Numerical Analysis of the Double Porosity Consolidation Model

Transcription

1 XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real septiembre 2009 (pp. 1 8) Numerical Analysis of te Double Porosity Consolidation Model N. Boal 1 F.J. Gaspar 1 F.J. Lisbona 2 P.N. Vabiscevic 3 1 Dpto. Matemática Aplicada Universidad de Zaragoza C.P.S. M a de Luna Zaragoza Spain. s: nboal@unizar.es fjgaspar@unizar.es. 2 Dpto. Matemática Aplicada Universidad de Zaragoza Facultad de Matemáticas Pedro Cerbuna Zaragoza Spain. lisbona@unizar.es. 3 Institute for Matematical Modelling RAS 4-A Moscow Russia. vab@imamod.ru. Poroelasticity double porosity staggered grids finite differences energy norm estimates. Key words: Abstract A family of finite difference metods for a two dimensional double porosity consolidation problem is considered. Stabilized discretizations using staggered grids in bot space and time are proposed. A priori estimates for displacements and pressures in discrete energy norms are obtained and te corresponding convergence results are given. 1. Introduction Soil consolidation teory addresses te time dependent coupling between te deformation of a porous matrix and te fluid flow inside. Te porous matrix is supposed to be saturated by te fluid pase and te flow is governed by Darcy s law. Te state of te continuous medium is caracterized by te knowledge of displacements and fluid pressure at eac point. Te consolidation process under one dimensional conditions was first investigated by Terzagi [17] and a model for a rater general situation was proposed and analyzed by Biot [2] in tree dimensions. Tese autors assumed an elastic response of te soil skeleton to te loads. Under te previous assumption a cange in stress will generate a deformation and an excess of pore pressure. Te dissipation of te pressure will result into a final deformed state. Naturally fractured rock formations form important subsurface flow systems wit a ig degree of local eterogeneity. Aquifer or reservoir systems wic contain bot open fractures and interconnected pore spaces are often modelled as a double porosity medium (Barenblatt et al [1]). 1

2 N. Boal F.J. Gaspar F.J. Lisbona P.N. Vabiscevic Te double porosity model produces fluid pressure and displacement istories at early times wic cannot be adequately modelled using equivalent single-porosity assumptions. Te magnitude of te difference between te single and double porosity results is determined by te degree of interaction between te fracture and porous matrix. Te constitutive equations can be formulated by extending Biot s concepts of poroelasticity to double porosity media. Here it is assumed tat te composite rock/pore/fracture/ fluid mixture is bot omogeneous and isotropic. Te governing equations of te double porosity consolidation problem in te incompressible case take te form µ u (λ + µ) grad div u + β 1 grad p 1 + β 2 grad p 2 = g β 1 t (div u) κ 1 η p 1 κ (p 2 p 1 ) = f 1 β 2 t (div u) κ 2 η p 2 + κ (p 2 p 1 ) = f 2 were u is te displacement vector of te solid skeleton ( p 1 is te fluid ) pressure for pores and p 2 is te fluid pressure for fissures. Te operator 0 = is te vector Laplace 0 operator in cartesian coordinates. Te constants λ and µ are te Lamé coefficients and te constants β 1 and β 2 measure te cange of porosities due to an applied volumetric strain. Te constants κ 1 and κ 2 are te permeability of te respective pores and te constant µ denotes te viscosity of te pore fluid. Te last term in equations (1) 2 and (1) 3 is a fluid flow term proportional to te difference in pressure between te fracture and matrix and were te constant κ measures te transfer of fluid from te fissures to te pores. We consider te system of equations (1) on te unit square Ω = (0 1) (0 1). For simplicity in te analysis we assume tat Ω is rigid and permeable so we ave a pure Diriclet problem wit boundary conditions given by u(x t) = 0 p 1 (x t) = 0 p 2 (x t) = 0 x Ω 0 < t T. (2) Due to te incompressibility of te fluid at te initial time te u(x 0) = 0 x Ω is satisfied. Moreover we suppose existence and uniqueness of a sufficiently regular solution in (0 T ] Ω. It is immediate to note tat coosing β 2 = k 2 = κ = 0 one obtains te governing equations of te Biot s consolidation problem wit single porosity. On te oter and neglecting deformation effects i.e. taking β 1 = β 2 = λ = µ = 0 problem (1) reduces to Barenblatt s equations for fluid flow troug non-deformable media wit double porosity. Te numerical solution of Biot s problems is usually approaced using finite element metods see for instance te monograp by Lewis and Screfler [11]. Problems were te solution is smoot are satisfactorily solved by standard finite element discretizations. Neverteless wen strong pressure gradients appear tese metods are unstable in te sense tat strong nonpysical oscillations appear in te approximation of te pressure field. It is well known tat tis penomenon appears at te beginning of te consolidation process wen a load is applied on a part of te boundary. Te same beavior of te numerical metods occurs wen te double porosity model is considered. Te oscillations in te FEM can be minimized if stabilized metods are used. As for Stokes problems approximation spaces for te vector and te scalar fields satisfying LBB (1) 2

