Eirik Keilegavlen

A vertically integrated model with vertical dynamics for CO2 storage

Conventional vertically integrated models for CO2 storage usually adopt a vertical equilibrium (VE) assumption, which states that due to strong buoyancy, CO2 and brine segregate quickly, so that the fluids can be assumed to have essentially hydrostatic pressure distributions in the vertical direction. However, the VE assumption is inappropriate when the time scale of fluid segregation is not small relative to the simulation time. By casting the vertically integrated equations into a multiscale framework, a new vertically integrated model can be developed that relaxes the VE assumption, thereby allowing vertical dynamics to be modeled explicitly. The model maintains much of the computational efficiency of vertical integration while allowing a much wider range of problems to be modeled. Numerical tests of the new model, using injection scenarios with typical parameter sets, show excellent behavior of the new approach for homogeneous geologic formations.

Sufficient criteria are necessary for monotone control volume methods

Control volume methods are prevailing for solving the potential equation arising in porous media flow. The continuous form of this equation is known to satisfy a maximum principle, and it is desirable that the numerical approximation shares this quality. Recently, sufficient criteria were derived guaranteeing a discrete maximum principle for a class of control volume methods. We show that most of these criteria are also necessary. An implication of our work is that no linear nine-point control volume method can be constructed for quadrilateral grids in 2D that is exact for linear solutions while remaining monotone for general problems.

Domain decomposition strategies for nonlinear flow problems in porous media

Domain decomposition (DD) methods, such as the additive Schwarz method, are almost exclusively applied to linearized equations. In the context of nonlinear problems, these linear systems appear as part of a Newton iteration. However, applying DD methods directly to the original nonlinear problem has some attractive features, most notably that the Newton iterations now solve local problems, and thus are expected to be simpler. Furthermore, strong, local nonlinearities may to a less extent affect the numerical algorithm. For linear problems, DD can be applied both as an iterative solver or as a preconditioner. For nonlinear problems, it has until recently only been understood how to use DD as a solver. This article offers a systematic study of domain decomposition strategies in the context of nonlinear porous-medium flow problems. The study thus compares four different approaches, which represents DD applied both as a solver and preconditioner, to both the linearized and nonlinear equations. Our model equations are those obtained from a fully implicit discretization of immiscible two-phase flow in heterogeneous porous media. In particular we emphasize the case of nonlinear preconditioning, an algorithm that to our knowledge so far has not been studied nor implemented for flow in porous media. Our results show that the novel algorithm is up to 75% faster than the standard algorithm for the most challenging problems for a moderate number of subdomains.

Physics‐based preconditioners for flow in fractured porous media

Discrete fracture models are an attractive alternative to upscaled models for flow in fractured media, as they provide a more accurate representation of the flow characteristics. A major challenge in discrete fracture simulation is to overcome the large computational cost associated with resolving the individual fractures in large-scale simulations. In this work, two characteristics of the fractured porous media are utilized to construct efficient preconditioners for the discretized flow equations. First, the preconditioners are tailored to the fracture geometry and presumed flow properties so that the dominant features are well represented there. This assures good scalability of the preconditioners in terms of problem size and permeability contrast. For fracture dominated problems, numerical examples show that such geometric preconditioners are comparable or preferable when compared to state-of-the-art algebraic multigrid preconditioners. The robustness of the physics-based preconditioner for less favorable fracture conditions is further demonstrated by a systematic degradation of the fracture hierarchy. Second, the preconditioners are physics preserving in the sense that conservative fluxes can be computed even for an inexact pressure solutions. This facilitates a scheme where accuracy in the linear solver can be traded for efficiency by terminating the iterative solvers based on error estimates, and without sacrificing basic physical modeling principles. With the combination of these two properties a novel preconditioner is obtained which bridges the gap between multiscale approximations and iterative linear solvers.

