Automatic History Matching in Petroleum Reservoirs Using the TSVD Method
History matching is an important inverse problem extensively used to estimate petrophysical properties of an oil reservoir by matching a numerical simulation to the reservoir’s history of oil production. In this work, we present a method for the resolution of a history matching problem that aims to estimate the permeability field of a reservoir using the pressure and the flow rate observed in the wells. The reservoir simulation is based on a two-phase incompressible flow model. The method combines the truncated singular value decomposition (TSVD) and the Gauss-Newton algorithms. The number of parameters to estimate depends on how many gridblocks are used to discretize the reservoir. In general, this number is large and the inverse problem is ill-posed. The TSVD method regularizes the problem and decreases considerably the computational effort necessary to solve it. To compute the TSVD we used the Lanczos method combined with numerical implementations of the derivative and of the adjoint formulation of the problem.
KeywordsReservoir simulation History Matching Optimization TSVD Adjoint formulation
Unable to display preview. Download preview PDF.
- 1.Tavakoli, R., Reynolds, A.C.: History Matching With Parameterization Based on the SVD of a Dimensionless Sensitivity Matrix. SPE J. SPE-118952-PA (2009)Google Scholar
- 2.Rodrigues, J.R.P.: Calculating derivatives for history matching in reservoir simulators. In: SPE Reservoir Simulation Symposium (2005)Google Scholar
- 6.Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc users manual. Technical Report ANL-95/11 - Revision 2.1.5, Argonne National Laboratory (2002)Google Scholar
- 7.Komzsik, L.: The Lanczos Method, Evolution and Application. Society for Industrial Mathematics, Philadelphia (1987)Google Scholar
- 8.Englezos, P., Kalogerakis, N.: Applied Parameter Estimation for Chemical Engineers. Marcel Dekker, New York (2001)Google Scholar
- 9.Bjork, A.: Numerical Methods for Least Square Problems. Society for Industrial and Applied Mathematics, Philadelphia (1996)Google Scholar
- 10.Hansen, P.C.: Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. Society for Industrial and Applied Mathematics, Philadelphia (1998)Google Scholar