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Pressure transient analysis of arbitrarily shaped fractured reservoirs

Pressure transient analysis of arbitrarily shaped fractured reservoirs 2007 VolA No.2 Petroleum Science Gao Huimei', He Yingfu', Jiang Hanqiao' and Chen Minfeng' (1. School ofPetroleum Engineering, China University ofPetroleum, Beijing 102249, China) (2. Institute ofPorous Flow & Fluid Mechanics, CNPC & Chinese Academy ofSciences, Langfang, Hebei 065007, China) (3. Center for Enhanced Oil Recovery, China University ofPetroleum, Beijing 102249, China) Abstract: Reservoir boundary shape has a great influence on the transient pressure response of oil wells located in arbitrarily shaped reservoirs. Conventional analytical methods can only be used to calculate transient pressure response in regularly shaped reservoirs. Under the assumption that permeability varies exponentially with pressure drop, a mathematical model for well test interpretation of arbitrarily shaped deformable reservoirs was established. By using the regular perturbation method and the boundary element method, the model could be solved. The pressure behavior of wells with wellbore storage and skin effects was obtained by using the Duhamel principle. The type curves were plotted and analyzed by considering the effects of permeability modulus, arbitrary shape and impermeable region. Key words: Deformable reservoir, double-porosity reservoir, boundary element method, transient pressure, type curve permeability modulus, arbitrary shape and impermeable 1. Introduction barriers. Many methods use radial and circular systems to 2. Mathematical model interpret unsteady state double-porosity reservoir flow problems (Chen and Jiang, 1980; Mavor and Cinco-Ley, For single-phase slightly compressible fluid flow in 1979; Zhang and Zeng, 1992), but very little information is deformable double-porosity media, n, including n; available for arbitrarily shaped reservoirs. For many cases, sinks of strengths ql, the corresponding equations may however, the double-porosity reservoir drainage shape is be written as: too complicated to be approximated by a circular shape. The existence of one or more impermeable regions further a~~ +a~~ -p[(::r +(:: n= complicates the problem. So a numerical means is required (Britto,and Grader, 1987; Kikani and Home, 1988; Liu, et al., 2001; Liu and Duan, 2004; Masukawa and Home, PPID e [w OPID + (1- w) opmn) + (1) 1988; Paulo and Abraham, 1988; Sato and Home, 1993a; at at D D 1993b; Wang, et al., 2000; Yin, et al., 2005; Zhang and n, PP ID Zeng, 1992). Numerical techniques for solving partial e L qD/o(x - XD/ )O(YD - YD/) differential equations describing various physical I;' processes can be categorized into two distinct classes: the domain methods and the boundary methods. Finite difference and finite element methods fall in the first class, and the boundary element methods (BEM) constitute the (3) second. The BEM is superior to the domain methods in several ways. The most notable advantage is the high degree of accuracy that results from its sound aPmn = apID = 0 (4) mathematical foundations. Flexibility in defining boundary anD anD geometries and conditions is another feature to be emphasized. However, the conventional boundary element (i=2,3, .. ·,m) (5) method is not applicable to the problem of fluid flow in porous deformable media. In this paper, under the assumption that permeability varies exponentially with pressure drop (Kikani and Pedrosa, 1991; Ning, et al., (6) 2004), transient pressure response of wells in deformable double-porosity reservoirs was obtained. The type curves where it is assumed that the flow is in a horizontal plane were developed and analyzed by considering effects of and that the Darcy's law