Modeling Phenomena of Flow and Transport in Porous Media

The chapter starts by defining a porous medium and discusses the continuum description of phenomena and flow and transport of extensive quantities such as mass, momentum and energy at the microscopic and macroscopic continuum levels. The derivation of the unscaled continuum description at the macroscopic level by volume and mass averaging is presented and discussed, although the book is based primarily on deriving the continuum description of transport in porous media by the phenomenological approach. Scales of description are discussed for inhomogeneous porous media. The general procedure for mathematical modeling is presented.

The chapter introduces some fundamentals of thermodynamics of phases and chemical species that are required for the understanding and constructing the flow and transport models considered in this book. Phases and chemical species are defined. Among the concepts defined and discussed are the pressure, density, chemical and other potentials, Gibbs function, internal energy, enthalpy and capillary pressure. Equations of state and their role in modeling are discussed, as well as the concepts of phase Behavior. The concepts of a tensors, stress, strain ar introduced. They will be used in modeling flow in deformable porous media, as well as for the discussion on poromechanics and deformation in Chap. 9. Onsager?s Theory of Coupled Processes is briefly reviewed. No effort is made to present a complete review of the considered subjects as these can easily be found in texts on Thermodynamics. In this chapter, as in the entire book, we use the International System of units.

This chapter starts by discussing the concepts of a point, velocity and flux. The general balance is developed for any extensive quantity, for a point at the microscopic level, i.e., at a point within a fluid phase present in the void space, and at a point in the solid matrix. Then, the phenomenological approach is used to present this balance equation at the macroscopic level. The two approaches are briefly described. Then, the balance equations for the transport at the macroscopic level of mass, momentum and energy are derived. The Advective and diffusive flux appearing in both the microscopic and macroscopic balance equations are introduced, as well as e dispersive flux appearing only the latter. Special attention is devoted to the coefficients appearing the macroscopic balance equations. The macroscopic balance equation is also obtained by volume and mass averaging. Interphase transfers and sources appearing in the balance equations are also discussed, with their appropriate expressions and coefficients. Special attention is devoted to the meaning and role of coefficients. Dimensionless numbers and are discussed and used for determining non-dominant effects.

This chapter is devoted to the macroscopic fluid motion equation, emphasizing its origin as a momentum balance equation. Starting point with the momentum balance equation of a phase, and introducing certain simplifying assumptions, this equation is reduced to the well-known (linear) Darcy’s law. Other non-Darcian forms of the motion equation—Forchheimer, Kozeny and Klinkenberg- are also presented.

This chapter is devoted to modeling mass transport of a single fluid phase, liquid or gas, that completely occupies the void space. The core of such model is the mass balance equation of the phase. This (macroscopic) equation is obtained from the general macroscopic balance equation when applied to mass, or phenomenologically. The specific storativity is introduced to account for fluid and solid matrix compressibility. Boundary and initial conditions are presented, leading to a well-posed flow (= mass transport) model. A separate model is developed for a constant density fluid and essentially horizontal flow, commonly used to describe flow in aquifers. The entire discussion in this chapter is under isothermal conditions (non-isothethermal conditions are presented I Chap. 8). Two additional subjects are discussed in this chapter. One is flow in a deformable porous medium, primarily as associated with storage of water in aquifers. The general subject of flow and other phenomena of transport in deformable porous media are discussed in detail in Chap. 9. The other subject is an introduction to flow in fractured porous medium domains. Throughout the chapter, we make use of the concepts of stress and shear which are second rank tensors, assuming that the reader is familiar with these concepts. Some introductory remarks about stress and strain are presented at the beginning of the chapter. The last section in this chapter is an introduction to flow in fractured rocks and in fractured porous rock domains.

In this chapter, we construct mass transport models for the case of multiple (i.e., two or three immiscible) fluid phases that occupy the void space simultaneously. The objective is to lead to complete, well-posed mathematical models, taking into account the coupling that takes place between the phases. The fluids occupy disjoint (microscopic) subdomains that together fill up the entire void space. Because of surface tension phenomena, one of the fluids, called the wetting fluid, tends to adhere to the solid, while the other, called the nonwetting fluid, stays farther from the solid surface. The capillary pressure and its relationship to saturation is discussed. The case of drainage and imbibition of a wetting fluid in an unsaturated vertical column in the unsaturated zone is presented. Another case is that of oil water and gas in a petroleum reservoir. Thus, the material in this chapter should be of interest to those who deal with the unsaturated zone in the subsurface, e.g., in connection with irrigation and drainage in agricultural engineering. It should also be if interest to reservoir engineers, to those who plan the disposal of supercritical CO2 in deep brine-containing geological formations, and to those who consider the injection of air or natural gas into depleted oil and gas reservoirs for storage purposes.

