Efficient numerical solution of the Biot poroelasticity system in multilayered domains

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1 Efficient numerical solution of the Biot poroelasticity system Anna Naumovich Oleg Iliev Francisco Gaspar Fraunhofer Institute for Industrial Mathematics Kaiserslautern, Germany, Spain Workshop on Model Concepts for Fluid-Fluid and Fluid-Solid Interactions Freudenstadt-Lauterbad, March 0-, 006

2 Outline Theory of poroelasticity and its practical applications Biot model of poroelasticity Layered domains, interface problem Finite volume discretization Multigrid method Multigrid for problem with discontinuous coefficients. Operator-dependent prolongation Numerical results Summary

problem for flow and stress fields Practical applications: -geomechanics(land subsidence, borehole

3 Poroelasticity Poroelasticity change in fluid flow = elasticity + diffusion deformation of the elastic porous material change in stress of the solid fluid flow inside the pores In general case coupled problem for flow and stress fields Practical applications: -geomechanics(land subsidence, borehole problems, construction of embankments, etc.) - biomechanics - industrial applications (e.g. filter manufacture) etc...

4 Biot model Momentum balance equation Diffusion equation T ( ) p 0 μ u+ u + λ u I + α = n p κ β α p f( x, t) t + u η = additional stress in the structure due to the fluid pressure additional fluid content due to the deformation u = p ( uvw) - displacement of the solid - fluid pressure unknowns parameters n k λ, μ β η f ( xt, ) α - porosity - permeability - Lame coefficients of the solid skeleton - compressibility of the fluid - viscosity of the fluid -sourceterm - Biot-Willis constant (suppose = further on) Parabolic-type system * The solution exists and it is unique ** * & ** R.E. Showalter. Diffusion in poroelastic media. Jour. Math. Anal. Appl., 000

Interface problem porous medium parameters: λ μ ϕ κ Interface Γ porous medium parameters: λ μ ϕ κ Interface conditions for the Biot system: Material parameters of the media λ and λ, μ and μ κ and κ,

5 Interface problem porous medium parameters: λ μ ϕ κ Interface Γ porous medium parameters: λ μ ϕ κ Interface conditions for the Biot system: Material parameters of the media λ and λ, μ and μ κ and κ, ϕ and ϕ may differ several orders of magnitude and, S [ p] Γ = 0 [ u] = 0 Γ [ V n] Γ = 0 [ Sn ] Γ = 0 κ where V = - p η = μ u+ u T + λ u ( ( ) ) ( ) - fluid pressure continuity - displacement continuity - continuity of the normal fluid flux - continuity of the normal effective stress I - fluid velocity (relative to the solid) - effective stress

6 3D model with horizontal plane interface z = ξ Interface : λ μ λ μ κ κ z = ξ n n elasticity part [ u ] = 0 z =ξ u μ z v μ z u λ x w + x w + y + v y [ v ] = 0 [ w ] = 0 + z =ξ z =ξ z =ξ = = 0 0 w z z =ξ ( λ + μ ) = 0 z =ξ diffusion part [ p ] = 0 k p z z =ξ z =ξ = 0

7 Finite volume discretization. Staggered grid We use staggered grid to discretize the system * - the domain is divided into 4 sets of control volumes staggered grid in 3D The finite volume discretization is derived: - using the polinomials, which are piecewise linear, extended with one special bilinear term in respective direction (for displacement components) - using piecewise - trilinear polinomials (for fluid pressure) All these polinomials must satisfy interface continuity conditions - discretization near the interface resulted in some specific averaging expressions for coefficients; - numerical experiments showed that this discretization provides second order of convergence for basic variables as well as for fluxes of the problem in maximum norm. pressure (in the vertices) u component of the displacement v component of the displacement w component of the displacement * following the work of P. Vabischevich, F. Lisbona, F. Gaspar A FD analysis of Biot s consolidation model

8 Multigridmethodand itscomponents MG is a general strategy, which exploits the fact that a problem can be solved on different scales of resolution (grids, levels) MG is an optimal order method. Offers a possibility to solve problems with N unknowns with O(N) work and storage for a large class of problems MG convergence speed does not depend on a discretization mesh size h MG COMPONENTS: Type of the MG cycle Smoother Coarsening Coarse grid operator Restriction Prolongation! The components of a MG method have to be adjusted to particular class of problems

9 Multigrid for problems with strongly discontinuous coefficients Convergence of a multigrid solver is strongly influenced by the jumps of the coefficients For strongly jumping coefficients convergence may depend on the size of the jump and its location with respect to the grid lines. Even divergence may occur. IMPORTANT Smoother Restriction Coarse grid operator Prolongation operator PROLONGATION OPERATORS: - tri- (bi-) linear interpolation - bicubic interpolation, etc. - operator-dependent interpolation - Accounts for the jumps of coefficients Rely on the continuity of the errors and their gradients - Rely on the interface continuity conditions (not continuity of the gradients)

