆

Fig. 7.43 Russell afterflow analysis Exercise 7.9)

The value of (p

(∆t) - p

(t))/(1−1/C∆t) at ∆t = 1 hour can be read from the linear plot

as 1329 psi. Therefore the skin factor can be calculated using equ. (7.79) as

1329 19.4

S 1.151 log 3.23 4.7

151

.2 .6 20 10 .09

−

æö

=− +=

ç÷

×× × ×

èø

2) McKinley Analysis

To apply the McKinley method it is necessary to plot the closed in time ∆t(minutes)

versus the pressure buildup ∆p= p

, (∆t)−p

(t) (psi) on transparent paper using the

same log-log scales as on the McKinley type curve chart. When this plot is moved

laterally across the McKinley chart, keeping the abscissae of both charts together, the

early part of the buildup is found to match the type-curve which has the parametric

value T/F = 2500, fig. 7.44. Selecting a match point ∆t = 60 minutes, ∆p = 750 psi, the

corresponding value on the abscissa of the McKinley chart is ∆pF/q = 0.14. Therefore,

OILWELL TESTING 235

MATCH POINT

t = 60 mins

p = 750 psi

∆

= 0.14

∆

-2

-1

100

∆

1000

= 5000

= 2500

(mins)

∆

Fig. 7.44 Match between McKinley type curves and superimposed observed buildup

(Exercise 7.9).

−

−−

− match for small ∆

∆∆

∆t (T/F = 2500)

•

••

• −

−−

− match for large ∆

∆∆

∆t (T/F = 5000)

pF T pF

0.14 2500 350

qF q

∆∆

×= = × =

OILWELL TESTING 236

and since q = 300 rb/d, then the wellbore transmissibility is

3500 300

T 140 mD.ft / cp

750

== =

and therefore

k8.4mD=

For values of ∆t greater than 150 minutes the actual buildup curve breaks away from

the parametric curve for T/F = 2500, indicating the presence of a positive skin factor.

For large values of ∆t the buildup matches the McKinley curve with parametric value

T/F = 5000, fig. 7.44. Even for this latter match, however, the buildup continues to be

dominated by afterflow. Therefore, a minimum value of the formation transmissibility

can be estimated as

()

T/F

=×

giving

5000

k 8.4 16.8mD

2500

=×=

Thus the comparison between the Russell and McKinley techniques is quite

reasonable in this case.

REFERENCES

1) van Everdingen, A.F. and Hurst, W., 1949. The Application of the Laplace

Transformation to Flow Problems in Reservoirs. Trans. AIME. 1 86: 305-324.

2) Earlougher, R.C., Jr., 1971. Estimating Drainage Shapes from Reservoir Limit

Tests. J. Pet. Tech., October: 1266-1268.

3) Ramey, H.J., Jr. and Cobb, W.M., 1971. A General Pressure Buildup Theory for

a Well in a Closed Drainage Area. J. Pet. Tech., December: 1493-1505.Trans.

AIME.

4) Horner, D.R., 1951. Pressure Build Up in Wells. Proc., Third World Petroleum

Congress. E.J. Brill, Leiden. ll, 503.

5) Odeh, A.S. and Jones, L.G., 1965. Pressure Drawdown Analysis, Variable Rate

Case. J. Pet. Tech., August: 960-964. Trans. AIME.

6) Matthews, C.S. and Russell, D.G., 1967. Pressure Buildup and Flow Tests in

Wells. SPE Monograph: 130-133.

7) Matthews, C.S., Brons, F. and Hazebroek, P.,1954. A Method for the

Determination of Average Pressure in a Bounded Reservoir. Trans.

AIME.201: 182-191.

OILWELL TESTING 237

8) Cobb, W.M. and Dowdle, W.L., 1973. A Simple Method for Determining Well

Pressures in Closed Rectangular Reservoirs, J. Pet. Tech., November:

1305-1306.

9) Dietz, D.N., 1965. Determination of Average Reservoir Pressures from Build Up

Surveys. J. Pet. Tech., August: 955-959. Trans. AIME.

10) Earlougher, R.C., Jr., Ramey, H.J., Jr., Miller, F.G. and Mueller, T.D., 1968.

Pressure Distribution in Rectangular Reservoirs. J. Pet. Tech., February:

199-208. Trans. AIME.

11) Ramey, H.J., Jr., Kumar, A. and Gulati, M.S. 1973. Gas Well Test Analysis Under

Water-Drive Conditions. American Gas Assn., Arlington, Va.

12) Pinson, A.E., Jr., 1972. Concerning the Value of Producing Time in Average

Pressure Determinations from Pressure Buildup Analysis, J. Pet. Tech.,

November: 1369-1370.

13) Kazemi, H., 1974. Determining Average Reservoir Pressure from Pressure

Buildup Tests. Soc. Pet. Eng. J., February: 55-62. Trans. AIME.

14) Odeh, A.S. and Selig, F., 1963. Pressure Buildup Analysis, Variable Rate Case.

J. Pet. Tech., July: 790-794. Trans. AIME.

15) van Poollen, H.K., Breitenback, E.A. and Thurnau, D.H., 1968. Treatment of

Individual Wells and Grids in Reservoir Modelling. Soc. Pet. Eng. J.,

December: 341-346.

16) Earlougher, R.C., Jr., 1972. Comparing Single-Point Pressure Buildup Data With

Reservoir Simulator Results. J. Pet. Tech., June: 711-712.

