Ramey_ Wellbore Storage II

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An Investigation of Wellbore Storage and Skin Effect in

Finite Difference Treatmentnsteady Liquid Flow: II.

ROBERT A. WATT ENLARGER*

H.J. RAMEY, JR,

MEMBERS AIME

ABSTRACT

An inuest iga tiono/ the ef fect o{ w el lbore s torage

and sk in ef / ect on trans ien t f low w as conducted

us ing a / i n it e-d i ff er en ce solu t ion to the basicL - —...l J ..J / . ——. . ._ 7purLzuL azf [eren lza t equa t ion . The concept 0/ sk in

effect w as genera l ized to include a da rnaged

arrnu la r region ad jacen t to the w el lbore (a compos ite

reservoir ), The numer ica l solu t ions w ere compared

w ith ana ly tica l solu t ions for cases w ith the usua l

steady -s ta te sk in ef fect . It w as found tha t the

solu t ions for a fin i te-capacity sk in ef fect cornp a red

closely w ith ana ly iica i’ .sot ’u iions a i shor i i imes

(w ellbore storage con trolled ) and a t long t imes

a f ter the usua l stra igh t line w as reached . For

in termed ia te t imes, presence of a fin ite-capacity

sk in ef fect caused sign if ican t depar tu res from tbe

in f in itesima l sk in solu t ions. Tw o stra igh t l ines

occurred on the d raw d ow n plot for cases of la rge

rad ius of d amage. The f irst bad a s lope

character ist ic of the f low capacity of the damaged

region; the second stra igh t l ine bad a slope

cbaracterr ’st ic 0 / the f low capacity of the

undamaged region . Resu lts a ;e presen ted both in

tabu la r form and as log-log plots of d imension less

pressures us d imension less t imes. Tbe log-log

plot may be used in a ty pe-curve match ing

p?oce lTg ~Q CZy ra ly ze ShQrf-f j rne (be{cvg ?zClrma

straight line) well- tes t d ata .

INTRODUCTION

Skin effect was defined by van Everdingen 1 and

Hurst’2 as being an impediment to flow that is

caused by an infinitesimally thin damaged region

around the wellbore. The additional pressure drop

through this skin is proportional to the wellbore

flow rate and behaves as though flow through the

skin were steady-state.

Original manuscript received in Society of Petroleum Engineers

office J an. 16, 1969. Revised manuscript of paper SPE 2467

received Sept. 19, 1969. @ Copyright 1970 American I nstitute

of Mining, Metallurgical, and Petroleum Engineers, I nc.

preferences given at end of paper.

“Presently with Scientific Software Corp., E nglewood, Colo.

This paper will be printed in T r a n sa ct ion s vo l um e 249,

which will cover 1970.

MOBIL RESEARCH 8 DEVELOPMENT CORP.

DALLAS, TEX.

STANFORD U.

STANFORD, CALIF.

Wellbore storage is caused by having a moving

liquid level in a wellbore, or by simply having a

volume of compressible fluid stored in the wellbore. 3“,, —.-,,L ---wnen surface fiow rates change abruptiy, weiuzore

storage causes a time lag in formation flow rates

and a corresponding damped pressure response.

A recent study ’1 was made to determine the

combined effects of infinitesimally thin skin and

wellbore storage. Analytical methods were used

along with numerical integration of a Laplace

transformation inversion integral. Tabular and

gra~hical resuits were presented for various cases.

It was recognized during the study that this

representation of skin was oversimplified; that

skin effect should be thought of as a result of

formation damage or improvement to a finite region

adjacent to the wellbore.It was suggested that a skin effect could arise

physically in a number of ways. One simple example

would be to assume that an annular volume

adjacent to the wellbore is reduced uniformly to a

iower permeability than “~re originai ‘-- ‘“--alue. This

would be similar to the composite reservoir

problem. Perhaps a better example would be to

assume that the permeability increases continuously

from a low value at the wellbore to a constant

value in the undamaged reservoir. In either case,

the damaged region would have a finite storage

capacity and would lead to transient behavior

within the skin region.

A negative skin effect could arise from an

increase in permeability within an annular region

adjacent to the wellbore. This might physically

result from acidizing. But it is believed that cases

of more practical importance are those in which

negative skin effects are caused by hydraulic

fracturing:. —-. —.- A high- perrneabiiit~- fracture

communicating with the wellbore gives the

appearance of a negative skin effect.

For the purposes of this study, it was decided

to represent a skin effect, either positive or

negative, as an annular region adj scent to the

wellbore with either decreased or increased

permeability. This then is the composite reservoirproblem wherein a permeability kl exists from the

SEPTEMBER,970 5457 291



.

well radius to a radius of the damaged region, rl.

For composite reservoirs. there are an infinite

number of pairs of values, rl and kl, which

correspond to a value of skin effect. s. The main

purpose of this study was to investigate the

behavior of a well during initial transient flow in

the presence of a finite-capacity skin effect with

the presence of wellbore storage. The goal was to

provide information which might be useful in

interpretation of short-time well-test data (either

buildup or drawdown).

