Analysis of Simultaneously Measured

Pressure and Sandface Flow Rate in

@Transient Well TestingF. Ku?uk, SPE, Schl.mbc%er Well Services

L. Ayestaran, SPE, Schl.mberger Well Services

SUrmnary

New well test interpretation methods are presented that

eliinate wellbore storage (aftertlow) effects. These new

methods use simultaneously measured sandfuce flow r“te

and weffbore pr~ure Ma. It is shown that formation

behavior without storage effects (unit response or in-

fluence function) can be obtained from deconvolution of

Jkmdface flow rate and wellbore pressure data. The

storage-free formation behavior canbc analyzed to iden-

ti& the system (reservoir flow pattern) that is under testing

and to 6stimate its parameters. Convolution (radial multi-

rite) methods for reservoir parameter estimation and a

few synthetic examples for deconvolution and convolu-

tion also are presented.

Introduction

Welj testing with measured sandface flow rate can be

traced to tbe beginning of reservoir engineering. The rate

must + measured over time to calcuk and{or approx-

imate constant rate to obtain even a single reservoir

parameter from pressure measurements. Tfds approiimatc

.comstant rate has been sufficient for estimatingptmneabili-

ty, skin, and initial formation pressure during the ra&d

infinite-acting period. During tbk period, the well should

produce at? conkant rate at the sandface or at a zero rate

if a buildup test is condu~ed. Because of compressible

fluid in the production string (weflbore storage effects),

it (alces a long @e to reach the radial infinite-acting

period. The effect of outer boundaries also may stat

before the end of the wellbore storage effects.

In gener~, the storage ~pacity of the wellbore,

wellbore geometry, rim-wellbore complexities, and ex-

ternal boundaries affeqt transient bebavior of a well. Dur-

ing the. aualysis of pressure-time data, each of these

phenomena and its dtition must be recognized for the

aPPEcatiOn Of SeMilog and typecurve techniques to deter-mme formation flow capacity (k/z), @mage skin, and

average formation pressure. The influence of these

phenomena on transient behavior of a wg?fl progresses over

time. For the sake of convenience, the test time can be

divided into ~ periods according to which phenomenon

is affecting the pxessuze. These periods are defined as

follows ,

EarIy-T@ Period. The combmed effects of wellbme

storage, +nmge skin, and pseudoskin (which include par-

mwmt f9.33 sww .f P.t,.h+ EWW.IS

REBRU.4RY 1985

5R!5 lal’71

tkd. penetration, perforation, aciduing, fractures, nOn-

Darcy flow, and permeability reduction caused by gas

saturation around the wellbore) dominate pressure

behavior. The stratification and dud porosity also may

affect wellbore pressure during this period.

Middfe-Time Period. During thii period, radial flow is

es@blished. Conventionally, semilog techniques are used

to determine formation kh and initial pressure and skin.

Late-Time Period. During tbk period, outer boundary

effects start to distort the semilog straight line. For ex-

ampIe, the gas cap shows a curve-flattening effect on log-

log and Homer plots.

Sometimes the separation of these periods from each

otier is impossible; particularly, the effects of bottom-

water influx andtor gas cap may start during the. middle-

time period. Thus, the semilog approach sometimes can-

not be app~,ed at all.

Furtbeirnore, the drawdown or buildup tests as con-

ducted today tend to homogenize tie reservoir behavior.

In other words, ” most of the reservoirs behave

homogeneously during the storage-free radial intinite-

acting period because most of the heterogeneous behavior

takes place during the early-time period.

The &pe-cume approaches have been introduced to

overcome some of these problems. The theories, applica-

tions, and elaborations of the type-curve methods, as welf

as WY” references, can be found in Ref. 1. In 1979,

Gringarten ez al. 2 introduced new type-curves that use

different pammetetiation than the earlier ones, namely

Rhmey, 3 Agar.val et al., 4 McKinley, 5 and Eafkmgher

and Kemch6 types. All the type curves presented by these

au~ors, and many others, were developed under tie

assumption that ~e fluid compressibility (density) in the

tubing and anmdus remains constant during @e test period.

During the early time, pamicularly for buildup tests, shut-

in pressure incrkases ve~ rapidly; thus, the compresiibJl-

ty is usually higher than the compressibility of the fluid

in the reservoir for producing wells. Since the piessure

in the wellbore is a fgnction of the depti, the com-

pr&ibfity of the fluid at the wellhead can be 10 or even

100 tink geater than the compressibility of the fluid at

the @ttOm. T@, the assumption that the ‘wellbore storage

coefficient is constant during the drawdown, and pal-

ticukrly during buildup, may not be correct. A variable

weflbore storage coefficient alone makes the application

323

of type-curve methods afmost impossible. The combina-

tion of variable or even constant wellbore storage with

wellbore “geometry timber complicates the type-curve

~tching process. Moreover, wellbore pre.wure &ta afone

tiY not indicate changing wellbore storage.

Some of the problems inherent with the use of type-

cume methods can be efiiated by the sbnuftaneous use

of measured sandface flow rate and pressure data.

‘fhe purpose of this work is to study the use of the

measured sandface flow rate in a broad sense with regard

@ transient weU”testing. Furthermore, we explore the use

of convolution and deconvolution $1 the interpretation of

pressure beliavior of a well with afterflow (buildup case)

or wellbore storage (drawdown case).

Background

The use of tie sandface flow rate in transient testing is

not riew. To oti knowledge, van Everdingens and

Hurst7 were the first to estimate and use the sandface

flow rate to calc~ate the wellbore pressure. To do this,

they approximate the stidface flow rate by the forr+nda

qsf=(’-’-”o$. . . . . . . . . . . . . . . . . . . . . ...(1)

where 13 is a positive constant. These authors stated that

the constant, ~, cm be determined from well aad res?r-

voir parameters. Using tie above formula and the con-

volution integd, van Everdingen7 and Hursts presented

an expression for the weUbore pressure with a variable

wellbore storage effect. Gladfefter et al. g presented’ a

method to determine the formation kh from pressure and

aftertlow ikita. The afterflow &ta were obtained by

measuring the rise in the liquid level in the ivellbore.

Ramey 10 applied the Gladfelter approach to gas’ well

buiklup tests.

A considerable amount of work afso has been done on

multirate (vimiible) rate tests dining the last 30 yeys.

However, these are basically sequential constant-rate

dratidowns; only transient pressure is measured and rate

is assuined. constant during each drawdown test. The

techniques related to thk type of multirate tests afso can

be found in Ref. 1. AU the ivork mentioned so far de#s

with the direct problem. In other words, the constant-rote

solution (the influence or the unit response function) is

convolved (superimposed) with the the-dependent inner

@~dary condition to obtain solutiom to the diffusivity

equation. This process is calfed “convolution.”

Hutchison and .%kora, 11 Katz et af., 12 and Coates er

af. 13 presented methods for determ.inin g the influence

function ~ectly from field data for aquifers. The proc-

ew of determining g fie influence function is cafled’ ‘decon-

volition. ”

Jargon and van Poolen 14 were perhaps the first to use

the deconvolution of variable rate and pressure data to

compute the constant-rate pressure behavior (the inffnence

fiction) of the formation’in well testing. Bostic et al. 15

used a dwonvolution tecbni@e to obtain a constant-rate

solution from a variable rate history with a known pressare

history. They also extended ~e deconvolution technique

to combine production and buldup data as a single test.

