A reduced fracture model for two-phase flow with different rock types

Ahmed, Elyes; Jaffr, Jrme; Roberts, Jean E.

doi:10.1016/j.matcom.2016.10.005

Cited by 57 publications

(57 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The saturation is set equal to 0.9 on Γ D,in = {(x, y, z) ∈ ∂Ω | x = 0 and 0.4 ≤ y ≤ 0.6}. On the outflow boundary Γ out = {(x, y, z) ∈ ∂Ω | x = 1}, the saturation at time t n+1 is set equal to that inside the closest cell at time t n (see [2]). We assume homogeneous Neumann boundary conditions on the remaining part of the boundary.…”

Section: Example 1: Two Rock Types With a Straight Interfacementioning

confidence: 99%

A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types

Ahmed¹,

Hassan²,

Japhet³

et al. 2019

The SMAI Journal of Computational Mathematics

Self Cite

View full text Add to dashboard Cite

We consider two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. This is a nonlinear and degenerate parabolic problem with nonlinear and discontinuous transmission conditions on the interface. We first describe a space-time domain decomposition method based on the optimized Schwarz waveform relaxation algorithm (OSWR) with Robin or Ventcell transmission conditions. Complete numerical approximation is achieved by a finite volume scheme in space and the backward Euler scheme in time. We then derive a guaranteed and fully computable a posteriori error estimate that in particular takes into account the domain decomposition error. Precisely, at each iteration of the OSWR algorithm and at each linearization step, the estimate delivers a guaranteed upper bound on the error between the exact and the approximate solution. Furthermore, to make the algorithm efficient, the different error components given by the spatial discretization, the temporal discretization, the linearization, and the domain decomposition are distinguished. These ingredients are then used to design a stopping criterion for the OSWR algorithm as well as for the linearization iterations, which together lead to important computational savings. Numerical experiments illustrate the efficiency of our estimates and the performance of the OSWR algorithm with adaptive stopping criteria on a model problem in three space dimensions. Additionally, the results show how a posteriori error estimates can help determine the free Robin or Ventcell parameters.2010 Mathematics Subject Classification. 65M08, 65M15, 65M50, 65M55, 76S05.Keywords. two-phase Darcy flow, discontinuous capillary pressure, finite volume scheme, domain decomposition method, optimized Schwarz waveform relaxation, Robin and Ventcell transmission conditions, linearization, a posteriori error estimate, stopping criteria.

show abstract

Section: Example 1: Two Rock Types With a Straight Interfacementioning

confidence: 99%

A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types

Ahmed¹,

Hassan²,

Japhet³

et al. 2019

The SMAI Journal of Computational Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…These discontinuities play a major role and should be accurately captured in many important processes such as capillary driven imbibition in oil recovery or capillary barrier effects in oil migration and gas storage. This is particularly enhanced in the case of Discrete Fracture Matrix (DFM) models which exhibit highly contrasted permeabilities and capillary pressure curves between the fracture network and the surrounding matrix domains [9,35,42,38,34,12,13,2,31,14,1,3,45,4].…”

Section: Introductionmentioning

confidence: 99%

“…Numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3D large Discrete Fracture Matrix (DFM) models. 1 discretizations like the Mixed Hybrid Finite Element (MHFE) method [34,2] or the Hybrid Finite Volume scheme [1]. The additional asset of nodal discretizations compared with cell centered discretizations is to include d.o.f.…”

mentioning

confidence: 99%

Vertex Approximate Gradient discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media

