A FORMULATION FOR HYDRAULIC FRACTURE PROPAGATION IN A POROELASTIC MEDIUM

Transcription

1 A FORMULATION FOR HYDRAULIC FRACTURE PROPAGATION IN A POROELASTIC MEDIUM J. L. Antunes O Sousa, P. R. B Devloo Faculdade de Engenharia Civil M Utiyama Faculdade de Engenharia Mecanica Universidade Estadual de Campinas Abstract This paper draws on the simulation of hydraulic fracturing processes combining a model for fluid flow in the fracture with Blot's' model of coupled poroelasticity for the rock behavior. The theoretical background for the computational implementation is discussed. INTRODUCTION The strategy for the simulation of hydraulic fracturing consists basically in a model that considers the variation of fracture opening as a function of the fracture pressure distribution, taking into account poroelastic^ behavior of the rock. On the other hand, the fracture pressure distribution is obtained based on the assumption of a flow regime of a Newtonian fluid between two parallel plates. This paper describes the theoretical background for the computational implementation of this strategy and presents some preliminary results. FLUID FLOW IN THE FRACTURE Fluid flow in the fracture is modeled by Equation 1, based on the assumption of flow between two parallel plates'". "P (i)

2 488 Offshore Engineering where q : unit flow w : fracture opening p :fluidpressure jj, : fluid viscosity For a given control volume in the fracture, mass conservation is described by Equation 2: d w + q,=0 (2) where j : leakoff rate t : time The corresponding weak formulation is given by where O : domain co : test function Jdivq co dfl+ f m dq + jqj ca dfl - 0,^ n ndt o w; Introducing Equation 1, the constitutive relation between q and Vp, into Equation 3, results in: J-div(-^ Vp) co do+ J co dq + jqj co dq = 0,., n 12M- Qdt Q (4) Integrating Equation 4 by parts : J -co ( Vp)-ndSn + J-( Vp) -VwdO + 12u + J- co dd-f Jq, co dq = 0 /.\ Subdividing the fracture boundary dq. in dq\ and d&2 such that and replacing the boundary values into Equation 5 (weak formulation), results in:

3 Offshore Engineering 489 Q f dw Vp) VCD di2 + I Qdt CD O wdq = 0 I/) Assuming, as a first approximation, proportionality between pressure in a given cross section and the corresponding fracture opening, results: P (8) where k is a spring constant. Thus, Equati Lion 7 results: w^, dw J - q n (s) co ddq -f I - Vffldn+^-Ndn.-Jq,«dn = o 00-) O In Equation 9, the only independent variable is p. Assuming that the fracture is one-dimensional, and there is no leakoffto the formation, Equation 10 is obtained. dp dec 1 I dp dx dx k A dt (10) ^ This simplified formulation was used for preliminary tests performed with a straight fracture. Results are presented in Figure 1. Quadratic elements were used, with four integration points per element r Abscissa along fracture path (m) Figure 1 - Evolution of fracture opening profile Rewriting this simplified formulation by replacing Equation 8 by a function taking into account poroelastic effects, and including a term accounting for leakoff, an adequate formulation is obtained for the two-dimensional simulation of a hydraulic fracturing process in a poroelastic medium.

4 490 Offshore Engineering POROELASTIC RESPONSE OF THE ROCK The poroelastic response of the rock is governed by the equations of equilibrium and mass conservation: diva = where a : total stress(tensor) C : variation of fluid contents The relation between total stress a and effective stress a' is described by o = a'-apl Combining the elastic constitutive equation for the solid framework Equation 12, results: - T 2Gv a = G(Vu + Vu*) + (divu)l-apl (13) where u : displacement field G : shear modulus v : Poisson's ratio a : Riot's poroelastic constant The relations governing mass conservation are: q = -KVp % 2, 1 ^P - = a-(d,vu) + -- where K : permeability coefficient Q : constant associated with the compressibility of the fluid The semi-discrete form of these equations is: divo"+* =0 where the superscripts n and n+1 indicate, respectively, the previous and the current time interval. To these equation, a weak formulation and a finite element approximation are applied, resulting

5 Offshore Engineering 491 The matrix [K] follows directly from the weak formulation from Equation 13. The elements of vector {d} are grouped in (u,v,p). The matrix [L] results from the terms related to the previous time step n. The vector {F} contains boundary conditions and possible leakoff (incoming flow positive). Note that [H] depends on the previous time step n. CONDENSATION OF THE DISCRETE EQUATIONS FROM POROELASTICITY ONTO THE FRACTURE Separating the discretization functions in internal functions (subscript 0) and function on the fracture (subscript 1), the equation system can be rewritten as KOI HO HI (17) Applying static condensation of the internal functions on the interface functions: where : (18) Reordering the equations on the fracture in (u,v,p), results: Kup ^vu ^pu K PP. (20) It should be noted tha H\ includes a term which is dependent on the pressure in the fracture surface. This term results from the integration by parts applied to Equation 13: Hy ~ Hy + Mypp (L\) Considering the fracture is horizontal, another static condensation of u can be applied on (v,p), resulting in: K, ired2 r >, (ij } red2 "VP rl rv,_,red2 Kpv H, (22)