10 Operator-dependent prolongation We want the operator-dependent prolongation to be consistent with our FV discretization Matrix form of the FV discretization of the Biot system: A A f uu up u u = A A f p pp p u p A A A A A A A uu A A A CASES: uu uv uw = vu vv vw wu wv ww A uu A pp p discrete elasticity operator (second order) A u and are first order coupling operators. Interpolation of each variable does not depend on the others (when blocks and dominate, and A uu in the block subblocks A uu A vv A up A up = Avp and dominate) ww. Interpolation of the elasticity variables and diffusion are independent (when blocks and dominate) A A wp discrete diffusion operator (second order) 3. Completely consistent interpolation of elasticity and diffusion variables (no assumption concerning block dominance) A uu A uu Ap u A pp A pp considered we did not consider

11 Operator-dependent prolongation for diffusion part (block App) Valid for the cases.,. STEPS,, 3, 4: Prolongation is based on the trilinear interpolation STEP 5: 5 Operator-dependent prolondation in z-direction p exploits continuity of the k across the interface z 4 3 h ( L p) h i, j,k+ 0.5k 0.5k p +, 0.5 i, j, k p θ i, j,k+ ( θ) k + θk ( ) θ k + θk = 0.5k 0.5k p + p, θ > 0.5 i, j, k i, j,k+ ( θ) k + θk ( θ) k + θk * * The formula above can be derived in different ways: see, e.g. works of Wesseling, de Zeeuw, Knapek This formula also follows from the polinomials, used in the FV discretization of the problem

12 Operator-dependent prolongation for elasticity part (block Auu) DIFFICULTIES: location of the coarse ( ) and fine ( ) grids for the first displacement component:. Components of the displacement u, v, w, depend on each other. u, v, w are defined on different grids (staggered) 3. Fine and coarse grids for each of the components are non-nested Altogether 6 (3 coarse + 3 fine) different grids with no common vertices...

13 Operator-dependent prolongation for elasticity part. Case. Assumption: A uu in the block subblocks Auu A vv A and dominate ww Interpolation of each displacement component does not depend on the others For the considered interface problem we have to interpolate in the following way: Standard linear interpolation in x - and y - directions Operator-dependent interpolation in z - direction e.g., interpolation in z direction of the w-component: λ + μ λ + μ wxyz (,, k hz) = 0.5 wk 0.5( xy, ) wk+ 0.5( xy, ) λ μ + λ + μ λ+ μ λ+ μ wxyz (,, k hz) = 0.5 wk 0.5( xy, ) wk+ 0.5( xy, ) λ + μ λ + μ θ < 0.5 where μ + λ + μ = (0.5 θ)( λ + μ ) + (0.5 + θ)( λ + μ ) = ( θ ) μ θμ Rem: for u- and v- components the prolongation is done in a similar way

14 Operator-dependent prolongation for elasticity part. Case. No assumption about the subblock dominance in Auu ( θ 0.5)( λ λ ) For the considered interface problem we have to interpolate in the following way: Standard linear interpolation in x - and y - directions Operator-dependent interpolation in z - direction e.g., interpolation in z direction of the w-component: λ + μ λ + μ wxy (,, z h ) = 0.5 w ( x, y) 0.5 w ( x, y) λ μ + + λ + μ k 0.5 z k 0.5 k h h hy 4 λ + μ z x + uxk, x, y + vyk, x, y λ + μ λ + μ w( x, y, zk hz) = 0.5 wk 0.5( x, y) wk ( x, y) λ + μ λ + μ h 4 ( θ 0.5)( λ λ ) interpolation of displacement components depend on each other + h hy,, λ + μ z x u xk, x y + vy, k x y Account for components u,v during the interpolation of w-component Account for components u,v during the interpolation of w-component Rem: for u- and v- components the prolongation is done in a similar way

15 Numerical experiment. MG* convergence. Domain: [0;]x[0;]x[0;] Time: [0;0^-] Coefficients: (jumps one order of magnitude) ξ = 0.50 λ =, λ = 0 μ =, μ = 0 k =, k = 0 Based on the standard prolongation Cycle type: F Nr of pre-smoothing steps: 3 Nr of post-smoothing steps: Smoother: collective alternating lexicographic line GS Based on the operator-dependent prolongation * the MG code is an extension (for the case of discontinuous coefficients) of the MG poroelasticity code, written by C.W. Oosterlee and F. Gaspar

16 Numerical experiment. MG convergence. Domain: [0;]x[0;]x[0;] Time: [0;0^-] Coefficients: (jumps 7 orders of magnitude) ξ = λ =, λ = 0 7 μ =, μ = 0 7 k =, k = 0 Cycle type: F Nr of pre-smoothing steps: 3 Nr of post-smoothing steps: Smoother: collective alternating lexicographic line GS Based on the standard prolongation Based on the operator-dependent prolongation * * the prolongations and for this test example give identical results

17 Summary Biot model with discontinuous coefficients was considered Finite volume discretization, which provides second order of convergence for the primary as well as flux variables was derived Operator-dependent prolongation was derived in order to provide MG covergence in the case of discontinuous coefficients A set of numerical experiments was carried out

Aspects of Multigrid

Aspects of Multigrid Kees Oosterlee 1,2 1 Delft University of Technology, Delft. 2 CWI, Center for Mathematics and Computer Science, Amsterdam, SIAM Chapter Workshop Day, May 30th 2018 C.W.Oosterlee (CWI)