Unsteady-State Behavior of Naturally Fractured Reservoirs

Transcription

1 Unsteady-State Behavior of Naturally Fractured Reservoirs A. S. ODEH MEMBER AIME SOCONY MOBIL OIL CO., INC. DALLAS, TEX. ABSTRACT A simplified model was employed to develop mathematically equations that describe the unsteadystate behavior of naturally fractured reservoirs, The analysis resulted in an equation of flow of radial symmetry whose solution, for the infinite case, is identical in form and function to that describing the unsteady-state behavior of homogeneous reservoirs. Accepting the assumed model, for all practical purposes one cannot distinguish between fractured and homogeneous reservo irs from pressure build-up and/or drawdown plots, INTRODUCTION The bulk of reservoir engineering research and techniques has been directed toward homogeneous reservoirs, whose physical characteristics, such as porosity and permeability, are considered, on the average, to be constant. However, many prolific reservoirs, especially in the Middle East, are naturally fractured. These reservoirs consist of two distinct elements, namely fractures and matrix, each of which contains its characteristic porosity and permeability. Because of this, the extension of conventional methods of reservoir engineering analysis to fractured reservoirs without mathematical justification could lead to results of uncertain value. The early reported work on artificially and naturally fractured reservoirs consists mainly of papers by Pollard,l Freeman and Natanson,2 and Samara. 3 The most familiar method is that of Pollard. A more recent paper by Warren and Root showed how the Pollard method could lead to erroneous results. 4 Warren and Root analyzed a plausible two-dimensional model of fractured reservoirs. They concluded that a Horner-type pressure build-up plot of a well producing from a fractured reservoir may be characterized by two parallel linear segments. These segments form the early and the late portions of the build-up plot and are connected by a transitional curve. Original manuscript received in Society of Petroleum Engineers office May 22, Revised manuscript received Nov. 2, Paper presented at SPE 39th Annual Fall Meeting held in Houston, Tex., Oct , References given at end of paper. In our analysis of pressure build-up and drawdown data obtained on several wells from various fractured reservoirs, two parallel straight lines were not observed. In fact, the build-up and drawdown plots were similar in shape to those obtained on homogeneous reservoirs. Fractured reservoirs, due to their complexity, could be represented by various mathematical models, none of which may be completely descriptive and satisfactory for all systems. This is so because the fractures and matrix blocks can be diverse in pattern, size, and geometry not only' between one reservoir and another but also within a single reservoir. Therefore, one mathematical model may lead to a satisfactory solution in one case and fail in another. To understand the behavior of the pressure buildup and drawdown data that were studied, and to explain the shape of the resulting plots, a fractured reservoir model was employed and analyzed ma thematically. The model is based on the following assumptions: 1. The matrix blocks act like sources which feed the fractures with fluid; 2. The net fluid movement toward the wellbore obtains only in the fractures; and 3. The fractures' flow capacity and the degree of fracturing of the reservoir are uniform. By the degree of fracturing is meant the fractures' bulk volume per unit reservoir bulk volume. Assumption 3 does not stipulate that either the fractures or the matrix blocks should possess certain size, uniformity, geometric pattern, spacing, or direction. Moreover, this assumption of uniform flow capacity and degree of fracturing should be taken in the same general sense as one accepts uniform permeability and porosity assumptions in a homogeneous reservoir when deriving the unsteadystate fluid flow equation. Thus, the assumption may not be unreasonable, especially if one considers the evidence obtained from examining samples of fractured outcrops and reservoirs. Such samples show that the matrix usually consists of numerous blocks, all of which are small compared to the reservoir dimensions and well spacings. Therefore, the model could be described to represent a CChomogeneously" fractured reservoir. In this paper, the fundamental equation of flow SOCIETY OF PETROLEUM ENGINEERS JOURNAL