The initial-value problem is a special case of a

composite reservoir problem wherein porosity,

viscosity and compressibility are the same for

both damaged and undamaged formation regions,

but with the Permeability changing at the boundary––

of the two regions; the complication of wellbore

storage is also added. The skin effect, .s, does not

appear explicitly in the formulation of the problem,

but may be inferred from the steady-state pressure

drop through the damaged annular region adjacentto the wellbore.

The initial-value problem may be stated as

follows. For the “damaged” or skin region,

a2pl D + J_ap, D3P1D “k —. .

—= —

arD2 ‘D arD ‘1 atD ‘

l~r D<r, D....,..

For the undamaged formation,

a2P2D+ ~ ::2D _ ap2D ;

?lr D2 ‘D D atD

‘l D<rD~ m-”””””

Inner boundary condition,

(1)

(2)

. . . . . . . . . . . . . . . . (3)

PwD(tD) t)p,D(b D

for all tD . . . . (4)

Interface conditions between skin region and

formation,

‘ID= Pm; (r, D, tD) . . . . (5)

aplD k

k—= ~ ; (I-ID, tD) ~ (0

1 arD D

292

Outer boundary condition,

1 im

‘D_’O’P 2D= 0.. . .. .. . 7

And the initial condition is,

‘1~ (rD}O)=p2D(r D,0) = O . . . . (~)

The dimensionless variables have usual definitions,

‘D=r/rw’ ””””””””””. (9)

kt lo‘D=—’”””-”””””” “

q~c rw

p~D(r DJtD) = & (pi-p~, t) ;

r ~r < r, . . . . . . . . . . .(11)w

‘2D(r D>tD) = & (Pi-pr, t) ;

‘1<r~ ~, . . . . . . . . . . (12)

PwD tD) = pi-pw) “ “ “ ’13)

t=c (14)

21-rhqcr z’””””””””””w

where C represents the fluid storage capacity in

the wellbore, cc/atm.

The equivalent steady-state skin effect, s, can

be expressed as a hrnction of k, kl, r l and rw.

That is,

s= ~- l l n ~

i w

= (~ - 1) lnr, D”. .. (15)

Thus, it is possible to select appropriate values

of ~1~ and (kIkl) to provide a specific value ofl-- -1.:- -I L--- T1-- .-l-.: .––l-– ~.. —.-– .L -L1lC SYK1ll CI ICC-L. J ne relaclonsnip Derween tne skin

effect and the pressure drop attributable to the

skin is

2rrkhs = ~@ Skin . . . . . .. (16)

Strictly speaking, the above problem may be

solved to provide the pressure at any time andradial location within either region. But since the

SOCIETY OF PETROLEUM ENGINEERS J OURNAL



main obiective was information for well-testTABLE 1- VALUES OF rlD AND (kI/k) USED FOR VARIOUS

SKIN EFFECTSnalysis, only pressures at the well were sought.

All results reported in the study are for

dimensionless pressures at the well which are

denoted as pwD (tf)).

Although the subject problem may be handled

analytically with ease, the analytical result would

require numerical integration to provide finaltabulations of the dimensionless variables.

Fortunately, a finite-difference computer program

was available that could readily handle the

problem. The program was a one-dimensional

radial model prepared to solve real gas flow with

damage and wellbore storage. The details of this

program have been described previously. s, ~

Solutions for liquid flow were obtained by employing

constant-fluid physical properties. The

finite-difference solution represented a finite-radius

reservoir. But the reservoir radius selected was

large enough so that pressure a t the ou ter boundary

was not affected for producing times of tD x 108.

A check of the outer boundary pressure was madefor each run. In addition, all solutions were found

to agree closely with the infinite-reservoir

analytical solution for the longest producing times

run. Since solutions to the diffusivity equation

were obtained in terms of pressure, the usual

assumption of small pressure gradients throughout

the flow system was made.

Give n a value of equivalent skin effect,

corresponding values of ?lD and (k I/k) were used

in the model. Values of rlD = 1, 10, 100 and 1,000

were used to give a range of conditions. The case

of rlD = 1 is equivalent to the van Everdingen-Hurst

infinitesimally thin damaged region. When no

wellbore storage is present, the solution to thiscase can be obtained by adding the skin effect to

the s = O solution. When wellbore storage is

present, the solutions to this case are equivalent

to those solutions given by Agarwal et a l, 4

A negative skin effect in a composite reservoir

implies that the region around the wellbore has

improved permeability, k.l > k. It should be noted

that for a given radius of permeability improvement,

rlD, there is an upper iimit to the magnitude of

negative skin effect. This limit is reached as the

p 5 5 f 1 m ~=bi i i ~~ -GL2 2 3 d~~ le ~ , e i i ~e r e ~p p ~~~~~~ ~

Skin-Zone

Radius, rlD (kI/k) for Skin Effects, s, of

–5 +5 +20

10 0.3153 0.1032

100 0.4794 0.1872

1,000 3.6209 0.5801 0.2567

TABLE P — p,”” (?.) VS tD FOR * O

10 100 1,000 10,000 100,000k

;51. 104

2 X 1 0’

5. 10’I . 102

2. 10’

5. 1021 x 10’

2. 103:: ;;:

