Beschreibung:
We analyze the problem of injection of water with micro-organisms into an underground porous medium containing another fluid (oil or gas). The microbes produce a metabolite (a surfactant) that changes capillary and wetting properties between the fluids, which increases the oil mobility. We analyze the Riemann problem for balance equations, which has been reduced to a hyperbolic system of fourth degree. The fractional flow function (F) is assumed to be discontinuous with respect to the surfactant concentration, which provides us the opportunity to develop a qualitative theory of the process and even to obtain the analytical solution. We have determined explicitly the characteristic speeds of continuous waves, several contact jumps, and shock waves, along with a non-classical element, such as a triple jump, the initial state of which is a thorn of saturation. We have shown that a triple jump and a thorn are not numerical artifacts, but true physical objects that satisfy the mass balance and the entropy conditions. Physically, a triple jump means the fast variation of wetting, which leads to the formation of an oil bank and a water wall in front of it. We have revealed a complete qualitative scenario for the propagation of saturation and concentrations, which contains several steps. This information makes it possible to control the correctness of numerical simulation of the process. The subsequent numerical analysis was based on four methods: Godunov's, MacCormack's, upwind and an implicit scheme with small diffusion. Their free parameters were calibrated to reproduce all the stages of the scenario identified analytically. We analyze the ability of various numerical schemes to capture triple jumps and thorns, comparing with the analytical solution.