3 Numerical Analysis of te Double Porosity Consolidation Model stability condition [3] can be used. Tis approac as been analyzed in [ ] by Murad et al. and by Mira et al. [12] for te quasi-static Biot s model. Neverteless tese metods still present small oscillations in pressure approximation wen very sarp boundary layers occur; ceck for example Taylor Hood metod on te 1D Terzagi problem [17]. Oter autors as Korsawe [10] use a combination of te least square mixed finite element metod and local refinement of te grid near te load boundary in order to obtain non-oscillatory solutions. Tis local refinement may imply an excess of computational effort. Naturally as for finite elements standard finite difference scemes may suffer te same unstable beaviour in pressure approximation. In [4] a reason of tis instability for one-dimensional problems as been identified and tat leads to use staggered grid discretizations for poroelasticity problems [6]. In tis way second order convergent scemes in discrete energy norms for displacements and pressures ave been obtained. New stabilization tecniques based on reformulation of te problem and oters based on te addition of artificial terms to te original equations can be seen in [7] and [8] respectively. Our approac to te numerical approximation of double porosity is based on te finite difference metod on staggered grids. To tis respect we would like to mention tat very often if te problem of interest permits it finite difference metods on staggered grids lead to discretizations tat mimic te continuous problem very well so tat te main properties of te continuous problem are preserved in te discrete case. Note also wen structured grids are used te finite difference metods can easily be combined wit fast solvers as efficient geometric multigrid metods. For instance in [5] and [18] efficient algoritms for poroelasticity equations ave been proposed. In tis work we investigate a family of finite difference scemes to solve double porosity consolidation problems. A space discretization on staggered grids is proposed. For time discretization a weigted two level sceme on a time staggered mes as been adopted for time stepping wic is coerent wit te lack of initial condition for te pressures. A priori estimates in discrete energy norms are obtained for displacements and pressure and te corresponding convergence results are given. 2. Finite Difference Discretization 2.1. Grids and grid operators Following [6] in order to produce a stabilized finite difference metod we will use staggered grdis for vector unknowns. We start giving te different grids and defining te discrete operators used in te space discretization. For simplicity in notation we will consider uniform grids ten let N a positive integer all te grids will ave = 1/N as te mes size in eac spatial direction. For te scalar fields p 1 p 2 we work in te following uniform grid (see Figure 1) ω p = {(i j) / i j = 0... N}. (3) Let us denote ω p te set of inner nodes and ω p te set of boundary nodes of ω p. Associated wit tis mes we define H ωp as te space of scalar grid functions over ω p wit te inner 3