This chapter is devoted to the transport of chemical species dissolved in the fluid phases that occupy the void space, with or without chemical reactions. Solute fluxes are due to diffusion, dispersion, and advection. Various sources and sinks, as well as chemical reactions and interphase transfers are discussed and integrated in the species (first microscopic and then macroscopic) mass balance equation. Like any other model of transport, the core of the solute transport model is the mass balance of the considered chemical species. The discussion leads to a well-posed model of the solute transport problem. (mass balance equations for the chemical species, initial and boundary conditions, constitutive relations, etc. To facilitate the discussion, a review of selected topics of chemistry that describe source/sink and interphase exchange phenomena that occur within the considered fluid(s) is presented in this chapter. One such topic is chemical reactions that occur within the fluids that occupy the void space. Adsorption, solid dissolution and precipitation are also modelled. The presentation considers also sinks and sources in the form of extraction and injection of solute carrying fluids through wells. Obviously, the presentation of the chemical aspects should be considered merely as a brief introduction to some of the essentials and to the employed terminology. Following the methodology presented in this book, once phenomena are understood at the microscopic level, where they really occur, they are incorporated in the macroscopic ones The discussion in this chapter includes the effects of temperature, but the subject of flow and transport under non-isothermal conditions is discussed in Chap. 8. The objective is to derive models that describe solute transport not only at laboratory scale domains, but also at large natural domains, primarily in heterogeneous geological formations. The material presented in this chapter should be of use for those dealing with phenomena of transport in geological formations. However, the material will also be useful to chemical engineers who design chemical reactors in the Chemical Engineering industry. Appendix A discusses chemical reactors, and the various phenomena of transport that occur in them.

The chapter deals with modeling non-isothermal mass momentum, and energy transport. Here, an additional variable is added-the temperature. To obtain the temperature distribution in the fluid and solid phases that occupy the porous medium domain, we have to write and solve the energy balance equation, obviously with appropriate initial and boundary conditions, as well as with temperature-dependent constitutive relationships. This equation, with appropriate initial and boundary conditions are considered in this chapter. The assumption that underlies this chapter is that thermal equilibrium exists between all phases present at point in the porous medium domain. The presentation in Chaps. 5, 6, and 7, focused on flow and transport under isothermal conditions. Under non-isothermal conditions, all constitutive relations involve also temperature an additional variable, and the complete model should also include the energy balance equation. The solute transport models considered in Chap. 7, may now involve exogenic and endogenic reactions. Within a porous medium domain, thermal energy may be transported by four mechanisms: (1) Advection by a fluid (or fluids) moving in the void space, (2) conduction in all solid and fluid phases (overlooking advection in a deformable solid), (3) mass diffusion in the fluid phases, and (4) thermal dispersion in the fluid phase(s). All these modes of energy transport are discussed in this chapter. The effects of natural thermal gradients, produced by solar radiation at ground surface, on water and water vapor movement, as well as on the chemical and biological behavior in the subsurface, may serve as examples of interest to soil scientists. In dealing with contaminated groundwater, a number of remediation techniques are associated with heating the soil, and injection of steam. Other interesting cases that require knowledge of heat and mass transport in porous media are the storage of energy in aquifers, or in the unsaturated zone, the production of geothermal energy, the disposal of \(\text {CO}_{2}\) in deep geological formations, the geological storage of high-level nuclear waste, and the thermally enhanced production of petroleum. Finally, in the chemical industry, most processes that take place in reactors occur under non-isothermal conditions. Because of its importance, we have added an appendix (App. A) that presents and discusses modeling of phenomena of transport that take place in chemical reactors. Coupling between the transport of mass and heat is also due to the fact that both the fluid’s density and its viscosity are temperature dependent. Most partitioning and equilibrium coefficients, discussed in Chaps. 6 and 7, are strongly temperature dependent.

The chapter deals with cases of transport of mass and energy in which the solid matrix, as the entire porous medium domain, undergoes deformation. Models of consolidation as a consequence of construction, and models of land subsidence due to heavy pumping, are developed and presented. Also, the propagation of waves in a porous medium domain, in which soil deformability plays an essential role, is briefly discussed. Models of consolidation as a consequence of construction, and models of land subsidence due to heavy groundwater pumping, are developed and presented. Also, the propagation of waves in a porous medium domain, in which soil deformability plays an essential role, is briefly discussed. The chapter focusses on cases in which fluids are extracted from or injected into geological formations, possibly under non-isothermal conditions. The presented models describe phenomena of deformation of porous medium domains in aquifers in response to imposed stresses. Examples are: (1) Land subsidence as a consequence of pumping water from aquifers, (2) ground surface upheaval, as a consequence of injection, (3) development of fractures as a consequence of injecting fluids into a tight geological formation (4) induced seismicity as a result of fluid injection into confined formations, and (5) soil liquefaction. Change in the flow regime, associated with fluid pressure changes, causes changes within a considered porous medium domain causing formation deformation. The chapter starts by introducing the concepts of stress and strain in a single phase continuum-first at the microscopic level and then at the macroscopic one. Then, following the phenomenological approach, the complete model for stress-strain analysis is developed for the saturated (or multiphase) elastic porous medium, leading to the strain and deformation within the considered porous medium domain.