is applicable; r, is the external Vol,4 No.2 Pressure Transient Analysis ofArbitrarily Shaped Fractured Reservoirs 67 boundary of reservoir, r _ U r _ and r _ _ = 0; r, Further assuming {Jrt«l (Ning, et al., 2004), by l 1 l 2==r l 1 Inr l 2 (i > 1) is the boundary of impenneab1e regions, r;nr = using the regular perturbation method (Kikani and 0, (i=l=-j); m is the number of boundaries; r5 is Dirac delta Pedrosa, 1991) we obtain the zero order perturbation function. Subscripts f and m present fracture and matrix. equation 2'70 2'70 The following dimensionless properties are introduced 0 + 0 = w 0'70 + (1- w) oPmD + in the mathematical model: Ox ;),,2 ot ot D vY D D Pi -P (9) PD==--- n, Po l5 L qD/ ( X - X )l5(YD - YDt) D D t X Y t~1 X == .fA' YD=.fA Using the same method to manipulate Eq. (2), we have Kfit t D == ----"'--- oPmD (¢C) f+m j.iA J.(PmD -'7o)+(l-w)--= 0 (10) ot j.iA qD=- Kf,po qt Transforming the corresponding equations to those in Laplace space, we obtain K mt J.=c A Kfi 021fo 021fo f()­ --+--- s S'7 Ox 2 ;)" 2 0 (¢C)f D VYD w=-----"- (11) (¢C)f+m 1 n, = - L qD/l5(x - x )l5(YD - YDt) D D t s t~1 p== por _ J. 1 oK (12) PmD= J.+ (1- w)s '70 r==-- K op where Pi is initial formation pressure, Pa; Po is reference pressure, Pa; A is problem region area, nr'; f.i is fluid viscosity, Pa-s; K is permeability, m'; Kfi is initial GPmD = o1fo = 0 fracture permeability, rrr'; qt is the strength of source onD onD well I, S-I; c iscompressibility, Pa- ; e is interporosity flow shape factor, m'; J. is the interporosity flow coefficient, i.e. the dimensionless matrix fracture permeability ratio; ca is the dimensionless fracture storage parameter; P is dimensionless permeability modulus; rjJ is porosity. where f(s) = (1- w)(ili' + J. and s is the Laplace Eq. (1) is not written in a convenient form to be (1- w)s + J. solved by using the boundary element method. With parameter. Pedrosa's substitution (Ning, et al., 2004) 3. Boundary element method Pm == - P In(1- P'7) (7) The zero order perturbation Eq. (11) is associated and after some algebraic manipulation, Eq. (1) can be with the modified Helmholtz operator. The transformed into corresponding boundary integral equations in terms of the transformed variable rto can be expressed as - ( ) ~ r (G o1fo - eo \AT' a'7o X ' YD = L.. ~. -0 - '70 -0J'.ll- (8) i~I' n nD n, (16) L qD/l5(x - x )l5(YD - YDt) D D t t~1 where rt is a dimensionless dependent variable. where a = e/2Jr and e is the internal angle; G is the 68 2007 Petroleum Science fundamental solution for the modified Helmholtz equation. ", a/foi = L (V;ij7Jonlj + ~ij7JOnaj+l - ~ij7JOj - W 7JO+ l ) 2 ij j~1 (18) 1 ", ­ -- LGi/qol S 1=1 where (X'o,yo) and (x ' Yo) are the arbitrary points where over n ; r is the dimensionless distance, r = [(x - X'D)2 + (YD - YD)2 r ; and J is the zero o D VI" = 1 (r j:Gdf - j: 1 r Gdf) y ;: . _ ;: . .lIT '" I '" J+.hi I order modified Bessel function of the second kind. '" ]+1 '" J J J In order to evaluate the contour integral involved in the boundary integral equation, the boundary I' is discretized V .. = 1 (_ r J:r.'.df + J' r Gdf) l) j: _ j: .hr. ",~, "'J.hi I nb elements. Nodes are allocated at the edges of into '" j+l '" j J ) elements, and boundary values are interpolated linearly in between. The node-numbering direction for outer boundary is the counterclockwise, and the clockwise direction is chosen for the inner boundary. A local (';,0 coordinate system is introduced for convenience's sake (Fig. 1). The origin of coordinates is at point P, from which the I; axis is set parallel to the boundary element ~fj and in the opposite direction of node numbering. The ( axis is defmed by the right-hand rule. Eq. (18) can be solved for unknown boundary values in Laplace space by using the