Pascal 1s also used deconvolution techniques to obtain a

constant-rote solution from variable rate (measured at the

surface) and pressure measurements of a drawdown test.

More recently, Metier et al. 17 have used sandface

flow measurements with pressure data fo? buildup test

amafysis. This has been the first successful attempt to use

direct measurements of sandface flow rate data in weU

testing. They showed that the Homer method can be

modilied to ‘&ach a semifog stmight line earlier km type-

curves or the 1 I%-cycle @e irdcates.

Theoretical Developments

During the last 4 or 5 decades, many solutions have been

developed for transierit fluid, flow through porous media.

me superposition theorem (Dahamel’s tieorem) has been

used @derive solutions for time-dependent boundary con-

ditions from time-independent boundary conditions. For

example, the multiple-rate testing is a special application

of the superposition theorem.

fn their classic paper on unsteady-state flow problems,

van Everdingen ~d Hurst’s presented the dimensionless

wellbore pressure for a continuously varying flow rate as

PwD(b)=mJ(f9Pso (b)

+ jtDqD‘(T)p@ (f~ - ~)dr. . .

0“

(2a)

An alternative form to E.q. 2a can be obtained by an in-

tegration by parts as

PwD(tD) ‘qD(tD)psD@)

+~fZD(7h’D&r~)dC . . .. . . . . . ..(zb)

0

where

pwD = *1%-PM$l!

0.0002637kt~D =

qipctr$

psD = PD+L$,

PD(tD) = the dimensiodws sandface pressure for

the constant-rate case without wellbore

storage and sf& effects,

S = steady-state skin factor,

qD (tD) = !hf(tD)fq, ,

qi(b) = dqD(tD)/dtD ,

q, = reference flow rate—if the stabilized

constant rate is available, then q,

should be replaced by q~,

q~t) = variable sandface flow rate (flowmeter

readings) , and

qD(2Li) = dheUSiOdeSS sandface rate.

324 JOURNAL OF PETROLEUM TECHNOLOGY

1.0.

shut-m Time, AI, hrs

Fig. I—Dimensionless sandface flow rates for constant

tiel Ibore storage and exponential decfi ne cases.

Although traditionslfy the skin effect is considered a

dimensionless quantity different from the dimensionless

formation pressure, the skin effect will be treated here

as part of the imer boundmy condition for the solution

of a unit rate production case. This boundsry condition

is known as the homogeneous boundary condition of the

thiid kind..

It shoufd be emphasized that Eqs. 2a and 2b csm be ap-

plied for many reservoir engineering problems. The

linesrity of the diffusivity equation allows us to use Eqs.

2a and 2b for fractored, layered, anisotropic, snd

heterogeneous systems as long as the fluid in the reser-

voir is single phase. F@. 2a and 2b cm be applied to both

dmwdown and buifdup tests if the initial conditions sre

known. For a reservoir with m idkd constant snd uni-

form pressure distribution [pD(0)=O], Eqs. 2a and 2b

can be expressed as

‘D

PWD(fD)=f q~(7)p@(tD–T)dT . . . . . . . . . . . .(3a)

0

=SgD(tD)+ fDqD(T)p,’D(tD –~)dr. (3b)

0

Furthermore, Eq. 3a sfso can be expressed as

PWD(tD)= ~’DqA(iD–T)P@(r)dT. . . .’ . . . . . . . . (3c)

.0

fn Eqs: 3a tid 3c, it is assumed that q~(rD) exists. If

qD(2D) is constant, then Eq. 3b must be used. ls@. 3a

snd 3C we known as a Volterra integral equation of the

first kind aod the convolution type.

Akhough it is assumed that PO is a constant-rate solu-

tion without stomge effect, mathematically and physical-

ly it can be a solution of a constant-storage case. In

pmctice, p$D afways will be affecti by the welfbore fluid

that occupies the volume below the flowmeter u,!dess the

ssndface rate is measured through perforations. However,

the volume below the flowmeter will lx sms.lf for most

wells, since the flowmeter and the pressure gauge usual-

ly csn be placed just above perforations. Therefore,

throughout this paper, we will assume that p,D is not af-

fected by the fluid volume below the flowmeter.

FEBRUARY 1985 ““

TABLE l—HOMOGENEOUS RESERVOIR ROCK

AND FLUID DATA

8, bbl/STS0,, psi-’

ccl

h. ft

p, Cp

.6

1.0,..5

1,000

100

40

2,666,94

3,000

0.35

10,000

0.76 x10-4

0.8

0.2

The purpose of the test welf interpretation, as stated by

Gringarten et al., z is to identify the system and deter-

mine its goverrdng psmrneters from measured dsts in the

wellbore and at the wellhead. Thk problem is known as

tie inverse problem. Tbe solution of the inverse problem

usuafly is not unique. As Gringarten et al. 2 pointed out,

if the number snd tie range of measurements increase,

the nonuniqueness of the invers.k problem will be reduced.

Thus, combining sandface flow rate with pressure

measurement will enhance the conventional (including

type-curve) weU test interpretation methods.

AS m inverse problem, the ssndface pressure, psD(tD),

hss to be determined by the deconvolution of the integral

in Eqs. 3a, 3b, and 3c. As stated previously, p,D(tD) is

the solution for the constant-flow-rate (sandface) case.

Taking the Lsplace transform of Eq. 3a and solving for

~@(s)* yields

~~D ($)= ~:D ‘s)=, . . . . . ..8. . . . . . . . . . . . . . . . . . (4)

where s is the Laplace transform variable.

The Laplace Esnsform of P,D in Eqs. 3b and 3C will

be the same m Eq. 4, keeping in mind that p,D(o) =S.

Thu3, we hzve only one operational form of the convolu-

tion intograf given by Eqs. 3a, 3b and 3c. The superposi-

tion thmrem io this case is notliig more thsn the

convolution of P,D (tD) aod qD (tD). The Laplace tmns-

form of the convolution intepal allows us to express the

convolution integral in msny different forms. Further-

more, the kemef solution csn be a solution of constant-

rate or comtsnt-pre-wure case for the convolution integral.

If the wellbore storage is constant, the dmensiordes:

sandface flow rste can be expressed as4,1s

dp ~,D (tD)q~(tD)=l–c~—, (5). . . . . . . . . . .. . . . . .

dtD

where

qD(tD) ‘q,f(t)fqr.

As a special application of Eq. 4, the dimensionless

wellbore pressure solution for the constant-storage case

cao be written directly from Eqs. 4 and 5 as

~sD ($)

~wD (s)=

1+ CDS2F,D(S)’ ““””’”’””””””’”””””

(6)

Vhmughout lhls paper, the runclian ?(.9 w be Mod the Wlace tram form.! tha

f.ncsion F(t).

325

/’-...O.W9

,,,.”.., !.!

k&,~.. ,~.z , ~.,10“

shut-to n.., M, hr.

Fig. 2—Shut-in pressures for constant wellbore storage and

exponential decline cases.

where ~$D (s) is the dimemionIess sandface pressure for

the constant-rate case witbout storage effect but including

skin.

Van Everdingen and Hurst 18 presented an equatiOn

simifa to Eq. 6, and Aganval et al. 4 presented the same

equation as an integrodifferentid form for radial systems.

