Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.

Flow in porous media I: A theoretical derivation of Darcy's law

Boundary and inertia effects on flow and heat transfer in porous media

In this paper, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed. Both macroscopically homogeneous and heterogeneous rocks are considered. The latter are characterized by large-scale spatial variations and correlations in their effective properties and include rocks that may be characterized by several distinct degrees of porosity, a well-known example of which is a fractured rock with two degrees of porosity---those of the pores and of the fractures. First, the diagenetic processes that give rise to the present reservoir rocks are discussed and a few geometrical models of such processes are described. Then, measurement and characterization of important properties, such as pore-size distribution, pore-space topology, and pore surface roughness, and morphological properties of fracture networks are discussed. It is shown that fractal and percolation concepts play important roles in the characterization of rocks, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Next, various structural models of homogeneous and heterogeneous rock are discussed, and theoretical and computer simulation approaches to flow, dispersion, and displacement in such systems are reviewed. Two different modeling approaches to these phenomena are compared. The first approach is based on the classical equations of transport supplemented with constitutive equations describing the transport and other important coefficients and parameters. These are called the continuum models. The second approach is based on network models of pore space and fractured rocks; it models the phenomena at the smallest scale, a pore or fracture, and then employs large-scale simulation and modern concepts of the statistical physics of disordered systems, such as scaling and universality, to obtain the macroscopic properties of the system. The fundamental roles of the interconnectivity of the rock and its wetting properties in dispersion and two-phase flows, and those of microscopic and macroscopic heterogeneities in miscible displacements are emphasized. Two important conceptual advances for modeling fractured rocks and studying flow phenomena in porous media are also discussed. The first, based on cellular automata, can in principle be used for computing macroscopic properties of flow phenomena in any porous medium, regardless of the complexity of its structure. The second, simulated annealing, borrowed from optimization processes and the statistical mechanics of spin glasses, is used for finding the optimum structure of a fractured reservoir that honors a limited amount of experimental data.

Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing

Important features of multiphase flow in porous media that distinguish it from single-phase flow are the presence of interfaces between the fluid phases and of common lines where three phases come in contact. Despite this fact, mathematical descriptions of these flows have been lacking in rigor, consisting primarily of heuristic extensions of Darcy's law that include a hysteretic relation between capillary pressure and saturation and a relative permeability coefficient. As a result, the standard capillary pressure concept appears to have physically unrealistic properties. The present paper employs microscopic mass and momentum balance equations for phases and interfaces to develop an understanding of capillary pressure at the microscale. Next, the standard theories and approaches that define capillary pressure at the macroscale are described and their shortcomings are discussed. Finally, an approach is presented whereby capillary pressure is shown to be an intrinsic property of the system under study. In particular, the presence of interfaces and their distribution within a multiphase system are shown to be essential to describing the state of the system. A thermodynamic approach to the definition of capillary pressure provides a theoretically sound alternative to the definition of capillary pressure as a simple hysteretic function of saturation.

http://www.soilmech.polito.it/content/download/169/887/version/1/file/Hass_Gray_WRR93_Basis+of+Pc.pdf

Thermodynamic basis of capillary pressure in porous media

Publisher Summary This chapter discusses the mathematical formulation of the problems of convective heat transfer in geothermal systems. Geothermal reservoirs may have numerous near-vertical faults and relatively impermeable intrusive interspersed in the aquifers. Both theoretical and experimental investigations of heat transfer in geothermal systems are reviewed. A qualitative understanding of the large-scale convection processes in a geothermal reservoir and the prediction of reservoir behavior under production can be obtained by idealizing it as a saturated porous medium. The identification of a viable geothermal reservoir and the estimation of its capacity remain major problems in the utilization of geothermal resources. Thermal anomalies in geothermal areas can be detected by surface manifestations, aerial infrared surveys, geochemical analyses, or exploratory drillings. Many of the analyses are applicable to a wide range of engineering problems whenever they can be idealized as convection in a porous medium. These include the problems of the secondary recovery of oil by thermal methods, the use of fibrous materials for thermal insulations, the design of aquifers as an energy storage system, and the deposition of mineral ore in the subsurface formation. Results from short-duration well testing are used to determine reservoir characteristics.

Heat Transfer in Geothermal Systems

Local volume averaging of the equations of continuity and of motion over a porous medium is discussed. For steady state flow such that inertial effects can be neglected, a resistance transformation is introduced which in part transforms the local average velocity vector into the local force per unit volume which the fluid exerts on the pore walls. It is suggested that for a randomly deposited, although perhaps layered, porous structure this resistance transformation is invertible, symmetric, and positive-definite. Finally, for an isotropic porous structure (the proper values of the resistance transformation are all equal and are termed the resistance coefficient) and an incompressible fluid, the functional dependence of the resistance coefficient is discussed with the Buckingham-Pi theorem used for an Ellis model fluid, a power model fluid, a Newtonian fluid, and a Noll simple fluid. Based on the discussion of the Noll simple fluid, a suggestion is made for the correlation and extrapolation of experimental data for a single viscoelastic fluid in a set of geometrically similar porous structures.

Flow of viscoelastic fluids through porous media

The local volume average of the equation of motion is taken for an incompressible fluid flowing through a porous structure under conditions such that inertial effects may be neglected. The result has two terms beyond a pressure gradient: g, the force per unit volume which a flowing fluid exerts on a porous structure, and the divergence of the local volume-averaged extra stress tensor (viscous portion of the stress tensor).

Constitutive equations for g are examined with the aid of the principle of material indifference. When g is assumed to be a function of the velocity of the fluid relative to the solid as well as various scalars, the usual results for a nonoriented (isotropic) porous structure are obtained. When g is assumed to be a function of the local porosity gradient as well, we derive a new expression for g applicable to oriented (anisotropic) porous structures.

For a Newtonian fluid with a constant viscosity, the divergence of the local volume-averaged extra stress tensor is proportional to the Laplacian of the averaged velocity vector. Boundary conditions for the averaged velocity vector are discussed. Three problems are solved for the flow of an incompressible Newtonian fluid in a nonoriented permeable medium. These solutions, as well as an order-of-magnitude analysis, suggest that we may often neglect both the Laplacian of average velocity and the boundary conditions for the tangential components of averaged velocity at an impermeable wall.

Two specific constitutive equations for g are proposed for the flow of incompressible Noll simple fluids in nonoriented porous structures. Flow through a porous medium bounded by an impermeable cylindrical surface is solved for these two constitutive equations, and the results are compared with previously available experimental data.

Single-phase flow through porous media

Local volume averaging of the equations of continuity and of motion over each phase in a porous medium is discussed. For a Noll simple material, inertial effects may be neglected with respect to viscous effects as the product of a Reynolds number with a Weissenberg number goes to zero (in the limit of a Newtonian fluid, inertial effects may be neglected as the Reynolds number goes to zero). When inertial effects can be neglected, a resistance transformation for each phase is introduced which in part maps the local volume-averaged velocity vector into the local force per unit volume which that phase exerts on the pore walls and the moving phase interfaces. Capillary pressure is defined through a local surface average of the jump balance for momentum. The shape of the curve denoting capillary pressure as a function of saturation in a two-phase flow may vary with the contact angle. For an isotropic multiphase flow of incompressible materials, the functional dependence of the resistance coefficient for a Noll simple material and for a Newtonian fluid is discussed by means of the Buckingham-Pi theorem.

John C. Slattery

Papers

Flow of viscoelastic fluids through porous media

Single-phase flow through porous media

Multiphase viscoelastic flow through porous media

Flow of Viscoelastic Fluids Through Porous Media