Richard E. Ewing

An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation

Abstract Many numerical methods use characteristic analysis to accommodate the advective component of transport. Such characteristic methods include Eulerian-Lagrangian methods (ELM), modified method of characteristics (MMOC), and operator splitting methods. A generalization of characteristic methods can be developed using an approach that we refer to as an Eulerian-Lagrangian localized adjoint method (ELLAM). This approach is a space-time extension of the optimal test function (OTF) method. The method provides a consistent formulation by defining test functions as specific solutions of the localized homogeneous adjoint equation. All relevant boundary terms arise naturally in the ELLAM formulation, and a systematic and complete treatment of boundary condition implementation results. This turns out to have significant implications for the calculation of boundary fluxes. An analysis of global mass conservation leads to the final ELLAM approximation, which is shown to possess the conservative property. Numerical calculations demonstrate the behaviour of the method with emphasis on treatment of boundary conditions. Discussion of the method includes ideas on extensions to higher spatial dimensions, reactive transport, and variable coefficient equations.

The Mathematics of reservoir simulation

Problems arising in the modeling of processes for hydrocarbon recovery Finite element and finite difference methods for continuous flows in porous media A front tracking reservoir simulator Five-spot validation studies and the water coning problem Statistical fluid dynamics: The influence of geometry on surface instabilities Some numerical methods for discontinuous flows in porous media.

Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics

A nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium. Conservation of mass for the mixture leads to an elliptic equation for pressure, and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid. A sequential implicit time-stepping procedure is defined, in which the pressure and Darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a Galerkin finite element method and the method of characteristics. Optimal-order convergence in L2 is proved. Time-truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation; temporal and spatial errors are lessened by direct computation of the velocity in the mixed method, as opposed to differentiation of the pressure. Several extensions of these results are outlined.

On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials

We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

Accurate multiscale finite element methods for two-phase flow simulations

In this paper we propose a modified multiscale finite element method for two-phase flow simulations in heterogeneous porous media. The main idea of the method is to use the global fine-scale solution at initial time to determine the boundary conditions of the basis functions. This method provides a significant improvement in two-phase flow simulations in porous media where the long-range effects are important. This is typical for some recent benchmark tests, such as the SPE comparative solution project [M. Christie, M. Blunt, Tenth spe comparative solution project: a comparison of upscaling techniques, SPE Reser. Eval. Eng. 4 (2001) 308-317], where porous media have a channelized structure. The use of global information allows us to capture the long-range effects more accurately compared to the multiscale finite element methods that use only local information to construct the basis functions. We present some analysis of the proposed method to illustrate that the method can indeed capture the long-range effect in channelized media.

A summary of numerical methods for time-dependent advection-dominated partial differential equations

Abstract We give a brief summary of numerical methods for time-dependent advection-dominated partial differential equations (PDEs), including first-order hyperbolic PDEs and nonstationary advection–diffusion PDEs. Mathematical models arising in porous medium fluid flow are presented to motivate these equations. It is understood that these PDEs also arise in many other important fields and that the numerical methods reviewed apply to general advection-dominated PDEs. We conduct a brief historical review of classical numerical methods, and a survey of the recent developments on the Eulerian and characteristic methods for time-dependent advection-dominated PDEs. The survey is not comprehensive due to the limitation of its length, and a large portion of the paper covers characteristic or Eulerian–Lagrangian methods.

Simulation of Miscible Displacement Using Mixed Methods and a Modified Method of Characteristics

Numerical dispersion and grid orientation problems with adverse mobility ratios are two of the major difficulties in the numerical simulation of enhanced recovery processes. An efficient method for modeling convection-dominated flows which greatly reduces numerical dispersion and grid orientation problems is presented and applied to miscible displacement in a porous medium. The base method utilizes characteristic flow directions to model convection and finite elements to treat the diffusion and dispersion. The characteristic approach also minimizes certain overshoot difficulties which accompany many finite element methods for problems with sharp fluid interfaces. The truncation error caused by the characteristic time-stepping technique is small, so large stable time-steps can be taken as in fully-implicit methods without the corresponding loss in accuracy.

Superconvergence of the velocity along the Gauss lines in mixed finite element methods

Superconvergence for the velocity along the Gauss lines is derived for the mixed finite element method for the second-order elliptic equation in rectangular domains. These estimates are based on the rectangular finite elements of Raviart–Thomas type.

Finite volume element approximations of nonlocal reactive flows in porous media

In this article, we study finite volume element approximations for two-dimensional parabolic integro-differential equations, arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called NonFickian flows and exhibit mixing length growth. For simplicity, we consider only linear finite volume element methods, although higher-order volume elements can be considered as well under this framework. It is proved that the finite volume element approximations derived are convergent with optimal order in H1- and L2-norm and are superconvergent in a discrete H1-norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in L∞- and W1,∞-norms. These results are also new for finite volume element methods for elliptic and parabolic equations.

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.