Cmco-Ley and Mmaniego 19 {for fractured reservoirs),

and Kufmk and Kirwan20 (for partially penetrated wells)

presented the same expression for the dimensionless

weflbore pressure as in Eq. 6.

on the other hand, the dimensionless sandface flOW raE

can be obtained directly from Eqs. 5 and 6 in terms of

the dimensionless formation pressure as

1~~(s)= . . . . . . . . . . . . . (7)

St 1 + CDS2ESD (s)1

Fig. 1 presents vafues of qD wdculated from Eqs. 1 and

7 as a function of real time for a buildup test using reserv-

oir and fluid properties given in Table 1. As can be seen

tlom this figure, for the exponential&clime case, the samd-

face tkw rate declines faster than the constant-wellbore-

storage cwe. IZamey and Agarv.m121 also pcesentd values

of qD(tD) as a function of tD for various skin and storage

constants:

Another important application of the convolution in-

tegral given inF.qs.’3a and 3C was resented by van Ever-

dingen, 7 Hurst, g and Ramey*J for calculating the

wellbore pressure by using Ea.. 1 and the line source

solution.

For a finite wellbore radius, tie dimensionless wellbore

pressure solution afso can be written directly from Eqs.

1 and 4 for the exponential sandface rate decline case as

~wD(.,=y —. ,., ,.,. .,, ,,, ,. .,, . . . . . . . . (8)

‘fhe exponential constant, i3, is giVeII M V~ Eve@gen7

as

(3=m#yic,rw210.000264 k,

where u can be determined from welfbore pressure data,

such as the wellbore storage constant. Note that (3 is a

dimensionless constant like CD.

326

As Ramey 10 noted, i3 cannot be estimated as re@ily

as the weflbore storage constant, CD. However, in prin-

ciple, Eq. 9 has a much more ~ediate cO~~tiOn wi~

the real systems. In fact, the pressure increwes more

rapidly during the early-time buildup tests; then it slows

down during the transition period and boilds up vay sfow-

ly during the semilog period. Because of the rapid change

of pressure, the wellbore storage wilf decrease condnuoLls-

Iy except in the case of phase redistribution.

In many cases, it is difficult to recognize changing

wellbore storage effdcts because it is a gradual and con-

tinuous change. Furthermore, tie fifi~ clOsing time Of

the welffread valve also will affect pressure at tie same

time.

Most of the work thus Em in early-time analysis has been

directed toward the construction of type curves from the

solutions of Eq. 7 for a constant wellbore storage for dif-

ferent wellbore geometries, such as fractured wells, par-

tial penetration, etc. Type curves for Eq. 8 for different

vafues of O and skin also can be developed and used for

graduafly decreasing wellbore storage cases to determine

skin snd W?.

Fig. 2 presents a semilog plot of p ~, (At) vs. At,

calctiated from Eqs. 7 and 8 by using reservoir and fluid

parameters given in Table 1. A8 shown by thk figure,

tle exponential decline case approaches a semilog straight

line earlier than the constant-wellbore-storage case.

The most important point to be made frO.m tbe abO~e

discussion and buildup data prc%ented in Fig. 2 is that the

principle limitation of the type-curve analysis stems from

the lack of information about the sandface flow rate

behavior. Thus, the type-mu-w analysis usuafly is used

for quabtive answers and supported by the semilog

analysis. For quantitative analyses of the early-time data,

it is necessary to measure the sandface flow rate. Fur-

thermore, the use of measured sandface flow rate also can

improve the semilog anzfysis. In the following s&tiOns,

the convolution and deconvolution of simultaneously

measured sandface flow rate and pressure data will be

shown to obtain the formation pressure and parameters.

Convolution (Superposition)

Continuous Multirate Method. The simplest approach

to solving the convolution integral is to assume a psD

function in Eq. 3a. This psD tb~On cO~d ~ a line-

source solution, infinite conductivity vertical fractured

solution, etc., for the constant-flow-rate case or the cOn-

stant-pressure case. The chosen pa function can be con-

volved with qD (sandface flow rate) by using the

convolution integril (Eqs. 3a tkougb 3c) to modify the

time function or pwD. For example, the cOnvOlutiOn

(superposition) of variable rate with the log approxima-

tion for tbep,D function and wellbore press~e CCJMMOR-

ly is used for the anafysis of muldrate tests. The same

technique also can be used for the analysis of buildup or

drawdown tests with measured sandface flow rate data.

USing the 10g approtition for P.D in f%. 3a! fie

change in measured pressure in oilfield units can be ex-

pressed as

Af)&t) =m@(,)@og(t-r) +~]d~, . . . . . . . . (9)”

0

JOURNAL OF PETROLEUM TECHNOLOGY

where

Ap ~f) ‘pi ‘p tit),

‘=”’’’+’”g(*)-’’’”

and

162.6 @q

~.—

kh

Eq. 9 can be rewritten as

where b= ,~m.

Letusapproximate the integral in Eq. 9 by the Riemamr

sum, which yields

AP.jc(/n) =

qD (tn)

where t. is the measured (discrete) time point. This equa-

tion has been presented elsewhere for multiple-rate

analysis. Eq. 10 gives the condnuous form of the muhirate

(variable-rate) equation. Any integration techniques, as

well as the Riemarm sum used in Eq. 11, can be used to

evaluate the integral given in Eq. 10. A plot of the left

side vs. the first term of the right side of Eqs. 10 or 11

will yield a straight line with a slope m and an intercept

b. For buildup tests, qD(f) should be replaced by

1 –qD (At) and AP ~f(At) by Ap ~, (At) =p ~, (At)

–p~At=O) in Eqs. 9 through 11.

The major advantage of continuous variable-rate test

(rate measure just above the perforation) over the con-

vention multirate testis tit the wellbore storage effects

are minimized. The wellbore storage effect has not been

discussed in the literature for the conventional mtddrate

tests. The wellbore measured pressure used in the con-

ventional malysis should be taken from the storage-free

in ftite-acting period. Thus, the end of the storage effect

should be determined. In other words, the sandface rate

should be equal to the constant-surface rate for each

drawdown or buildup period.

The second problem with the conven!iond multirate test

is the surface step rate change cannot be taken as a step

rate change for the sandface. Neglecting the continuous

rate change from one rate to another will affect the result

of the conventional multirate test analysis.

PEBRUARY 1985

As noted earlier, the continuous variable-rate test sug-

gested here will not eliminate completely the effect of the

wellbore storage, since there is a finite volume between

the bottom of the well and the flow meter, but it will

minimize the wellbore storage effect.

The method suggested by Gladfelter et al. 9 further

simplifies Eq. 12. However, it does not improve the

muhirate method. The disadvantage of the Gladfelter et

al. 9 method over the mukirate method is that here is a

pxsibfity of the existence of one to three different straight

lines.

Motiled Homer Method. Using the measured sandface

rate &ta, Meunier et al. 17 presented a modification of

the Homer time ratio, which they named “the rate-

convolved buildup time function. ” They showed that the

time required for the start of the semilog straight line can

be reduced consi&rably by using the rate-convolved

buildup time function instead of of the conventional

Homer time ratio. Metier et al. 17 gave detailed ex-

planations on how to modify the Homer dme ratio if the

sandface rate measurements are available. In this section,

the fundamental mture of tie Homer and modified Horner

methods will be examined.

The buildup test is perhaps the most popular transient

testing practiced by the oil industry over tie last 3 decades.

The Horner method used for the analysis of the buildup

test is appealiig for its simplicity, generality, and ease

of application. The reliability of kh, skin, and the ex-

trapolated pressure estimated from the Homer method

depends on the slope of the Homer semilog straight line.