4 N. Boal F.J. Gaspar F.J. Lisbona P.N. Vabiscevic product (p q) ωp = j i p ij q ij j=0 i=0 were k = for k = 1... N 1 and 0 = N = /2 and H ωp is te subspaces of H ωp of grid functions vanising on ω p. For te approximation of vector field of te displacements u = (u v) we consider a staggered grid (see Figure 1) defined by ω u = {( (i ) j) / i = 1/ N 1 N 1/2 j = 0... N } ω v = {( i (j )) / i = 0... N j = 1/ N 1 N 1/2 }. (4) ω u ω v te set of internal nodes and ω u ω v te set of boundary nodes of ω u ω v respec- Figure 1: Grids for displacement u = (u v) u and for v (left picture) and grid for pressures p 1 and p 2 (rigt picture) tively. Also H ωu H ωv denote te spaces of grid functions defined in ω u and ω v respectively wit scalar products (u w) ωu = (v w) ωv = j=0 N 1 j=0 N 1 i=0 j u i+1/2j w i+1/2j + i v ij+1/2 w ij+1/2 + i=0 j u 0j w 0j + j=0 i v i0 w i0 + i=0 j u Nj w Nj j=0 i v in w in. i=0 H ωu and H ωv are te subspaces of H ωu and H ωv respectively of grid functions vanising on ω u ω v. Finally Hωu = H ωv H ωv is te space given by te discrete vector valued functions u = (u v) wit scalar product and norm defined as (u 1 u 2 ) = (u 1 u 2 ) ωu + (v 1 v 2 ) ωv u = (u u) 1/2 and H ωu = H ωu H ωv te corresponding subspace of grid functions null on te boundary. After giving te meses and te spaces of grid functions we carry on wit te semi discretization process introducing te discrete space operators. We start defining a discrete divergence of displacements on points of ω p and a discrete gradient of pressure wose components are located on displacements points suc tat te discrete gradient operator 4

5 Numerical Analysis of te Double Porosity Consolidation Model be te negative adjoint to te discrete divergence operator. Ten let be D : H ωu H ωp te discrete divergence operator given by (Du) ij = (u x ) ij + (v y ) ij wit (u x ) ij = u i+1/2j u i 1/2j (v y ) ij = v ij+1/2 v ij 1/2 for te internal points and on te boundary (u x ) 0j = u 1/2j u 0j 0 = 2 (u 1/2j u 0j ) j = 0... N (u x ) Nj = u Nj u N 1/2j N = 2 (u Nj u N 1/2j ) j = 0... N (v y ) i0 = v i1/2 v i0 0 = 2 (v i1/2 v i0 ) i = 0... N (v y ) in = v in v in 1/2 N = 2 (v in v in 1/2 ) i = 0... N. After tat te discrete gradient operator G : H ωp H ωu is Gp = (G x p G y p) H ωu H ωv were for j = 0... N p i+1j p ij i = 0... N 1 (G x p) i+1/2j = p 0j i = 1/2 p Nj i = N 1/2 and for i = 0... N p ij+1 p ij j = 0... N 1 (G y p) ij+1/2 = p i0 j = 1/2 p in j = N 1/2. Te discrete elasticity operator A : H ωu H ωu is defined as wit Au = µ u (λ + µ)gd u = ( u 0 0 v were u and v are te discrete five points Laplace operators on H ωu and H ωv respectively. Now we ave (Au v) = (u Av) for all u v H ωu i.e. A is selfadjoint in H ωu and (Au u) µ( u u) consequently te discrete operator A is positive definite on H ωu. Finally we define te diffusion operator connected wit te pressures in te pores and in te fissures respectively. Let be B : H ωp H ωp defined by Bp = p were = DG is te usual five point stencil approximation for te Laplace operator on H ωp. So B = B and B δ B E were δ B > 0 independent of and E is te identity operator i.e. B is symmetric and positive definite on H ωp. 5 )

