Magne S. Espedal

An ELLAM Scheme for Advection-Diffusion Equations in Two Dimensions

We develop an Eulerian--Lagrangian localized adjoint method (ELLAM) to solve two-dimensional advection-diffusion equations with all combinations of inflow and outflow Dirichlet, Neumann, and flux boundary conditions. The ELLAM formalism provides a systematic framework for implementation of general boundary conditions, leading to mass-conservative numerical schemes. The computational advantages of the ELLAM approximation have been demonstrated for a number of one-dimensional transport systems; practical implementations of ELLAM schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper. Extensive numerical results are presented to compare the ELLAM scheme with many widely used numerical methods and to demonstrate the strength of the ELLAM scheme.

Rate of Convergence of Some Space Decomposition Methods for Linear and Nonlinear Problems

Convergence of a space decomposition method is proved for a class of convex programming problems. A space decomposition refers to a method that decomposes a space into a sum of subspaces, which could be a domain decomposition or a multilevel method when applied to partial differential equations. Two algorithms are proposed. Both can be used for linear as well as nonlinear elliptic problems, and they reduce to the standard additive and multiplicative Schwarz methods for linear elliptic problems.

Macrodispersion for two-phase, immiscible flow in porous media

Abstract Two-phase immiscible flow in a heterogeneous porous media is studied. The permeability is given as a log-normal random function with a given mean, variance and correlation. A set of up-scaled model equations with a new effective macrodispersion tensor is derived for such models. The macrodispersion solution is compared with high-resolution solutions of the original equations, based on several realizations in the permeability. It is shown that the average solution gives a very accurate approximation of the flow. A model with a two-scale permeability field is also studied where the variation on the largest scale is kept in the averaged equations. Here too the macrodispersion solution is compared with high-resolution solutions and the result is very promising. Capillary diffusion effects on the macrodispersion is also discussed.

Multiphase Flow Simulation with Various Boundary Conditions

Multiphase ow simulation with various nonhomogeneous boundary conditions in ground water hydrology and petroleum engineering is considered The phase ow equations are given in a fractional ow formulation i e in terms of a saturation and a global pressure It is shown that most commonly used boundary conditions for groundwater hydrology and petroleum engineering problems can be incorporated into the pressure saturation formu lation

Meshing of domains with complex internal geometries

This paper presents a meshing algorithm for domains with internal boundaries. It is an extension of the gridding algorithm presented by Persson and Strang. The resulting triangulation matches all boundaries, and the triangles are all nearly equilateral. Equilateral triangles are beneficial for a finite volume discretization, as fluid flow between elements of very different size is only possible at small timesteps. The mesh generator is compared with the well regarded Triangle programme, where both element quality and simulation performance are checked. It is shown that our mesh generator consistently delivers better meshes. Copyright

Convergent iterative schemes for time parallelization

Parallel methods are usually not applied to the time domain because of the inherit sequentialness of time evolution. But for many evolutionary problems, computer simulation can benefit substantially from time parallelization methods. In this paper, we present several such algorithms that actually exploit the sequential nature of time evolution through a predictor-corrector procedure. This sequentialness ensures convergence of a parallel predictor-corrector scheme within a fixed number of iterations. The performance of these novel algorithms, which are derived from the classical alternating Schwarz method, are illustrated through several numerical examples using the reservoir simulator Athena.

Reservoir description using a binary level set model

We consider the inverse problem of permeability estimation for two-phase flow in porous media. In the parameter estimation process we utilize both data from the wells (production data) and spatially distributed data (from time-lapse seismic data). The problem is solved by approximating the permeability field by a piecewise constant function, where we allow the discontinuity curves to have arbitrary shape with some forced regularity. To achieve this, we have utilized level set functions to represent the permeability field and applied an additional total variation regularization. The optimization problem is solved by a variational augmented Lagrangian approach. A binary level set formulation is used to determine both the curves of discontinuities and the constant values for each region. We do not need any initial guess for the geometries of the discontinuities, only a reasonable guess of the constant levels is required.

Symmetry and M-Matrix Issues for the O-Method on an Unstructured Grid

More sophisticated discretization methods than the traditional control-volume finite-difference methods, have been proposed by Aavatsmark et al. in recent papers for solving the mass balance equations for porous media flow. These methods are based on a local representation of fluxes across cell-edges of control volumes (CVs). This paper will focus on mathematical properties of the discrete operator that arises when an elliptic term of the form −∇⋅(K∇p) is discretized based on these discretization principles.

Applications of a Space Decomposition Method to Linear and Nonlinear Elliptic Problems

This work presents some space decomposition algorithms for a convex minimization problem. The algorithms has linear rate of convergence and the rate of convergence depends only on four constants. The space decomposition could be a multigrid or domain decomposition method. We explain the detailed procedure to implement our algorithms for a two-level overlapping domain decomposition method and estimate the needed constants. Numerical tests are reported for linear as well as nonlinear elliptic problems.

A Convergent Algorithm for Time Parallelization Applied to Reservoir Simulation

Parallel methods are not usually applied to the time domain because the sequential nature of time is considered to be a handicap for the development of competitive algorithms. However, this sequential nature can also play to our advantage by ensuring convergence within a given number of iterations. The novel parallel algorithm presented in this paper acts as a predictor corrector improving both speed and accuracy with respect to the sequential solvers. Experiments using our in house fluid flow simulator in porous media, Athena, show that our parallel implementation exhibit an optimal speed up relative to the method.