An application of the Reduced-Basis Method for Darcy flows

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Submitted on 23 Oct 2018

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An application of the Reduced-Basis Method for Darcy flows

Riad Sanchez, Sébastien Boyaval, Guillaume Enchéry, Quang Huy Tran

To cite this version:

LA_{TEX TikZposter}

An application of the Reduced-Basis Method for Darcy flows

Riad Sanchez

†

, S´ebastien Boyaval

‡

, Guillaume Ench´ery

†

, Quang Huy Tran

†

†

_{IFP ´}

_{Energies Nouvelles,}

‡

_{Laboratoire Saint-Venant (ENPC - EDF R&D - CEREMA) & Matherials (INRIA)}

An application of the Reduced-Basis Method for Darcy flows

Riad Sanchez

†

, S´ebastien Boyaval

‡

, Guillaume Ench´ery

†

, Quang Huy Tran

†

†

_{IFP ´}

_{Energies Nouvelles,}

‡

_{Laboratoire Saint-Venant (ENPC - EDF R&D - CEREMA) & Matherials (INRIA)}

Context and motivations

• Subsurface flow modeling (history-matching, optimization of field performance, . . .). • Constraints :

– Long simulation times (grid’s size, time iterations, . . .)

– Simulations should be rerun for different input parameters.

Two-phase flow model

• Unknowns : water saturation S, pressure P . • Parameter : water viscosity µ ∈ _{P ⊂ R}+. • Equations in Ω ⊂ R2: div v = 0, (1) v = −λ_T S, µ
K∇P, (2) φ ∂S ∂t + div  f_w(S, µ)v = 0, (3)

subject to inflow-outflow boundary conditions.

Objective and difficulties

Build a reduced model for the pressure problem (1) - (2) at the final time step of the simulation.

Difficulties :

• Nonlinear coupling between P and S.

• No affine parameter dependance of λ_T with respect to µ.

Finite volume discretization

Discret finite volume approximation of the Darcy fluxes:

if σ = K|L : F_K,σint = λ_σ(µ)K_σD_σP_Nn+1, (4) if σ ∈ E_KDir : F_K,σDir = λ_T(S_Kn , µ)K_σD_σP_Nn+1, (5) where D_σP_Nn+1 = P_{N ,K}n+1 − P_{N ,L}n+1 if σ = K|L and D_σP_Nn+1 = P_{N ,K}n+1 − P_σDir if σ ∈ E_KDir.

Discret conservation of the volume: X

σ∈E_Kint

F_K,σint + X

σ∈E_KDir

F_K,σDir = 0. (6)

Reduced model for the pressure problem

• _PGreedy := {µ_n}N_n=1 ⊂ _P.

• Q_N = span {P_N (·; µ_n) | µ_n ∈ P_Greedy} an N-dimensional subspace of Q_N . • Z_N := {ζ_n}N_n=1 be an orthonormalized basis of Q_N.

• Algebraic form of RB-approximation

AN pN = fN, (7) where AN
n,m = X σ∈Eint λ_σ(µ)K_σD_σζ_nD_σζ_m + X σ∈EDir λ_T S_K, µ
K_σζ_n,Kζ_m,K, fN
n = X σ∈EDir λ_T S_K, µ
K_σP_σDirζ_m,K.

Visual comparison

Figure 1: Comparison between truth solution (left), reduced basis approximations for N = 20 (middle) and relative error (right) for µ = 27 cP

A posteriori error estimation

Define D R_N P_N(µ)
, q E = X σ∈E λ_σ(µ)K_σD_σP_N(µ)D_σq. (8) Then PN (µ) − P_N(µ) 1,N ,µ∗ ≤ ∆ N 1,µ∗(µ) := 1 α_N (µ) RN P_N(µ)
−1,N ,µ∗,

Efficient RB approximation

Use the Empirical Interpolation Method (EIM) to approximate λ_T S, µ
by λJ_T(µ) = J X j=1 Θλ_j (µ)ξ_j. (9) One obtains  _XJ j=1 Θλ_j(µ)AJ_N,j  pJ_N(µ) = J X j=1 Θλ_j (µ)f_N,jJ , (10) where AJN,j
n,m = X σ∈Eint ξ_j,σK_σD_σζ_nD_σζ_m + X σ∈EDir ξ_j,KK_σζ_n,Kζ_m,K, f J N,j
m = X σ∈EDir ξ_j,KK_σP_σDirζ_m,K.

Convergence of the greedy algorithm

Figure 2: Convergence of the reduced-basis approximation for different values of J (layer 85 of the SPE10 test-case)

Conclusions

• Reduction of the pressure problem with a non-affine dependence in the parameter at the final time step.

• Certification of the RB model (a posteriori error estimate in energy norm).

• Efficient implementation of the RB method, i.e. the reduced problems can be constructed in complexity independent of N .