conventional BEM. Then interior solutions in Laplace space can be obtained. The Laplace space interior solution can be inverted to real space by using the Stehfest algorithm (Stehfesh, 1970). By using Duhamel principle (Kikani and Pedrosa, 1991; Yang and Zhao, 2002), the dimensionless bottom hole pressure considering wellbore storage and skin effect can be expressed as: 7JOw =[ _ s + S2 C (19) Dl)-1 s170 + 8 (20) Fig. 1 Local (,;, S') coordinate system where 7Jow is the Laplace space interior solution of In the local (,;, S') coordinate system, boundary values are interpolated as zero order which does not consider wellbore storage and skin effect; L- is Laplace inversion transform operator; PwD is dimensionless bottom hole flowing pressure. 8 = 8 qll ,and 8 is skin factor; 2JrK fhpo fhp or; ';j ::::; ,; ::::; ';j+l C = C 2JrK and CD is dimensionless D1 0 qJ1A According to the previous instruction, the wellbore storage coefficient, rw is well radius, m; q is 3/s; discretized form ofEq. (16) becomes well flow rate, m h is formation thickness, m. VolA No.2 69 shaped boundary. Fig. 3a shows the transient pressure 4. Pressure transient analysis responses for arbitrarily shaped stress-sensitive A comparison of the analytical solution and reservoir with A, ==400 and OJ ==0.04. The shape of numerical solution obtained with the boundary element reservoir and well location are shown in Fig. 3b. From method for the double-porosity reservoir with a closed Fig. 3a it can be deduced that the boundary condition of circular boundary is presented in Fig. 2, which shows a the reservoir has a great effect on pressure behavior. very good agreement between the analytical and Compared with a constant pressure boundary, the numerical solutions. piecewise constant pressure and piecewise closed - BEM solution boundary delay the decline time of time derivative of • Analytical solution pressure. The pressure-derivative type curves are not on P"o the 0.5 horizontal lines. As a result of the stress sensitivity of permeability, the derivative curves rise up and the slop increases with the increase in permeability dp"p/dlnrp variation coefficient. Fig. 4a shows the effect of a single impermeable -2 '---"---......-----.......---'----'---' region on the pressure responses for stress sensitive -7 -6 -5 -4 -3 -2 -I o reservoir with A, ==200 and OJ ==0.02. The corresponding shape of reservoir and well location are Fig. 2 Typical curves for pressure behavior in a closed shown in Fig. 4b. It can be seen that the impermeable circular deformable reservoir region has a great influence on the pressure behavior, One of the advantages of the boundary element making the time of rise-up of the derivative curve method is the flexibility for the treatment of arbitrarily earlier than expected in the case without impermeable regions. 1.2 " -I -2 1-.._-'--_........_--'-__'--_""--..1,.11..-'-1-........ -3 -2 -6 -5 1.5 tv Fig. 3b Shape of deformable reservoir without Fig. 3a Type curves for pressure behavior in arbitrarily impermeable regions and well location shaped deformable reservoirs 1.2 0.8 :-,°0.6 i:5 0.4 I. p'~O.5 wnh a impenneable region ~ -1 2. "p'~O.5 withonrimpenneable region 3. "j.1~O.2 wrth a impermeableregion 0.2 4. ]J~O.2 wllhout impermeable regioo -2'--'---'---'---"---"---"---"---"-----' Ol--_"-_........,.:~-~.-:::;;."..