The assumption required for the Homer method is that

the sandface flow rate becomes zero during the semilog

period. Flom a theoretical point of view, as long as the

measured pressure increases at the wellbore, the sand-

face rate never will be zero unless the fluid in the wellbore

is incompressible. Thus, rhe effect of the decaying sand-

face flow rate on the Horner semilog straight line wifI

be investigated in this section.

The convolution integral given by Eq. 3b can be writ-

ten for buildup tests as

PDS(~pD +AtD)=%D(ArD) +PD(t@ +AtD)

-~[1-qD(T)@;(MD-~)d~, . . . . . . . ..(12)

o

where

and q is the constant rate. before shut-in.

For tie sake of simplicity, let us assume that after shut-

in, the afterflow rate decliies exponentially, as suggested

by van Everdingen6 and Hurst7 so that Eq. 1 becomes

qD(At) =e ‘-&, (13)

where a is determined from measured sandface flow rate

data. For e~ple, if sandface rate measurements are

available, a in Eq. 13 can be determined by the least-

square.s curve fitting of Eq. 13 to the sandface rate data

327

Horner & Modified Horner Time Functions

Fig. 3-Modified Horner and Horner plots for exponential

dec~ne sandface rate,

Fig. 4—Functions for the start of the Horner and modified

Horner semilog straight tines.

Substitution of the exponential integral solution forp~

and Eq. 13 for qD in Eq. 12 (the details of the deriva-

tion are given in Appendix A) yields

pi-pw’’’)=~[’”g(%)+al (14)The constant a in Eq. 14 can be considered an afterflow

parameter. The term l/2.303cYAt will modify the Horner

&me ratio, (tP +At)/At. A semilog plot of pw~ (At) vs. An-

tilog {log[(tP +At)/&] + l/2.303aAt} wilJ yield a ~ght

liie with a correct slope. This straight line starts much

earlier than the Homer “setnilog straight line, as shown

in Fig. 3 (approximately one-half cycle earlier). In fact,

Chen and Brighamz found that the correct semilog

straight liie on the Homer plot is obtained after an ex-

Fig. 5—OimenslonIess times for the start of the Horner and

modified Horner semilog straight lines as a function

of 0.

tremely long time period. It is obvious from Eq. 14 that

as At increases, 1/2. 303cYAi becomes smaller, but log

[[tP +Ar)/At] decreases as well. Thus, the Homer plot,

as it is seen in Fig. 3, asymptotically approaches the cor-

rect semilog straight line.

If the term (ZP +Ar)/At is very large compaed to the

term l/2.303aAt, then the Horner and modified Homer

straight lines are almost identical at large At. In other

words, if tp is veiy large compaed to the maximum shut-

in time, then the correction caused by the aftertlow

becomes almost negligible as At becomes large.

Before determining g an approximate formula for the stint

of the modified Homer sefiog straight line, it will be

interesting to determine an approximate formula for the

start of the Horner sem-ilog straight line. Theoretically

speaking, the Homer semi30g straight line never yields

the correct straight hne. However, we can define an er-

ror criterion between the correct and computed Homer

slopes. Then, we. take. tie time at which the error criterion

is satisfied as the start of a Homer semfiog straight line.

As developed in Appendix B, m approximate formula

for the start of the Homer semilog straight line is given by

12ctPD –AtD[(rF +At)D] @ ‘btiD [–in p –27

1+-In 4+EXJ3AtD)+2S]- 1/AtD =0, . . . . . .. .(15)

where

an error criterion,

%., = slOpe of correct Horner semi-log straight

lie,

m~~~ = slope of computed Homer semilog stmaight

line, and

tpD = dimensiOIdess producing time.

For a given e (relative error), 6, skin, and producing time,

zeros of Eq. 15 with respect to AtD will give the sw

of the Homer semilog stmight line. It is interesting to

328 JoURNAL OF PETROLEUM TECHNOLOGY

—-—

observe that the start of the semilog Homer straight line

is a function of the”producing time, as expected.

A simple formula cannot be derived for the start of the

Homer semifog straight tie from Eq. 15 because it is

a transcendental function. Fig.. 4 presents values of9

15 as a” function” of dimensionless time for tPD = 10 ,

(3=10-4, S=0.0, and E= O.01. As can be seen from Fig.

4, E41. 15 has two roots (zeros) for the values of tPD, 6,

S, &d e given previously. The first zero resufts from the

early-time period, which can be seen easily from Fig. 2.

The second zero results from the radial inkinite-acting

period..l%e upper curve in Fig. 5 presents dimensionless

time for me start of the Homer semilog straight line as

a function of P for tPD = 10 6, S=0.0, and 6=0.1. As Cm

be seen from Eq. 15, the dimensiordess time for the stwt

of the Homer semilog S-might line is a very wesk func-

tion of skin. It is basically a function of the production

time and IS. This had been observed by Chen and

Binghamzz for tie constaht-wellbore-storage case. Eq.

15 slso can be used for the constant-we~bore-storage case

by substituting lICD for 13. However, UCD is a ve~

crpde approximation for “B.

Eq. 15 cm be quite useful for the design of buildup tests

for an optin@ value of a producingdme to achieve a cer-

tain accuracy for the Homer semilog straight line. Par-

ticularly, Eq. 15 will be usefuf for driflstem tests, since

production time is limited by production facilities.

For large producing tie, tPD, Eq. 15 can be simPlifi~

to

AfD=~. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. (16)

l+q. 16 also can be derived from drawdown solutions

by using the same principle given in Appendix B.

If p i.iapproximated by I/CD, Eq. 16 then can be writ-

ten as

AfD=:CD. . . . . . . . . . . . . . . . . . . . . . . . . . . ..(17)

As noted previously, Eq. 17 will yield very optimistic

values for tie stat of Homer semilog straight line if in-

deed the weflbore storage remains constant during the test.

The dimensionless time for the start of the motied

Homer semilog stmight line (details of derivations are

given in Appendm B) also can be given by

e[2@(tPAt)D +(tP+At)D] -b’2AtD2[(tP+AZ)Dl

.e-8~D(–1” &7-27+fII 4+2S)=0. .(18)

As in the Homer case, tie second root of Eq. 16 will give

tie dimensionless time for the sw of the modified Homer

semilog straight lime as a function of 8. Fig. 4 presents

vslues of Eq. 18 as a function of dimensionless time for

tPD=’106, “13=10 -4> S=0.0, wid 6=0.01.

The lower curve in Fig. 5 presents dimensionless time

for the start of the modified Horner semilog straight line

w a fUIItiWI of j3 for rPD=106, S=0.0, and c= O.01. The

sw of modified Homer semilog stmigbt we is also a

weak function of skin.

For large producing @ne, tPD, @. 18 cm 5iMP~i~ ~

AtD=:, . . . . . .. . . . . . . . . . . . . . . . . . . . . .. (19)

4@3

The modified Horner semilog straight line star@ at least

one-half cycle earfier than the Homer straight fine. fn

FEBRUARY 1985

other words, the time required for the stat of the modified

Homer stmight line is half of the time required for the

Homer straight line, as can be seen from Eqs. 16 and 19.

Depending on the formation and fluid parameters, hours

could be saved on the testing time.

Very simple qD and pD functions me used to explore

the effect of afterflow on the Horner analysis. It must be

recognized that the sandface flow rate (afterflow) has to

be measured to obtain accurate and reliable restdts from

the modified Homer analysis.

