Analysis of matching conditions at the boundary surface of a fluid-saturated porous solid and a bulk fluid: the use of Lagrange multipliers

Fluid Shock Wave Generation at Solid-Material Discontinuity Surfaces in Porous Media

“…In (Coussy and Bourbie (1984), Coussy et al (1998), Coussy (1985-1986), a variational approach is used to study some wave propagation phenomena of interest in oil industry: the boundary conditions proposed by Deresiewicz (1963) are examined there with a view towards the applications. In Kubik and Cieszko (2005) a dissipative Rayleigh surface potential is proposed which is suitable to produce a particular form of Beavers-JosephSaffman boundary conditions: many versions of them are compared in Alazmi and Vafai (2001).…”

“…in Baek and Srinivasa (2004) (the spirit of which is very similar to the one adopted here). Several versions of "surface balance of force" or "kinematical boundary conditions" can be found in Deresiewicz (1963), Ciezko and Kubik (1998a-b), Debergue et al (1999), Goyeau et al (2003), Haber and Mauri (1983), Kuznetsov (1997), Le Bars and Grae Worster (2006), Levy and Sanchez-Palencia (1975), Rajagopal and Tao (1995), Sharma (2008), Ochoa-Tapia and Whitaker (1998). Our boundary conditions seem suitable to describe (macroscopically) the behavior of solid material interfaces open to fluid flow.…”

Macroscopic description of capillary transport of liquid and gas in unsaturated porous materials

“…We use the dissipation inequality of mechanical energy (43) to formulate a consistent set of constitutive equations for the proposed model of the capillary transport of liquid and gas in an unsaturated porous medium. This will be done by application of a similar procedure to that used in phenomenological thermodynamics, based on the entropy inequality and the concept of Lagrange multipliers [45,52,71,72].…”

“…(31), (33) and (34). Equation (31), due to immobility of the capillary liquid, defines the volume force p cs of its interaction with the skeleton, which is not present in the dissipation inequality (43). The momentum balance Eqs.…”

“…However, they have to be taken into account when the fluid is flowing along the boundary of a porous material, where the fluid strain rate is the greatest [11,43].…”