-_......... -; -3 o 2 -7 -6 -2 -I to -0.2 0.3 0.8 J.3 Xl) Fig. 4a Effect of impermeable region on type curves Fig.4b Shape of reservoir with a single impermeable for pressure behavior in deformable reservoirs region and well location 70 2007 Petroleum Science composite gas reservoir. Journal of Daqing Petroleum 5. Conclusions Institute, 28(2), 34-36 (in Chinese) Paulo R. B. and Abraham S. G (1988) The effects of size, shape 1) The transient pressures of a circular reservoir with a and orientation of an impermeable region on transient constant pressure boundary and a closed boundary are pressure testing. SPE, 16376 analyzed by the boundary element method. Compared with Sato K. and Home R. N. (1993a) Perturbation boundary element the analytical solutionthis method is proved to be correct. method for heterogeneous reservoir: Part 1- Steady-state flow 2) The impermeable region has a great influence on problems. SPE, 25299 pressure and pressure-derivative type curves, making Sato K. and Home R. N. (1993b) Perturbation boundary element the time of rise-up of derivative curve earlier than method for heterogeneous reservoir: Part 2- Transient flow expected in the case without impermeable regions. problems. SPE, 25300 3) Compared with constant pressure boundary, the Stehfest H. (1970) Algorithm 368: Numerical inversion of boundary conditions of piecewise constant pressure and Laplace transforms. D-5 Communications of the ACM, 13(1), piecewise closed boundary delay the decline time of 47-49 Wang N. T., Ma X. M. and Huang B. G (2000) Difference derivative of pressure. between approximate and precise mirror reflections in transient well test. Journal of Southwest Petroleum Institute, References 22(2),32-34 (in Chinese) Britto P. R. and Grader A. S. (1987) The effects of size, shape and Yang D. Q. and Zhao Z. S. (2002) The Boundary Element Theory orientation of an impermeable region on transient pressure and Its Application. Beijing: Beijing Institute of Technology testing. SPE, 16376 Press, 1-35 (in Chinese) Chen Z. X. and Jiang S. L. (1980) The exact solution for equation Yin H. J., He Y. F. and Fu C. Q. (2005) Pressure transient analysis group of double-porosity reservoir. Science in China Series of heterogeneous reservoirs with impermeability barriers by A-Mathematics, 10(2), 152-165 (in Chinese) using perturbation boundary element method. Journal of Kikani J. and Home R. N. (1988) Pressure transient analysis of Hydrodynamics, Series B, 17(1), 102-109 arbitrarily shaped reservoirs with the boundary element Zhang W M. and Zeng P. (1992) A boundary element method method. SPE, 18159 applied to pressure transient analysis of irregularly shaped Kikani J. and Pedrosa O. A. Jr. (1991) Perturbation analysis of double-porosity reservoirs. SPE, 25284 stress-sensitive reservoirs. SPE Formation Evaluation, 379­ About the first author Liu Q. G, Li X. P. and Wu X. Q. (2001) Analysis of pressure transient behavior in arbitrarily shaped reservoirs by the Gao Huirnei was born in 1979 boundary element method. Journal of Southwest Petroleum and received her MS degree from Institute, 23(2), 40-43 (in Chinese) Daqing Petroleum Institute in 2001. Liu Q. S. and Duan Y. G (2004) Application of boundary element She now is studying for a doctoral method to unsteady state flow. Petroleum Geology & Oilfield degree in China University of Development in Daqing, 23(2), 36-37 (in Chinese) Petroleum (Beijing) with her Masukawa J. and Home R. N. (1988) Application of the research interests in oil/gas field boundary integral method to immiscible displacement development. problems. SPE Reservoir Engineering, 1069-1077 E-mail: huimeigao@126.com Mavor M. J. and Cinco-Ley H. (1979) Transient pressure behavior of naturally fractured reservoirs. SPE, 7977 (Received June 12, 2006) Ning Z. F., Liao X. W, Gao W L., et al. (2004) Pressure transient response in deep-seated geothermal stress-sensitive fractured (Edited by Sun Yanhua) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Petroleum Science Springer Journals

Pressure transient analysis of arbitrarily shaped fractured reservoirs

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Publisher
Springer Journals
Copyright
Copyright © 2007 by China University of Petroleum