Even though the modified Horner anafysis improves the

semilog anafysis, it cannot be applied to very early-time

pressure data. The otier methods, such as continuous

mukirate, require prerequisitepD functions. Thus, decon-

volution methls will be used to obtain p,D functions (in-

fluence functions) and formation parameters in the

following section..

Deconvolution

Deteminin g wellbore geometries and reservok types

(fractured, layered, composite, etc.) is an important part

of well testing. The reservok engineer must have stiffi-

cient information about the system being analyzed. For

example, if w curves for fully penetrated wells ae used

for partially penetrated wells, both kh and damage skin

will be underestimated. Thus, the system identilcation

becomes a very important part of well testing. For in-

stance, identifying a one-half slope on a log-log plot of

tie pressure data will indicate. a vefically fractured well,

as two paral!el straight limes on a Horner graph wi13 in-

dicate a fractured resemoir. However, either wellbore

storage or’afterflow” usually dominates these chamcteristic

bebaviors of we.ffs and reservoirs durtig the early-time

period. Thus, the pressure behavior of formation without

the weflbore storage effect must be cahdated or the

wellbore storage effect on the formation pressure should

be minimized for the conventional identification of the

system. Tbis is merely the deconvolution of F@ 3a

tlqough 3C to calculate p,~(r~) from P WD (tD) and

qD (rD). seve~ graphs Of PsD(t~) VS. tD , such 2$ liim,

sphericsl, etc., will provide information about a given

wellbore geomet~ and reservoir.

This approach to system idendi5cation is not genersl,

but it uses our conventional knowledge about well and

resemoir behaviors. In general, the problem of the system

identification is much more complex because often we do

not fmow the governing differhisl equation.

It is worth repeating that the fluid flow in the forma-

tion is described by the linear dh%sivity equation for the

deconvolution methods given next.

There are several methods for tie deconvolution of Eqs.

3a through 3c. These metiods will be discussed in the

following section.

Lineariza tion of the Convolution Integral. In this sec-

tion, the “convolution equation,” Eq. 3c, will be solved

directly by using the limaiization niethod. Eq. 3C can be

discretized asn

PWZND”+l)= z ~“mlti%+l -~)i-o ~Dt

“ps~(~)d~. . . . . . . . . . . . . . . . . . . . ..(20)

329

il’’’’”F”’-’F‘“:”’’’””:;:L:L---- “-”-”

~’”:~,~l ,,!! ~~~~~~~~~

,o- ,0-’ , 0“

slut-l. n.,, At, hrs

Fig. 6—Calculated formation pressure drop using linearization

method and wellbore” pressure drop.

By using the trapezoid rule for integration in Eq. 20,

n

PwD(tDfI+l)= X bSD(tDt+I)Qi(tDn+I ‘tDi+l)

i=!l

+P.D(tDi)q~(2Dm+ I –2Di)l(tDi+I ‘2Di)/2. . . (21)

Eq. 21 givea a system of linear algebmic equations. The

coefficient inairiix of this system of equations is a lower

triangular niatri~ that is, all its nonzero elements aze in

tie lower right of the mat@ The system of equations

can be solved easily ‘by forward substitution.

For field data, p ~D sho~d be replaced by fp i –p wj)

and (p w, -PJ for drawduwn and b~dup tests, respec-

tively; and p,D should be replaced hy (pi –pw)f and

Q% –P&, ~bich we s@dface press~e +fferenc=without storage effecm (formation pressure drop). qD fur

drawdown and (1 –qD) for buildup must be used to

replace qD in Eq. 21 for ~eld case ch. If tie flOW ~S

radial, formation kh can be determined from the slupe of

the straight line if the (p ~, –pM)f vs. log’Af plot. Skin

factor also can be “determined from me conventiowf skin

formula. However, the trapezoidal method used in Eq.

21 gives osciffatory result.% As can be seen in Fig. 6, at

very early times, the cal~ulated values of @w. –P ~)f

oscillates. Furthermore, iugher-urder mefiods, inclu~g

the SiniPson rule, yield divergent results for the integral

in Eq. 20. ..” .’ .. .Hamminz23 sumzested a stible integration scheme for

evaluation ~f the ‘%wolntion integrul~. He also showed

that direct integration (discussed as the linearization

method) of convolution integmfs usually will r@t in w

owiflation.

The intigral in Eq. 20 can be approximated as

‘PSD(~Di+%)~1q~(2Dn+l-7)dT. . . . . . . ..(22)

r~i

The right side of Eq. 22 can he integrated directly.

Substitution of the integration resuhs in Eq. 20, and solv-

ing for p,D yields

p,D(tDn+,A)=~”~(f~+l)–s~ ,. .,.... . . . (23)

fZD(tDn+ I ‘tD.)

33U

,6,0-

=:

:,200

i . . . . . .“..,. !. ah”..!.

‘.”., !,. s,..,.!.

:

, m, -

+ ./”/., ./. .0,

/“: i“

--. -”.-.0 . .

,

10 ,o- ,0- 1,.’ d

SW-i- ‘m., .!, b

Fig. 7—Calculated formation pressure drop using Hamming

method and wellbore Pre?sure drop.

where “sum” is equal to

n—1

,2 psD(bi+fi)[qD(tDrI+l ‘b)

‘qD(tDn+I ‘tDi+I)l,

and the fist value ofp ,D is given by

PsD(tD%)=:;;;; — . . . . . . . . . . .. . . .. . . . . . ...(24)

As can be seen from Fig. 7, Eq. 23 gives a stable and

nonosdlatory integration scheme for the convolution in-

tegral given in Eq. 20. Furthermore, the derivative of the

sandface flow rate data, q~(tD), is not needed in Eq. 23.

However, for Eq. 21, q;(t) must be known. A iinite dif-

ference appromixation also can be used for q&D), but

some accufacy would be lost.

The semilog plot of @ ~, –p ,,.,+)f vs. At, given in Fig.

7 for radi+ synthetic buildup, test data, yields a straight

line starting kom a veiy early time (At=O.05 hours) @tb

a correct slope. The lower curve in the same figure is tbe

plot of (pW, –p;), which includes the we~bOre it~rage

effect.

Laplace Transform Decunvulntiun. The convolution of

RI. 4 yields

‘D

pSD(tD)= \ K(T)PWD(tD –7)d~, . . . . . . . . . . . . (25)~

where

“1

Mlr(z~)=c-1— . ................. (26)

s?D ($)

K(rD) can be cumputed eifher from the Laplace

transforms of qD(tD) data or a curve-fitted equation uf

qD(tD) data. However, it would be @e consuming to

invert all the gD(tD) data in Laplace space and transform

it back to real space in accordance with Eq. 26. Thus,

aPProhtion functions must be used for qD (tD) data.

Once an approximation is obtained, it will be easy m com-

pute K(tD) and integrate Eq. 25 to determine p,D(tD).

A few types of approximation functiuns can be used to

approximate qD (tD). We have tried rational functions,

power series, and ex~nentid functions. Expmentid func:

tions give a good representafiori of qD (tD) data. Ex-

JOURNAL OF PETROLEUM TECHNOLOGY

.—

—

TABLE 2—FRACTURED RESERVOIR ROCK

ANO FLUID DATA

B, bbl/STB 1.0

~, psi-l,..5

1,000

h,Dft 100

k,, md 400

pti, psia 4,216.94

q, STB/0 3>000

rw 0.35

R 10

tp, hours 10,000

k (spherical matrix used),.-5

~, Cp 0.8

+, 0.2

. 0.06

ponentiel functions approximation for qD can be

expressed as

qD(tD)=cleB1tD +c2e82tD +...cne8~tD, .:. (27)

where i= 1,2. .,.n.