6 492 Offshore Engineering where TV- red 2 TV- TV- 1^1^ \A ^vp ~ *^vp ^vu^uu^up **iyp rrred2 _ uo ^ -! (23) v - v ~ ^vu^uu^u Equation 22 allows a relation between pressure p and fracture opening for the cases in which v is normal to a fracture on a plane of symmetry in the direction x In this case, the fracture opening is obtained as w=2v. When dealing with fractures with arbitrary geometry, Equation 22 should be rewritten to give w as a function of u and v. COUPLING OF THE FLUID FLOW IN THE FRACTURE TO THE POROELASTIC RESPONSE OF THE ROCK' >* From Equation 22, given a pressure p, v can be computed as: v w vp (24) The leakoff term can be obtained in the same way by Equation 25:,TT \red2 _ r^ -,red2,». r^,red2,» {ttp} -ikpyj {v}+[kppj {p} (25) Applying a finite element approximation to the weak formulation for the flow in the fracture, Equation 9, the following algebraic expression can be obtained : =0 where w=2v for the case of a fracture in the plane of symmetry. The matrices in Equation 26 are defined by : Iwldv,/,^, ~ dx dx ** 1 I I Qli =/Vi Qdx i Vjdx (27) The vector {Q, } corresponds to the leakoff term from the problem of fluid flow in the fracture. The vector {Hp} is the corresponding vector in the formulation offluidflow in the porous medium. Thus, Qi can be replaced by

7 Offshore Engineering 493 Q _ fu ired2 rr, -,red2, ^rj^ ired2 /, P pv PP (28) The nonlinear formulation for the flow in the fracture, with w=2v, becomes Symbolically, Equation 29 can be written as a residue equation, such that the solution for a given time step corresponds to a zero residue. " + ' n" + h = 0 (30) Equation 24 can be replaced into Equation 29 to work only with the pressure variable: = Res(p"+') Thus, Equation 31 leads to the tangent matrix and load vector for the coupled problem: The solution process is iterative. Thus, a convergence optimization is desirable. Replacing the variable p"*' from Equation 31 by p" +Ap, results ired2 /r^ -,red2 >,-! (r^ -,red2\/ n,. > vp] ){p +Ap}- (33) ^ red 2 This equation can be rewritten as

8 494 Offshore Engineering Res(p" (34) Thus, as Res(p" + Ap)is the residue to minimize, Equation rewritten as: 34 can be -Res(p") (35) atrix [A] depends on the displacements. However, the same matrix computed based on the previous step was considered, although the displacements change. The approximation signal in Equation 35 is due to this fact. Rewriting Equation 35 to make the computation of the pressure increment explicit, results: {Ap} = Equation 24 expresses the relation between pressure and vertical displacements. Applying the same procedure to obtain the displacement field increment, results: Ap} (37) (38) CONCLUSION The procedure discussed in the previous sections allows the pressure and displacement fields to be obtained at the end of the propagation process. These fields satisfy the elasticity equations for the rock matrix, and fluid flow in the porous medium and in the fracture. Compatibility in the interfaces of the models is satisfied by these fields. Preliminary results, for a simplified model were presented. The computational implementation for twodimensional problems is underway, and is expected to be published in the near future.

9 Offshore Engineering 495 ACKNOWLEDGMENTS The authors would like to acknowledge the continuous support from Petrobras S. A., CNPq-Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (Grants PQ /92-6 e PQ /94-3) and FAPESP- Funda$ao de Amparo a Pesquisa do Estado de Sao Paulo (Grants e ). REFERENCES [1] Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys.,v. 12, pp , [2]Devloo, P. R. B.; Sousa, J. L. A. 0.; Siqueira, C A. M., Sobre a implementapao da condicao inicial em problemas de poroelasticidade acoplada, Anais do CNMAC, Curitiba, SBMAC, pp , [3] Murad, M. A.: Modelagem e analise numerica de escoamentos saturados em meios porosos rigidos e elasticos lineares, Tese de Doutorado, Pontificia Universidade Catolica do Rio de Janeiro, Rio de Janeiro, RJ, [4] Siqueira, C A M., Um sistema orientado por objetos para analise numerica da poroelasticidade acoplada pela tecnica dos elementos fmitos, Disserta$ao de Mestrado, Universidade Estadual de Campinas, Campinas, SP, [5] Sousa, J. L. A. O; Devloo, P. R B.; Siqueira, C. A. M., Um sistema orientado por objetos para a simulacao acoplada de efeitos poroelasticos em rochas de reservatorio petroliferos, Anais da I Workshop sobre Engenharia de Reservatorios, Universidade Estadual de Campinas, Campinas, SP, pp , [6] Devloo, P. R B.; Santana, M. L. M., Desenvolvimento de algoritmo de sub-estruturacao para elementos finitos. Proceedings VI ENCYT/VI LATCYM, Florianopolis, SC, pp , 1996.