2 h for the selected model is derived and solved for the infinite case. This case applies to a very large reservoir and to all reservoirs in the early stages of their depletion when the boundary effects have not become appreciable. DERIVATION OF UNSTEADY-STATE FLOW EQUATION Consider a small radial element. of length dr and angle a. This element consists of fractures and matrix blocks. The fractures mayor may not contain infill. Let f3 be the bulk volume of fractures per unit bulk volume of the element, h be the total productive thickness of the sand, CPf be the porosity of the fractures and m be the porosity of the matrix blocks. f and m are defined respectively as the ratio of pore volume of the fractures and matrix blocks to their associated bulk volumes. CPf is unity if the fractures do not contain infill. Thus, V f = /3epf a rhdr = the pore volume of fractures of the element and, V m = (1-/3)epm a rhdr= the pore volume of the element. Let qml rand qml r+dr be the mass rate of fluid flow in the fractures at rand r+dr respectively. Then from mass balances one can write V m qmlr+dr - qml r + f3 f V f dpm a rhdr(jt= -,B f a where /3ep V m was substituted for (1- mep 1 which f V f m follows from the definition of V m and Vf' Pf and Pm are respectively the fluid densities in fractures and matrix. Expanding q I in Taylor's series, retaining m r+dr only the first two terms, substituting - ~ for qmlr' and simplifying yields: Or (1) ak f hr/3pf apf rhdr -a-t- --'-----=f.l cm(p -P )-cf(pf-p ) The exponential term e moo can be approximated by 1- (c m 6P m - C f 6P ) f Both C m and and cf are in the range of 16~5 and Moreover, cm ~ cf and ~Pm ~ ~Pf' so that cm~pm.:: cf ~Pf' Thus, one can drop the (em 6P - c m f 6P ) term in the f above equation to arrive at Eq. 3 describes the unsteady-state flow of a slightly compressible liquid in the previously described model of fractured reservoirs. It is identical to the heat conduction equation except for an extra term which represents the matrix block's contribution. The solution of Eq. 3 requires a functional relationship between PI and Pm' For this purpose, it is assumed that the rate of feed to the fractures by the matrix is described by quasi-steady-state behavior and is given by where a is a shape factor as described by Warren and Root 4 and has the dimension of reciprocal area. The assumption that the flow in the matrix blb'cks is described by quasi-steady state is quite adequate. This can be shown by employing any of the formulas which describe the initiation of quasi-steady-state flow. Such formulas, for radial flow, have been reported by Muskat,S Swift and Kiel,6 Jones 7 and others. For linear flow, one can derive similar formulas. Thus calculations show that for a matrix block of 0.01 per cent porosity, 1 md permeability, 2 ft long and for an oil of l-cp viscosity and 15 x 10-6 psi -1 in compressibility, quasi-steady-state flow obtains within 10 seconds regardless whether the flow is assumed to be linear or radial. (3) SOLUTION AND DISCUSSION Subscripts m and f refer respectively to matrix and fractures. Substituting P e c(p-po ) for P in Eq. 2, differo 2 o. 0 entiating, dropping t e (dp -_ f ) term, an d d'!v!'do!ng by C (p - P ) give d r P e f f 0 o (2) Eqs. 3 and 4 are to be solved for a well of infinite radius of drainage producing at a constant rate q. Thus, Pm =Pf = Pi' t =0 - ~~I f.lf or -q, r =rw t > 0 t > (5) where A f is the area of flow available to the fractures at the wellbore and is equal to (2rrhf3r w ) MARCH,

