Paper Welltest.pdf

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WELLBORE STORAGE: HOW IT AFFECTS PRESSURE BUILDUP AND PRESSURE DRAWDOWN TESTS W. John Lee, Mississippi State University

ABSTRACT Pressure buildup and drawdown tests, including drill stem tests, are frequently used to estimate formation characteristics such as permeability and wellbore damage. Unfortunately, differences between sandface and surface flow rates during the initial part of these tests cause serious discrepancies, called wellbore storage effects, which cause serious interpretation problems. Difficulties arise for two main reasons: (1) wellbore storage effects in a buildup or drawdown test delay or prevent the appearance of the ideal straight line region on a plot of test data--and the slope of this line, which is related to formation permeability, is the key to proper analysis of the test. (2) “False” straight lines appear in plots of the data distorted by wellbore storage. If the analyst attempts to estimate formation permeability from the slope of a “false” straight line, serious error can result. This paper presents techniques which allow the test analyst to distinguish between “false” straight lines caused by wellbore storage effects, and straight lines which truly reflect formation permeability.

INTRODUCTION Pressure buildup tests, including drill stem tests, have become increasingly popular as formation evaluation techniques in recent years. These tests are used to estimate the extent of damage around a wellbore, the effectiveness of a stimulation treatment of a well,. and the permeability of the formation. Pressure drawdown tests have also been used for these same purposes, but to a lesser extent. Interpretation of each of these types of tests requires that we make a large number of assumptions about a formation. In particular, flow must be essentially radial throughout the test; the formation must be reasonably uniform; and, in particular, reservoir boundaries must not influence the test before sufficient data “ have been obtained to estimate formation permeability accurately. Under these ideal conditions, a pressure buildup test Is modeled adequately by the theory presented to the petroleum industry by Hornerl. Horner suggested that the following equation should describe the buildup of pressure in a well

Pw

=

Pj-

162,6 #

log (~-)

(1)

l%is equation implies that if we plot sandface pressure, ~s, as recorded during a pressure buildup test against the time group, log (t+At)/At, a straight line should result with slope m = 162.6 qpB/kh and intercept pi (see Fig. 1).

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PWs

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ope m = 162.6 @ \

//

,/”’”

‘y I

1

I

100

10 t+At

(—) At

Fig. 1. Plot of buildup test as suqgested by Homer’s theory .We measure the slop% m, of such a curve and thus estimate formation permeability with the equation” k = 162.6 #

(2)

Further analysis 2 shows that we can use the data obtained during a buildup test to characterize the extent of damage or the degree of stimulation by calculating the skin factor, s, with the equation

S = 1.151 [(

P’ ‘S;

pwf) - log (-)

+ 3.23]

(3)

w where p$s is the pressure at any shut-in time At’. The pressure p+s must lie on the straight line whose slope is related to formation permeability by eq. (2). Frequently, At’ is arbitrarily chosen to be 1 hour and p’ws is The most important implication of our discussion to this ~~~~~e~spk~;~ If Homer’s ideal theory is obeyed, all data from the bufldup test should lie on a straight line if we plot the data as suggested by his theory; if part of the data deviates from a straight line, then the ideal theory does not describe this portion of the test. In practice when we run a pressure buildup test and, armed with the ideal theory, plot the data as suggested by Homer we usually observe a non-linear curve over at least part of the time range (Fig. 2). Logical questions are: Why don’t we see an exactly straight line in practice? In what ways is the real reservoir different from the ideal reservoir which was assumed by Homer in his analysis? Complete answers to these questions are quite complicated, but one important consideration is this: Homer’s analysis assumes that the when a well is shut in for a pressure flow at the sandface ceases immediately buildup test. If flow at the sandface does not cease immediately, then we