6 N. Boal F.J. Gaspar F.J. Lisbona P.N. Vabiscevic 2.2. Discrete approximation Te semi discrete approximations u (t) H ωu p 1 (t) p (t) H ωp of te solution of te continuous problem (1) are given by te difference differential system β 1 β 2 A u (t) + β 1 G p 1 (t) + β 2 G p (t) = g (t) d d t (D u (t)) + κ 1 η B p 1(t) κ (p (t) p 1 (t)) = f 1 (t) d d t (D u (t)) + κ 2 η B p (t) + κ (p (t) p 1 (t)) = f (t) for 0 < t T wit te initial condition Du (0) = 0. To obtain te fully discrete approximation we apply a time discretization to te Caucy problem (5) using a staggered grid in time. Let M be a positive integer = T/M te time step size and {t m = m} M m=0 {t m+1/2 = (m + 1/2)} M 1 m=0 te time levels were te displacements and te pressures are approximated respectively (see Figure 2). On tis time staggered mes we define te following sceme Du m+1 β 1 Du m+1 β 2 Du m Du m A u m+1 σ + β 1 G p m+1/2 1 + β 2 G p m+1/2 = g m+1 σ + κ 1 η B pm+1/2 1 κ (p m+1/2 p m+1/2 1 ) = f m+1/2 1 + κ 2 η B pm+1/2 + κ (p m+1/2 p m+1/2 1 ) = f m+1/2 were u m+1 σ = σu m+1 + (1 σ)u m wit 0 σ 1 and g σ(t m+1 ) is similarly defined. u m u m+1 p m+1/2 l time (5) (6) Figure 2: Staggered mes in time. Grids for displacement at time t m and for pressures (l = 1 2) at time t m+1/2. 3. Stability estimates and convergence Let us establis ere a priori estimates tat provide te stability of te difference sceme (6) wit respect to te initial data and te rigt and side. Stability estimates are given in discrete energy norms for bot displacement and pressures. Tese results of stability joint wit local error estimates are used for deriving te convergence of te sceme. Proposition 3.1 If σ 1/2 te approximation of displacements given by sceme (6) satisfy te following a priori estimate: u m+1 2 A 4 u 0 2 A + 4 g g m+1 2 A 1 A 1 m + 2 f k+1/2 C C B 1 2 f k+1/2 2 + C g k+1 g k 2 B 1 3 k=0 6 A 1

7 Numerical Analysis of te Double Porosity Consolidation Model for m = M 1 and were C 1 C 2 and C 3 are constants independent of and. Proposition 3.2 If σ 1/2 te approximation of pressures given by sceme (6) satisfy κ 1 p 1/2 η κ ( 2 p 1/2 B η 2 η f 2 1/2 B κ η ) f 1/2 1 B 1 κ B ( 1 4 g 1 2 σ + u 0 2 A 1 A + C 1 f 1/2 1 2 ) + C B 1 2 f 1/2 2 B 1 and κ 1 p m+1/2 η κ 2 p m+1/2 B η 2 p + κ m+1/2 p m+1/2 1 2 B ( η f m+1/2 2 κ η ) f m+1/2 1 B 1 κ B 1 m 2 g k+1 σ gk 2 σ f k+1/2 + C 3 1 f k 1/2 2 1 f k+1/2 k=1 A + C 4 f k 1/2 2 1 B 1 B 1 for m = 1... M 1 and were C 1 C 2 C 3 and C 4 are constants independent of and. Now we introduce te error functions for displacements and pressures δu m (x) = um (x) u(x t m) H ωu δp m+1/2 l (x) = p m+1/2 (x) p l (x t m+1/2 ) H ωp for m = M 1 and l = 1 2 wic solve te following difference equations Dδu m+1 β 1 Dδu m+1 β 2 Dδu m Dδu m A δu m+1 σ l + β 1 G δp m+1/2 1 + β 2 G δp m+1/2 = Ψ m+1 σ + κ 1 η B δpm+1/2 1 κ (δp m+1/2 δp m+1/2 1 ) = φ m+1/2 1 + κ 2 η B δpm+1/2 + κ (δp m+1/2 δp m+1/2 1 ) = φ m+1/2 were te discrete functions Ψ m+1 H ωu and φ m+1/2 l H ωp l = 1 2 are te approximation errors. For smoot solutions it is easy to sow tat Ψ m+1 σ (x) = O(2 + α ) for x ω u and φ m+1/2 l (x) = O( ) for x ω p were α = 2 if σ = 1/2 and α = 1 oterwise. Ten error estimates follow from te stability of te difference sceme and we can prove te following converge teorem. Teorem 3.3 Let (u(x t) p 1 (x t) p 2 (x t)) be te solution of te double porosity problem (1) and (u m+1 p m+1/2 1 p m+1/2 ) te numerical solution of te finite difference sceme (6) on te staggered mes given previously. If σ 1/2 and u 0 is an O(2 ) approximation of u(x 0) ten te sceme (6) is convergent and olds u m+1 u( t m+1 ) A + p m+1 1 p 1 ( t m+1/2 ) + B p m+1 p 2 ( t m+1/2 ) K ( 2 + α ) B were α = 2 if σ = 1/2 and α = 1 oterwise K being a constant independent of and. 7 (7)