2 lo~5. 1041. 10,

2. 1055. lo~I . 1062. 106

5. 1061. 107

2. 1075. 1071, 10,

0.0000099822O.oow 199640.WOC1499100.000099818

0.030199630.00049’3040.00099798

0.00199560.C+2498660.0099663

0.0199070.049592

0.098654

0.195340475040.91122

0.779351.00461.3493

16411

1.9513

2.38162.7164

0.090324

0.172340.W88

0.677091.10461,82072.28892.8794

0 0 0 9 s7 9 50019659

0048558

0,0955

0.185990435140,79441

0000997190.00199340.0049713

0.W99282

0.019821

0.0492060.097432

0.19146

0.457220.855751,5237

2,86684.OIXI

4.915J25.66176.0823646156,93827 i 91 97.6412

3.05s0

3,50883.8534

419844.654950008

5,3465580336.15016.49596.95261,29937,64518.10248.44368.7947

9.25179.5982

0.019880

0.0496670.098243

0.193990.668520.88987

1.6X293.19044.6539586196,77147,21697.6042

3,43223.81294 1 76 74.s4534,9958

5.34235.80216.14?56 4 9%

6.95257.29937.6451

1.6865

3.43515.18696.7373

8.10248.44348.7947

8.10228.44848,7966

9.2517

9,5982

8.10278.44768,7942

9.25159.598I

8.08568.4398

879029.2498

9.5972

7,27668.3504

8.747?

9.23259.5883

9.2517

9.5902

.F, ,e.plm p,ec, ,,m, WO, token f,m ,Im c.mptie, OU p,,, b“, doe, no, ,,wIY rho he

,.lti,.n h-. lht $ dew.. d OCCWmy.

TABLE 3 — p,, [,,,) VS ,n FOR s s, r ,,, 1,000

~0.334740,41377

0.52578

0.61522

0,70689

0.820440.9249 I1.0199

1.14s31.2412

1.3247

1,4623

1.5584

1 65S9

1.7803

1.8811

2.0053

2.2353

2.46692.7401

3.1472

3,4754

3.8125

4.26414.6084

J. @-

1

1

~0.0803730.144670.=5230.43189

0,50913

0.777060,39669I 0049

1,1392

i ,2371

1.3343

1,4625

1.5380

1.6537

1.7802

).88 10

2.0052

2.23572.4669

2.7402

3.1472

3.4754

3.8125

4,26414.6088

~ 1,000 10,000 100,000

0.00000998210.000019964

0.000049908

0,0000998120.0W19961

0.000498920.000997S40,0019962

0.0097524

0.019266

0.046705

0.0895710.16676

0.350690.55918

0.79531

0.000995880.0019893

0.0049575

0.0098715

0.019591

0.0430190.093311

0.17733

51. 10,

2. 10

5’. 101 102

2> 1025. 102

I x 1032. 103

5. 1031, lo,

2, 104

5> 10*1. 102

2. 105

0.0099141

0.0197230.063609

0.0951620.18322. . . . . .“ .4 ’ .

0.708571.0970

1.5772

1.0541

1.2002

1.3163

i.45431.5540

1.6s16

1.7793

0.388860.64782

0.96432

i.32741.5021

1.62771.7697

1.8805

2.0049

2 2 3 SS

2.66682.7401

3 1 47 1

3.4753

3.81254,26414.6088

1.8751

2.0012

2.2332

2.46S2

2.7391

3.1466

3.47503.8123

0.752451.2053

1.8477

2.25412.6154

3.086)

3.4608

5. lo%

I . lo~2> 106

5. 104

1. 10?2 k 10,

5. 1071 x 108

3.7934

4,255s4.6041

4.26404,6087

4.26334.6083

Infinity.

limiting

s =

Table

Ftom Eq. 15, it can be seen- ‘that the

negative skin effect (as k l + CO) isTABLE 4—OD(,DI VSIDFORS– +5, r,D - 1

A-

;51. 10

2> 108

5, 10’

1, 10’

2, 10’

5> 10’I. 103

2. 1035, 103

I, 104

2, 1045, 104

1, 105

2, 1055> lo,

I Y 10*2. 106

5, m

1, 10,

2 x lo,

5, lo?

1, 108

10 100 1,000 10,000

.— —

100,000

0.098841 0.0099724o.o199m

0.049695

0.0990W0.19650

0.48083

0.939901.7463

3.68265.7702

0.000998120.0019961

0.00498890.0099739

0.019933

0.0497240.099108

0.19692

0.000099818 0.00000998040.000 1W63 0,000019956

o.ooc14wr17 0.000049899

0,@32W811 0.0C029930Q

0.0019961

“nrl D.”” ”””” ””. ”( 17)

5,7705

5 9 %5 5

63405

6.6323

6.9425

7.37287.7076

8,0471

8. 2018.8446

9.18969.6660

9.992.2

10.338

0.794

11.141

11.487

11.944

12.291

12.636 3.094

13.444

13.786

14.243

14.389

0.196030.47808

0.923071.7112

3.5’2395.36577.036 I

& 2S568.74169.1299

%62609,9818

10,33210.792

0.00019962

0.00049906

0.000998120,w1W61

0.CH349S9S.