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P*

B f

Pw

,oo~ t+At (-) At Fig. 2. Typical “actual” buildup test plot. have no right to expect the ideal theory to apply. In practice, wells are ordinarily shut in at the surface and flow continues into the wellbore for some time. The additional fluid entering the wellbore compresses the liquid and gas that were present in the wellbore at shut-in. This continued production in a buildup test is called “afterflow”. Afterflow can cause serious interpretation problems, because it causes all sorts of brief linear trends to appear in the buildup test plot. The serious question thus arises in practice: Which one of these apparent straight lines (if any) reflects formation permeability and which of these lines has no relation at all to reservoir properties; i.e., which of these straight lines is caused by afterflow? Of course , phenomena other than afterflow also complicate the shape of pressure buildup tests. For example, if a buildup test is run for a long period of time, boundary effects eventually appear in the test; these boundary effects also cause deviations from the “ideal’’straight line whose slope is related to formation permeability and thus further confuse the analysis of the pressure buildup test. Experience and theory show that region A in Fig. 2 is dominated by afterflow; region B is modeled by ideal theory; and boundary effects dominate region C. When the buildup curve is extrapolated to (t+At)/At = 1, we call the pressure “p*’);this pressure is related to, but usually somewhat higher than, average pressure in the drainage area of the tested well. Drawdown tests also present difficulties similar to those encountered in pressure buildup tests. The equation describing drawdown tests (also based on Homer’s analysis) is

P~f = Pi - 162.6 #

[log ~4Wc~~

3.23+

0.87s]

(4)

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This equation suggests the flowing pressure at the sandface, pwf, be plotted against the logarithm of flowing time, t; a straight line should result (in the ideal case) with slope, m = 162.6 qpB/kh (Fig. 3). An assumption in this equation is that the well started flowing at a constant rate at zero time and

0.1

10

1

1

t-4 Fig. 3. Plot of drawdown test as suggested by theory. continued to flow at that rate. This constant sandface rate is difficult to achieve in practice. We can (with some exercise of control) produce at an essentially constant surface rate in a pressure drawdown test, but the sandface rate may not become constant for some time. The amount of fluid withdrawn at the surface is not equal to the amount of fluid entering the wellbore from the formation for some time after the drawdown test is begun. This phenomenon is known as “wellbore unloading”. Once again, we should expect deviation from ideal behavior at earliest times in drawdown tests , and we should expect boundary effects to cause deviations from ideal behavior at later times in the tests. Indeed, we see these deviations in practice (l?ig.4). Wellbore unloading dominates region A on the curve;

! Pwf

c

0.1

I 1

i 10

1

t~

Fig. 4. Typical “actual” drawdown test plot.

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“ideal” theory describes region B; and boundary effects dominate in region The observed perverse behavior of buildup and drawdown tests leaves us with serious interpretation problems; we will attempt to offer some guidlines for their solution in this paper.

c.

All these considerations lead us to the objective of this paper-to answer the following questions: How can we find the line whose slope reflects formation permeability? How can we eliminate from further consideration “false” straight lines dominated by wellbore storage effects? Unless we can find the correct line among the several straight lines that may appear on a buildup or drawdown test plot , we cannot estimate formation permeability accurately and we will be unable to estimate the extent of wellbore damage. (Damage is usually characterized by the skin factor, s; eq. (3) shows the correct straight line must be identified before s can be calculated.) A detailed discussion of the time at which boundary effects can be expected to appear in a pressure buildup or pressure drawdown test is beyond the scope of this paper; useful ideas on boundary effects are available in the literature3Sk. In this paper we will concentrate on techniques which will help the analyst find the time at which afterflow or wellbore unloading has ceased distorting pressure buildup or pressure drawdown tests.

TECHNIQUES FOR DETECTING WELLBORE-EFFECT DOMINATED DATA We have found several methods valuable for determining whether wellbore effects are distorting an apparent straight line on a pressure buildup or pressure drawdown test plot. No one of these techniques is foolproof or universally applicable, so we will outline each of the methods which we have used in practice and which we believe other buildup and drawdown tests analysts will also find useful. The first (and most rigorous) method is to use analytical solutions developed by Ramey 5 which predict, from theory, when wellbore storage effects should cease. For a gas well, Ramey found that afterflow in a buildup test or wellbore unloading in a drawdown test should both end at the time,-taft, given by the equation

t

(5)

aft =

For a liquid-filled wellbore (such as in a drill stem test or in an injectionfalloff test) , wellbore effects should end at time taft given by

t

aft

=

2X105

Pv +

Clw

(6)

For a partially liquid-filled wellbore (i.e., one in which thereis a rising liquid level), afterflow should end at a time taft given by

