We present an analysis of the slug test in a well surrounded by an annulus of altered material, which is treated as a skin of finite thickness. By assuming the skin has a thickness, the storage capacity of the altered material is included in the analysis. The problem is solved in the Laplace domain. The solution is found in terms of well-bore storage and the thickness, hydraulic conductivity, and specific storage of the skin. Type curves are generated by numerical inversion of the Laplace transform solution. We find that standard methods of analysis, involving a skin of infinitesimal thickness, are adequate for open-well or drill-stem tests. However, for pressurized tests the response may differ markedly from standard slug-test solutions

Hsieh, P. A.

Moench, A. F.

UNT Digital Library

.'PROCEEDINGS, Tenth Workahop on Ceothernal Rescrvuir Engineering Stanford University, Stanford, California, January 22-24. 1985 SGP-TR-84 SLUG TESTING IN WELLS WITH FINITE-THICKNESS SKIN A.F. Moench and P.A. Hsieh U.S. Geological Survey Menlo Park, CA 94025 ABSTRACT We present an analysis of the SlUg test in a well surrounded by an annulus of altered material, which is treated as a skin of finite thickness. By assuming the skin has i i  thickness, the storage capacity of the altered material is included in the analysis. The problem Is solved in the Laplace domain. The solution is found in terms of well-bore storage and the thickness, hydraulic conductivity, and specific storage of the skin. Type curves are generated by numerical inversion of the Laplace transform solution. We find that standard methods of analysis, involving a skin of infinitesimal thickness, are adequate for open-well or drill-stem tests. However, for pressurized tests the response may differ markedly from standard slug-test solutions. INTRODUCTION The slug test is known to be a valuable tool for evaluating the hydraulic properties of water-bearing formations. It has the advantage that it is simple and inexpensive to run and can usually be performed in a short period of time. It has the disadvantage, however, that only a small rFgion around the well bore is investigated. Consequently it is extremely sensitive to the hydraulic properties of the material in the immediate vicinity of the well. In many instances the hydraulic properties in this region have been altered by such factors as invasion of the pores or fractures of the formation by drilling mud, the buildup of scale due to chemical precipitation, or well-stimulation endeavors. Historically, this annulus of altered material has been treated as a skin of infinitesimal thickness. Ramey and Agarwal (1972) presented an analytical solution for a Slug test in a well with a skin of infinitesimal thickness and Ramey et al. (1975) showed that the solution could be approximated by a single set of type curves. In this paper, which is a revised and abbreviated version of a paper by Moench and Hsieh (1985a). we present an analysis of a slug test in a well with a skin of finite thickness. The solution was found in the Laplace domain and type curves were obtained by numerical inversion. Using realistic values of aquifer and skin hydraulic properties, we show the effects of a finite- thickness skin on slug-test responses f o r  both open-well, or  drill-stem, tests and pressurized tests. MATHEMATICAL MODELS Skin of Infinitesimal Thickness Except for the presence of skin the assumptions of Ramey and Agarwal (1972) are the sane as those of Cooper et al. ( 1  967). These assumptions are that the well fully penetrates a confined, homogeneous and isotropic aquifer of constant thickness and of infinite lateral extent. It is also assumed that the hydraulic properties of the skin and aquifer remain constant for the duration of the test. -169- CONF-850107-29DISCLAIMER  This report was prepared as an account of work sponsored by an agency of the United States Government.  Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.  Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.  The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER  Portions of this document may be illegible in electronic image products.  Images are produced from the best available original document.  In the notation of this paper, the solution in the Laplace domain can be written as where o is the skin factor, which is a constant of proportionality relating the flux of water through the akin to the difference in hydraulic head across the skin; p (=q2) is the Laplace transform parameter and is inversely related to the dimensionless time; and Y is the dimensionless well-bore storage. The mathematical notation is collected and defined at the end of the paper. Skin of Finite Thickness Figure 1 is a schematic diagram of a slug test conducted in an open well surrounded by a finite-thickness skin. Except for the presence of the finite-thickness skin, the assumptions are the same as those of Cooper et al. (19671, outlined above. The boundary value problem that describes the slug test depicted in Figure 1 is set up formally by Moench and Hsieh (1985a). The mathematical problem can be solved in a straightforward -Equation (2) cannot be easily inverted analytically. In this paper type curves are obtained for equations ( 1 )  and (2) by numerical inversion using the Stehfest (1 970) algorithm. The accuracy of the numerical procedure was checked for the case of a = 1 and 6 = 1 by comparison with tables published by Cooper et al. (1967). I WATER LEVEL AT TIME 0 I I I I  I IMPERMEABLE Figure 1. Schematic-diagram of a slug test conducted in an open well surrounded by a finite-thickness skin. manner by the Laplace transform method. Laplace transform solution for the dimensionless head in the well is The ( 2 )  RESULTS It is evident from the number of dimensionless parameters involved that numerous sets of type curves can be generated from equation ( 2 ) .  To illustrate the effect of the finite-thickness skin on well response, we present some typical type curves f o r  both open-well tests and pressurized tests. Open- well tests are distinguished from pressurized tests by the magnitude of the well-bore storage. For open-well tests the well-bore storage is dependent upon the cross-sectional area of the riser pipe: for pressurized tests it is dependent upon the fluid compressibility and compliance of the well-test equipment. Because the well-bore storage of a pressurized test may be four to five orders of magnitude smaller than that of an open-well test. the dimensionless well-bore storage parameter ( Y )  -170- may be four to five orders of magnitude smaller for pressurized tests than for open- well tests. For open-well tests, Y generally ranges from lo2 to l o 6  and for pressurized tests, Y generally ranges from lom2 to 102. Open-well tests In this section we compare type curves for a slug test in a well with a skin of finite thickness with corresponding type curves for a skin of infinitesimal thickness. Moench and Hsieh (1985a and 1985b) show that the solution for a finite-thickness skin with negligible skin storativity is equivalent to the solution for a skin of infinitesimal thickness provided 0 ,  t, and Y in the latter solution are replaced by 2 t* = t/rs 2 and Y* - Y / r S  Ramey et al. (1975) showed that the infinitesimally thin skin solution can be represented as a function of two dimensionless groups: t/Y and Ye2' . storativity is negligible, the finite- thickness skin solution should also be a function of two dimensionless groups: t * / V  (=t/Y) and Tie2'*. Thus, if skin Figure ( 2 )  shows type curves generated from the finite-thickness skin solution using realistic skin and aquifer parameters. A value of 6-1 is used because it is considered unlikely that the porosity and rock compressiblity in the altered region will differ significantly from that of the aquifer material. Values of Y*e2'" are shown in the inset. Using the definitions (3)-(5), we also generated type curves from the infinitesimally thin skin solution, equation (1) .  