Subject
Earth Sciences; Mineral Resources; Industrial Chemistry/Chemical Engineering; Industrial and Production Engineering; Energy Economics
ISSN
1672-5107
eISSN
1995-8226
DOI
10.1007/BF03187444
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See Article on Publisher Site

Abstract

2007 VolA No.2 Petroleum Science Gao Huimei', He Yingfu', Jiang Hanqiao' and Chen Minfeng' (1. School ofPetroleum Engineering, China University ofPetroleum, Beijing 102249, China) (2. Institute ofPorous Flow & Fluid Mechanics, CNPC & Chinese Academy ofSciences, Langfang, Hebei 065007, China) (3. Center for Enhanced Oil Recovery, China University ofPetroleum, Beijing 102249, China) Abstract: Reservoir boundary shape has a great influence on the transient pressure response of oil wells located in arbitrarily shaped reservoirs. Conventional analytical methods can only be used to calculate transient pressure response in regularly shaped reservoirs. Under the assumption that permeability varies exponentially with pressure drop, a mathematical model for well test interpretation of arbitrarily shaped deformable reservoirs was established. By using the regular perturbation method and the boundary element method, the model could be solved. The pressure behavior of wells with wellbore storage and skin effects was obtained by using the Duhamel principle. The type curves were plotted and analyzed by considering the effects of permeability modulus, arbitrary shape and impermeable region. Key words: Deformable reservoir, double-porosity reservoir, boundary element method, transient pressure, type curve permeability modulus, arbitrary shape and impermeable 1. Introduction barriers. Many methods use radial and circular systems to 2. Mathematical model interpret unsteady state double-porosity reservoir flow problems (Chen and Jiang, 1980; Mavor and Cinco-Ley, For single-phase slightly compressible fluid flow in 1979; Zhang and Zeng, 1992), but very little information is deformable double-porosity media, n, including n; available for arbitrarily shaped reservoirs. For many cases, sinks of strengths ql, the corresponding equations may however, the double-porosity reservoir drainage shape is be written as: too complicated to be approximated by a circular shape. The existence of one or more impermeable regions further a~~ +a~~ -p[(::r +(:: n= complicates the problem. So a numerical means is required (Britto,and Grader, 1987; Kikani and Home, 1988; Liu, et al., 2001; Liu and Duan, 2004; Masukawa and Home, PPID e [w OPID + (1- w) opmn) + (1) 1988; Paulo and Abraham, 1988; Sato and Home, 1993a; at at D D 1993b; Wang, et al., 2000; Yin, et al., 2005; Zhang and n, PP ID Zeng, 1992). Numerical techniques for solving partial e L qD/o(x - XD/ )O(YD - YD/) differential equations describing various physical I;' processes can be categorized into two distinct classes: the domain methods and the boundary methods. Finite difference and finite element methods fall in the first class, and the boundary element methods (BEM) constitute the (3) second. The BEM is superior to the domain methods in several ways. The most notable advantage is the high degree of accuracy that results from its sound aPmn = apID = 0 (4) mathematical foundations. Flexibility in defining boundary anD anD geometries and conditions is another feature to be emphasized. However, the conventional boundary element (i=2,3, .. ·,m) (5) method is not applicable to the problem of fluid flow in porous deformable media. In this paper, under the assumption that permeability varies exponentially with pressure drop (Kikani and Pedrosa, 1991; Ning, et al., (6) 2004), transient pressure response of wells in deformable double-porosity reservoirs was obtained. The type curves where it is assumed