Stephen L. Lyons

Derivation of the Forchheimer Law via Homogenization

In this paper we derive the Forchheimer law via the theory of homogenization. In particular, we study the nonlinear correction to Darcys law due to inertial effects on the flow of a Newtonian fluid in rigid porous media. A general formula for this correction term is derived directly from the Navier–Stokes equation via homogenization. Unlike other studies based on the same approach that concluded for the nonlinear correction to be cubic in velocity for isotropic media, the present work shows that the nonlinear correction is quadratic. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier–Stokes equation is obtained and is utilized to show the validity of the quadratic correction. Both incompressible and compressible fluids are considered.

An Approximation to Miscible Fluid Flows in Porous Media with Point Sources and Sinks by an Eulerian--Lagrangian Localized Adjoint Method and Mixed Finite Element Methods

We develop an Eulerian--Lagrangian localized adjoint method (ELLAM)-mixed finite element method (MFEM) solution technique for accurate numerical simulation of coupled systems of partial differential equations (PDEs), which describe complex fluid flow processes in porous media. An ELLAM, which was shown previously to outperform many widely used methods in the context of linear convection-diffusion PDEs, is presented to solve the transport equation for concentration. Since accurate fluid velocities are crucial in numerical simulations, an MFEM is used to solve the pressure equation for the pressure and Darcy velocity. This minimizes the numerical difficulties occurring in standard methods for approximating velocities caused by differentiation of the pressure and then multiplication by rough coefficients. The ELLAM-MFEM solution technique significantly reduces temporal errors, symmetrizes the governing transport equation, eliminates nonphysical oscillation and/or excessive numerical dispersion in many simulators, conserves mass, and treats boundary conditions accurately. Numerical experiments show that the ELLAM-MFEM solution technique simulates miscible displacements of incompressible fluid flows in porous media accurately with fairly coarse spatial grids and very large time steps, which are one or two orders of magnitude larger than the time steps used in many methods. Moreover, the ELLAM-MFEM solution technique can treat large mobility ratios, discontinuous permeabilities and porosities, anisotropic dispersion in tensor form, and point sources and sinks.

An ELLAM Approximation for Highly Compressible Multicomponent Flows in Porous Media

We develop an ELLAM-MFEM approximation to the strongly coupled systems of time-dependent nonlinear partial differential equations (PDEs) and constraining equations, which describe fully miscible, highly compressible, multicomponent flows through heterogeneous and compressible porous media with singular sources and sinks. An Eulerian–Lagrangian localized adjoint method (ELLAM) is presented to solve the transport equations for concentrations. A mixed finite element method (MFEM) is used to solve the pressure PDE for the pressure and Darcy velocity simultaneously, which generates accurate fluid velocities and minimizes the numerical difficulties occurring in standard methods caused by differentiation of the pressure and then multiplication by rough coefficients. The ELLAM-MFEM solution technique symmetrizes and stabilizes the governing transport PDEs and greatly reduces nonphysical oscillation and/or excessive numerical dispersion present in many large-scale simulators. Computational experiments show that the ELLAM-MFEM solution technique can generate stable and physically reasonable numerical simulations even if coarse spatial grids and very large time steps are used.

Integrated Two-Dimensional Modeling of Fluid Flow and Compaction in a Sedimentary Basin

In this paper we employ mixed finite elements and numerically study an integrated two-dimensional model of fluid flow and compaction in a sedimentary basin. This model describes a single phase incompressible flow in a two-dimensional section of a sedimentary basin with vertical compaction. At each time step, an iterative algorithm is used to solve this model. The determination of the grid movement is based on the mass conservation and movement of sediments in the basin, while the mixed method is utilized to solve the fluid flow over the moving grid. Numerical experiments are presented to verify this iterative algorithm and show representative solutions for the model under consideration.

Global Scale-up on Reservoir Models with Piecewise Constant Permeability Field

Global scale-up was first proposed in the 1980s, and its benefits are well-described in the literature. However, global scale-up has not been widely applied in practice due to significant technical challenges. In this paper, we analyze some of the theoretical and numerical difficulties and present practical resolutions. First, we derive the flux and energy formulations for scale-up by using a Multiscale Finite Element framework. These formulations can be applied to local, extended local and global scale-up. Then we present a new method to improve scale-up accuracy for geo-cellular models with piecewise constant permeability. On these models, large jumps in the permeability field lead to well-known singularities in the flow solutions, making them difficult to be calculated accurately. We show that inaccurate flow solutions can lead to large scale-up errors. To mitigate the effect of the singularities and improve the scale-up accuracy, we have developed a hybrid method for solving flows which utilizes the fact that the Continuous Galerkin (CG) finite elment method and the Mixed finite element method provide upper and lower bounds, respectively, for the effective permeability. The new method uses CG to compute the pressure solution followed by a weighted L2 projection of the numerical fluxes to enforce local mass conservation. Numerical examples are presented to demonstrate the effectiveness of the hybrid method.

Validation of Non-darcy Well Models Using Direct Numerical Simulation

We describe discrete well models for 2-D non-Darcy fluid flow in anisotropic porous media. Attention is mostly paid to the well models and simplified calibration procedures for the control volume mixed finite element methods, including the case of highly distorted grids.

An Eulerian-Lagrangian Formulation For Compositional Flow In Porous Media

We derive an Eulerian-Lagrangian formulation for two-phase, multicomponent compositional flow in porous media with sources and sinks. The formulation can be used by many Eulerian-Lagrangian methods in solving the component mass balance equations. It can be combined with different types of pressure solvers in solving the coupled systems of compositional models. A full thermodynamic flash calculation is carried out to determine phase stability and composition. Numerical experiments are presented to investigate the performance of the formulation and to compare it with widely used upwind method for compositional flow. These results indicate that the Eulerian-Lagrangian formulation generates stable and accurate solutions that resolve moving steep fronts and are physically reasonable, even if it uses a time step of at least two orders of magnitude larger than that used by the upwind method.

A Numerical Algorithm for Single Phase Fluid Flow in Elastic Porous Media

In this paper we consider an integrated model for single-phase fluid flow in elastic porous media. The model and mathematical formulation consist of mass and momentum balance equations for both fluid and porous media. We propose a mixed finite element scheme to solve simultaneously for the porous media displacement, fluid mass flux, and pore pressure. A prototype simulator for solving the integrated problem has been built based on a finite element object library that we have developed. We will present numerical and sensitivity results for the solution algorithm.

An Accurate Approximation to Compressible Flow in Porous Media with Wells

An Eulerian-Lagrangian localized adjoint method (ELLAM) is presented for compressible flow occurring in compressible porous media with wells. The ELLAM scheme symmetrizes the governing transport equation, greatly eliminates non-physical oscillation and/or excessive numerical dispersion present in many large-scale simulators widely used in industrial applications, and conserves mass. Computational experiments show that the ELLAM scheme can accurately simulate incompressible and compressible fluid flows in porous media with wells, even though coarse spatial grids and very large time steps, which are one or two orders of magnitude larger than those used in many numerical methods, are used. The ELLAM scheme can treat large mobility ratios, discontinuous permeabilities and porosities, anisotropic dispersion in tensor form, and wells.