When compared with one another, the two sets of type curves are found to be identical. For these parameters the same relationship was found to hold for all values of Y equal to or larger than lo2.  well slug tests, the skin storativity may be neglected, and the skin thickness may be considered as infinitesimally thin. This suggests that for open- 0.8 \ \  \ i a =  1 \ \l" \ I  0.6 t '  1 u- \ \  \ 4 10 2 . 6 ~  lo8 100 4 . 0 ~  Figure 2. Type curves for an open-well slug test for Y=103, rs-2, 601, and the indicated values of a. Inset shows the corresponding values of y*e20*. For open-well tests, in order to show a deviation from the solution for a skin of infinitesimal thickness, it is necessary to use a very large value of skin storativity. Figure 3 shows a comparison of type curves similar to those in Figure 2 but with a value of skin storativity that is unreasonably large. At early time, the finite-thickness skin solution shows a marked departure from the type curves for a skin of infinitesimal thickness. With an increase in rs it was found that the departure is even greater. Thus if 6 and/or rs are very large, the equivalence between the solutions may not hold. From the definition of the dimensionless quanitiies t and Y,  we find that Thus with known C. a match of slug test data to an appropriate type curve should yield an -1 71- estimate of the  t r a n s m i s s i v i t y  (K2L) of the tes t  i n t e r v a l .  For l a r g e  v a l u e s  of a, s i n g l e  d imens ion le s s  group,  a2 t .  S u b s t i t u t i n g  however, there is nonuniqueness i n  t h e  match and an a c c u r a t e  estimate of t r a n s m i s s i v i t y  cannot  be ob ta ined .  I n  e q u a t i o n  ( 7 )  h.J is a f u n c t i o n  of a t h e  d e f i n i t i o n s  of t h e  d imens ion le s s  q u a n t i t i e s .  we f i n d  a2t  = 4n2(rwv )2K1Ss1L2t’/C2 ( 8 )  We expec t  t h a t  f o r  very small Y,the a sympto t i c  s o l u t i o n  ( 7 )  w i l l  a c c u r a t e l y  approximate t h e  head r e sponse  of a p r e s s u r i z e d  s l u g  tes t .  Thus,  i f  s k i n  is p r e s e n t  and Y is s u f f i c i e n t l y  small, a match of s l u g - t e s t  data w i t h  t h e  s o l u t i o n  ( 7 )  w i l l  y i e l d  o n l y  t h e  p roduc t  KISsl .  and no in fo rma t ion  abou t  the a q u i f e r  h y d r a u l i c  p r o p e r t i e s  w i l l  be o b t a i n e d .  I n  t h e  absence  of  s k i n  ( 6  - 1 and a = 1 )  e q u a t i o n  ( 7 )  becomes i d e n t i c a l  t o  t h e  short-time s o l u t i o n  of Bredehoeft and Papadopulos ( 1  980). 103 102 101 100 101 102 103 t I Y  Figure  3. Type cu rves  for an open-well  s l u g -  t es t  for  Y=103, rs=2, 60100 and t h e  i n d i c a t e d  v a l u e s  of a. h, = ea2, erfc(/&) 08 P r e s s u r i z e d  Tests An i n c r e a s e  of complex i ty  is encoun te red  i n  a n a l y z i n g  a p r e s s u r i z e d  s l u g  test conducted i n  a w e l l  surrounded by a f i n i t e - t h i c k n e s s  s k i n .  Because wel l -bore s t o r a g e  ia small, the  e f f ec t  of t h e  wel l -bore s k i n  is enhanced. I n  g e n e r a l ,  the  s t o r a t i v i t y  of  t h e  s k i n  canno t  b e  n e g l e c t e d .  As a r e s u l t ,  the head i n  the  well may respond t o  s k i n  p r o p e r t i e s  a t  e a r l y  time and t o  a q u i f e r  p r o p e r t i e s  a t  a la ter  time. T h i s  r e sponse  is markedly d i f f e r e n t  from the  s t a n d a r d  t y p e  c u r v e s ,  and data a n a l y s i s  by cu rve  matching can be ex t r eme ly  d i f f i c u l t .  Because t h e  r e sponse  t o  p r e s s u r i z e d  tests may be as much as f i v e  o r d e r s  of magnitude t \ = . a = - i o  i F i g u r e  4. Type c u r v e s  f o r  a p r e s s u r i z e d  s l u g  tes t  f o r  Yp10-l .  