3 and h is total productive thickness of the sand. Let f1p = Pi - P, substituting in Eqs. 3, 4 and 5, and taking the Laplace transformation of the resulting equations gives 2- a.6.p f I a.6. P f - [ rcs ] p 8s+--. (5a) ar 2 r ar - f stc The solution to Eq. 5a that satisfies the boundary conditions is QK (~/2 r).6. 0 (6) P f = I I s>/2 K 0./2r ) I w where A-(\;:'S + rcs) Q - ql-'-,and K 0 and - 0 s+c' - 27Tk f h.8r w K 1 are respectively modified Bessel functions of the second kind of zero and first order. Thus, where Hi (-x) is defined by ds. (7) Solution 7 is the exact solution for the problem at hand, which is valid for all times. An approximate solution for dimensionless time T > 100, where T= k t 2 f, can be obtained by making the I-'- rw (cep)average usual substitutions 8 Ko(Z) = - (In Z/ ) and K 1 (Z) = 1/Z. Thus Eq. 6 becomes = Qr w, f<t+i 00 [ -1/2(lns8t In(s+ 8C~rC) 2-rr I 5 <T-ioo -In(s+Cl)+ln Jest ds s which gives, after inverting and substituting, q,u [ 4tk f Vf.6.p = In ---::----'--'---- f 4-rrk f h,b r 2,ucpf{cfV f +C m V m ) C Ei (-cl) +E i(-ci(i+~ ~~ )l]' (8) 00 Ei(-x)=-f ~-udu Eq. 8 is similar in functional form to Eq. 15 of Warren and Root.4 However, one must remember that the mathematical models that were employed x to arrive at the respective equations are basically different. Vmcm ) The value of -Ei(-Ct)+Ei (-Ct(I+VC) f f becomes negligible after a very short period of flow or of pressure build-up for two reasons. First, each term in itself becomes very small at early times, for Hi (-7).=:::.. o. The 'value of C appearing in.. ak m the arguments IS hard to determine. C=,.I.. c m 'fjmfl where a is a shape factor. However, for cubical matrix blocks of x = y = z = h, a~ 60/h 2 4 For a cubical matrix block of 0.01 per cent porosity, 0.1 permeability and a crude oil of 1 cp and an effective compressibility of 200 x 10-6 volume/volume/ atmosphere, C = 3 x 10 3 /h 2 in reciprocal seconds. Therefore, C is larger than one for an effective h < 55 cm. In any case, C is not very much less than unity. Thus, the values of Ei(-Ct) and V C ( 1+ V m m) )become negligible during the first few f c f minutes, if not the first few seconds, of flow, or during the time after flow effects obtain in pressure build-up cases. The second reason why we can neglect the Hi functions in the early period is because one of them is negative whil~ the other is positive. Thus, they tend to cancel each other out. For all practical purposes Eq. 8 becomes Ei (-Ct.6.p= q,u [In 4tk f Vf rrk f h,q r 2,uepf(c f V f +c m V m ) J ~ _-_q,u'------_ Ei (_ 2 r epf,u (cfvf+cmvm) ) 4-rrk f h,8 4k f V f t [Ei (_ r2,u(cf,8cpf+cm(i-,8)epm))~ q,u 4-rrk f h,8 4k f,8t ~ It is of interest to compare Solution 9 to the pressure behavior of a homogeneous reservoir. According to Horner9 (9) _ q,u' (_ r 2ep,uc)...6.p rrk h. Ei 4 kt (9a) Therefore, both solutions are of the same form except for two factors which characterize the fractured reservpirs. These are f3 and c f,bepf + Cm(l-,8)cp which! is (c ) average. Thus, in case ot a pre~sure drawdown or build-up analysis, a plot of lnt vs P and In [(t + 0)/0] vs p respectively results in a straight line identical to the one obtained on homogeneous reservoirs. This IS so because the effect of [-Ei (-CIl + Ei (-CI( 1+ ~:~: l)] 62 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

4 which could differentiate between the fractured and homogeneous reservoirs I plots, may be noticeable only during the very early period of the test. However, during this period one cannot usually obtain 'reliable data. The identity in shape of the above mentioned plots was true in all drawdown and build-up cases analyzed by the author. These cases consisted of several drawdowns and build-ups obtained on wells from various fractured reservoirs both in the United States and Middle East. Also, Dyes and Johnston reported pressure build-up plots of wells from Spraberry which are similar in shape to those obtained on homogeneous reservoirs. 