We study the motion of viscous miscible liquids in porous. Back front of polymer slug in surfactant-polymer flooding in oil fields is our prototype model. Injection of less viscous fluid to a more viscous one generates instabilities, which are often called “viscous fingers”. This phenomenon has a negative impact on various flooding schemes in oil fields. The model is a system of PDEs consisting of conservation of mass, incompressibility condition and Darcy’s law. We try to find rigorous estimates for the size of the mixing zone, where instabilities are located. In the talk I present result of numerical simulations and rigorous estimates of the mixing zone and compare them to other models popular in oil industry. Based on our approach we suggest a graded viscosity banks scheme, which allows to reduce amount of polymer used in surfactant-polymer flooding. The main idea is replacing the postflushing polymer slug by a serious of smaller slugs with gradually decreasing viscosity. The calculation of concentration of polymer and sizes of those slugs becomes an optimization problem. We provide an explicit formula for optimal solution in the case when number of slugs tends to infinity. The result suggest that the main impact is made by applying the scheme with 2-3 slugs and further improvements are minor.

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