Decay of the initial oil concentration discontinuity in 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 of time, the partition is removed and a non-negative suspension velocity is maintained at the injection wells. An accurate analysis of the unique solution to the Buckley-Leverett model shows that at the initial moment of time, oil begins to displace the suspension, resulting in the formation of a zone of mixing of oil and suspension. If the velocity of the suspension at the injection wells is high enough, then at some point in time the natural option of displacing oil by the suspension begins to be realized.}\keywords{Free boundary problems, transport equations, displacement of oil by suspension, strong discontinuity conditions.