10 oct en a. ~ 2200 LJJ 2100 :: ::l en en LJJ :: a I FIG. 1 - tc PRESSURE BUILD-UP. o Fractured reservoirs, described by the model analyzed in this paper, cannot be distinguished from homogeneous reservoirs on the basis of drawdown arid build-up curves, using field measured data. In addition, calculations for skin effect, for pressure drop as a function of time and distanc;:e, for the distance to a fault, and various others are identical to those of nonfracture_d reservoirs except for the proper insertion of (3 and (c )av as indicated by comparing Eqs. 9 and 9a. Finally it should be re-emphasized that all fractured reservoirs may not necessarily adapt to the above analysis. The assumption of homogeneous fracturing may not apply to every fractured reservoir. Consequently, the drawdown and build-up curves of these reservoirs to which the homogeneousfracturing assumption does not apply could have their special characteristics. The following field case is presented to illustrate the theoretical: results. DATA TABLE 1 - CONCLUSIONS EXAMPLE The pressure build~up data shown in Table 1 were obtained from a test on Well X producing from naturally fractured limestone. Other pertinent data are: stabilized rate of flow prior to shut in ::=: 905 STB/D, cumulative production from the well at the time of shut in = 5,800 STB, formation volume factor ::::: 1.085, viscosity at reservoir conditions 1.6 cp, and reservoir bubble-point pressure = 700 psi. Construct the pressure build-up curve and calculate the flow capacity of the reservoir. SOLUTION The modified time - t c was (5,800/905) x 24 PRESSURE BUILD-UP DATA Shut-in Time Pressure Shut-in Time Pressure (J (minutes) (psi a) (J (minutes) (psia) MAlteH, hours.(tc + tj)/e was then calculated and plotted on semilog paper, Fig. 1. e is the shut-in time in hours. The slope m of the straight-line portion is -7/cycle. Therefore, from the similarity between Eqs. 9 and 9a -m 7 which gives k f h( x 905 x x 1.6 k f h( x 905 x x = 36.5 x One can consider k f h(3 to represent the effective flow capacity of the reservoir. If one is interested in k f, then hand f3 must be known. It may be possible to make some estimate of the magnitude of f3 from caliper logs. ACKNOWLEDGMENT The author wishes to express appreciation to G. W. Nabor, E. E. Moreland and J. H. Halsey for useful discussions, and to the management of Socony Mobil Oil Co., Inc. for permission to publish this paper. REFERENCES 1. Pollard, P.: "Evaluation of Acid Treatment from Pressure Build-up Analysis", Trans., AIME (1959) Vol. 216, Freeman, H. A. and Natanson, S. G.: "Recovery Problems in a Fracture-Pore System - Kirkuk Field", Proc., Fifth World Petroleum Congress, Sec. 11 (1959) Samara, H.: "Estimation of Reserves from Pressure Changes in Fractured Reservoirs", presented at Second Arab Petroleum Congress, Beirut, Lebanon (Oct., 1960). 4. Warren, J. E. and Root, P. J.: "The Behavior of Natur1;llly Fractured Reservoirs", Soc. Pet. Eng. Jour. (Sept., 1963) Vol. 3, Muskat, M.: The Flow of Homogeneous Fluids Through Porous Media, J. W. Edwards, Inc. (1946). 6. Swift, G. W. and Kiel, O. G.: "The Prediction of Gas-Well Performance Including the Effect of Non-Darcy Flow", Soc. Pet. Eng. Jour. (ftllarch, 1962) Jones, L. G.: "Reservoir Reserve Tests", Jour. Pet. Tech. (March, 1963)

5 8. van Everdingen, A. F. and Hurst, W.: "The Application of the Laplace Transformation to Flow Problems In Reservoirs", Trans., AIME (1949) Vol. 186, Horner, D. R.: "Pressure Build-Up in Wells", Froc., Third World Petroleum Congress, The Hl?-gue, Netherlands (1951). 10. Dyes, A. B. and Johnston, O. C.: "Spraberry Permeability from Build-up Curve", Trans., AIME (1953) 198, 135. DISCUSSION J. E.WARREN p. J. ROOT MEMBERS AIME KUWAIT OIL CO. AHMA 01, KUWAI T GULF RESEARCH & DEVELOPMENT CO. PITTSBURGH, PA. The author has described and developed in great detail a model for the representation of a naturally fractured reservoir. He has used its behavior for a reasonable, but particular, set of parameters and some field observations as the basis for general conclusions. However, several of the points made by the author are misleading. 