(7)

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These equations are based on analytical work begun by van Everdingen and Hurstb. The equations are exact only for wells that are neither damaged nor stimulated; i.e. , wells in which the skin factor, s, is zero. The the permeability-thickness product, kh, equations can be applied only after is known for a tested well. This means that a likely-looking straight line in the buildup or drawdown test plot must be assumed to be the straight line whose slope is related to the permeability-thickness product of the formation surrounding the well. A check can then be made to see when wellbore effects should have ceased distorting the buildup or drawdown test data (assuming that the well is behaving like swell with zero skin factor). The assumed “correct” straight line may prove to be in the afterflow-dominated region; if so, another region of the test data must be examined. We have found a slight modification of Ramey’s equations to be useful in practice. Note that the group, kh/v, which appears in equations (5), (6), and (7) can be related to the productivity index for swell in the following way: For stabilized flow in a bounded region, a productivity index in reservo{r barrels per day per psi pressure drawdown is7 keh JR=

‘B P - Pwf = 141.2p[ln(re/rw)-0.5]

(8)

keh —= u

141.2[ln(re/rw)-0.5]JR = 1000JR

(9)

from which

Eq. (9) was developed assuming ln(re/rw) = 7.6; deviations from by more than 20 percent are rare. The equation can be used for oil well or gas well with satisfactory accuracy. Note that for and gas wells JR must be expressed in reservoir barrels per day unusual units for a gas well.

this value either an both oil per psi--

The permeability, ke, in eq. (8) is not the same as the permeability, k, ineqs. (l)-(7). ke is greatly influenced by damage or stimulation near the well; k is formation permeability away from the well. Thus, we alter eqs. (5)-(7) when we replace k by kc--but this alteration proves beneficial. In terms of productivity index, JR, the equation for a gas well becomes

t

aft =

(lo)

2X102 & ‘R

For a liquid-filled wellbore,

Vwclw taft = 2X102

(11)

‘R

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For a partially liquid-filled wellbore,

t

A = 5.25x103 ~ aft ‘RPIw

(12)

Let us point out some characteristics of these modifications of Ramey’s equations. First, the modifications, although non-rigorous, agree qualitatively with some later solutions published by Ramey and co-workers8’9 for wells with non-zero skin factors. In this later work, the authors’ results show that afterflow lasts somewhat longer for a damaged well than for a well with no damage. The modified equations also predict longer duration for wellbore storage effects in damaged wells than in undamaged wells, since the productivity index for a damaged well will be smaller than for a well with the same formation permeability but with no damage. For highly stimulated wells Ramey 5 has observed that wellbore storage is shorter-lived. Since productivity index is larger in a stimulated well than in an unstimulated well in the same formation,the modified equations predict shorter duration for wellbore storage effects. Thus , the equations agree qualitatively with Ramey’s observations. A second important characteristic of the modified equations is that the buildup or drawdown test is ever run to estimate they can be used before the duration of afterflow or wellbore unloading. To make this estimate, of course, we must know the productivity index for the well--but this information is frequently available. Third, the modified equations can be used after a test is run but before we have committed ourselves on a choice of the correct straight line. An example later in this paper illustrates use of the modified equations in analyzing a well test. Ramey’s equations or their modifications are not foolproof. They involve assumptions that may not always be satisfied in practice. Further, some of the data required in these equations may not be readily available; e.g., compressibility of the liquid in a liquid-filled wellbore can be quite difficult to estimate. Thus , it is helpful to have further techniques to help distinguish between straight lines caused by wellbore effects and straight lines which reflect formation properties. We have used at least three other techniques in practice. They have been successful enough that we can recommend them to the novice test analyst. First, we note that afterflow in a buildup test imparts a characteristic “S” shape to the buildup curve (Fig. 2, region A); in a drawdown test, a characteristic inverted “S” shape occurs in the test (Fig. 4, region A). This curve-shape recognition technique has an obvious flaw; part of the “S” may be missing in any given test. Second, we have frequently found it useful, in analyzing a buildup tests to extrapolate straight lines to (t+At)/At = 1. This extrapolated pressure will usually be close to a good estimate of current average reservoir pressure if the straight line reflecting formation properties has been chosen. The extrapolated pressure will often lie well above any reasonable estimate of current reservoir pressure if we have chosen a straight line