8 N. Boal F.J. Gaspar F.J. Lisbona P.N. Vabiscevic Acknowledgements Tis researc as been partially supported by te project MEC/FEDER MTM and by te Diputación General de Aragón. References [1] G.I. Barenblatt I.P. Zeltov I.N. Kocina. Basic concepts in te teory of seepage of omogeneous liquids in fissured rocks. Prikl. Mat. Mek. 24 (1960) [2] M. Biot. General teory of tree dimensional consolidation. J. Appl. Pys. 12 (1941) [3] F. Brezzi. On te existence uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Model Mat. Anal. Numer. 8 (1974) [4] F.J. Gaspar F.J. Lisbona P.N. Vabiscevic. A finite difference analysis of Biot s consolidation model. Appl. Numer. Mat. 44 (2003) [5] F.J. Gaspar F.J. Lisbona C.W. Oosterlee R. A. Wienands. Systematic compariso n of coupled and distributive smooting in multigrid for te poroelasticity system. Num. Lin. Algebra Appl. 11 (2004) [6] F.J. Gaspar F.J. Lisbona P.N. Vabiscevic. Staggered grid discretizations for te quasi static Biot s consolidation problem. Appl. Numer. Mat. 56 (2006) [7] F.J. Gaspar F.J. Lisbona C.W. Oosterlee P.N. Vabiscevic. An efficient multigrid solver for a reformulated version of te poroelasticity system. Comput. Metods Appl. Mec. Engrg. 196 (2007) [8] F.J. Gaspar F.J. Lisbona C.W. Oosterlee. A stabilized difference sceme for deformable porous media and its numerical resolution by multigrid metods. Comput. Vis. Sci. 11 (2008) [9] F.J. Gaspar J.L. Gracia F.J. Lisbona P.N. Vabiscevic. A stabilized metod for a secondary consolidation Biot s model. Numer. Metods Partial Differential Equations 24 (2008) [10] J. Korsawe G. Starke W. Wang O. Kolditz. Finite element analysis of poro-elastic consolidation in porous media: standard and mixed approaces. Comput. Metods Appl. Mec. Engrg. 195 (2006) [11] R.W. Lewis B.A. Screfler. Te finite element metod in te static and dynamic deformation and consolidation of porous media. Jon Wiley & Sons [12] P. Mira M. Pastor T. Li X. Liu. A new stabilized enanced strain element wit equal order of interpolation for soil consolidation problems. Comput. Metods Appl. Mec. Engrg. 192 (2003) [13] M.A. Murad J.H. Cusman. Multiscale flow and deformation in ydropilic swelling porous media. Int. J. Engrg. Sci. 3 (1996) [14] M.A. Murad A.F.D. Loula. Improved accuracy in finite element analysis of Biot s consolidation problem. Comput. Metods Appl. Mec. Engrg. 95 (1992) [15] M.A. Murad A.F.D. Loula. On stability and convergence of finite element approximations of Biot s consolidation problem. Internat. J. Numer. Metods Engrg. 37 (1994) [16] M.A. Murad V. Tomée A.F.D. Loula. Asymptotic beaviour of semid iscrete finite-element approximations of Biot s consolidation problem. SIAM J. Numer. Anal. 33 (1996) [17] K. Terzagi. Teoretical soil mecanics. Jon Wiley New York [18] R. Wienands F.J. Gaspar F.J. Lisbona C.W. Oosterlee. An efficient multigrid solver based on distributive smooting for poroelasticity equations. Computing 73 (2004)