An Implicit Smoothed Particle Hydrodynamics Multiscale Method for Porous Media Flow Cornelis Vuik, Owen Clark, Alexander A. Lukyanov Delft University of Technology, Netherlands SIAM Conference on Mathematical and Computational Issues in the Geosciences September 11—14, 2017, Erlangen, Germany
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An Implicit Smoothed Particle Hydrodynamics MultiscaleMethod for Porous Media Flow
Cornelis Vuik, Owen Clark, Alexander A. Lukyanov
Delft University of Technology, Netherlands
SIAM Conference on Mathematical and Computational Issues in theGeosciences September 11—14, 2017, Erlangen, Germany
Overview
Reservoir Simulation in a Nutshell
SPH in a Nutshell
Meshless Scheme for Fluid Flow in Porous Media
Multiscale, Multilevel, Multigrid and Deflation Methods
Figure: CPR-based FIM simulation framework: two stage CPR where AMG is used tosolve the pressure equation and ILU(0) for the second-stage full residual correction.
3: for j = 1, 2, . . . ,m do4: wj = A∗ppM−1vj5: for i = 1, . . . , j do6: hi,j = (wj , vi )7: wj = wj − hijvi8: end for9: hj+1,j = ‖wj‖2
10: if hj+1,j = 0 or converged then11: set m = j and go to 1512: end if13: vj+1 = wj/hj+1,j
14: end for
15: Fill Hm = hij for 1 ≤ i ≤ m + 1, 1 ≤ j ≤ m.16: Compute the minimizer um of ‖βe1 − Hmu‖2 and set xm = x0 + M−1Vmum.17: if converged then xm solution and return else set x0 = xm and go to 2
Multiscale, Multilevel, Multigrid and Deflation Methods
Galerkin Projection:
A coarse-scale system can be constructed by applying restriction R and prolongation Poperators (a number of times)
Ak+1 = RkAkPk ,
where k is the level of the appropriate step of multiscale, multilevel, multigrid, anddeflation methods, Ak+1 is the matrix on the next level (for the pressure system), Rk ,Pk are the restriction and prolongation operators at the level k.
Post-Galerkin Projection Steps:
Smoothing (e.g., Gauss-Seidel (GS) or ILU(k) or BILU(k) smoothing method orKrylov-space accelerator)
Figure: MSFV grid imposed on the given fine-scale grid (center): Nc coarse (solidlines) and Nd dual-coarse (dashed lines) grids. A coarse and a dual-coarse grid cell arehighlighted on the right and left, respectively.
Table: SPE9 model with capillary pressure: settings. Differencescan be seen in the smoothers and in the solver used on thecoarse scale pressure system.
Table: SPE9 with capillary pressure: the table shows the total numbers of nonlinearand of linear iterations and the total and the linear solver CPU time of each run.
Iteration Count CPU time (s)Runs Timesteps Nonlinear Linear Linear solver TotalRun 1 75 284 1888 759 1672Run 2 75 284 1867 1202 2130Run 3 75 281 1627 1105 2036
where m is the iteration index, [pF ]m is the pressure vector at the iteration m, V is theleft multiscale meshless based preconditioner defined as an operational object.
Meshless multiscale preconditioner
[zF ] = P (AC )−1 R [vF ] , AC = RM−1AF P
[wF ] = [zF ] + S−1γ · ([vF ]− AF [zF ])
where S−1γ is the smoothing operator (e.g., Gauss-Seidel (GS) or ILU(k) or BILU(k)
smoothing method or Krylov-space accelerator) applied γ times.
The black oil, iso-thermal and thermal compositional models with varying degree ofheterogeneity in the reservoir grid properties are considered in this paper to test theperformance of the Multiscale Meshless Based Method (MsMBM)
Fully Implicit SPH Based Multiscale Method is presented and allows to handlelow-frequency modes on the coarse level. High-frequency errors are then resolvedby employing a smoother on fine grid.
Restrictions and prolongation operators reduce to the subdomain-levelsetdeflation vectors, used in subdomain-levelset deflation method and MsRSB.
This method does not require a coarse partition and, hence, can be applied togeneral unstructured topology of the fine scale.
The SPH based multiscale method provides a reasonably good approximation tothe pressure system and speeds up the convergence when used as apreconditioner for an iterative fine-scale solver.
The method exhibits expected good (not ideal!) scalability during parallel fullyimplicit simulations.
J.H. van der Linden and T.B. Jonsthovel and A.A. Lukyanov and C. VuikThe parallel subdomain-levelset deflation method in reservoir simulationJournal of Computational Physics, 304, pp. 340-358, 2016
A. Lukyanov and C. VuikParallel Fully Implicit Smoothed Particle Hydrodynamics Based MultiscaleMethodECMOR XV - 15th European Conference on the Mathematics of OilRecovery, August 29 - September 1, 2016 Editor: J.D. Jansen EAGE,Houten, 2016DOI: 10.3997/2214-4609.201601748