0.0099760

0.0199410.0497700.099279

0.00698920.0099730

0.019923

0.0497490,099200.197a

0,48521

0.945721,8005

3.9259

1 shows the values of (k I/k), which

correspond to the various va lues of r lD and s in

the study. The first two entries for s = –5 are

blank because this negative skin effect is not

obtainable for rlf) = 10 or 100, according to fZq. 15.

RESULTS

Finite-difference solutions were generated for

a totai of 60 cases. ~-~1-- ‘-l -L-- .. ..h 1 1 -;.?- ~~~1i3U1C> L LLuuu~ll LL ~AVL

results of these computer runs . Each table gives

sEPTEMBER, 1970

0.48326

0.9s751,7767

3,8169

6.1196

8.5@3010,29611.002

7,80119.3341

9,871410.39110.772

0, 197s5

0.48672

0,951%1,8204

4.0193

6.6834

9,727012.46713.256

11.140

11.487

11,944

12,290

12.626 3,094

13.440

13.786

14,243

14,589

11.130

1 ,481

11,941

12.i89

6,4242

9.1463

11.41

12.131

12.365

13.066

13.426

13.779

1&‘m

13.708

14,213..—.14.ar44.568

293



.

TABLE 5 —pD (fD) VS t D FOR S= +S, rtD= 10the results for given values of s and r~D. In each

table the various wellbore storage cases are listed,

ranging from 77= O to ZT= 100,000.

Only the solutions at selected values of tD are

shown. Smaller time steps were taken between the

printed results to reduce the time truncation error.

A total of 816 time steps were taken for each case.

The cases of infinitesimally thin skin (r~D = 1)

were compared to the ‘‘analytical” solutions of

Ref. 4. Solutions in Ref. 4 were obtained by the

numerical integration of an inversion integral. The

agreement h =rv~~~ & ~~~ ~~~ca .,-. hi.:,..-. e-W. u ..””= al-id

finite-difference solutions was excellent. The

maximum differences between the two were about

~=~ 10 100 I ,w a 10,COOA.. . ——— 100,0001 1.5404 0.094963 0.0099317 0.00299772 0.WW99817

1

0.0WO09W23

2.0907 0,18566 0.019814 0.0019949 0. WO19963

5

0,00@319965

2.9418 o.439a4 0,0492S9 0,004984S 0,00049905

1,103

o.omo4991 1

3.7167 0.82430 0.097818 0.0099619 0. WW9802 0,000399821

2X1O’ 4.5849 1.4903 0.19336 0,0199W O.w 1W58 0.0C019964

Sxlof

1 x 102

2 x 102

s x 102

1 x 1032 x 103

5 x 103

1 x 104

2 x 104

s,842a

6.813s

7,6336

8.37~

0.78789.1627

9.6336

9,9m.9

10,33s

2.’W87

4,5972

6.3326

8.0045

8.66179.1087

9.6147

9,9764

10,320

0,47009

0,906811.6942

3.3776

5.66Il 17,6W7

9.3(E.9

9,8637

10,278

0.04 3608

0.098820

0,196250.68142

O.93XI61.7698

3,8060

6.1034

8.6872

0.0049%8 1

0.0099721

0.019934

0.0497%

0.0991630,19719

0.483W

0,94S31

0,W0499Q8

0.000996120.0019961

0,0069893

0.00997560.019940

0.04W67

0.099275

0.19754.7998

s x 1041 x lo~

2 x 10>5 x 105

I x . lo ~

2 x 106

:: ;;;

