On inverse problems in secondary oil recovery

We review simple 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 mainly due to the degeneracy of the system. The practical value of the problem justifies efforts to improve the methods for its solution. We formulate 'history matching' as a problem of identification of two coefficients of this system. We consider global and local versions of this inverse problem and propose some approaches, including the use of the inverse conductivity problem and the structure of fundamental solutions. The global approach looks for properties of the ground in the whole domain, while the local one is aimed at recovery of these properties near wells. We discuss the use of the model proposed by Muskat which is a difficult free boundary problem. The inverse Muskat problem combines features of inverse elliptic and hyperbolic problems. We analyse its linearisation about a simple solution and show uniqueness and exponential instability for the linearisation.