An analytical solution for simultaneous flow of two immiscible liquids through porous fissured media is presented. The derivation of the parameters and of the equations of flow is based on a schematic model of the medium and of the flow mechanism.
The equations of flow are solved in the case of one-dimensional horizontal displacement of a nonwetting liquid by a wetting one. The influence of the capillary pressure is neglected in the fissures, but not in the blocks.
The solution leads to the existence of a front (discontinuity) of saturation in the fissures, while in the blocks the saturation varies gradually.
In contrast with the Buckley-Leverett solution for an ordinary (nonfissured) porous medium, the front velocity and the saturation at the front decreases in time, due to the transfer of the wetting liquid from fissures to blocks.
The ratio between the volume of the recovered nonwetting liquid at the end of the column and the volume of the injected wetting liquid has been found by numerical integration in a particular case - as a function of the injection rate. The results are in agreement with laboratory experiments and field observations.
The porous fissured medium considered here consists of a network of fissures surrounding porous blocks (Fig. 1). In natural reservoirs of this porous blocks (Fig. 1). In natural reservoirs of this type the characteristic width of the fissures is of the order of 1/10 mm, while the characteristic length of the block is of the order of meters.
Since the fissures occupy only a small portion of the total volume, the medium as a whole has the following properties:
(1) The porosity is approximately that of the individual blocks and most of the liquid is contained in these blocks;
(2) The contribution of the blocks to the permeability may be of the same order of magnitude as that of the fissure network (in spite of the large permeability of the individual fissure in comparison to that of the individual block).
As an example, let us assume that we are dealing with an idealized porous fissured medium with cubic blocks in a cubic package. For flow parallel to one of the fissure package. For flow parallel to one of the fissure directions the permeability of the composite medium is given approximately by: k = (k L + k L )/(L + L ) = k + k, where L, is the width of the fissure, L the block dimension, k and k the permeability of the individual fissure and block, permeability of the individual fissure and block, respectively; k, = k L /(L + L ) and k = k L / (L + L ) are the permeabilities of the fissure and block system, respectively. If we consider limestone blocks (L = 1m, k = 100 md) and fissures with L = 0.1 mm, k = L /12 - 10 d, we obtain k - k - 0.1 d. Thus, one cannot generally neglect the contribution by the blocks for the over-all permeability. permeability. We consider here flow fields with characteristic length much larger than those of the blocks. Consequently, one can average different properties (porosity, permeability) and treat the medium as continuous and homogeneous. There are, however, substantial differences between the flow in a porous fissured medium and an ordinary one. For instance, in a water-drive oil reservoir the water reaches the wells earlier than in an ordinary medium due to the rapid flow through the fissures. This phenomenon was described first by Gibson. Later the research was focused on the study of imbibition, which was, recognized as an important factor in the recovery of oil from the blocks. A mathematical model of the simultaneous flow of immiscible fluids through porous fissured media was first given by Barenblatt.