that the flow is in a horizontal plane were developed and analyzed by considering effects of and that the Darcy's law is applicable; r, is the external Vol,4 No.2 Pressure Transient Analysis ofArbitrarily Shaped Fractured Reservoirs 67 boundary of reservoir, r _ U r _ and r _ _ = 0; r, Further assuming {Jrt«l (Ning, et al., 2004), by l 1 l 2==r l 1 Inr l 2 (i > 1) is the boundary of impenneab1e regions, r;nr = using the regular perturbation method (Kikani and 0, (i=l=-j); m is the number of boundaries; r5 is Dirac delta Pedrosa, 1991) we obtain the zero order perturbation function. Subscripts f and m present fracture and matrix. equation 2'70 2'70 The following dimensionless properties are introduced 0 + 0 = w 0'70 + (1- w) oPmD + in the mathematical model: Ox ;),,2 ot ot D vY D D Pi -P (9) PD==--- n, Po l5 L qD/ ( X - X )l5(YD - YDt) D D t X Y t~1 X == .fA' YD=.fA Using the same method to manipulate Eq. (2), we have Kfit t D == ----"'--- oPmD (¢C) f+m j.iA J.(PmD -'7o)+(l-w)--= 0 (10) ot j.iA qD=- Kf,po qt Transforming the corresponding equations to those in Laplace space, we obtain K mt J.=c A Kfi 021fo 021fo f()­ --+--- s S'7 Ox 2 ;)" 2 0 (¢C)f D VYD w=-----"- (11) (¢C)f+m 1 n, = - L qD/l5(x - x )l5(YD - YDt) D D t s t~1 p== por _ J. 1 oK (12) PmD= J.+ (1- w)s '70 r==-- K op where Pi is initial formation pressure, Pa; Po is reference pressure, Pa; A is problem region area, nr'; f.i is fluid viscosity, Pa-s; K is permeability, m'; Kfi is initial GPmD = o1fo = 0 fracture permeability, rrr'; qt is the strength of source onD onD well I, S-I; c iscompressibility, Pa- ; e is interporosity flow shape factor, m'; J. is the interporosity flow coefficient, i.e. the dimensionless matrix fracture permeability ratio; ca is the dimensionless fracture storage parameter; P is dimensionless permeability modulus; rjJ is porosity. where f(s) = (1- w)(ili' + J. and s is the Laplace Eq. (1) is not written in a convenient form to be (1- w)s + J. solved by using the boundary element method. With parameter. Pedrosa's substitution (Ning, et al., 2004) 3. Boundary element method Pm == - P In(1- P'7) (7) The zero order perturbation Eq. (11) is associated and after some algebraic manipulation, Eq. (1) can be with the modified Helmholtz operator. The transformed into corresponding boundary integral equations in terms of the transformed variable rto can be expressed as - ( ) ~ r (G o1fo - eo \AT' a'7o X ' YD = L.. ~. -0 - '70 -0J'.ll- (8) i~I' n nD n, (16) L qD/l5(x - x )l5(YD - YDt) D D t t~1 where rt is a dimensionless dependent variable. where a = e/2Jr and e is the internal angle; G is the 68 2007 Petroleum Science fundamental solution for the modified Helmholtz equation. ", a/foi = L (V;ij7Jonlj + ~ij7JOnaj+l - ~ij7JOj - W 7JO+ l ) 2 ij j~1 (18) 1 ", ­ -- LGi/qol S 1=1 where (X'o,yo) and (x ' Yo) are the arbitrary points where over n ; r is the dimensionless distance, r = [(x - X'D)2 + (YD - YD)2 r ; and J is the zero o D VI" = 1 (r j:Gdf - j: 1 r Gdf) y ;: . _ ;: . .lIT '" I '" J+.hi I order modified Bessel function of the second kind. '" ]+1 '" J J J In order to evaluate the contour integral involved in the boundary integral equation, the boundary I' is discretized V .. = 1 (_ r J:r.'.df + J' r Gdf) l) j: _ j: .hr. ",~, "'J.hi I nb elements. Nodes are allocated at the edges of into '" j+l '" j J ) elements, and boundary values are interpolated linearly in between. The node-numbering direction for outer boundary is the counterclockwise, and the clockwise direction is chosen for the inner boundary. A local (';,0 coordinate system is introduced for convenience's sake (Fig. 1). The origin of coordinates is at point P, from which the I; axis is set parallel to the boundary element ~fj and in the opposite direction of node numbering. The ( axis is