2 x 107

5 x 107

1 x 108

10.793

11,141

11,487

11,946

12290

12.626

13,094

13.46013.786

14.243

10,791

11,140

11,486

11,943

12.290

12,6%

13.094

13.660

13.706

14.243

10,771

11.122

11,431

11,941

12.~9

1Z626

13.093

13.439

13.736

14.243

10.392

11.002

11.423

11,919

12.278

12.630

13.091

13,4=

13,785

14.243

3.9245

6,4224

9,1448

11.440

12.131

12.565

13.046

)3.426

13.779

14.24U

14.388

0.686700.95134

1.8X33

4,0191

6.6832

9.7268

12.467

13.236

I 3.708

14.213

14.3744.5%9 14.389 14.389

TABLE 9 —pD(tD) M ,D FoR ,=+N, rD= 10TABLE 6 — ,,D – Im

10,000.—

0,0W0998160.00019963

0.W049903

0.W0997960,C419956

0.00498700.0099684

F= 0 10-—

0.093595

0.18161

0,42415

0.779731.3s75

2,5813

3.74s3

4,9155

6,26637.1110

7.9057

100 1,000.—

loo mo ?-0 10

0=5

0. 192*

0.44780

0,905601.7219

3.8s606.7178

10.915

17.755

10 0

0.009958 I

0.019890

0.049562

0.098823

0.19658

0.485260.936421,85s8

4,32097,8278

13,033

20.652

24,032

2s.112

25.706

26.08526.466

I ,wo

0.00099798

0.m19957

0.W249878

0.0099722

0.019933

0.0497710,099372

0,198170.49182

10,W3O

0.LWO09982U

0,Wi719964

0.W449’W8

0,W0998120.G121996’l

0,W498970.L?Q99777

0.019950

0.049837

0,099564

10 0mo-o.omm99823

0.WW19965

0.00W49911

0,mm9m220.00019964

0.000499 IO0,0W99818

0.0019962

0.00499040.0099796

3.1149

4.1034.2271

1,6117

2.2328

2.7824

0,00991680.019771

0.049064

0,000997570,0019965

0,0049025

0.00995S50.019879

0,0495020.0964%

0.19.9220.475s3

0.0LxXH299823

0.WQ1319965

0.00W49911

0,0000998210,W21W64

0.WQ$9’W70.00099809

0.001995.0

0.W49887

0.0099726

0.01W34

0.049742

o.099m7

0. 1972a

0.6862Q

0.95029

2

:1,10

2x10’

5,101

I , 102

2 h 10’5X 1021 x 103

2 x 103s x 103

51.10

2X1O’

s. 10’I x 102

2x 102

:: ;::

2. 103

; H m:

2 x 104

5 x 1041 x 10,

2 x 105

s x 1051 x 106

2 x 1065 x 106

I h 107

2 x lo~

5. ]o~

1. 10,

5,9241

7.7232

9.8762

13.21216.034

18.994

0.0971980,19135

0.660d7

0.873861.5999

3,23804.9438

3.sass

4.2446

4.925S

5.6231

6.%017.2757

7,9935

0.019917

0.049667

0.098965

0.196590.68260

0.939151,78S7

3,8917

6.2749

22.236

23.472

24.040

24.576

24.945

25.241

25.764

26.11326,440

26.9)7

27.264

27.610

20.0+7

28.413

28.759

29.21629.s43

21.884

23,726

24.514

24.917

2s.287

25,759

26.110

26.45326.916

27.264

27.610

28,067

28.413

23.759

29.216

29.563

0.972811.9068

4,4793

s3.134213,W7

21.386

25,165

26.266

0.917241.7203

2.650 I

5.8329

8, 18Q8

10.257

10.959

11.607

0.198700.49263

0,97722

1.9155

4.5156

8.226213,849

0.019955

0.049856

0.0996090. PW81

0.4940s

0.978571,91994:s372

8.305214.0S4

22,932

37.170

28s3429.151

29.532

6.60898,4575

9.396110,07S

10.706

11.10011.4s7

8.92929,6076

10.660

10.733

11.112

11.473

i L938

12.288

12.635

13.093

8.90139.sa94

10.152

10,730

11.111

11.473. . .

l.Y.M

12.288

12.635

13,093

13.439

I x 104

2 x 104

M:

2 x lo~

5X 10.

I. 10e

2x. 1065.106

1.w7

2. 107

5. 1071 x 108

9.0989., . . ., l..’.

12.127

12,564

13.065

13,425

13,779

14.240

14.s88

1,8182~,o;a

6.677)9,7210

12.464

13.2M

13,708

14.21316.574

26.911

27,Z1

27.608

28.06728.413

28.75929.216

29,%3

26.r537

27,23S

27,395

23.061

Z.41O

~.758

29.21629.563

22.2?5

26,183

27.601

28,m4

B.363

28.745

29.21029.560

--,I I., SJ

12.227

.....,,.,,.12,275

12,6%

13,090

13.4%

13.785

14.243

14.529

12.634

13.093

13.439

13.786

14.24314.369

13,439

13,786 13.786

14.243 14.243

14,589 14.389

TABLE 7 — ,eD(tD) vs ,0 FOR s= +5, ,,D= 1,000

F= o 10 100 I 000 lo,om 100,030—— —

TABLE 10 — m (*D) VS tD FOR ,=+22, rm = 106

75.0

2.16372.8709

4,1026

S.2671

6.6105

8.6171

10.265

11.989

14.261

16.153

17.982

20.410

22.246

23.939

25,605

2S.981

26.401

X , 895

27.2s3

27.604

~.06S

28.412

28.759

29.216

29.563

10 10 0 ),om-— —

lo , mo

0.000099819O.OOQ19963

0.00049906

0.00099808

0.0019960

0.&2498W

lW,C.20

0.00W0998230,000019+6S

0.0W”24W11

0.000099822

0.0W19964

0,0004W09

~

12

51.10,

2> 10,

5.10

1.102

2.10,

5x102

I. 103

2. 1035. 103lx 104

2. 104

5. 104

lx 101

2x lo~

5x lo~1X 106

2X 106

5x106

lxlo~

2.107

5X107lx lo~

0.0a66o. 179s50.415780.756181.3068

2.2S74

3.2684

4.3105

5.2678

6,C418

6.7088

7,s293

8.1330

8.74039.5321

10.131

10.7~

11,309

12,03E

12,526

13.054

13,421

13.77714.239

14.W

1.092S

1.42391,9643

2.4324

2.9415

3.64104,2a3

4.8076

5.5$40

6.1762

6.7700

7,53S2

8. 1S23

8,74809.5354

10,133

10,7B

11.. 0912.058

12,S26

13.054

13.421

13.777

14.23914.5S8

0.009W86

0,0197470.0489550.096850

0,19022

0.4S507

0,8%621.5474

3.035$

4.4972

5,8823

7.2335

0.00099768

0.00199430.0049814

0,00995180.019867

0.0496410.098245

0, 194=

0.47174

0.000099815

0,CJ20199620.00049902

0.000997920.00199S5

0.0449864

0,0399663

0.019910

0,04%27

0.WIOW199823

0.0000199650.000049911

0, W2WW200.00019964

[email protected]