1. The theoretical model employed by the author is based on a set of physical assumptions which, despite the author's claim to the contrary, is equivalent to that used by Warren and Root 1 in particular, see assumptions a, band c and their generalization in Appendix A. As a consequence, the behavior equation derived in the subject paper is not ~ ~ similar in functional form"; it is identical. This contention may not be obviously true; however, the confusion introduced by parameter (3 can be eliminated by making the following substitutions in Eq. II-I of Ref. 1: k 1 k m k 2 kt (3 1 (1 - (3) m 2 (3 cpt C 1 C 2 a S*-Sd C m C f (1 - (3) a a q)m m These substitutions cause the equation to become identical with Eq. 8 of the subject paper. 2. Since the matrix poro~ity m has been defined in exactly the same manner in both papers, Eq. 5 of Ref. 1, together with the relationship between 1 and m used to equate the behavior equations, it can be utilized to obtain the following results: m m (3 2 + (l - 2) (Swc)m <PI [1/ 1 +e~2<p2j (Swc)mJ It is evident that the three parameters in this paper can be defined in terms of two independent parameters. Therefore, f3 is not an independent parameter; furthermore, it is neither physically meaningful nor readily measurable. In a sense, f3 complicates, rather than simplifies, the physical model. 3. In the author's analysis of the behavior equation, he concludes that the two exponentialintegral terms, aside from that for the equivalent homogeneous reservoir, are negligible for practical purposes. This conclusion is based on the observation that each term approaches zero rapidly. Consider the following conditions: k m 1 md m 0.1 Swc 0.5 Cp C w = 3 x 10-6 (psi)-l Co 12 x 10-6 (psi)-l N ~ 450 WELL A... '" ll.i 440 :> 1Il 1Il l&i l&i 0 III 420 / ~...J l&i ~ 410 " t s+ t.t ~ / /, "": 5100 iii : ::l II) II) l&j : Q l&j CD ~..J 4800 l&j ;J: 4700 I WELL B FIG. D{ - FIELD BUILD-UP CURVE. FIG. D2 - FIELD BUILD-UP CURVE. 64 SOCIETY OF PETROLE1JM ENGINEERS JOURNAL

6 fl 0 = 10 cp 1 = 100 cm n = 1 From the above, C m = 21 x 10-6 (psi)-l; and, using the author's nomenclature, C = (sec)-l. Consequently, - E i (-C t ) IS approximately zero (C t ::>l 7) after 30 minutes. Furthermore, since :::; 10, the difference between the two terms may not be negligible. It should be noted that it was suggested in Ref. 1 that a bottom-hole shut-in tool should be used to minimize the effect of after-flow on the early portion of the build-up curve. Thus, for a choice of parameters different from Odeh's, the behavior of the model is distinguishable from that for a homogeneous reservoir. 4. The author's field ~evidence to support his conclusion that fractured-~reservoirs behave in a homogeneous manner consists of one documented case plus experience based on the analysis of several drawdown and build-up curves. In rebuttal, Figs. D-1 and D-2 are presented. Another example may be found in Ref. 2 and it is also plotted in Ref. 3. REFERENCES 1. Warren, ]. E. and Root, P. ].: "The Behavior of Naturally Fractured Reservoirs", Soc. Pet. Eng. Jour. (Sept., 1963) Vol. 3, No.3, Pollard, P.: "Evaluation of Acid Treatments from Pressure Build-Up Analyses", Trans., AIME (1959) Vol. 216, Matthews, C. S.: "Analyses of Pressure Build-Up and Flow Test Data U, Jour. Pet. Tech. (Sept., 1961) Vol XIII, 9, 862. AUTHOR'S REPLY TO J. E. WARREN AND P. J. ROOT 1. Warren and Root state that I used the behavior of the fractured reservoir model that was developed ttfor a reasonable, but particular, set of parameters and some field observatidns as the basis for general conclusions". Under ttconclusions" of my paper it is stated Ufractured reservoirs, described by the model analyzed in this paper... ". Also, paragraph 2 under ttconclusions" reads tlfinally it should be re-emphasized that all fractured reservoirs may not necessarily adapt to the above analysis". It is obvious that no generalization was intended or stated. 