Myron B. Allen

Numerical modelling of multiphase flow in porous media

Abstract The simultaneous flow of immiscible fluids in porous media occurs in a wide variety of applications. The equations governing these flows are inherently nonlinear, and the geometries and material properties characterizing many problems in petroleum and groundwater engineering can be quite irregular. As a result, numerical simulation often offers the only viable approach to the mathematical modelling of multiphase flows. This paper provides an overview of the types of models that are used in this field and highlights some of the numerical techniques that have appeared recently. The exposition includes discussions of multiphase, multispecies flows in which chemical transport and interphase mass transfers play important roles. The paper also examines some of the outstanding physical and mathematical problems in multiphase flow simulation. The scope of the paper is limited to isothermal flows in natural porous media; however, many of the special techniques and difficulties discussed also arise in artificial porous media and multiphase flows with thermal effects.

Numerical modeling in science and engineering

Basic Equations of Macroscopic Systems. Introduction to Numerical Methods. Steady-state Systems. Dissipative Systems. Nondissipative Systems. High-order, Nonlinear, and Coupled Systems. Appendix: Summary of Vector and Tensor Analysis. Index.

Macroscale Properties of Porous Media from a Network Model of Biofilm Processes

We demonstrate how a network model can predict porosity and permeability changes in a porous medium as a result of biofilm buildup in the pore spaces. A biofilm consists of bacteria and extracellular polymeric substances (EPS) bonded together and attached to a surface. In this case, the surface consists of the walls of the porous medium, which we model as a random network of pipes.Our model contains five species. Four of these are bacteria and EPS in both fluid and adsorbed phases. The fifth species is nutrient, which we assume to reside in the fluid phase only. Bacteria and EPS transfer between the adsorbed and fluid phases through adsorption and erosion or sloughing. The adsorbed species influence the effective radii of the pipes in the network, which affect the porosity and permeability.We develop a technique for integrating the coupled system of ordinary and partial differential equations that govern transport of these species in the network. We examine ensemble averages of simulations using different arrays of pipe radii having identical statistics. These averages show how different rate parameters in the biofilm transport processes affect the concentration and permeability profiles.

Well-conditioned iterative schemes for mixed finite-element models of porous-media flows

Mixed finite-element methods are attractive for modeling flows in porous media since they can yield pressures and velocities having comparable accuracy. In solving the resulting discrete equations, however, poor matrix conditioning can arise both from spatial heterogeneity in the medium and from the fine grids needed to resolve that heterogeneity. This paper presents two iterative schemes that overcome these sources of poor conditioning. The first scheme overcomes poor conditioning resulting from the use of fine grids. The idea behind the scheme is to use spectral information about the matrix associated with the discrete version of Darcy’s law to precondition the velocity equations, employing a multigrid method to solve mass-balance equations for pressure or head. This scheme still exhibits slow convergence when the permeability or hydraulic conductivity is highly variable in space. The second scheme, based on the first, uses mass lumping to precondition the Darcy equations, thus requiring more work per i...

Numerical Modeling of Multiphase Flow in Porous Media

The simultaneous flow of immiscible fluids in porous media occurs in a wide variety of applications. The equations governing these flows are inherently nonlinear, and the geometries and material properties characterizing many problems in petroleum and groundwater engineering can be quite irregular. As a result, numerical simulation offers the only viable approach to the mathematical modeling of multiphase flows. This chapter provides an overview of the types of models that are used in this field and highlights some of the numerical techniques that have appeared recently. The exposition includes discusssions of multiphase, multispecies flows in which chemical transport and interphase mass transfers play important roles. This chapter also examines some of the outstanding physical and mathematical problems in multiphase flow simulation. The scope of the chapter is limited to isothermal flows in natural porous media; however, many of the special techniques and difficulties discussed also arise in artificial porous media and multiphase flows with thermal effects.

Stochastic Analysis of Adsorbing Solute Transport in Two‐Dimensional Unsaturated Soils

Adsorbing solute transport in two-dimensional heterogeneous unsaturated soil was studied by means of stochastic numerical simulations. Heterogeneities in the soils hydraulic properties and in the adsorption isotherm were simulated using random fields having specified statistical structures. Macrodispersion was analyzed using the spatial moments of numerically generated solute plumes. Among different realizations of the heterogeneous soil, the discrepancies between second-order moments and macrodispersion coefficients were large. Macrodispersivities of unsaturated soils increased with decreasing water content. Also, heterogeneous adsorption of solute enhanced the solute spreading. When the adsorption coefficient was negatively correlated with the saturated hydraulic conductivity, solute spreading was greater than when adsorption was uncorrelated or positively correlated with the conductivity.

Mechanics of multiphase fluid flows in variably saturated porous media

Abstract This article proposes a set of flow equations governing the simultaneous movement of aqueous and nonaqueous liquids in variably saturated soils. The basic principles and balance laws of continuum mixture theory, along with thermodynamically admissible constitutive laws and simplifying kinematic assumptions, yield a formulation for isochoric multiphase flows through a nondeforming porous matrix. Cast in terms of familiar quantities, the governing equations are similar in form to the classic Richards equation for each liquid phase. The development suggests new rock-fluid properties that must be measured to characterize multiphase flows in the unsaturated zone.

Two-grid methods for mixed finite-element solution of coupled reaction-diffusion systems

We develop two-grid schemes for solving nonlinear reaction-diiusion systems , @p @t ? r (Krp) = f(x; p); where p = (p; q) is an unknown vector-valued function. The schemes use discretizations based on a mixed nite-element method. The two-grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton-like iterations to grids much coarser than the nal one, with no loss in order of accuracy. The iterative algorithms examined here extend a method developed earlier for single reaction-diiusion equations. An application to pre-pattern formation in mathematical biology illustrates the methods eeectiveness.

Parallel computing for solute transport models via alternating direction collocation

Abstract We examine algorithmic aspects of M. Celias alternating-direction scheme for finite-element collocation, especially as implemented for the two-dimensional advection-diffusion equation governing solute transport in groundwater. Collocation offers savings over other finite-element techniques by obviating the numerical quadrature and global matrix assembly procedures ordinarily needed in Galerkin formulations. The alternating-direction approach offers further saving in storage and serial runtime and, significantly, yields highly parallel algorithms involving the solution of problems having only one-dimensional structure. We explore this parallelism.

The finite layer method for groundwater flow models

The finite layer method (FLM) is an extension of the finite strip method familiar in structural engineering. The idea behind the method is to discretize two space dimensions using truncated Fourier series, approximating variations in the third via finite elements. The eigenfunctions used in the Fourier expansions are orthogonal, and, consequently, the Galerkin integrations decouple the weighted residual equations associated with different Fourier modes. The method therefore reduces three-dimensional problems to sets of independent matrix equations that one can solve either sequentially on a microcomputer or concurrently on a parallel processor. The latter capability makes the method suitable for such computationally intensive applications as optimization and inverse problems. Four groundwater flow applications are presented to demonstrate the effectiveness of FLM as a forward solver.