Brenner

Masson

Quenjel

2020

Journal of Computational Physics

View full text Add to dashboard Cite

In this article, a new nodal discretization is proposed for two-phase Darcy flows in heterogeneous porous media. The scheme combines the Vertex Approximate Gradient (VAG) scheme for the approximation of the gradient fluxes with an Hybrid Upwind (HU) approximation of the mobility terms in the saturation equation. The discretization in space incorporates naturally nodal interface degrees of freedom (d.o.f.) allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves. It is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term. Numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3D large Discrete Fracture Matrix (DFM) models. 1 discretizations like the Mixed Hybrid Finite Element (MHFE) method [34,2] or the Hybrid Finite Volume scheme [1]. The additional asset of nodal discretizations compared with cell centered discretizations is to include d.o.f. at rock type interfaces as long as the mesh is conforming with the heterogeneities of the porous medium. These nodal pressures and saturations unknowns will be used to enforce the Darcy normal flux continuity equations combined with the saturation jump condition [16,11,27]. In this work, we will consider the Vertex Approximate Gradient (VAG) discretization developed in [27,12,13] for two-phase Darcy flows in heterogeneous media. The VAG scheme is based on nodal d.o.f. like Control Volume Finite Element (CVFE) methods but it also includes the cell d.o.f. which are eliminated at the linear algebra level at each Newton iteration without any fill-in. These cell d.o.f. provide an additional flexibility in the choice of the control volumes in order to avoid mixing different rock types at nodal control volumes. It has been shown in [27] to be more accurate than usual CVFE discretizations.The time integration will be chosen implicit to avoid severe time step restrictions in high velocity regions such as fractures. It will be also fully coupled to account for the strong coupling between the pressure and saturation unknowns in the transmission conditions at different rock type interfaces. As noticed in [5], the pressure and saturation unknowns cannot be decoupled at such different rock type interfaces to preserve the stability of the discretization.The selected numerical method should also provide a robust nonlinear convergence of the Newton type solver to allow for large time steps, typically at the time scale of the matrix in DFM simulations. Let us refer to [41,37] for alternative linearization schemes to the Newton method which aim to the improvement of the nonlinear solver. In this work, the additional nonlinear solver robustness will be first achieved by extension of the Hybrid Upwind (HU) transport scheme to the VAG discretization framework. The HU transport scheme has been introduced in [24,28] as an alternative to the Phase Pot...

show abstract

“…The DD method in this paper allows reusing existing codes for different numerical schemes and domain decomposition techniques specialized to each component of the problem. Another advantage distinguishes our method is that it can integrate easily more complex problems; the method is extended to address reduced fracture model between different rock types presented in [4].…”

Section: Introductionmentioning

confidence: 99%

Splitting-based domain decomposition methods for two-phase flow with different rock types

Ahmed

2019

Advances in Water Resources

Self Cite

View full text Add to dashboard Cite

In this paper, we are concerned with the global pressure formulation of immiscible incompressible two-phase flow between different rock types. The aim is to develop for this problem a robust algorithm based on domain decomposition methods and operator splitting techniques, in which the numerical solution is achieved by solving sequentially reduced pressure, saturation-advection and saturation-diffusion problems posed on the interfaces between the rocks. This approach makes possible the use of specialized numerical methods for each sub-problem and different time steps for diffusion and advection as well as independent time steps for the advection in the different rocks. For the discretization, the advection problem is approximated in time, with the explicit Euler method where different time grids are employed to adapt to different time scales in the rocks, and in space with hybridized cell-centered finite volume method of first order of Godunov type. That of the diffusion problem is approximated in time with the implicit Euler method and in space with a hybridized mixed finite element method as is used for the pressure problem. Numerical experiments illustrate the performance and the flexibility of our domain decomposition algorithm on different model problems in three space dimensions.Key words: Two-phase flow in porous media; continuous capillary pressure; non-overlapping domain decomposition; operator splitting; hybrid mixed finite element; hybrid finite volume; non-conforming time grids. * This work was partially funded by the Hydrinv Inria Euro Med 3+3: HYDRINV project. It has also received funding from the EPIC project (within LIRIMA: http://lirima.inria.fr) and the Tunisian Ministry

show abstract

A reduced fracture model for two-phase flow with different rock types

Cited by 57 publications

References 42 publications

A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types

A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types

Vertex Approximate Gradient discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media

Splitting-based domain decomposition methods for two-phase flow with different rock types

Contact Info

Product

Resources

About