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from the afterflow-dominated portion of the buildup curve. A third technique is to calculate flow efficiency for a well. Flow efficiency can be defined as the ratio of the production rate of a well to the production rate that the well would have if it were neither damaged nor stimulated. Flow efficiency, E, can be calculated from knowledge of skin factor, s, with the equation F - Pwf - 0.87sm E= F - Pwf

=

P* - pwf -

0.87sm (13)

P* - Pwf

Eq. (13) shows that flow efficiency should be greater than unity for a stimulated well and less than unity for a damaged well. We frequently find that a “false” straight line from the afterflow-dominated region will lead to an apparent flow efficiency of 2 or greater for damaged wells. Such an inconsistency is a reliable indicator that the wrong straight line has been chosen. A new curve-fitting technique by Ramey1° may prove to be the most powerful method of all for dealing with wellbore effects. In addition to allowing us to be reasonably certain about when wellbore effects have disappeared, the technique also helps us overcome practical difficulties in estimating quantities such as the product of the fluid volume below the packer and the compressibility of the packed-off mud (Vwclw)--a fo~idable problem. The method deserves immediate evaluation by pressure test analysts.

APPLICATION TO BUILDUP TEST ANALYSIS We now consider an example buildup test to show how some of the ideas that we have discussed above can be applied. This buildup test is a slightly modified version of an example test presented by MatthewsIl. that our recommended We do not present this example in a attempt to prow techniques for dealing with wellbore effects are satisfactory; in fact, we had to make several assumptions about properties of this well just to make the example complete. The purpose of the example is to show how to appZy some of the techniques we recommend. Properties of the formation, reservoir fluid, and wellbore are given in Table 1; the buildup test is plotted in Figure 5. (A much larger graph of this buildup test, with grid, is given on p. 134 of Ref. 2.) There are three possible straight lines in Fig. 5; we wish to find which of these straight lines is most likely to provide information about formation permeability. We first estimate afterflow duration from eq. (12): 5.25x103Awb t

aft =

— ‘Rplw

(5.25X10 S)(2.18X10-2) (3.21x10-1)(5x101)

= ~ , ~r “

This shut-in time, At, of 7.1 hr occurs at a value of (t+At)/At = (13630+7.1)/7.1 = 1920; this is quite near the point at which the buildup curve changes slope abruptly. Pressures at values of (t+At)/At greater

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I

/

4600. .

If ------—4200. -

PWs ,psi 3800. . ~z100,000

10 ;000 1000 t+llt (—) At

1(

Fig. 5. Pressure buildup test with afterf1ow.

Table 1 Data for Example Buildup Test Analysis 4

= 0.039

c

= 1.7x10-5psi-l

P

= 0.8cp

rw

= 0.198 ft

= 0.0218 ft2 (2-inch I.D. A wb = 250 STB/D ~ B.

tubing)

= 1.136 RB/STB

Pwf = 3534 psi h

= 69 ft

JR

= 0.321 RB/D/psi

Plw = 50 lb/ft3 Reservoir pressure in recent survey = 4700 psi Producing time, t = 13630 hr Wellbore partially liquid filled (produces mostly oil a some gas) Well believed to be damaged

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than 1920 (i.e., shut-in time, At, less than 7.1 hr) are probably distorted by afterflow; pressures at values of (t+At)/At less than 1920 are probably free from the effects of afterflow. Thus, according to this test, the final straight line is the most likely candidate for estimating formation properties. A significant assumption remains: If we estimate formation permeability from the slope of the final straight line, we must assume that boundary effects have not yet appeared in this buildup test. One final note on this technique: One should by no means expect such remarkable agreement between calculated values of taft and the beginning the “true” straight line in all cases. In Fig. 5 we also note the S-shape at earliest shut-in times (largest values of (t+At)/At); this shape is characteristic of a test influenced by afterflow. Note, however, that if the very earliest data had been missing we would have had the case in which these were simply two straight lines, and the S-shape test would fail. If we extrapolate each of the straight lines to (t+At)/At = 1, and compare the extrapolated pressure, p*, with a recent estimate of average reservoir pressure (4700 psig), we find that the initial line extrapolates to p* = 4284 psi , which is low but reasonable; the second line extrapolates to p* = 11,480 psi , which is quite unreasonable; and the third line extrapolates to p* = 4585 psi, which is the most reasonable value of all. With this extrapolation technique, we can eliminate the second line, but we need other techniques to choose between the initial and final lines. To apply the flow efficiency check , we must first calculate apparent permeability, k, and skin factor, s. For the initial straight line, we note that