After determining Ci xnd L3i from qD(tD) data, apply-

ing the Lapktce transformation to Eq. 27 and substituting

the resulting expression in Eq. 26 yields “..

1.”i?(y)= ~ , “ ..!:.’: ””””.”.:.:”’: ’(2. !)4

AS SugEeSM bv Sneddo”. 24 if the inverse tramfOrma-

tion d~~s not &fist, Eq. “28 should be divided by”p until

its inverse’ trxnsform is possible. For each ~vision ofp,

P~D(tD) fi Eq. 25 has to be differentiated.

We have seen from vw Everdngen’s7 and Hurst’sg

works thxt qD cti be approximated by m exponential

fonction as

qD=l_e-@D ,. ..,. . . . . . . . . . . . . .. . . . . . . (29)

which is the simplest form of Eq. 27. For this case, P$D

cm be wriiten directly from Eq. 3 as

mm=?’”’”;’‘+PwD(tD). . . . . . . . . . . (30)

TABLE 3—COMPARISON OF

COMPUTED AND ANALYTICAL

PRE?SURE FOR A FRACTURED

RESERVOIR

At

0.02

0.04

0.06

0.08

0.10

0.20

0.30

0,40

0.50

0.60

0.70

0.80

0.90

1.00

2.00

3,00

4.00

5.00

6.00

7.00

8.00

9.00

~131.93

133.57

134.66

136.47

136.12

138.27

139.62

140.62

141.42

142.10

142,68

143.20

143.66

144.08

146.94

148.!36

149.92

150.88

151.66

152.%

152.85

153.33

.-131.96

133.78

134.86

135.64

136.25

138.27

139.55

140.53

141.33

142.02

142.62

143.15

143.63

144;07

146.98

148,70

149.92

160.86

151.63

152.29

152.85

153.35

*65.33

98.32

115.30

124.37

129.44

137.03

138.87

139.83

140.85

141.59

142.24

142.81

142.32

143.78

146.84

148.60

149.85

150.80

151.59

152.24

152,81

153.32

From Eq. 30 it is very simple to calculate p,D if qD dat6

cm be approximated suc.csss~y ,by .,,using ,:.Eq. 29..

qD (tD ) data ds.o can be approximated by piecewise ex-

ponential fun~om for a selective interval for Eq. 26.

Curve-fit App:oximmtioiis’o fp,D”. p~~(tD) in Eqs. 3a

and 3:..c~wb&approximated by choosing suitable func-

tion~ approximations. These ~ctionel approximations

copld be power series, continued functions, rational func-

tions, or exponential functions. The success of this method

depends on how well the approximation function

represents the behavior of p,D. As noted earlier, the 10g

apPrOfimatiOn was used to approximate p,D (tD). In tis

case, the natural choice will be the power series of in tD

such tbatzz

m

“’””” >sD(tD)= x C@ I tD)i-l. . . . . . . . . . . . . . . ..(31){=1.. . . . . .—,

,.,

Substitution of Eq. 31 into Eq. 20 yields n number of

equations and “m number of unknown pxmneters,

.=(cI ,c2 .:.c~)T. In fact, obtxining these paramters, .;,

becomes an unconstrained opdmiiation problem. We have

appEed a fotirtb-degree polynomial, &f. 31, to weIfbore

pressure and rxte data (synthetic) that were obtained from

a fmctured reservoir for the resewoir and fluid pam.meteis

gjven in Table 2. The second column of Table 3 presents

the cxkdated values of Apti from curve-fit approxima-

tion for the fluid and reservok date given in Table 2, while

the third column of Table 3 presents the vxlues of Ap,f

calculated directly from q amlyticel solution without

storzge effect. The fourth column in Table 3 presents

&~, with the wellbore storage effect.

As seen from Table 3, dMerences in Ap,f from the

curve-fit approximation and the anxlydcel solutions are

xlmost identicel. The relative error decreases for lerge

times.

C0nc1u8i0nx

1. The convolution integml (superposition theorem) is

used for the analysis of continuously varying wellbore

flow rate and pressure. This analysis is ve~ similxr to

the conventional mukirxte meth0d3.

2. Wellbore storage (afterflow) effects can be present

to a significant degree in the Homer semi-log strxight Iiue.

The Horner an~ysis is modified by using measured sxnd-

face flow rate data to obtain a correct semilog straight

liie. The modified Horner semilog straight line steits”xt

last one-half cycle earfier tlmn the conventional Horner

FEBRUARY 1985 331

straight line. Approximate formulas afe presented for the

stat of the modified Homer and Homer semilog straight

limes as a function of sandface rate decline, production

time, and the relative error between the correct and com-

puted s30pes.

3. The formation pressure (influence function) can be

cakulated ffom the deconvolution of measured wellbore

pressure and sandface ffow rate data. Some new decon-

volution techniques are introduced to compute the format-

ion pressure (iiuence function) without wellbore storage

(atlertlow) effects. W&bore and reserv@r geometries ca

be identified from this computed fofmation pressure. Fur-

thei’more,, the conventional methods can be used to analyze

this computed formation pressure to detefmine formation

parameters.

4. Deconvolution of synthetic data from a homogeneous

and ffactured rekervoir shows tiiat it is possible to com-

pute the formation pressure from tbe”beghing of the test.

‘IW computed pressure reveak the characteristic behavior

of homogeneous and fractured reservoirs.

Acknowledgment

We thank Scblumberger We! Services for its peffnission

o pub3ish OdS paper aid acknowledge the en~u~gement

and help of Gerard Catala of Servic& Techniques

Scblumberger during the initial stage of this work. We

alio t@& H.J. Ramey Jr. for providing a solution for

the integral in Appendm A.

Nomencfatnre

B = oil fofmation volume factor, RBISTB

[res m3/stock tank m3]

ct = system total compressibility, psi-1

ma-l]

C = wellbore ,storage coefficient, bbllpsi

[m3/kPa]

CD = wellbore storage constant, dimensionless

h = formation tilckness, ft [m]

k = fofmation permeability, md

K = kernel of the convolution integraf

p: = deferential of pD

PDS = shut-in pressure drop, dimensionless

PD(tD) = fOfnJatiOn pressure, dimensionless

pi = initial pressure, psi [kPa]

PSD = PD +S = fofmation pressure includlng

skin, dmensi0n3ess

P WD = pressure drawdown, dmensicmless

PW = bOttotiole flowing pressure, psi [!-@a]

P~. = bottomhole shut-in pressure, psi ~]

q = stabflkid constant ate, STB/D

[stock-tank m3/d]

qD = s@face flow rate, dimensimdezs

qr = reference flow rate, B/D [m3/d]

qR = resewoti flow rate, BID [m3 /d]

q~ = smdface flow raw, B/IJ [~3/d]

– wellbore radius, ft [m]~w —

s = Laplme m~~fO~ vfiable

S = skin factor

t = time, hops

t~ = time, dimensionless

tp = production time, hours

332

tPD = dimensionless production t@e

T = transpose

At = running testing time, hours

ArD = shut-in time, dimensionless

a = O .000264kLWpc,r;

# = a positive constant

.y = 0.57722 . . . = Euler’s constant

p = viscosity, cp pa-s]

f = rate-preasufe convolved time function

~ = dwmny integration variable

+ = porosi~, fraction

- = LaPlace transform of

References

1. Earlo.gher, R.C. JI.: Advances in well Test Analysis, Monograph

Serk, SPE, Richardson, TX (1977), 5.