rs=2, 6-1, and t h e  i n d i c a t e d  v a l u e s  of a. The a sympto t i c  s o l u t i o n  f o r  shor t  time is a l s o  shown for comparison. F i g u r e  4 shows t y p e  c u r v e s  f o r  more r a p i d  than  t h e  r e s p o n s e  t o  open-well  tests, it is advantageous t o  seek t h e  s h o r t -  time asympto t i c  approximation t o  e q u a t i o n  (2). T h i s  is given by Moench and Hsieh (1985a)  i n  real  s p a c e  as p r e s s u r i z e d  s l u g  tests wi th  Y = 10-l .  short-time s o l u t i o n  given by ( 7 )  is a l s o  shown f o r  comparison. Note t h a t  t h e  short-time s o l u t i o n  a c c u r a t e l y  describes t h e  type  cu rves  ove r  a l a r g e  p e r i o d  of t h e  head response.  If Y is decreased t o  or l ess ,  t h e  t y p e  cu rves  w i l l  be i n d i s t i n g u i s h a b l e  from t h e  s h o r t - t i m e  s o l u t i o n .  The hw - e x p ( a 2 t )  e r f c [  (a2t)1’2] ( 7 )  where a2 = 6 / ( a Y 2 )  -172- 0.6 3 - 0.4 0.2 0.0 10-3 10-2 io-' ioo a2t Figure 5. Type curves f o r  a tes t  f o r  Y = 1 ,  rs=2, 6 = 1 ,  and va lues  of a. The asymptotic s h o r t  time is a l s o  shown f o r  Figure 5 shows tha t  the  101 102 lo3 pressurized s l u g  t h e  ind ica ted  s o l u t i o n  f o r  comparison. aqui fe r  p r o p e r t i e s  begin t o  a f f e c t  t h e  shapes of t h e  type curves as Y is increased.  For a tenfo ld  increase  i n  Y ,  t h e  type curves show dramatic d i f f e r e n c e s  from those of Figure 4. I n i t i a l l y ,  t h e  type curves fol low t h e  shor t -  time s o l u t i o n ,  i n d i c a t i n g  t h a t  t h e  head response i n  the well is pr imar i ly  c o n t r o l l e d  by the  hydraul ic  p r o p e r t i e s  i n  t h e  immediate v i c i n i t y  of t h e  wel l .  As time increases ,  t h e  type curves d e v i a t e  from the short- t ime s o l u t i o n  a s  w e l l  as from one another. For a > 1 ,  t h e  type curves f a l l  below the  a - 1 (no s k i n )  curve,  i n d i c a t i n g  a f a s t e r  head response due t o  the higher hydraul ic  conduct iv i ty  in the a q u i f e r  material away from t h e  well. For a < 1 ,  t h e  type curve d e v i a t e s  above t h e  a .I 1 curve,  i n d i c a t i n g  a slow down i n  head response due t o  the lower hydraul ic  conduct ivi ty  i n  the  aqui fe r .  I f  Y is increased by another  t e n  or a hundred f o l d ,  t h e  s t o r a t i v e  e f f e c t  of the s k i n  a t  early time w i l l  be e n t i r e l y  o b l i t e r a t e d ,  and we approach t h e  case of an open-well test. Figure 6 shows t h e  e f f e c t  of changing t h e  s k i n  radlus, rs, whi le  keeping t h e  value of Y a t  10-l. These curves show s i m i l a r  behavior as those  i n  Figure 5. The explanat ion f o r  t h i s  behavior is simple. As rs is decreased, t h e  e f f e c t  of t h e  s k i n  s t o r a t i v i t y  is a l s o  diminshed, and t h e  type curves a r e  c o n t r o l l e d  more and more by the hydraul ic  p r o p e r t i e s  of t h e  a q u i f e r .  1 .o 0.8 0.6 3 - 0.4 0 .2  0.0 I ' """'I a """'I ' """'I ' ' " " 1 ' 1  ' """ 0 hw = ea2' e r f c ( G 1  - Figure 6. t e s t  fo r  Y ~ 1 0 - ~ ,  rs=l.l, 6 - 1 ,  and the  ind ica ted  va lues  of a. The asymptotic s o l u t i o n  f o r  s h o r t  time is a l s o  shown f o r  comparison. Type curves f o r  a pressur ized  s l u g  DISCUSSION I t  appears t ha t  f o r  r e a l i s t i c  values  of skin and a q u i f e r  parameters t h e  conventional theory of Ramey and Agarwal (1972) and Ramey -- e t  a l . (1975) f o r  open-well t e s t s  is adequate. The reason f o r  t h i s  is t h a t  the  r e l a t i v e l y  l a r g e  volume of water involved i n  open-well t e s t s  tends t o  overwhelm t h e  s t o r a t i v e  e f f e c t s  of t h e  f i n i t e - t h i c k n e s s  skin. Thus, d a t a  a n a l y s i s  may y i e l d  reasonable  estimates of the  t r a n s m i s s i v i t y  of t h e  test i n t e r v a l .  