defmed by the right-hand rule. Eq. (18) can be solved for unknown boundary values in Laplace space by using the conventional BEM. Then interior solutions in Laplace space can be obtained. The Laplace space interior solution can be inverted to real space by using the Stehfest algorithm (Stehfesh, 1970). By using Duhamel principle (Kikani and Pedrosa, 1991; Yang and Zhao, 2002), the dimensionless bottom hole pressure considering wellbore storage and skin effect can be expressed as: 7JOw =[ _ s + S2 C (19) Dl)-1 s170 + 8 (20) Fig. 1 Local (,;, S') coordinate system where 7Jow is the Laplace space interior solution of In the local (,;, S') coordinate system, boundary values are interpolated as zero order which does not consider wellbore storage and skin effect; L- is Laplace inversion transform operator; PwD is dimensionless bottom hole flowing pressure. 8 = 8 qll ,and 8 is skin factor; 2JrK fhpo fhp or; ';j ::::; ,; ::::; ';j+l C = C 2JrK and CD is dimensionless D1 0 qJ1A According to the previous instruction, the wellbore storage coefficient, rw is well radius, m; q is 3/s; discretized form ofEq. (16) becomes well flow rate, m h is formation thickness, m. VolA No.2 69 shaped boundary. Fig. 3a shows the transient pressure 4. Pressure transient analysis responses for arbitrarily shaped stress-sensitive A comparison of the analytical solution and reservoir with A, ==400 and OJ ==0.04. The shape of numerical solution obtained with the boundary element reservoir and well location are shown in Fig. 3b. From method for the double-porosity reservoir with a closed Fig. 3a it can be deduced that the boundary condition of circular boundary is presented in Fig. 2, which shows a the reservoir has a great effect on pressure behavior. very good agreement between the analytical and Compared with a constant pressure boundary, the numerical solutions. piecewise constant pressure and piecewise closed - BEM solution boundary delay the decline time of time derivative of • Analytical solution pressure. The pressure-derivative type curves are not on P"o the 0.5 horizontal lines. As a result of the stress sensitivity of permeability, the derivative curves rise up and the slop increases with the increase in permeability dp"p/dlnrp variation coefficient. Fig. 4a shows the effect of a single impermeable -2 '---"---......-----.......---'----'---' region on the pressure responses for stress sensitive -7 -6 -5 -4 -3 -2 -I o reservoir with A, ==200 and OJ ==0.02. The corresponding shape of reservoir and well location are Fig. 2 Typical curves for pressure behavior in a closed shown in Fig. 4b. It can be seen that the impermeable circular deformable reservoir region has a great influence on the pressure behavior, One of the advantages of the boundary element making the time of rise-up of the derivative curve method is the flexibility for the treatment of arbitrarily earlier than expected in the case without impermeable regions. 1.2 " -I -2 1-.._-'--_........_--'-__'--_""--..1,.11..-'-1-........ -3 -2 -6 -5 1.5 tv Fig. 3b Shape of deformable reservoir without Fig. 3a Type curves for pressure behavior in arbitrarily impermeable regions and well location shaped deformable reservoirs 1.2 0.8 :-,°0.6 i:5 0.4 I. p'~O.5 wnh a impenneable region ~ -1 2. "p'~O.5 withonrimpenneable region 3. "j.1~O.2 wrth a impermeableregion 0.2 4. ]J~O.2 wllhout impermeable regioo -2'--'---'---'---"---"---"---"---"-----' Ol--_"-_........,.:~-~.-:::;;."..