2. Assumption eta" of Warren and Root1 paper states ttthe material... and is contained within a systematic array of identical, rectangular parallelopipeds". The first paragraph following Assumption 3 of my paper states that ttassumption 3 does not stipulate... the matrix blocks should possess certain size, uniformity, geometric pattern, spacing or direction". Thus, Assumptions a and 3, which are the key assumptions, are not equivalent. Eq. 5 of Warren and Root's 1 relates b 2 and m as they define them. The equation is If one chooses to define 1 = (1 - (3) m' then from the above equation or, if Swc = 0, 2 = f3. Therefore, Warren and Root cannot arbitrarily MARCH, 1965 substitute f3 t for 2,since this is not consistent with their Eq. 5. CPt may not be equal to unity if the fractures contain infil!. 3. Granted, it is possible to relate' by some function f3 and 2 and show that {3, CPt and 2 are not independent. Whether one should characterize fractured reservoirs in terms of f3 m or f3 av, or m 2 is a matter of choice. I elected to characterize them by an average and f3. This, in my opinion, does not lead to any complication. f3 is as readily measurable as CP2' An inconsistency should be pointed out in Warren and Root's definitions. Under Item 1 they took 2 = f3 t However, under Item 2 they took f3 = 2 + (1-2) (Swc)m' or f3 = 2 if Swc = O. In my paper f3 is defined to represent a specific parameter. Because of this, Warren and Root cannot manipulate the definition of f3 as the circumstances call for to prove their point. 4. In my paper I concluded that for a certain matrix block size and physical characteristics the two exponential integrals of Eq. 8 become negligible in a very short flow time. It was calculated that, generally, this is true for a matrix block of an effective dimension ~ 55 cm, and of reasonable physical characteristics. Obviously if the effective lengths.of the matrix blocks are large, the relative importance of the two exponential integrals may be enhanced. Thus, we are in agreement that the relative importance of the two functions will depend on several factors. The most important, I feel, is the dimensions of the matrix blocks. As a minor point, I would like to point out two errors in the calculations of Warren and Root example, under Item 3. The first is in the value of C. C as defined in my paper is given by C = akm/cm mfl' For a cube of length L. a = 60/L2, 65

7 Using the data of Warren and Root and remembering that k m' cm' m, fl and L are respectively in darcy, val/vol/atm, fraction, cp and em, one arrives at c 60 x x 0.1 x 10 x 21 x 14.7 ~ Therefore C = 0.02 in place of , and the exponential integral is approximately zero after about six minutes in place of 30 minutes of flow time. In actuality the exponential integral becomes relatively negligible at a much shorter time. In my opinion, Warren and Root's example tends to prove my point better than it tends to prove their contention, even though the dimension of the matrix block used in the e:xample is about twice as long as the upper limit stated in this paper. The second error is in the calculation of Cm If the rock compressibility is not taken into account, then C m = (3 x x 0.5) 10-6 = 7.5 x 10-6 vol/vol/psi. If the rock compressibility is accounted for, then for = 10 per cent, cf) "'" 5 x 10-6 and C m = 12.5 x Both of these values are different from 21 x 10-6 used by Warren and Root. If the correct value of C m is used, the flow time of six minutes is reduced to about two to three minutes. The two examples presented by Warren and Root strengthen my contention that all fractured reservoirs may not yield the same characteristic plots. I have presented one field example to show that some fractured reservoirs behave in a homogeneous manner. In fact this homogeneous behavior manifested itself in several build-up and drawdown cases that were studied but are not available for publication. Moreover, Dyes and Johnston 3 reported pressure build-up plots of wells from Spraberry which are similar in shape to those obtained on homogeneous reservoirs. REFERENCES 1. Warren, J. E. and Root, P. J.: "The Behavior of Naturally Fractured Reservoirs", Soc. Pet. Eng. Jour. (Sept., 1963) Vol. 3, Hall, H. N.: "Compressibility of Reservoir Rocks", Trans., AIME (1953) VoL 198, Dyes, A. B. and J ohnston,o. C.: "Spraberry Permeability from Build-up Curve", Trans., AIME (1953) Vol. 198, 135. *** 66 SOCIE1'Y OF PETROLEUM ENGINEERS JOURNAL