WaterSubject

This paper is the fourth in a series of efforts intended to yield complete, rigorous, closed models that describe transport phenomena in multiscale porous medium systems using the thermodynamically constrained averaging theory (TCAT) approach. The first paper provides an overview of the general TCAT approach, which is built on averaged conservation and thermodynamic equations that constrain an entropy inequality. The second paper provides the mathematical fundamentals and theorems that are used to generate needed macroscale equations. The third paper illustrates the application of the method for single-fluid-phase, single-species flow in a porous medium. In the present work, we develop additional fundamental components of the theory to enable the subsequent building of rigorous, closed models for multispecies systems.

The composition of a phase is of central importance for modeling many porous medium systems. Example applications include saltwater intrusion; contaminant fate, transport, and remediation; irrigation and fertilization; aquifer storage and recovery of treated drinking water; and analysis of the effects of nuclear waste disposal. Models used to describe such systems are typically based upon the advective–dispersive equation, assuming a Fickian form of the dispersion process. These models are typically posited directly at the macroscale and are not usually thermodynamically constrained. It is also commonplace for the variables that appear in such macroscale equations to lack precise definitions and connections to microscale quantities.

Of further significance is the general consensus that heterogeneity at the macroscale, typical of most natural systems, leads to the limited usefulness of the advective–dispersive equation for many problems of interest. However, we draw a distinction between the formulation of macroscale models and the upscaling of these macromodels to an even larger scale where the form of the model, and precise meaning of the parameters, should not be expected to be consistent with the underlying macroscale model. Thus multiple levels and types of upscaling are ultimately of concern for many situations of interest.

The TCAT approach ensures a precise connection to both microscale quantities and thermodynamic constraints, which are in turn used to guide the development of macroscale closure relations. The TCAT approach differs in another important respect from typical model formulation approaches: it includes specific conservation and balance equations for interfaces and common curves. Given the importance of problems that involve composition, the extension of the TCAT framework to compositional macroscale systems is a reasonable next step in the evolution of this model formulation approach. This extension will require conservation and balance equations for species in entities and averaged thermodynamic relations, which have not yet appeared in the literature.

The overall goal of this work is to advance components needed to formulate TCAT-based macroscale models to describe compositional multispecies, multiphase porous medium systems. The specific objectives of this work are: (1) to develop species conservation and balance equations for phase volumes, interfaces, and common curves; (2) to formulate macroscale thermodynamic relations for compositional systems based upon averaged microscale relations; (3) to outline a flexible constraint approach for connecting conservation equations to a system entropy inequality to yield a range of different models; and (4) to discuss ways in which the fundamental tools developed in this work can be utilized in the formulation of closed macroscale models of compositional multiphase porous medium systems.

The preceding work details the formulation of important components needed to advance more sophisticated TCAT models than those formulated to date. In particular, the multiphase species conservation and the entropy balance equations for phase volumes, interfaces, and common curves are now complete. The averaged forms of CIT for these entities has been formulated as well, and the conditions that must hold at equilibrium have further been detailed.

These components can be used to formulate a range of models, such as species transport in single-fluid-phase systems, multiphase flow, and multiphase flow and species transport. Because these fundamental components will not change as a function of the application of concern, these results can be used for any of these applications without modification. The TCAT framework that has been developed and the components detailed herein thus form the basis upon which hierarchies of models of various levels of sophistication can be built. This leaves the important future work to focus on model closure methods and detailed validation by comparison to microscale experimental observations and highly resolved simulations.

Further theoretical work is certainly possible, which would require the derivation of additional modeling components. We would advocate such approaches if models built upon the existing theoretical components proved to be inadequate. Advances that might warrant consideration would include the following:

• species internal energy form of ACIT—the averaged thermodynamics in this work is based upon the internal energy for an entity; an extension to this approach would be to develop the averaged thermodynamics for a species-entity combination; such an approach might prove to have utility in the future for more sophisticated models than those considered to date;

• alternative averaged thermodynamic basis—TCAT requires a thermodynamic representation at the microscale be averaged to the macroscale; to date we have relied upon classical irreversible thermodynamics; other approaches are possible and may be required to describe certain systems; such work would lead to alternative forms for the relationship between the material derivatives of internal energy, entropy, densities, entity measures, and other quantities that are intensive quantities at the microscale;

• consideration of systems that contain more than two separated length scales—the work accomplished to date has considered a microscale and a macroscale, with the intent being to develop macroscale models that are connected to the microscale; systems of concern may include additional scales of interest, which will require modifications to the basic TCAT methodology developed to date; and

• even more fundamentally, the lack of existence of an REV for certain systems of interest would require a detailed theoretical investigation of the underlying conservation equations; we acknowledge that many natural systems of concern have such characteristics and ultimately this issue should be considered, but doing so is beyond the scope of our current endeavors.