Transport in chemically and mechanically heterogeneous porous media. I: Theoretical development of region-averaged equations for slightly compressible single-phase flow

Abstract Transport processes in heterogeneous porous media are often treated in terms of one-equation models. Such treatment assumes that the velocity, pressure, temperature, and concentration can be represented in terms of a single large-scale averaged quantity in regions having significantly different mechanical, thermal, and chemical properties. In this paper we explore the process of single-phase flow in a two-region model of heterogeneous porous media. The region-averaged equations are developed for the case of a slightly compressible flow which is an accurate representation for a certain class of liquid-phase flows. The analysis leads to a pair of transport equations for the region averaged pressures that are coupled through a classic exchange term, in addition to being coupled by a diffusive cross effect. The domain of validity of the theory has been identified in terms of a series of length and timescale constraints. In Part II the theory is tested, in the absence of adjustable parameters, by comparison with numerical experiments for transient, slightly compressible flow in both stratified and nodular models of heterogeneous porous media. Good agreement between theory and experiment is obtained for nodular and stratified systems, and effective transport coefficients for a wide range of conditions are presented on the basis of solutions of the three closure problems that appear in the theory. Part III of this paper deals with the principle of large-scale mechanical equilibrium and the region-averaged form of Darcy's law. This form is necessary for the development and solution of the region-averaged solute transport equations that are presented in Part IV. Finally, in Part V we present results for the dispersion tensors and the exchange coefficient associated with the two-region model of solute transport with adsorption.

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