m = 157 psi/cycle

k = 162.6$=

(1.626x102 )(2.5x102)(0.8)(1.136) = s ~1 md . (1.57x102)(6.9x101)

From the buildup test plot, we read p~s = 3620 psi at At’ = 1 hr, or (t+At)/At = 13631. Then (P;s - Pwf) s = 1.151

[

m-

= 1.151 (+-109 [

~) 10g ‘$ucr~ (

+3.23

1

(3.41)(1) (3.9x10-2)(0 .8)(1.7x10-5)(3 .92x10-2)

= -5.10

3.23 )+ 1

Then p*-pwf-0.87ms E=

P*-Pwf

= 4284-3534-(0.87)(157)(-5.10) = , ~ . 4284-3534

Similar calculations show that apparent values of E are 2.2 for the second straight line and 0.631 for the final straight line. Since E must be less

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than one for a damaged well, and since the tested well is believed to be damaged, only the third straight line leads to physically reasonable results. Calculations described above are summarized in Table 2.

Table 2 Sunmary of Results of Example Buildup Test Analysis Apparent Value from Straight Line Parameter

Initial Line

2160

157

m, psi/cycle

3.41

k, md s

-5.10

E

1.9

Final Line

70

0.25 -4.9 2.2

4284

p*, psi

Second Line

7.65 6.37 0.631 4585

11,480

This particular test would offer no great challenge to the professional buildup test analyst. He would, by inspection; probablyrecognize that the earliest data are dominated by afterflow. However, certain real life tests are by no means obvious by inspection; in our experience we have seen several tests misinterpreted by competent reservoir engineers. Therefore, we feel that our suggestions for determining whether afterflow dominates a particular portion of a buildup or drawdown test are of considerably more than academic interest--particularly to the novice pressure transient test analyst. An example of a commonly misinterpreted test (Fig. 6) will help make

!

PWs

1000

100

10

(*) Fig. 6.

Buildup test with afterflow dominating early times.

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this point. In Fig. 6, regions Al and A2 are both dominated by afterflow; region B reflects formation permeability. Many analysts attempt to interpret this curve in the following way: They assume the slope of the curve in region Al determines formation permeability; region A2 is assumed to reflect some boundary, such as a sealing fault , near the well (many immediately estimate the distance to the fault, particularly if the slope of the straight line in region A2 is close to double that in region Al--this is the classic behavior of a buildup test in a well near a fault). Region B, if present, is assumed to be caused by some other reservoir hetereogeneity or, perhaps, the beginning of the region in which pressure levels out to its static value. Such a misinterpretation could be avoided if eqs. (10), (11), or (12) were applied.

CONCLUSIONS False straight lines during the period in which wellbore storage effects dominate the shape of a buildup or drawdown test can offer serious obstacles to the beginning or even the experienced well test analyst. Aids in finding the times at which wellbore effects have ceased distorting buildup and drawdown tests include (1) Ramey’s analytical equations which assume no damage or stimulation in a well; (2) modified versions of Ramey’s equations which include damage and stimulation effects in a non-rigorous but qualitatively correct way; (3) characteristic “S” shape or inverted “S” shape that wellbore-storage-effect dominated tests often exhibit; (4) extra– polation of buildup test plots to infinite shut-in time and compassion of the pressure p* at infinite shut-in time to an estimate of current static reservoir pressure; (5) calculation of flow efficiency, E, and assessment of the reasonableness of the result.