O,W 19959o.m49a83

0,w99721

0.019923

0.049712

0.099097

0.19698

0.6S416

0.942761.7989

3,96826.3643

9.3482

12,348

13.325

13.696

14.209

14.572

0 ,096322 0 .W996610.18919 0.019856

0,45SW 0.049428

0,86832 0.098379

.6145 0.19517

3.42S0 0.47870

0.002997860.0019954

0,0049864

0.0099676

0,019919

0,049701

25

1X1OI

2X1095X 104

1 x 102

2 x 102

s x 102I. 103

2 x 103

s x 103

1 x lo~2.104

5 x 1041 x 10s

2 x lo~

5 x lo~1 x 10*

2 x loo

s x lo~

1 x 107

2 x lo~

5. 107

lXICF

S.6514

8,4988

12.491

15,153

17.453

30.18022.124

23.88325,393

23,977

26,400

26,894

27.253

27.604

2U.W.528,412

i9.759

29.216

39.563

0.93297

1,7865

4.0154

6,9544

11,035

16.980m.62u

23,243

2s.260

25,960

0.099137

0.19737

0,68775

0.95960

1,8629

4.3123

7,7255

12.797

m,6m

24.672

0.0099754

0.019942

0.049796

0.099423

0.19824

0.49162

0.97146

1,9003

4,4707

0,0W99816

0.0019962

0. W49900

0.0099782

0.019950

o.049a3s

0.099530

0.19865

0.49352

0.904321,6778

3.66e6

5.3747

7.2963

9.0704

9,9275

10.638

1}.669

12.02+

0.0%8 18

0.19606

0.479730.92903

1.7517

3.75W

6,0415

8.52-?310,867

11.a22

7.9929

8.6671

9.S212

10.I15

10.72Q

11.506

12.0%

12,526

13.053

8.1S11

13.72U

22.152

26.12a

37.394

=.002

‘m.282

2s.744

J2.97738

1,9174

4,5311

8.2945

14.039

22,918

27.164

28,533

26.345

26.889

27.2%

27.603

2s.064

29,412

22.7s9

26.13a

26.831

27.223

27.3W

29.059

Z.609

26.79

12.518

13.051

12.435

13.023

13.426

13.76914,236

14.586

13.42013.777

14.23914.588

13.41913.776

14.239

14.SS729.216

29.563

29.216

29.%2

29.210

29.560

29.1s1

23.531

TABLE 8 — p= (,Dl VS tD FOR s= +%, rm= 1TABLE 11 —p.D (tDl VS ,D FOR ‘=+X( rm= l,OW

?= 10 100A___ ———

,000 lo,mo lW,WO F= o 10 100 I,W2 Io,ow 100,WO—. —— _

1 20,744 0,099558 0. W99796 0.00399819 0.00W998152

0.GfKI0099765

20.%9 0.19865 0.019954 0.0019963 0.00019961

s 21.3140.000419942

0.49306 0,049830 0,0049904 0.0W4990S 0.00W369870

lxlo~ 21,606 0,97667 0,099583 0,0099797 0, W099808 0.00W1997%

2XF2* 21,916 1,9047 0.19870 0,01W5S 0,0019962 0.00019954

5xml~ 22,366 4.4528 0.49334 12.049853 0, W49W4 O.W049899

1,7887

2,2707

3,3%3

4,2706

5.2044

6.8217

8.05109.3265

I 1.036

12.384I3.720

15.493

16.83

18.184

0.0955S4

0.18705

0,64633

0.84322

1.5429

0,0099379

0,019832

0.069337

0.098063

0.19415

0,47387

0,917iU1,7353

3.79m

6,33079.5684

13.687

16.WO17,770

}9.796

0J30QW778

o.m~995~

12.oo49a53

0.W99644

0.0 IW08

0,049649

0.098961

0, 1%77

0 ,0 w 0 w 8 ) q

0.0001W63

o.om4990s

0.0009W05

0,001W59

0.0049345

0,00997%

0.019935

0,049763

0.000WW823

o.omowms

7MOO0499 t1

0.0CK099822

0.00019964

o.om49909

0.00099314

0.00199620.0049896

o.m9977 I

0.019946

3,1747

5.0234

7,216110.023

11.83813.431

15.356

16.771

18.147

i x 102

2 x 102

:;}:

