Enhanced Estimation of Reservoir Parameters Using Decline Curve Analysis - Ebrahimi, Mohsen (2010)

SPE 133432Enhanced Estimation of Reservoir Parameters Using Decline Curve Analysis M. Ebrahimi, SPE, ACECR-Production Technology Research Institute Copyright 2010, Society of Petroleum Engineers This paper was prepared for presentation at the Trinidad and Tobago Energy Resources Conference held in Port of Spain, Trinidad, 27–30 June 2010. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract Traditional decline curve analysis is an empirical procedure used mainly to predict recoverable reserves and future production rates, based on the boundary dominated declining rate. Modern (typecurve) analysis, however, are partially (Fetkovich) or fully derived analytically, based on reservoir fluid flow equations and assuming some simplifying conditions. Such typecurves are generally used for predicting reserves and future production rates as well as reservoir parameters. In the current study, two cases have been analyzed using traditional and modern decline curves in an attempt to estimate key reservoir parameters. In the first case, an implicit reservoir simulator has been used for generating a set of declining rates for a well operating at ideal conditions, i.e. a homogenous cylindrical reservoir with isotropic permeability producing single phase oil at a constant bottomhole pressure. The generated data ware then used as an input for a decline curve analysis software to investigate the reliability and accuracy of estimated permeability, skin, and drainage area. In the second case, the declining rate of a real oil well is analyzed in terms of the previously cited parameters by the same software. The results of both cases indicate a good agreement between the actual and estimated parameters, with the Blasingame typecurve as the most accurate decline curve analysis technique. Introduction When sufficient production data are available and production is declining, a curve fit of the past production performance can be done using certain standard curves. This curve fit is then extrapolated to predict future performance. This procedure is called decline curve analysis in which all factors influencing the curves in the past are assumed effective (unchanged) throughout the producing time (Satter and Thakur, 1994). Traditional decline curve analysis is an empirical procedure used mainly to predict recoverable reserves and future production rates, based on the boundary dominated declining rate. Due to its empirical nature, traditional analysis can be used for almost any situation, on single fluid streams or multiple fluid streams, on reservoirs with pressures below or above bubble point and on constant or variable flowing bottom hole pressures. Modern (typecurve) analysis, however, are partially (Fetkovich) or fully derived analytically, based on reservoir fluid flow equations and assuming some simplifying conditions. Such typecurves are generally used for predicting reserves and future production rates as well as reservoir parameters. Although simplifying assumptions used in deriving such curves might imply their applicability to some certain conditions, but still a very good estimation of reservoir parameters is possible even when the producing conditions are not ideal. In this paper, after a brief review of the traditional and modern decline curve analyses, two cases will be analyzed in an attempt to determine key reservoir parameters. In the first case, a synthetic model is used to obtain reservoir parameters when producing conditions are ideal. The second case is a real well assumed to be pumped off at a constant bottomhole pressure. Results of both cases show the feasibility of decline curve analysis for obtaining reservoir parameters. Furthermore, results of the two cases reveal Blasingame typecurve as the most accurate technique compared with the other typecurves. Theoretical Background Traditional decline curve analysis (Arps, 1945; Arps, 1956) Traditional decline curve analysis is not grounded in fundamental theory but is based on empirical observations of production decline. Three types of decline curves have been identified, namely, exponential, hyperbolic and harmonic. When analyzing rate decline, two sets of curves are normally used. The flow rate is plotted either against time or against cumulative production. These curves provide a direct estimate of the ultimate recovery at a specified economic limit. 