17) Cobb, W.M. and Smith, J.T., 1975. An Investigation of Pressure Buildup Tests in

Bounded Reservoirs. J. Pet. Tech., August: 991-996. Trans. AIME.

18) Denson, A.H., Smith, J.T. and Cobb, W.M., 1976. Determining Well Drainage

Pore Volume and Porosity from Pressure Buildup Tests. Soc. Pet. Eng. J.,

August: 209-216.

19) Brons, F. and Marting, V.E., 1961. The Effect of Restricted Fluid Entry on Well

Productivity. J. Pet. Tech., February: 172-174. Trans. AIME.

20) Russell, D.G., 1966. Extensions of Pressure Buildup Analysis Methods. J. Pet.

Tech., December: 1624-1636. Trans. AIME.

21) McKinley, R.M.,1971. Wellbore Transmissibility from Afterflow-Dominated

Pressure Buildup Data. J. Pet. Tech. July: 863-872. Trans. AIME.

22) Ramey, H.J., Jr., 1970. Short-Time Well Test Data Interpretation in the Presence

of Skin Effect and Wellbore Storage. J. Pet. Tech., January: 97-104. Trans.

AIME.

OILWELL TESTING 238

23) Earlougher, R.C., and Kersch, K.M., 1974. Analysis of Short-Time Transient Test

Data by Type-Curve Matching. J. Pet. Tech., July: 793-800. Trans. AIME.

CHAPTER 8

REAL GAS FLOW: GAS WELL TESTING

8.1 INTRODUCTION

The first part of this chapter describes how the basic differential equation for radial fluid

flow, equ. (5.1), can be approximately linearized for real gas flow. This is achieved

using the real gas pseudo pressure function

pdp

m(p) 2

and subsequently, all equations in the chapter are expressed in terms of m(p) functions

rather than real pressures. The constant terminal rate solution of the radial diffusivity

equation is then presented in dimensionless form, equivalent to the p

functions for

liquid flow, and the solution is applied to the analysis of gas well tests. A similar

approach is used for analysing pressure buildup tests in solution gas drive reservoirs,

below bubble point pressure.

8.2 LINEARIZATION AND SOLUTION OF THE BASIC DIFFERENTIAL EQUATION FOR

THE RADIAL FLOW OF A REAL GAS

By assuming mass conservation, Darcy's law and applying the definition of fluid

compressibility, the basic equation for the radial flow of a single phase fluid in a porous

medium was derived in chapter 5 as

1k p p

rr r t

φρ

æö

∂∂ ∂

ç÷

∂∂ ∂

èø

(5.1)

This equation was linearized for liquid flow by deletion of terms, assuming that

- µ was independent of pressure

∂

was small and therefore

∂

æö

ç÷

∂

èø

was negligible

- c was small and constant so that cp << 1

which resulted in the radial diffusivity equation

1pcp

rr r k t

φµ

∂∂ ∂

æö

ç÷

∂∂ ∂

èø

(5.20)

Because this equation is linear for liquid flow, simple analytical methods could be

applied to describe stabilized inflow (Chapter 6) and the constant terminal rate solution

(Chapter 7). The assumptions made in linearizing equ. (5.1) are inappropriate when

applied to the flow of a real gas. In the first place, gas viscosity is highly pressure

dependent. Secondly, the isothermal compressibility of a real gas is

REAL GAS FLOW: GAS WELL TESTING 240

11 1

ppp

∂Ζ

=− ≈

Ζ∂

(1.31)

which again is highly pressure dependent and automatically violates the above

condition that cp << 1.

These problems, although severe, are not insurmountable. Nevertheless, it was not

until the mid sixties that reliable analytical solutions of equ. (5.1 ) were developed. Two

separate solution methods were published almost simultaneously in 1966; these are

- the Russell, Goodrich et. al., p

formulation

- the Al-Hussainy, Ramey and Crawford, real gas pseudo pressure formulation

Both techniques will be described in this chapter although the latter, for reasons

explained in the text, is preferred. To illustrate the difference in approach the radial

semi-steady state inflow equation, equivalent to equ. (6.12), will be derived in secs. 8.3

and 8.4, using both methods. Having thus established an analogy between liquid and

real gas flow equations, the constant terminal rate solution for gas is stated by

inference and its application described in detail in the remainder of the chapter.

Because of the great disparity between gas rates measured at the surface (Q) and in

the reservoir (q) it has become conventional to express gas flow equations using

surface rates, at standard conditions, with all parameters expressed in field units. This

practice will be adhered to in this chapter, using the following units

Q - Mscf/d µ -cp(= µ

)

(at 60°F and 14.7 psia) Z - dimensionless

t - hours p - psia

k - mD T - °R (460+°F)

h, r - ft

In all equations µ and Z are evaluated at some defined reservoir condition. The basic

derivations of the flow equations in sections 8.3 through 8.8 will still be performed in

Darcy units, with conversion to field units being made upon achieving the desired form

of equation.

8.3 THE RUSSELL, GOODRICH, et. al. SOLUTION TECHNIQUE

The authors approached the problem by making the initial assumption that it was

possible to linearize equ. (5.1) for real gas flow in precisely the same manner as for

liquid flow, described in Chapter 5, sec. 4. Admittedly, this approach should yield

inaccurate results. However, Russell and Goodrich also designed a numerical model of

a single well draining a radial volume element which itself was subdivided into finite

grid blocks as shown in fig. 8.1.