2 x 103

5 x lo~

I x 104

2 x lob

5X104

22,68 I

23.021

23,474

23.818

24.163

24.6m

24,966

25,311

25.768

8.W7413.163

20.273

23.1 IO

24.mo

24,565

24,939

25,2W

2S,763

0.97s I

1.9086

4.4757

8.0899

13.608

X.%624,149

2s, I33

2S.711

0.099593

0.198740.49360

0.976691.9127

4.4979

8.165513.632

21.621

0,0099798

0.01 W55

0.049856

0.099604

0.19879

0.4928s

0.977671.9164

0.W099807

0.0019962

0.W49904

0.00W7W

0.0 199s

0.0498390.099614

0.19882

0.68462

0.94352

1,8255

4.1312

7.176111.2M

o.owm80.19783

0.6893s

0.963201.8713

4.3% I

7,721512.6V19,995

0.049812

0.099665

0.19834

0,49181

0,971521,8984

4,4514

8,092513.633

4.5176 0,49408 19.962

21,313

22.660

24.424

25.76S

26.W3

27.869

za.330

2U.721

29.202

19.947

21.205

22.b55

24.437

25,764

26.W8

27,869

2a.320

23.72129.m2

29,5s

17.031

a. 10022.KZ

24.231

2S.664

26.867

27,Mo

2%.336

28.719

29.X7139.555

1 x 1032 x 10s

s x 105I x 106

2 x 106

5 x 1061 x lo,

2 x 107

5 x 1071 x 108

26,115

26,6$1

26,917

27.264

27.610

m ,J67

=.413

28.759

29,216

29,563

26.112

26.439

26.91727,2S4

27,61028.067

28.413

m.759

29.216

29.563

2b.087

26.44726.912

27.26}

27.608

28067

~.413

28.759

29,216

29.563

25.178

36.269%.89

27.235

w .596

23.061

28.410

23.73029.216

m’,363

8.2396

13,854

22.279

26.105

27,601

23,m4

B,ra3

m.745

29.21039.s0

0,97862

1.92u3

4.5374

8.2055

14.055

22.933

27,171

23.534

39.15129.532

21.227

22.615

24.419

25.75S

24.904

2T,868

23,330

23.72129.202

29.53s

24.059

26.32627.761

ZU.293

2U.705

29.19s29,553

22.281

26.866

33,43

29. 2429,524?.556

294



‘“~

4tt4LyT C&L —

F,N, TE OIFFERENCE -

I

tc

FIG. 1 — /J L) ( D) VS tD FOR ANALYTICAL AND

F INI TE -DI FF ER ENCE S OLUTIONS S= O .

0 .5 percent, with the analytical solutions being

higher than the finite-difference solutions. This

maximum difference occurred for the larger values

of ? at the transition between the wellbore

storage-controlled period and the period in which

wellbore storage was not importanr. On Q log-iog

plot of p~(t~) vs tD this is the region of greatest

curvature. At other points on the curve the solutions

are almost identical.

Fig. 1 shows a comparison of the aiiaiyti~ai

solutions from Ref. 4 (solid lines) and

finite-difference solutions (points). The differences

between the two are not noticeable on a graph. All

of the curves in Fig. 1 are for no skin effect, s = O.

Fig. 2 presents results for skin effects of –5, O

and +20. The skin effect of +5 was not plotted to

make Fig. 2 more readable. Dimensionless pressures:~,~a ~ii e

nlnrfed with skin.v---- effect,

d imension less radius of the damaged zone, and the

dimensionless wellbore storage constant as

parameters. The results for the skin effect of -5

are only for a dimensionless ‘ ‘damaged” zone

radius of 1,000. It was impossible to achieve a

negative skin effect as large as -5 with ~~D values

less than 150. Thus, the data for a skin effect of

+20 on Fig. 2 are best for studying the effect of

the radius of the damaged region upon short- time

well-test data. Results for a skin effect of +5 were

simiiar.

DISCUSSION

AS shown in Fig. 2, by comparison of results

for different rl~ values, representation of a skin

effect as an annuiar region of aitered permeabilityl-=, -l @cart .> ..* sigcificarrt chan~es in the early

pressure-time history, with or without wellbore

storage. (All lines on Fig. 2 for rl~ = 1 represent

infinitesimally thin skin cases, or previous

analytical cases. ) The lines for the finite-radius,

damaged regions fall successively below the

infinitesimally thin skin case, but finally join the

infinitesimally thin skin case at times which

increase as the damage radius squared, as would

be expected. If there is significant wellbore

storage, a separation is noticed within the

transition region from storage control to outer

formation control, as damaged-zone radius increases.

The separation diminishes with increasing weiibore

storage constant and is essentially negligible for

storage constant of 100,000. This results because

the finite-storage constant cases must join the

zero storage case. “--:.-.:- -. .helnus P~. i?ViOiiS LIIKlla LUl L1lC

duration of the wellbore storage effect a requires

some modification. If the damaged-zone radius is

great, transients caused by the large volume of the

damaged region may last longer than those caused

by weHbore storage. In extreme cases, this could

result in an early period caused by wellbore storage,

followed by two straight Iines on a conventional

~emli]og ~e]].te~~ P]QC — the first having a sloPe

indicative of fie permeability of the damaged region;

the second having the correct slope indicative of

the undamaged formation. This is shown on Fig. 3.