2 SPE 133432 Traditional decline curves represent production from the reservoir under boundary dominated flow conditions. This means that during the early life of a well, while it is still in transient flow and the reservoir boundaries have not been reached, decline curves should not be expected to be applicable. All decline curve theory starts from the definition of the instantaneous or current decline rate, D, as follows: q t q t q q D ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∆ ∆ − = ∆ ∆ − = (1) Exponential decline occurs when the decline rate, D, is constant. If D varies, the decline is considered to be either hyperbolic or harmonic, in which case, an exponent “b” is incorporated into the equation of the decline curve, to account for the changing decline rate. Exponential decline is given by: Dt e i q q 1 = (2) Hyperbolic decline is described by: ( )b t i bD i q q 1 1 1 + = (3) Where D i is the decline rate at flow rate q i , and b is an exponent that varies from 0 to 1. When b equals 1, the curve is said to be Harmonic. When 0 < b < 1, the curve is said to be Hyperbolic. When b=0, this form of the equation becomes indeterminate, but it can be shown that it is equivalent to Exponential decline. Arps type curves If the hyperbolic decline equation is presented in graphical form it can be seen to encompass the whole range of conditions from exponential decline (b=0) to harmonic decline (b=1) where each value of q i , D i , and b will produce its own unique curve. Arps (1945 and 1956) generalized these curves by making the equation dimensionless; He defined a dimensionless rate as q Dd = q (t) / q i , and a dimensionless time as t Dd = D i t. The resulting dimensionless equation used to generate a set of type curves is: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = b Dd bt Dd q 1 1 1 (4) Through type curve matching the values of b, q i and D i are obtained. Other relevant equations for oil phase are presented in the appendix. Fetkovich type curves Fetkovich (1980 and 1987) used analytical flow equations to generate type curves for transient flow, and he combined them with the Arps empirical decline curve equations. Accordingly, the Fetkovich type curves are made up of two regions, the right- hand side which is identical to the Arps type curves and the left-hand side derived from the analytical solution to the flow of a well in the centre of a finite circular reservoir producing at a constant wellbore flowing pressure. Matching of the late-time data gives an indication of the reserves, which in turn is a direct function of r e . Armed with this knowledge of r e and the match of the transient data, the r e /r w parameter can then be used to calculate the effective wellbore radius, r we , from which the skin factor, s, can be obtained using the equation r we = r w e -s . Blasingame type curves The production decline analysis techniques of Arps and Fetkovich are limited in that they do not account for variations in bottomhole flowing pressure in the transient regime, and only account for such variations empirically during boundary SPE 133432 3 dominated flow (by means of the empirical depletion stems). In addition, changing PVT properties with reservoir pressure are not considered, for gas wells. Blasingame and Lee (1986) and Palacio and Blasingame (1993) have developed a production decline method that accounts for these phenomena. The method uses a form of superposition time function (material-balance-time), that only requires one depletion stem for typecurve matching, the harmonic stem. Conceptually, the material-balance-time is defined as the ratio of cumulative production, Q, to instantaneous rate, q. In effect, Blasingame’s typecurves allow depletion at a constant pressure to appear as if it were depletion at a constant flow rate. In Blasingame typecurve analysis, three rate functions can be plotted against material balance time, namely: normalized rate, rate integral, and rate integral derivative. The last two functions help in smoothing the often-noisy character of production data, and in obtaining a more unique match. wf p i p q p q − = ∆ Normalized rate (5) t t dt p q i p q ∫ ∆ = ∆ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 0 Rate Integral (6) c t c dt i p q d c t d i p q d id p q ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∆ − = ∆ − = ∆ ) ln( Rate Integral Derivative (7) Agarwal-Gardner type curves Agarwal and Gardner (1998) have compiled and presented new decline typecurves based upon the work of both Fetkovich and Palacio-Blasingame, utilizing the concepts of the equivalence between constant rate and constant pressure solutions. Agarwal and Gardner present new typecurves with dimensionless variables based on the conventional well test definitions, as opposed to the Fetkovich and Blasingame dimensionless variables. They also include primary and semi-log pressure derivative plots (in inverse format for decline analysis). The horizontal axis is material balance time, and the vertical axis is Normalized Rate (equation 5) and inverse semi-log derivative of pressure normalized rate as defined in equation 8. ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∆ ∂ = ) ln( 1 1 c t q p DER (8) NPI (Normalized Pressure Integral) The Normalized Pressure Integral was initially developed by Blasingame (1989). The NPI provides an analysis method that is similar to Blasingame, but that is based on pressure, rather than rate. The main purpose is to present an analysis that is very similar to a well test analysis. Normalized pressure along with normalized pressure integral (equation 9) and pressure integral derivative (equation 10) are used to interpret decline curves in NPI type curve. Normalized pressure is the inverse of definition for normalized rate and is defined as ΔP/q. In some cases, the data is already very noisy and difficult to read because of many fluctuations. The normalized pressure integral can be used to smooth the data so that trends in the data may be interpreted easier. In most cases, a Pressure Derivative was found to be too sensitive to be useful. The Pressure Integral Derivative is less sensitive and so can be interpreted more handily, while still providing detail about the character of the data. ∫ = DA dt D P DA t DI P 1 (9) 4 SPE 133432 ) ln( DA t d DI dP DID P = (10) Case studies Synthetic model An implicit reservoir simulator has been used to generate a set of production data for an ideal case. This case is called ideal because all the assumptions used for generating Fetkovich and other type curves have been honored. In other words, a well is draining a circular area with a constant bottomhole pressure set to a value just above bubble point pressure so that only a single phase (oil) is produced. Table 1 shows the reservoir, well bore and fluid properties used to build the simulation model and Figure 1 shows the production behavior of the model. Table 1: Properties of the model used for generating an ideal rate response Porosity 30% Permeability (K r , K θ , K z ) (md) 10 r e , r w (ft), reservoir area (acre) 2050, 0.25, 295.74 NR, Nθ, NZ 10, 1, 1 Depth to tops of formation (ft) 5500 DZ, Dθ (ft, degree) 500, 360 Datum depth 5750 Datum pressure (psia) 4550 Oil, Water and gas densities (lb/cu ft) 53.97, 64.79, 0.06692 Rock compressibility @ 4550 psia (psi -1 ) 0.31E-05 Water compressibility (psi -1 ) 3.13E-6 (assumed constant) Skin 0 Bubble point pressure (psia) 2944 Figure 1: Production behavior of the synthetic model In the first step, a traditional analysis is performed as illustrated in figure 2. An area of 326.47 acres is calculated with a b value of 0.018 which indicates an almost exponential decline. In the next step, other type curves matching including Fetkovich, Blasingame, Agarwal-Gardner and Normalized Pressure Integral (NPI) are performed as shown in figures 3 to 6. The results of the above mentioned analyses are compared in table 2. Figure 2: traditional declining rate versus time analysis for the synthetic model SPE 133432 5 Figure 3: Fetkovich type curve analysis- synthetic model Figure 4: Blasingame type curve analysis- synthetic model Figure 5: Agarwal Gardner rate versus Time type curve analysis- synthetic model Figure 6: NPI type curve analysis- synthetic model Table 2: the comparison of different analysis results Traditional Fetkovich Blasingame Agarwal&Gardner NPI K S A K S A K S A K S A K S A N/A N/A 326.5 10.76 0.29 335.7 10.76 0.37 292.4 10.75 0.22 291.9 10.45 0.38 294.8 6 SPE 133432 As illustrated in figures 3 to 6, the generated data set matches properly with different typecurves. Considering the actual permeability, skin, and drainage area shown in table 1, Blasingame results show better agreement with the actual values but on the whole, one can easily find out the applicability of decline curve analysis for predicting key reservoir parameters, especially when the results of more than one typecurve is available. Real oil well The second case is an oil well assumed to be pumped off in a constant bottomhole pressure of 600 psig. Table 3 shows the required properties for decline curve analysis of this well. Table 3: properties of the real well Bottom hole flowing pressure (psig) 600 Initial gas oil ratio (scf/stb) 483 Initial Bottom hole pressure (psia) 5300 Reservoir temperature (F) 200 Gas specific gravity 0.65 Reservoir thickness (ft) 20 Connate water saturation (%) 20 Oil gravity ( o API) 40 Porosity (%) 20 Well radius 0.25 Bubble point pressure (psia) 2300 The well has produced with a relatively constant gas oil ratio of 483 SCF/STB for about 27 month, before a work over is performed on July, 2002 to clean out the well. Well test results have shown the true values of K= 1.09, s= -1.240 and A=68.91. Figures 7 to 12 show different type curve analyses for this case. Figure 7: Multi phase fluid production- second case Figure 8: Traditional rate versus time analysis- second case SPE 133432 7 Figure 9: Fetkovich type curve analysis- second case Figure 10: Blasingame type curve analysis- second case Figure 11: Agarwal Gardner rate versus time type curve analysis- second case Figure 12: NPI type curve analysis- second case 8 SPE 133432 Table 4 compares the results of different decline curve analyses with the results of well test and presents the differences in relative error. Table 4: the comparison of different analyses results- second case Method Parameters Quantity Relative Error (%) Traditional K (md) N/A N/A S N/A N/A A (acre) 87.47 26.9 Fetkovich K (md) 1.0802 0.9 S -1.365 10.1 A (acre) 85.71 24.4 Blasingame K (md) 1.0124 7.1 S -1.358 9.5 A (acre) 65.95 4.3 Agarwal-Gardner K (md) 1.004 7.9 S -1.595 28.6 A (acre) 70.09 1.7 NPI K (md) 1.0254 