Models of flux in porous media with memory

Some data on the flow of fluids in rocks exhibit properties which may not be interpreted with the classic theory of propagation of pressure and of fluids in porous media [Bell and Nur, 1978; Roeloffs, 1988] based on Darcy's law, which states that the flux is proportional to the pressure gradient. Concerning the fluids, some may react chemically with the medium enlarging the pores; some carry solid particles, which may obstruct some of the pores; and finally, some may precipitate minerals in the pores diminishing their size or even closing them as in geothermal areas. These phenomena create a spatially variable pattern of mineralization and permeability changes that can be localized. In order to obtain a better representation of the flux and of the pressure of fluids, Darcy's law is modified, introducing general memory formalisms operating on the flow as well as on the pressure gradient, which imply a filtering of the pressure gradient without singularities. We also modify the second constitutive equation of diffusion, which relates the density variations of the fluid to the pressure, by introducing a rheology in the fluid also represented by memory formalisms operating on the pressure as well as on the density variations. The memory formalisms are then specified as derivatives of fractional order. The equations used here for the diffusion of fluids are different from the classic ones; however, the equation governing the diffusion of the pressure is the same as that of the flux, as in the classic case. For technical reasons the majority of the studies on diffusion is devoted to the diffusion of the pressure of the fluid rather than to the flux; in this paper we shall devote our attention to studying the flux and its spectral properties in a practical example seeing that the memory used implies a low-pass filtering of the flux or a band pass centered in the low-frequency range. A half space is considered where the boundary values are applied to the plane limiting it, and the problems solved are the computation of the Green function of the flux in the cases when (1) a pressure constant in time is applied on the boundary plane and (2) a periodic pressure is applied to the boundary plane while the half space is initially at zero pressure in both cases. We found closed form formulae for the flux and its spectrum. A discussion follows concerning the mode of determination of the parameters of memory formalisms ruling the diffusion using the observed pressure and/or the flux at several frequencies in problem 2. Concerning the flux, it is tentatively seen that when the medium is oil rock and the fluid is water, for a pressure of 10(5) Pa at the boundary and derivative of order 0.1, the flux at a distance of 0.1 km from the boundary plane is 1700 kg h(-1).