Fig. 3 presents results in a conventional semilog

plot of the pressure-time data for a skin effect of

+5, a storage constant of 1,000, and two different

damaged-zone radll: ~lD of 1 and 1,000. pressure. .

data to a time of about 2.5 x 103 are almost

S*2O

-C. O rlo=i,

/ il Kr

1. .

I// 1 1 1,“

I 10 102,. 3

1 0104 105

,06 107

FIG. 2 — SHORT-TIME SOLUTIONS WITH WELLBORE STORAGE AND FORMATION DAh lAGE

sEP’YI:MIII:I{. lv70 295



cQ.rn.pje.@7 ,-L-,m;., c.cl by : ~,e “ alll..-..a-..”,.=..”,..l., c.. &u WL. .”” c

-’u’’=~~

effect. This can be seen better on a log-log plot;

the data form near perfect straight lines with a unit

slope on a log-log plot. At times of about 2 x 105,

both cases reach a straight line. For the

infinitesimally thin skin case, 71D = 1 the slope

is the correct value of 1.151 and is indicative of

the formation permeability. For the large-radius

damaged region case, the slope is 1.984, which is

indicative of the damaged zone permeability (see

Table 1). At a time of about 4 x 106, the

large-radius damaged region case finally reaches

the proper straight Iine. As can be seen on Fig. 3,

the proper straight line for the TID = 1,000 case

could be easily misinterpreted. It is far easier to

interpret this case with a Iog-log plot and type-curve

matching procedures, q and misinterpretation is far

less likely. Type-curve interpretation procedures

are discussed in detail in Ref. 7.

Several comments regarding well-test

interpretation may be made utilizing Fig. 2. First,storage constants for oil well tests often are of the

order of 1,000 or Qreater, nsrwction CIf Fig. ~.– —.-. =-------

indicates that the radius of the damaged region

would have to be greater ban 100rw to cause a

significant effect on the pressure-time history.

Thus, previous interpretation methods (such as

described in Refs. 4 and 7) should be valid. In

other words, the infinitesimal skin concept is valid

under these conditions. Second, storage constants

for gas well tests may be much less than 1,000.

Inspection of Fig. 2 indicates that the radius of

the damaged region may play an important role in

pressure-time data for these conditions. Thus Fig. 2

represents an additional type-curve which may beparticularly useful for interpretation of well-test

data when storage constants are small. In our

opinion, type-curve matching procedures are

extremely useful and will find increasing application

in well-test analysis, Finally, it is apparent that

Fig. 2 should also be useful in forecasting results

of production from a composite reservoir by proper

interpretation of the skin-effect parameter.

:.

:6 .

c0N01710N5

5.+5

T . ,000

4kl/k .059

I NI TI AL P fR ,0 0 TR , W3,,,0N mow

5TORACE CONTROL 4 ’

WELL BORE STWAGE ) T O%l RA , G ” T . L , N E.

2 -

0102 K)3 104

to105 ,06 I07

FIG. 3 — EXAMPLE OF BUILDUP DISTORTION WITHWELLBORE STORAGE AND F ORMATION DAMAGE .

------ .rm. n.,

~UN~L U>lUN

The infinitesimally thin skin concept of

van Everdingen and Hurst is applicable if wellbore

storage is significant for damaged-zone

dimensionless radii from 1 to about 100. If the

damaged-zone radius is as large as l, OOOrw, two

straight lines will be evident on a semi-logarithmic

plot. If the damaged-zone radius is small, it is not

possible to detect the value of the radius from

well-test data.

If the damaged-zone radius is greater than 100rw,

short-time well-test data should be interpreted by

means of solutions for a skin region of finite

storage capacity. One way to accomplish this end

is a type-curve matching procedure employing

plots such as Fig. 2.

If wellbore storage is not significant, a

damaged- or skin-region radius of IOYW or greater

may distort the early pressure data significantly.

In this event, two straight lines may appear upon

semi-logarithmic plots.

c=

c.

c-

l?=

k =

k l =

p=

i

plD =

P2LJ=

~w D t D) =

q=

r=

T1 =

T~.

TD =

‘lD =

s=

t=

tD =

=p.

NOMENCLATURE

total system compressibility, atm - 1

wellbore storage capacity, cc/arm

dimensionless wellbore storage con-

stant, Eq. 14

formation thickness, cm

undamaged formation permeability,

darcies

permeability of “damaged” region,

darcies

pressure, atm

initial pressure, atm

dimensionless pressure in ‘ ‘damaged”

region, Eq. 11

dimensionless pressure in ‘ ‘undamaged”

reservoir, Eq. 12

dimensionless pressure in wellbore

surface flow rate, cc/second

radial distance, cm

radius of “damaged” region, cm

wellbore radius, cm

dimensionless radius, Eq. 9

dimensionless radius of ‘‘damaged”

region: rll’rw

dimensionless skin effecttime, seconds

dimensionless time, Eq. 10

porosity, fraction

viscosity, cp

ACKNOWLEDGMENT

The computer time required for this study was

provided by Stanford U. Financial support was

received from Stanford and the National Science

Foundation. The

this assistance.

authors gratefully acknowledge

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Ramey_ Wellbore Storage II

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