Lynn S. Bennethum

A primer on upscaling tools for porous media

Over the last few decades a number of powerful approaches have been developed to intelligently reduce the number of degrees of freedom in very complex heterogeneous environs, e.g. mathematical homogenization, mixture and hybrid mixture theory, spatial averaging, moment methods, central limit or Martingale methods, stochastic-convective approaches, various other Eulerian and Lagrangian perturbation schemes, projection operators, renormalization group techniques, variational approaches, space transformational methods, continuous time random walks, and etc. In this article we briefly review many of these approaches as applied to specific examples in the hydrologic sciences.

A wildland fire model with data assimilation

A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients that can be approximated from prior measurements of wildfires. An ensemble Kalman filter technique with regularization is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one.

Multiscale, hybrid mixture theory for swelling systems—I: balance laws

A three-scale problem involving swelling media is considered. The structure of the system (e.g. clay, polymers, etc.) is assumed to be characterized by particles composed of more than one phase (usually solid and liquid) which may swell or shrink. The swelling (or shrinkage) is the result of mass transfer from (to) a multicomponent solution which is in contact with the particles. Mesoscopic equations for the particles themselves have been obtained elsewhere. The macroscopic equations for the combination of swelling particles and bulk solution as well as their interfaces, considered as a mixture, are obtained here. Constitutive theory for the system is the subject of a subsequent manuscript.

Macroscale Thermodynamics and the Chemical Potential for Swelling Porous Media

The thermodynamical relations for a two-phase, N-constituent, swelling porous medium are derived using a hybridization of averaging and the mixture-theoretic approach of Bowen. Examples of such media include 2-1 lattice clays and lyophilic polymers. A novel, scalar definition for the macroscale chemical potential for porous media is introduced, and it is shown how the properties of this chemical potential can be derived by slightly expanding the usual Coleman and Noll approach for exploiting the entropy inequality to obtain near-equilibrium results. The relationship between this novel scalar chemical potential and the tensorial chemical potential of Bowen is discussed. The tensorial chemical potential may be discontinuous between the solid and fluid phases at equilibrium; a result in clear contrast to Gibbsian theories. It is shown that the macroscopic scalar chemical potential is completely analogous with the Gibbsian chemical potential. The relation between the two potentials is illustrated in three examples.

Multiscale, hybrid mixture theory for swelling systems. II: Constitutive theory

Abstract In this second part of a three-part paper we derive constitutive theory for a multiphase, multicomponent, three-scale, swelling system which includes interfaces. In Part I, the governing field equations and the definitions of all mesoscopic and macroscopic variables therein were defined in terms of microscopic variables. In this paper, we choose the independent variables and derive constitutive restrictions for two cases of a dual-porosity multiple-component swelling media: one which assumes no interfacial effects, and one which includes interfacial effects. For each case, the entropy inequality is fully derived using a Lagrange multiplier technique. Novel definitions for macroscopic pressures and chemical potentials are given, and generalized Darcys and Ficks laws are presented. Although these models are developed with a clay soil in mind, the results hold for any medium which assumes the same set (or subset of) independent variables as constitutive unknowns, e.g. swelling biopolymers.

Multicomponent, Multiphase Thermodynamicsof Swelling Porous Media with Electroquasistatics:I. Macroscale Field Equations

A systematic development of the macroscopic field equations (conservation of mass, linear and angular momentum, energy, and Maxwells equations) for a multiphase, multicomponent medium is presented. It is assumed that speeds involved are much slower than the speed of light and that the magnitude of the electric field significantly dominates over the magnetic field so that the electroquasistatic form of Maxwells equations applies. A mixture formulation for each phase is averaged to obtain the macroscopic formulation. Species electric fields are considered, however it is assumed that it is the total electric field which contributes to the electrically induced forces and energy. The relationships between species and bulk phase variables and the macroscopic and microscopic variables are given explicitly. The resulting field equations are of relevance to many practical applications including, but not limited to, swelling clays (smectites), biopolymers, biological membranes, pulsed electrophoresis, and chromatography.

A multi-scale theory of swelling porous media: I. Application to one-dimensional consolidation

A theory is developed which describes flow in multi-scale, saturated swelling media. To upscale information, both the hybrid theory of mixtures and the homogenization technique are employed. In particular, a model is formulated in which vicinal water (water adsorbed to the solid phase) is treated as a separate phase from bulk (non-vicinal) water. A new form of Darcys law governing the flow of both vicinal and bulk water is derived which involves an interaction potential to account for the swelling nature of the system. The theory is applied to the classical one-dimensional consolidation problem of Terzaghi and to verify Lows empirical, exponential, swelling result for clay at the macroscale.

Multicomponent, Multiphase Thermodynamics of Swelling Porous Media with Electroquasistatics: II. Constitutive Theory

In Part I macroscopic field equations of mass, linear and angular momentum, energy, and the quasistatic form of Maxwells equations for a multiphase, multicomponent medium were derived. Here we exploit the entropy inequality to obtain restrictions on constitutive relations at the macroscale for a 2-phase, multiple-constituent, polarizable mixture of fluids and solids. Specific emphasis is placed on charged porous media in the presence of electrolytes. The governing equations for the stress tensors of each phase, flow of the fluid through a deforming medium, and diffusion of constituents through such a medium are derived. The results have applications in swelling clays (smectites), biopolymers, biological membranes, pulsed electrophoresis, chromotography, drug delivery, and other swelling systems.

Generalized Forchheimer Equation for Two-Phase Flow Based on Hybrid Mixture Theory

In this paper, we derive a Forchheimer-type equation for two-phase flow through an isotropic porous medium using hybrid mixture theory. Hybrid mixture theory consists of classical mixture theory applied to a multiphase system with volume averaged equations. It applies to media in which the characteristic length of each phase is ‘small’ relative to the extent of the mixture. The derivation of a Forchheimer equation for single phase flow has been obtained elsewhere. These results are extended to include multiphase swelling materials which have nonnegligible interfacial thermodynamic properties.

Three Pressures in Porous Media

In a thermodynamic setting for a single phase (usually fluid), the thermodynamically defined pressure, involving the change in energy with respect to volume, is often assumed to be equal to the physically measurable pressure, related to the trace of the stress tensor. This assumption holds under certain conditions such as a small rate of deformation tensor for a fluid. For a two-phase porous medium, an additional thermodynamic pressure has been previously defined for each phase, relating the change in energy with respect to volume fraction. Within the framework of Hybrid Mixture Theory and hence the Coleman and Noll technique of exploiting the entropy inequality, we show how these three macroscopic pressures (the two thermodynamically defined pressures and the pressure relating to the trace of the stress tensor) are related and discuss the physical interpretation of each of them. In the process, we show how one can convert directly between different combinations of independent variables without re-exploiting the entropy inequality. The physical interpretation of these three pressures is investigated by examining four media: a single solid phase, a porous solid saturated with a fluid which has negligible physico-chemical interaction with the solid phase, a swelling porous medium with a non-interacting solid phase, such as well-layered clay, and a swelling porous medium with an interacting solid phase such as swelling polymers.