5.9 S -1.628 31.3 A (acre) 73.71 7 Conclusion Two cases were selected for studying the feasibility of decline curve analysis usage for determining key reservoir parameters. The first case was an ideal synthetic model built in a manner that all the assumptions considered for deriving type curves is honored. This case was analyzed to evaluate the degree of reliability and accuracy of decline curve analysis methods in determining reservoir parameters. As was detailed in the paper, Blasingame and the other curves are capable of predicting such parameters with an acceptable accuracy. The second case was analyzed to see if the decline analysis is still correct for a real oil well. As pointed out in the paper, the degree of reliability for permeability and reservoir drainage area is quite good, but skin is of more uncertainty. Furthermore, Blasingame type curve is shown to be the most accurate model. It is recommended to use decline curve analysis for obtaining key reservoir parameters especially when reservoir pressure is above bubble point. Nomenclatures Greek Symbols Δ = difference Δp = pressure difference, psi μ = viscosity, cp Φ = porosity English Symbols A = drainage area, acre B = formation volume factor, bbl/STB c t = total compressibility, psi -1 D = instantaneous or current decline rate, D -1 D i = decline rate at flow rate q i , D -1 h = reservoir thickness, ft k = permeability, md N = oil in place, STB × 10 3 P di = pressure integral P did = pressure integral derivative p i = initial pressure, psi p wf = flowing bottomhole pressure, psi q = flowrate, STB/d Q = produced oil, STB q Dd = dimensionless flowrate based on decline rate q i = initial flowrate, STB/d r e = drainage radius, ft r w = wellbore radius, ft r wa = damaged zone radius, ft r we = effective wellbore radius, ft SPE 133432 9 s = skin factor s o = oil saturation t c = superposition time (Q/q), day t Dd = dimensionless time based on decline rate Subscripts c = superposition d = derivative D = dimensionless DA = dimensionless based on area Dd = dimensionless based on decline rate (D) i = initial References 1. Agarwal, R.G, Gardner, D.C., Kleinsteiber, S.W, and Fussell, D.D. 1998. Analyzing Well Production Data Using Combined Type Curve and Decline Curve Analysis Concepts. Paper SPE 57916. 2. Arps, J.J. 1945. Analysis of Decline Curves. Trans., AIME, 160, 228. 3. Arps, J.J. 1956. Estimation of Primary Oil Reserves. Trans., AIME, 182-191. 4. Blasingame, T.A., Johnston, J.L., and Lee, W.J. 1989. Type Curve Analysis Using the Pressure Integral Method. Paper SPE 18799 presented at the SPE California Regional Meeting, Bakersfield, CA, 05-07 April. 5. Blasingame, T.A, Lee, W.J. 1986. Variable-rate Reservoir Limits Testing. Paper SPE 15028. 6. Fekete Associates Inc. FAST RTA., http://www.fekete.com 7. Fetkovich, M.J. 1980. Decline Curve using Type Curves. JPT, 1065. 8. Fetkovich, M.J, Vienot, M.E, Bradley, M.D, and Kiesow, U.G. 1087. Decline Curve Analysis Using Type Curves - Case Histories. SPEFE, 637. 9. Palacio, J.C and Blasingame, T.A. 1993. Decline Curve Analysis Using Type Curves Analysis of Gas Well Production Data. Paper SPE 25909. 10. Satter, A. and Thakur, G.C. 1994. Integrated Petroleum Reservoir Management: a team approach. Penn Well Books. Appendix Table A-1 shows the different equations used for modern decline curve analysis. Table A-1: Different equations used for type curve analysis Method Parameter Equation Fetkovich k ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = 4 3 ln ) ( 2 . 141 match wa r e r wf p i p h B match Dd q q k µ r wa & s ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 4 3 ln 1 2 2 1 006328 . 0 match wa r e r match wa r e r t c k match Dd t t wa r φµ & ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = wa r w r s ln A ) 43560 ( o hs o NB A φ = Blasingame k ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∆ = 2 1 ln 2 . 141 match wa r e r h B match Dd q p q k µ r wa & s ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 2 1 ln 1 2 2 1 006328 . 0 match wa r e r match wa r e r t c k match DA t c t wa r φµ & ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = wa r w r s ln r e 2 2 1 2 . 141 006328 . 0 wa r match Dd q p q Dd t c t t c h B e r + ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∆ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = φ 10 SPE 133432 Agarwal& Gardner k match D q p q h B k ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∆ = µ 2 . 141 r e match DA t c t t c k e r ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = πφµ 00634 . 0 r wa & s match wa r e r e r wa r ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = & ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = wa r w r s ln NPI k match q P D P h B k ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∆ = µ 2 . 141 r e match DA t c t t c k e r ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = πφµ 00634 . 0 r wa & s match wa r e r e r wa r ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = & ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = wa r w r s ln