William G. Gray

General conservation equations for multi-phase systems: 1. Averaging procedure

Abstract A systematic procedure for averaging continuum equations over representative regions of multiphase systems is developed. Mass and areal averages as well as volume averages are defined to ensure that the averaged quantities are physically meaningful. The averaging procedure is then applied to a general balance equation, and the averaged general equation along with the appropriate macroscopic jump condition useful for the description of multiphase systems are derived.

Thermodynamic basis of capillary pressure in porous media

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 Darcys 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.

Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries

Abstract The main purpose of this work is to develop a macroscale thermodynamic theory to describe two-phase flow in porous media. Full thermodynamic properties are assigned to the boundary surfaces separating the phases at the microscale. Macroscopic equations of balance for mass, momentum, and energy for each phase and interface along with the averaged entropy inequality are employed as the starting point. A constitutive theory is developed resulting in balance equations and thermodynamics appropriate for modelling multiphase flow in porous media. Volume fractions of phases and areal fractions of interfaces are explicitly included in the theory. Incorporation of the interface equations into the theory allows for a complete description of the problem. The manipulations provide explicit functional dependence of the capillary pressure. An extended form of Darcys law for multiphase flow is obtained from the macroscopic equations of momentum balance. An additional term which accounts for non-uniform fluid saturation at equilibrium appears in the result.

A derivation of the equations for multi-phase transport

Abstract A general form of the equations for multi-phase transport is derived and applied to the particular case of mass transport. In the development, particular attention is paid to the proper allocation of the transport mechanisms between convection and diffusion. Furthermore, a new definition of the tortuosity vector is proposed which accounts for a decreased diffusion rate due to the geometry of the system.

General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow

Abstract Equations which describe single phase fluid flow and transport through an elastic porous media are obtained by applying constitutive theory to a set of general multiphase mass, momentum, energy, and entropy equations. Linearization of these equations yields a set of equations solvable upon specification of the material coefficients which arise. Further restriction of the flow to small velocities proves that Darcys law is a special case of the general momentum balance.

A wave equation model for finite element tidal computations

Abstract A shallow water wave equation is developed from the primitive two-dimensional shallow water equation. A finite element model based on this equation and the primitive momentum equation is developed. A finite difference formulation is used in the time domain which allows the model to be implicit or explicit while still centered in time. Results obtained with linear triangles and quadratic quadrilaterals are reported, and compare well with analytic solutions. The model incorporates all of the economical advantages of earlier models, and errors due to short wavelength spatial noise are suppressed without recourse to artificial means.

On the theorems for local volume averaging of multiphase systems

Abstract Proofs of the theorems of local volume averaging which relate averages of derivatives to derivatives of averages are presented. A distribution function whose derivatives are proportional to the Dirac delta function is used in the development and the proofs demonstrated are intended to be simpler than those found in the literature.

Unsaturated Flow Theory Including Interfacial Phenomena

The macroscopic porous medium equations for mass, momentum, and energy transport for air, water, and solid phases and the interfaces between these phases are examined in light of the second law of thermodynamics. Attention is focused on the momentum balance for the water phase. Appropriate forms of the momentum balance are obtained, in general, for the slow flow situation and for the case when the water phase completely wets the solid. This last case suggests that the relative wettability of the water and air phases is an important dependent thermodynamic variable which contributes to the hysteretic nature of the capillary pressure versus saturation curve.

High velocity flow in porous media

Experimental observations have established that the proportionality between pressure head gradient and fluid velocity does not hold for high rates of fluid flow in porous media. Empirical relations such as Forchheimer equation have been proposed to account for nonlinear effects. The purpose of this work is to derive such nonlinear relationships based on fundamental laws of continuum mechanics and to identify the source of nonlinearity in equations.Adopting the continuum approach to the description of thermodynamic processes in porous media, a general equation of motion of fluid at the macroscopic level is proposed. Using a standard order-of-magnitude argument, it is shown that at the onset of nonlinearities (which happens at Reynolds numbers around 10), macroscopic viscous and inertial forces are negligible compared to microscopic viscous forces. Therefore, it is concluded that growth of microscopic viscous forces (drag forces) at high flow velocities give rise to nonlinear effects. Then, employing the constitutive theory, a nonlinear relationship is developed for drag forces and finally a generalized form of Forchheimer equation is derived.

Computational methods in water resources X

1. Groundwater and Flow in Porous Media. 2. Subsurface Transport. 3. Scaling and Heterogeneity. 4. Geostatistics. 5. Reactive Flow. 6. Fractured Porous Media. 7. Parameter Estimation. 8. Remediation and Optimization. 9. Subsurface Multiphase Flow. 10. Saltwater Intrusion. 11. Shallow Water Equations. 12. Flow and Transport in Rivers. 13. Navier--Stokes Equations. 14. Coastal Flow. 15. Sediment Transport. 16. Algebraic Methods. 17. Software Development. 18. Parallel Methods.