Analysis of Control Volume Heterogeneous Multiscale Methods for Single Phase Flow in Porous Media

The standard approximation for the flow-pressure relationship in porous media is Darcys law that was originally derived for infiltration of water in fine homogeneous sands. Ever since there have been numerous attempts to generalize it for handling more complex flows. Those include upscaling of standard continuum mechanics flow equations from the fine scale. In this work we present a heterogeneous multiscale method that utilizes fine scale information directly to solve problems for general single phase flow on the Darcy scale. On the coarse scale it only assumes mathematically justified conservation of mass on control volumes, that is, no phenomenological Darcy-type relationship for velocity is presumed. The fluid fluxes are instead provided by a fine scale Navier--Stokes mixed finite element solver. This work also considers several choices of quadrature for data estimation in the multiscale method and compares them. We prove that for an essentially linear regime, when the fine scale is governed by Stokes...

Dual Virtual Element Method for Discrete Fractures Networks

Discrete fracture networks is a key ingredient in the simulation of physical processes which involve fluid flow in the underground, when the surrounding rock matrix is considered impervious. In this paper we present two different models to compute the pressure field and Darcy velocity in the system. The first allows a normal flow out of a fracture at the intersections, while the second grants also a tangential flow along the intersections. For the numerical discretization, we use the mixed virtual finite element method as it is known to handle grid elements of, almost, any arbitrary shape. The flexibility of the discretization allows us to loosen the requirements on grid construction, and thus significantly simplify the flow discretization compared to traditional discrete fracture network models. A coarsening algorithm, from the algebraic multigrid literature, is also considered to further speed up the computation. The performance of the method is validated by numerical experiments.

Auxiliary variables for 3D multiscale simulations in heterogeneous porous media

The multiscale control-volume methods for solving problems involving flow in porous media have gained much interest during the last decade. Recasting these methods in an algebraic framework allows one to consider them as preconditioners for iterative solvers. Despite intense research on the 2D formulation, few results have been shown for 3D, where indeed the performance of multiscale methods deteriorates. The interpretation of multiscale methods as vertex based domain decomposition methods, which are non-scalable for 3D domain decomposition problems, allows us to understand this loss of performance. We propose a generalized framework based on auxiliary variables on the coarse scale. These are enrichments of the coarse scale, which can be selected to improve the interpolation onto the fine scale. Where the existing coarse scale basis functions are designed to capture local sub-scale heterogeneities, the auxiliary variables are aimed at better capturing non-local effects resulting from non-linear behavior of the pressure field. The auxiliary coarse nodes fits into the framework of mass-conservative domain-decomposition (MCDD) preconditioners, allowing us to construct, as special cases, both the traditional (vertex based) multiscale methods as well as their wire basket generalization.

Multidimensional upstream weighting for multiphase transport on general grids

The governing equations for multiphase flow in porous media have a mixed character, with both nearly elliptic and nearly hyperbolic variables. The flux for each phase can be decomposed into two parts: (1) a geometry- and rock-dependent term that resembles a single-phase flux; and (2) a mobility term representing fluid properties and rock–fluid interactions. The first term is commonly discretized by two- or multipoint flux approximations (TPFA and MPFA, respectively). The mobility is usually treated with single-point upstream weighting (SPU), also known as dimensional or donor cell upstream weighting. It is well known that when simulating processes with adverse mobility ratios, SPU suffers from grid orientation effects. An important example of this, which will be considered in this work, is the displacement of a heavy oil by water. For these adverse mobility ratio flows, the governing equations are unstable at the modeling scale, rendering a challenging numerical problem. These challenges must be addressed in order to avoid systematic biasing of simulation results. In this work, we present a framework for multidimensional upstream weighting for multiphase flow with buoyancy on general two-dimensional grids. The methodology is based on a dual grid, and the resulting transport methods are provably monotone. The multidimensional transport methods are coupled with MPFA methods to solve the pressure equation. Both explicit and fully implicit approaches are considered for time integration of the transport equations. The results show considerable reduction of grid orientation effects compared to SPU, and the explicit multidimensional approach allows larger time steps. For the implicit method, the total number of non-linear iterations is also reduced when multidimensional upstream weighting is used.

A multiscale multilayer vertically integrated model with vertical dynamics for CO2 sequestration in layered geological formations

Efficient computational models are desirable for simulation of large-scale geological CO

Postinjection Normal Closure of Fractures as a Mechanism for Induced Seismicity

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