Movement of Two Liquids in a Layer Porous Medium

In this paper, we consider the problem of displacement of one incompressible fluid by another in a particular model of a linear layered bounded reservoir consisting of two layers separated by a small permeable bridge. This problem is extremely important both practical and theoretical.

In the two-dimensional formulation for the rectilinear case, the problem reduces to solving a system of two elliptic equations for the pressures in each of the media separated by an unknown moving boundary under given boundary conditions and satisfying the conditions of equality of pressures and normal components of the filtration rates of displaced and displacing liquids at the movable interface.

The process is considered to be isothermal, non-deformable porous medium, immiscible and chemically mixed not reacting with each other, phase permeability, capillary and gravitational forces are not taken into account. In this way the real picture of repression is idealized, and it is believed that injection fluid completely pushes back the previously filled reservoir fluid and between them there is a clear interface. In such In this case, the entire flow area $D$ is divided into two zones: the $D_{1}$ zone, occupied by the invading fluid, and the $D_{2}$ is the area of the fluid being displaced, separated from one another by a clear moving border of section $Γ$, the law of motion of which is to be determined. Unfortunately, it is impossible to present a solution in analytical form.

To simplify the investigation of the system, its equations are averaged over the thickness of the upper layer. The main feature of this averaging is the specification of the vertical component of the filtration rate in the form of a linear function from the vertical coordinate so that the boundary conditions on the roof and the base of the formation are satisfied. Within the framework of these assumptions, the problem is reduced to solving a system of averaged equations in the upper layer containing functions that describe the interface between two liquids in the upper layer and in the bridge. Integration of the equations in the resulting system in a closed form is not possible without strong restrictions on the law of changing the interface of liquids in the interlayer. In connection with this, two limiting schemes for the displacement of liquids are proposed. The first scheme assumes that there is no overflow of the displacing liquid, which corresponds to the acceleration of the advance of the displacement front in the upper layer. The second scheme assumes a complete overflow of the displacing liquid, which corresponds to the slowing of the displacement front in the upper layer. With such a scheme of motion, the position of the interface in the bridge does not significantly affect the distribution of average pressures in the upper formation. To find the law of displacement of each of the required boundaries, we obtain the Cauchy problems for ordinary differential equations. Numerical calculations have shown that the limit schemes considered differ little from each other, and consequently from the true solution concluded between them. The difference in the times of the complete displacement of one liquid by another, corresponding to these limiting schemes, does not exceed 5 percent.

Keywords: incompressible fluid, elliptic equations, filtration rate, averaging, overflow displacing, moving boundary, limiting schemes.