On inverse problems in secondary oil recovery

In oil production it is crucial to find out properties of the ground from various measurements of geophysical fields. In secondary oil recovery oil is extracted by pumping in water (through injecting wells) and creating pressure which pumps out oil through production wells. Water/oil pressure at wells is monitored providing with an information to determine two important characterictics of medium: permeability and porosity. These characteristics can indicate location of oil, and hence give valuable recommendations for drilling new wells and finding pumping regimes to optimize oil recovery. Looking for permeability and porosity can be viewed as an inverse problem for (system of) quasilinear partial differential equations describing fluids in porous media. This inverse problem is called ”history matching”. Despite its obvious applied importance this inverse problem was studied only numerically, in most cases by using routine least squares minimization [12], [14], [27]. Due to large size of the problem and its nonconvexity these methods are not efficient and not reliable. In particular, there is no uniqueness and stability analysis. We review simplest models of oil reservoirs and suggest some ideas for theoretical and numerical study of this important inverse problem. These models are formed by a system of an elliptic and a parabolic (or first order hyperbolic) quasilinear partial differential equations. There are and probably there will be serious theoretical and computational difficulties with both direct and inverse problems for this system mainly due to its degeneration. However, practical value of the problem justifies efforts producing any progress. At present, in inverse problems for elliptic and parabolic equations there is theoretical and numerical progress [16]. We believe that this progress can generate new efficient mathematical methods in oil recovery.