Capillary Pressure

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). \(\begin{gathered} p_c \left| {_y = \left\{ {p_c } \right\}^c } \right|_x + \left( {\rho _\gamma - \rho _\beta } \right)g \cdot \left( {y - \left\{ y \right\}^c } \right) + \Omega _\gamma \cdot [\left( {y + b_\gamma } \right) - \left\{ {y + b_\gamma } \right\}^c ] - \Omega _\beta \cdot [\left( {y + b_\beta } \right) - \left\{ {y + b_\beta } \right\}^c ] + \\ + \tfrac{1}{2}\nabla \Omega _\gamma :[\left( {yy + D_\gamma } \right) - \left\{ {yy + D_\gamma } \right\}^c ] - \tfrac{1}{2}\nabla \Omega _\beta :[\left( {yy + D_\beta } \right) - \left\{ {yy + D_\beta } \right\}^c ] + \\ + [\left( {\mu _\gamma A_\gamma - \mu _\beta A_\beta } \right) - \left\{ {\mu _\gamma A_\gamma - \mu _\beta A_\beta } \right\}^c ]\frac{{\partial \left\{ { \in _\beta } \right\}*}}{{\partial t}} + \\ + [\left( {\mu _\gamma c_\gamma - \mu _\beta c_\beta } \right) - \left\{ {\mu _\gamma c_\gamma - \mu _\beta c_\beta } \right\}^c ] \cdot \nabla \frac{{\partial \left\{ { \in _\beta } \right\}*}}{{\partial t}} + \\ + \mu _\gamma (E_\gamma - \left\{ {E_\gamma } \right\}^c ):\nabla \Phi _\gamma - \mu _\beta (E_\beta - \left\{ {E_\beta } \right\}^c ):\nabla \Phi _{\beta \cdot } \\ \end{gathered} \) Herep c ¦y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p c }¦x represents the large-scale capillary pressure evaluated at the centroid. In addition to{p c } c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as \(\left\{ {p_c } \right\}^c = \left\{ {\left\langle {p_\gamma } \right\rangle ^\gamma } \right\}^\gamma - \left\{ {\left\langle {p_\beta } \right\rangle ^\beta } \right\}^\beta ,\) , \(p_c \left| {_y = \left\{ {p_c } \right\}^c } \right|_{x \cdot } \) , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.

This paper discusses the role of the structural disjoining pressure exerted by nanoparticles on the spreading of a liquid film containing these particles. The origin of the structural disjoining pressure in a confined geometry is due to the layering of the particles normal to the confining plane and has already been traced to the net increase in the entropy of the system in previous studies. In a recent paper, Wasan and Nikolov (Nature, 423 (2003) 156) pointed out that the structural component of the disjoining pressure is strong enough to move a liquid wedge; this casts a new light on many applications—most notably, detergency. While the concept of spreading driven by the disjoining pressure is not new, the importance of the structural disjoining pressure arises from its long-range nature (as compared to the van der Waals' force), making it an important component of the overall force balance near the contact line. In this paper, we report on a parametric study of the spreading phenomena by examining the effects of nanoparticle size, concentration and polydispersity on the displacement of an oil–aqueous interface with the aqueous bulk containing nanoparticles. The solution of the extended Laplace–Young equations for the profile of the meniscus yields the position of the nominal contact line under the action of the structural disjoining pressure. Simulations show that the displacement of the contact line is greater with a high nanoparticle volume fraction, small particles for the same volume fraction, monodispersed (in size) particles rather than polydispersed particles and when the resisting capillary pressure is small, i.e., when the interfacial tension is low and/or the radius of the dispersed phase drop/bubble is large.

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two

A new, closed-form hysteretic model of the capillary pressure-saturation and relative permeability-saturation relationship has been implemented into ITOUGH2. The hysteretic capillary pressure function is based on the van Genuchten model, with a modified version of the dependent domain model of Mualem to describe the scanning curves. Hysteresis in the relative permeability relations is considered to be mainly a result of nonwetting fluid entrap- ment. The hysteresis model was used in combination with inverse modeling techniques to examine the potential of a simple drainage- imbibition experiment to determine hysteretic hydraulic properties.