For pressur ized  tests,  on the  o ther  hand, t h e  w e l l  response may show s i g n i f i c a n t  depar ture  from convent ional  theory.  Th i s  is due t o  t h e  f a c t  t h a t  the  very small volume of water involved i n  t h e  s l u g  t e s t  can be easi ly  d i s s i p a t e d  by the s t o r a t i v e  e f f e c t s  of the f i n i t e - t h i c k n e s s  skin. Consequently, pressurized test d a t a  may y i e l d  only t h e  product of hydraul ic  conduct iv i ty  and s p e c i f i c  s torage  of t h e  sk in .  -173- Because of the large number of parameters involved it may be difficult to separate effects of the skin from those of the aquifer. There is a trade-off between the time available to perform the test and the volume of material investigated. We recommend that open-well tests be performed whenever time permits. We also recommend, particularly when performing pressurized tests, that the flexibility in varying Y be exploited and that two or more tests be carried out on a given test interval using different values of Y. In this manner it may be possible to unravel effects of skin parameters from aquifer parameters. NOTAT I ON aquifer thickness, EL1 well-bore storage coefficient, cL21 c = n(rc')2 Por open well C VwpwgCsys for pressurized well compressibility of the combined fluid-well system, [LT2M-' 1 acceleration of gravity, CLT-21 dimensionless head in skin or aquifer hydraulic head in skin or aquifer, CL1 dimensionless head in well hydraulic head in well, E L I  initial hydraulic head in well, [L] modified Bessel function of the first kind and order zero modified Bessel function of the first kind and order one modified Bessel function of the second kind and order zero modified Bessel function of the second kind and order one hydraulic conductivity, [ L T - l I  thickness of test interval, [L] Laplace transform variable dimensionless radial distance from center of well radial distance from center of well, CLI radius of riser pipe, [L] well radius, [ L ]  distance from center of well to outer boundary of skin, [L] specific storage, CL-' I dimensionless time time, [ T I  volume of fluid in a pressurized well, C L ~ I  density of fluid in the well, [ML-3] argument of the Bessel functions skin factor for skin of infinitesimal thickness Subs cr i p t s : 1 skin 2 aquifer Dimensionless groupings: -1 74- REFERENCES Bredehoeft, J . D . ,  and S.S. Papadopulos, 1980. A method f o r  determining t h e  hydraul ic  p r o p e r t i e s  of t i g h t  formations,  Water Resour. Res.. 1 6 ( 1 ) ,  233-238. -- Cooper, H.H. .  J . D .  Bredehoeft, and I.S. Papadopulos, 1967, Response of a f i n i t e -  diameter well t o  an instantaneous charge of water ,  Water Resour. Res., 3(1) ,  263-269. Moench, A. F . ,  and P.A. Hsieh,1985a, Analysis of s l u g  test data i n  a well w i t h  f i n i t e -  th ickness  s k i n ,  Proceedings of the I A H  17 th  I n t e r n a t i o n a l  Congress on t h e  Hydrogeology of Rocks of Low Permeabi l i ty ,  Jan. 7-12, 1985, Tucson, AZ. Moench, A. F. and P.A. Hsieh,1985b, Comment on "Evaluation of Slug Tests i n  Wells Containing a Finite-Thickness Skin" by C .  R. Faust and J. W. Mercer, Water Resour. Res ( i n  p r e s s ) .  Ramey. H . J . ,  Jr . ,  and R.C. Agarwal, 1972, Annulus unloading r a t e s  a s  inf luenced by wellbore s torage  and s k i n  e f f e c t ,  Trans. Soc. Pet. Eng. AIME, 253, 453-462. ----Ramey, H . J . ,  Jr., R.G. Agarwal. and I .  Mart in ,  1975, Analysis of  ' s l u g  tes t '  o r  DST flow period d a t a .  J. Can. P e t r o l .  Technol., 37-47. S t e h f e s t ,  H . ,  1970, N w e r i c a l  invers ion  of Laplace t ransforms,  Commun. ACM. 1 3 ( 1 ) ,  47-49. -175- 1 7 ($ -- 

Slug Testing in Wells With Finite-Thickness Skin

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Slug Testing in Wells With Finite-Thickness Skin

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