-_......... -; -3 o 2 -7 -6 -2 -I to -0.2 0.3 0.8 J.3 Xl) Fig. 4a Effect of impermeable region on type curves Fig.4b Shape of reservoir with a single impermeable for pressure behavior in deformable reservoirs region and well location 70 2007 Petroleum Science composite gas reservoir. Journal of Daqing Petroleum 5. Conclusions Institute, 28(2), 34-36 (in Chinese) Paulo R. B. and Abraham S. G (1988) The effects of size, shape 1) The transient pressures of a circular reservoir with a and orientation of an impermeable region on transient constant pressure boundary and a closed boundary are pressure testing. SPE, 16376 analyzed by the boundary element method. Compared with Sato K. and Home R. N. (1993a) Perturbation boundary element the analytical solutionthis method is proved to be correct. method for heterogeneous reservoir: Part 1- Steady-state flow 2) The impermeable region has a great influence on problems. SPE, 25299 pressure and pressure-derivative type curves, making Sato K. and Home R. N. (1993b) Perturbation boundary element the time of rise-up of derivative curve earlier than method for heterogeneous reservoir: Part 2- Transient flow expected in the case without impermeable regions. problems. SPE, 25300 3) Compared with constant pressure boundary, the Stehfest H. (1970) Algorithm 368: Numerical inversion of boundary conditions of piecewise constant pressure and Laplace transforms. D-5 Communications of the ACM, 13(1), piecewise closed boundary delay the decline time of 47-49 Wang N. T., Ma X. M. and Huang B. G (2000) Difference derivative of pressure. between approximate and precise mirror reflections in transient well test. Journal of Southwest Petroleum Institute, References 22(2),32-34 (in Chinese) Britto P. R. and Grader A. S. (1987) The effects of size, shape and Yang D. Q. and Zhao Z. S. (2002) The Boundary Element Theory orientation of an impermeable region on transient pressure and Its Application. Beijing: Beijing Institute of Technology testing. SPE, 16376 Press, 1-35 (in Chinese) Chen Z. X. and Jiang S. L. (1980) The exact solution for equation Yin H. J., He Y. F. and Fu C. Q. (2005) Pressure transient analysis group of double-porosity reservoir. Science in China Series of heterogeneous reservoirs with impermeability barriers by A-Mathematics, 10(2), 152-165 (in Chinese) using perturbation boundary element method. Journal of Kikani J. and Home R. N. (1988) Pressure transient analysis of Hydrodynamics, Series B, 17(1), 102-109 arbitrarily shaped reservoirs with the boundary element Zhang W M. and Zeng P. (1992) A boundary element method method. SPE, 18159 applied to pressure transient analysis of irregularly shaped Kikani J. and Pedrosa O. A. Jr. (1991) Perturbation analysis of double-porosity reservoirs. SPE, 25284 stress-sensitive reservoirs. SPE Formation Evaluation, 379­ About the first author Liu Q. G, Li X. P. and Wu X. Q. (2001) Analysis of pressure transient behavior in arbitrarily shaped reservoirs by the Gao Huirnei was born in 1979 boundary element method. Journal of Southwest Petroleum and received her MS degree from Institute, 23(2), 40-43 (in Chinese) Daqing Petroleum Institute in 2001. Liu Q. S. and Duan Y. G (2004) Application of boundary element She now is studying for a doctoral method to unsteady state flow. Petroleum Geology & Oilfield degree in China University of Development in Daqing, 23(2), 36-37 (in Chinese) Petroleum (Beijing) with her Masukawa J. and Home R. N. (1988) Application of the research interests in oil/gas field boundary integral method to immiscible displacement development. problems. SPE Reservoir Engineering, 1069-1077 E-mail: huimeigao@126.com Mavor M. J. and Cinco-Ley H. (1979) Transient pressure behavior of naturally fractured reservoirs. SPE, 7977 (Received June 12, 2006) Ning Z. F., Liao X. W, Gao W L., et al. (2004) Pressure transient response in deep-seated geothermal stress-sensitive fractured (Edited by Sun Yanhua)

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Petroleum ScienceSpringer Journals

Published: Jun 16, 2010

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