2. Gri”gamen, A.C. er .1.: ; ‘A Cmnpmism Bctwee. Different Skin

and Wellbore Storage Type Cm’ves for Early-Time Transient

~Y@,” pawr SpE 8205 p=nted at tie 1979 SpE AIInuaJ

Tedmicaf Confer . . . . and Exhibition, Las Vegak, Sept. 23-26.

3. Ranmy, H.J. Jr.: “ShorGTime Well Test Dm Inteqxctation in e.

time of S!dn Effyt and WelJbore Storage,,, J. Pet. Tech. (Jan.

1970) 97-10% Tram, AJME., 249.

4. A&u-waJ, R. G., AJ-Hussainy, R., and Rarmy, H.J. Jr.: “AD 3n-

vesdgatim of Wellbore Storage and Skin Effect in Unsteady Liq-

uid Ffmw 1. AmJMcaJ Treatment,!, L%., A% EW J. (Sept. 1970)

279-90; Tram, , AJME, 249.

5. McKinley, R. M.: ‘YWeUbore TransmissibiShy from Afterflow-

Dominat&J Pressure Buifdup Data,” J. Pa Tech. (July 1971)

S63-7% Twns. , AIME, 251.

6. Ead.mgk, R.C. Jr. and Kersch, K. M.: “AnaJysis of Shofl-Time

Transient Test Data bv TVWCIUW Matching,,> J. Pet. Tech (JuJY

1974) 793-SC@ Tr&. ,-&JME, 257. -“

7. van EverdimgeII, A. F.: ‘“i% S!dn Effect and Its Infken.x on the

Prcducdve CPpacity of a Well, x,J, Pet. Tech. (June 1953) 171-76;

Tram. , Am 1 QR

8. Hurst, M

t. Fluid Flow into a WdJ Bore, ,7 Pet. E.g. (Oct. 1953) B6-B16.

,9. Gladfelwr, R. E., Tracy, G. W. , and Wikq, L.E, : ‘ ‘Selecting Well%

Which V?ii Res.mmd to Fmducticm3dmulation Tresxnent ,’, Drill.

and Prod. fi!lC. , API, DaJlas (1955) 117-29.

.-.. -, . . .IV.: ‘ ‘Establishmerd of the Skin Effect a“d Its kopediment

10. Ranw, H.J. Jr.: “NomDaIcy Flow and Wdl&me Storage Effects

in Pressure Build-Up and Drawdown of Gas Wells,,, J. Pet, Tech.

(Feb. 1965) 223-33; Trcms. , AIME, 234.

11, H.tchimm. T.S. and Sikora. V.J,: . ‘A Generalized Water-Drive

Am fvsii., 9 J. Pet. Tech. (Jui 1959) 169-77: Tram.: AfME. 216.

12. X& D.i., Tek, M.R., kd.Joms,’ S. C,: “i Gene&ized Mcdel

for Predicting the Pa’fomce of Gas Reservoirs S.bjec4 ta Water

Drive,>> papr SPE 42S, presented at 1962 SPE Anmd Meeting,

Los .@eles, Oct. 7-10.

13. Coats, K,H. et al.: “Determimtion of Aquifer fnfluence Functions

From FieJd Data,’- J. Pet. Tech. (Dec. 1964) 1417-24 Tram.,

AJME, 231.

14. Jargon, 1.R. and van PcalJen, H.K.:. “Unit ResFon% Funcdon From

V&nz.R?@ Dam. ‘y J. Pet. Tech. (Aug. 1965) 965-69; “Trans.,

tiE,-234.

15. Bred., J.N. et af.; ‘, Combined Analysis of Pmtfracmrin’g Perfor-

mance and Pressure BtdJdup Dam for Evaluating an MHF Gas

Well,,, J. Pet. Tech (Oct. 1980) 1711-19.

16, PascaJ, H.: ‘ ‘Advanm in EvaJuatinE Gas Well Deliverability Us-

ing Vtible Rate Tests under No”-Darcy Flow,’ 3 paper SPE 9841

presented at tie 1981 SPEIDOE LOW Perrneabili@ Symposium,

Denver, May 27-29.

17., Meunier, D., Withna@ M.J., ed Steward, G.: ‘Tnterpretatio.

of pressure Buildup Test Using IreSim Measnrmnert of Afterglow,”

J. Pet. “TecJa, (Jan. 19S51 143-52.

Trans format& to FlrJw Pmble& in R&voirs,x, Tram., .&E

(1949) 1S6, 305-24.

19, CkO.bY, H. ~d sanxmieso, F.: ‘aPressure Transient AmlY$is

for Naturally Fractured Reservoirs,, s paper SPE 11026 presented

at the 1982 SPE AnnuaJ Te&dcaI Co” femnca and Exhibition, New

Orleans, Sept. 26-29.

JOURNAL OF PETROLEUM TECHNOLOGY.

—

20. K@uk, F. and Kirwan, P. A.: “New skin and Wellbore StoraSe

TYCX CUweS for PardallY Penetrated Wells, ” paper SPE 11676

pms.nmd at fie 1983 SpE c~.~ Region~ Meeting. venti~,

hfarcb 23-25,

21, Kamey, H.J. Jr. and A&val, R,G. : ‘ ‘Annulus Unloading Rates

as Influenced by W.lhm WOW. and Skin Effect,’+ .%c. Pet. Ew.

J. (Oct. 1972) 453-62.

22, Chin, H.K. and Brigham, W,E.:’ ‘Pressure Buildup for a Well with

Storase and Skin in a Closed Square,,, J, Pet. Tech. (Jan. 197S)

14146.

23. Hammm g, R. W.: Numerical Mefiods for Sckvu&s anAEn@zeers,

McGwv-HiU, NW York City (1973) 375-77.

24. %eAdcm, I, M.: & Use of huq’ml Trmqfom, McGmw-3fiU, New

York City (1972) 207–14.

25. Stehf..st, H.: “Ntuncricaf fnversion of Lapk.ce Transforms,” Com-

numications of the ACM (Jan. 1970) 13, No, 1, &OriIhUl 368.

APPENDIX A

Substitution of the exponential integral solution for pD

and Eq. 13 for qD in Eq. 12 yields

where

y= O.5772

and

Et(aAt)= \8M:du.

-m

Neglecting the imaginary term .i, the van Everdingen7

and Hursts forms are obtained as

&(AtD)= –%e-@MD [–III j3-2-y+fn 4

+Ei(@tD)]. . . . . . . . . . . . . . . . . . . . . ..(A-7)

[

1

pSD[(tP+&)D]’ ‘~~i ‘—

4(tP +At)D 1

.%E,(.&)-j””e-Bexp[ -&]0

dr.—+SqD(AtD), . . . . . . . . . . . .. . .. (A-l)

2(AtD -,)

~-pw(~)=~[l”g(%)+~(A’)l,-(A-8)

The values of integral :(&D) from f3kf. A-7 were com-

psred with the vafues computed from the Stehfest z

munericaf Laplace trsnsform inversion technique using

Eq. A-3. When AtDs 30, the difference between two

vslues of @tD) becomes less than 1%. The difference.

becomes smaller as AtD increases. Substitution of the log

approximations for the exponential integrafs given in

Eqs. A-I and A-7 for the integral yields (in pmcticd units)

where

~D=e —d==-LmrD. . . . . . . . . . . . . . . . . . . . ..(A-2)

Thus, tie two forms of gD wifl be used interchangeably.

