The Structure of all Possible Solutions to the Buckley-Leverett Model
Abstract
We consider a free boundary problem for a one-dimensional system of Buckley-Leverett equations, describing the displacement of oil by a suspension. For this problem we formulated conditions for the strong decay of the discontinuity of the initial oil concentration. We will prove that the phenomenological Buckley-Leverett model does not adequately describe the physical process under consideration. To do this, we will study the problem of the decay of a discontinuity in the initial concentration of oil, when at rest in one half of the domain there is oil, and in the other half of the domain there is a suspension, and these domains are separated by an impenetrable partition. At the initial moment in time, the partition is removed, which initiates the movement of the fluids. A precise analysis of the unique solution to the corresponding initial-boundary value problem for the Buckley-Leverett model shows that there are several different configurations of movement for oil, suspension, and mixture, depending on the initial and boundary data. We will prove that for all these configurations of initial and boundary data, the model describes unrealistic fluid motion. For example, the mixture begins to displace oil and suspension.