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NOMENCLATURE

‘Wb = Area area

of wellbore containing fluid (area of tubing if packed off; of tubing plus annulus if not packed off), ft2

B

= Formation volume factor, bbl/STB

c

. Fluid compressibility, psi-l

Cw

=

CIW

=

E

= Flow efficiency, dimensionless

h=

Net formation thickness, ft

Average compressibility of gas in wellbore, psi-l

‘R k=

=

Average compressibility of liquid in wellbore, psi-l

Productivity index based on reservoir production rate, bbl/psi/day Formation permeability, md

ke

= Effective or apparent reservoir permeability (including damage or stimulation near well), md

m=

Absolute value of slope of portion of buildup or flow test curve reflecting formation permeability, psi/log cycle 10

Pi

= Initial reservoir pressure, psi

Pwf

= Bottom-hole flowing pressure, psi

PWs

. Bottom-hole pressure during pressure buildup test, psi

P’ Ws

= Pressure on ideal straight line of pressure buildup test, psi

6=

Average drainage-area pressure for tested well, psi

p*

= Pressure obtained when ideal straight line is extrapolated to (t+At)/At = 1, psi

q

= Production rate of well, STB/D

r w

= Wellbore radius, ft

s

= Skin factor, dimensionless

t

= Time of flowing, hours

= Duration of afterflow, hours t aft At

= Closed–in time, hours

At’

= Closed-in time corresponding to pressure p~s, hours

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Vw

= Volume of wellbore in communication with formation, bbl

P

= Formation fluid viscosity, cp

plW +

= Average density of liquid in wellbore, lb/ft3 = Porosity, fraction

REFERENCES 1.

Homer, D. R.: “Pressure Build-Up in Wells” Proc. Cong., E. J. Brill, Leiden (1951) ~, 503.

2.

Buildup and FZou Tests Matthews, C. S. and Russell, D. G.: Pressure in Wells, Monograph Series, Society of Petroleum Engineers, Dallas, Texas (1967) ~.

3.

Odeh, A. S. and Nabor, G. W.: “The Effect of Production History on Determination of Formation Characteristics from Flow Tests”, J. Pet. Ted-z. (Ott.,1966) 1343-1350.

4.

Miller, C. C., Dyes, A. B. and Hutchinson, C. A., Jr.: “The Estimation of Permeability and Reservoir Pressure from Bottom Hole Pressure Buildup Characteristics”, l’Pans., AIME (1950) 189, 91-104.

5.

Ramey, H. J., Jr.: “Non-Darcy Flow and Wellbore Storage Effects in Pressure Build-up and Drawdown of Gas Wells,r’ J. Pet. Tech..(Feb., 1965) 223-233.

6.

van Everdingen, A. F. and Hurst, W.: “The Application of the Laplace Transformation to Flow Problems in Reservoirs”, Trans. , AIME (1949) 186, 305-324.

7.

Reservoir Engineering, Craft, B. C. and Hawkins, M. F.: Applied Prentice-Hall, Inc., Englewood Cliffs, N.J., 1959.

8.

Wattenbarger, R. A. and Ramey, H. J., Jr.: “An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: II. Finite Difference Treatment”, SOG. pet, Eng. J. (Sept., 1970) 291-297.

9.

Agarwal, R. G., A1-Hussainy, R. and Ramey, H. J., Jr.: “An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment”, See. pet. Eng. J. (Sept.,1970) 279-290.

10.

Ramey, H. J., Jr.: “Short Time Well–Test Data Interpretation in Presence of Skin Effect and Wellbore Storage”, J. Pet. Tech..(Jan.,

1970) 11.

Third World Pet.

97-104.

Matthews, C. S.: “Analysis of Pressure Build-Up and Flow Test Data”, J, pet. l’~~h. (Sept. 1961) 862-870.

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AUTHOR

John Lee

John Lee graduated in 1962 from The Georgia Institute of Technology with a Ph.D. degree in chemical engineering. He then joined Humble Oil and Refining Company’s Production Research Division, which later was From 1962 to 1967 he was merged into Esso Production Research Company. engaged in research and technical service work on well testing, fluid flow in porous media, fracturing, and reservoir engineering. In 1967-68 he was a reservoir engineer in Humble’s Kingsville Production District. In September, 1968, he joined the Mississippi State University faculty He is where he is now an Associate Professor of Petroleum Engineering. buildup and drawdown analysis in the currently a lecturer on pressure Society of Petroleum Engineers Traveling Lecture Series.

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