The integral in Eq. A-1 cannot be integrated readily and,

unfortunately, van Everdingen7 and Ramey 10 dld not

present the &tds of integration. Ramey* provided me

a heurirtic derivation that wss edited out of his paper. 10

Rafney’s integration method is given below. The LapIace

transform of the integraf given in Eq. A-1 is

Z(S)= *KO(J). . .

The long-time approximation for KO(~

. . . . .. (A-3)

is

(vKoc~)=–lny+y. . . . . . . . . . . . . . . . ..(A4)

Substitution of Eq. A4 in Eq. A-3 yields

1j(s) = –— (h s–in 4+2T). . . . . . . . . ..(A-5)

2(s+.0)

The inverse Laplace transform of Eq. A-5 yields

@ro)= –fie-@tiD [–in &~i-2y+fn 4

+Ei(6AtD)], . . . . . . . . . . . . . . . . . . . . ..(A-6)

‘Ranw H.J, Jr.: P+rsmal eommunkallon S$anfwd U., Stanford, CA (Mm+l 1984).

FEBRUARY 19S5

where

i(At)=*e ‘aAr[–fn f3-27+fn 4+ Ei(aAf)

+~. . . . . . . . ... . . . . . . . . . . . . . . . . . . . ..(A-9)

The long-time approximation for g(At) maybe obtained

by substituting limiting fbims of ci+tain terms in Eq.’ A-7

for long times. For lsrger values of At, the term

e ‘@(-bI j3-27+ln 4+2$ in Eq. A-9 approaches

zero. The term e ‘aNEi(aAt) can be approxinsted as

e-a&Ei(aAt)=~, . . . . . . . . . . . . . . . . . ..(A-1O)

when &+l.

Substituting Eq. A-10 in Eq. A-9 yields

~(At) =~ (A-11)2.3026dr” ““”””’”””””’’”””””””

Further substitution of Eq. A-11 in Eq. A-8 gives Eq. 14

in the main text.

APPENDIX B

The dimensionless form of Eqs. A-8 and A-9 cm be writ-

ten as

PD(A’)=05[”[-1+’(A’D)]w333

where

~(AtD)= %. ‘~ND [–In (3-2y+ln 4+ Ei@AtD)

+2,g. . . . . . . .. . . . . . . . . . . . . . . . . . ...03-2)

The differentiation of Eq. B-1 with respect to

In{[(tp +At)D]/AfD} yields

‘com=0.5++[g(Ad],....... ...........@-3)

where. mcom is the computed dimensionless slope

of the Homer semifog straight line and x equals

ln[(f,n +At)D/AtD] .

Let us &fine a relative error for tie computed slope as

~= I =Ij ...... .... .........

where

mcom> 0.5.

Substitution of Eq. BzI in Eq. B-3 yields

. (B-I)

df(AtD)~= ~~—.. . .. . . . . ... .. . . . . . . . . . . . . . . . . . (B-5)

Ax

Taking the &rivative of $(AtD) with respect to x and

substituting in Eq. B-5 yields

AtD[(tP +M)DI~=

[

A3e-~&D [–1. 13-2-y+ln 4

2tpD

.1+.Ei((3AtD)+2S]-~ . . . . . . . . . ...@-@

Eq. B-5 gives “tie dimensionless dme for the start of the

Homer semilog straight line as a function of e, 0, fPD,

and S.

A formula for the dimensionless time for the start of

the modified Homer semilog straight line can be found

from Eqs. A-1 and A-2 as follows. Let us rewrite Eq.

A-1 in terms of the modified Homer time

f’D@@=”4w%31”1

+— I +%$(AtD), . .

26AtD

,. (B-7)

where

$(AtD)=!4e-5ti.c (–fn fl-2y+ln 4+2S). .: .(B-S)

lA$(AtD) in Eq. A-7 is the ordy remaining term that is

not included in the modified Horner time. Thus, the

relative error of slope of the computed modified Homer

semilog straight line (as in the Homer case) can be ex-

pressed as

df(AtD)~. —...- . . . . . . .. . . . .. . . . . . . . . . . . .

Ax

(E-9)

where

‘=1”[(%%]++ ‘@-’o)

Differentiation of Eq. B-8 with respect to x and substhw

tion of the result in Eq. B-9 yields

.(–in 6–2y+ln 4+2S). . . . . . . . . . . . .(B-11)

Eq. B-11 gives the dimensionless time for the start of the

modified Homer semiIog stiaigbt line as a fiction of e,

o, tPD, and skin.

S1 Metric Conversion Factors

bbl X 1.589873 E–01 = m3

Cp x 1 .“* E–03 = Pas

ft X 3.048* E–01 = m

psi x 6.894757 E+OO = kPa

.Gmversl.m Iaclor is exact JP’f

0ri@n=4 rmnusuiDt rec.stved in the Satiety of Pe,,%um Engineers ollb Od, 5, ? 9S3.

Pew accaPted for P.MWO. M.,ch 11. I ss,. ~~vi,ed manuscript ,ece~ved ‘e*

14, 19%. PaPer (SPE 121771 first Prem.ted at the 19S3 SPE Annual T@ch”icnl Con.

term. and mhlbitlo. held in S.. Franckm 0~. ~s.

334 JOURNAL OF PETROLEUM ~CHNOLOGY

4aoo.o

4a@o.o

4280.0

4970.0

4ZSOm0

4260.0

4240,0

4aao,o

4220,0

4210s0

4aoo.o

lb’HORNER & MODIFIED

Fig, 3-Modlfled Horner and Horner plots

\

Y‘\

LEQENDhornermo e_ ,dKd l!w!!,e~

\\

\

\.

—.

.

—

—

\

‘\

decllne sandfaco rate,

I

—

—

.

T

—

)2/77

1800.0

1700,0

1600,0

1600,0

1400.0

Iaoo.o

1200s0

1100.0

1000.0

IIII

TTll

.

1

.

Ill

111

P

.

“

LEGENDG wdlbOre DKM3SUWi

_m!M?lm.EE !E!M!!u0

Fig, 4-Calculated formation pressuredropusingIkwkatkmmethod and wellbore pressure drop,

1’

1A/7,7

1600,0

?Emh n

1400,0

1s00.0

1200,0

1100,0

[ m II’

100

Fig,

0—

0—

/

/

.01--110-8

5-Calculated formation pressure drop

TIME, HOUR8

using Hamming method and

10’

wellbore pressure

II

drop,

10R

aooooo

2800.0

2600.0

z S400.Oa

w“eg 2200,0

$

aan

z2000.0

Tt=sx@

1600.0

~

dwa

1600.0

1400.0

1200.0

1000.Q

——

——

LEQENDo formation pre8sun3

LY?s!!!l ,ore ixessurp_

—

—

—

—

—

—

—

—

Ii)-’

—

>

—

—

—

—

—

—

.

r

.

.

i)

.

@@”

—-

—

—

6

—

—

—

—

—

—

-

I

Fig, 6-Calculated formation pressure drop using polynomial approximation and welibore pressure for afractured reservoir,