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Unformatted text preview: Conditioning Geostatistical Models to Two-Phase Production Data Zhan Wu, SPE, A.C. Reynolds, SPE, and D.S. Oliver, SPE, U. of Tulsa Summary A discrete adjoint method for generating sensitivity coefficients related to two-phase flow production data is derived. The proce- dure is applied to calculate the sensitivity of wellbore pressure and water-oil ratio to reservoir simulator gridblock permeabilities and porosities. Using these sensitivity coefficients, an efficient form of the Gauss-Newton algorithm is applied to generate maximum a posteriori estimates and realizations of the rock property fields conditioned to a prior geostatistical model and pressure and/or water-oil ratio data obtained under two-phase ~ oil and water ! flow conditions. Introduction To the best of our knowledge Jacquard and Jain 1 presented the first procedure for numerically computing sensitivity coefficients for history matching purposes. They applied their method to the estimation of permeability in a two-dimensional reservoir from pressure data. The procedure was based on an electric circuit ana- log, but later, following the basic ideas of Jacquard and Jain, Carter et al. 2 presented an elegant derivation of a method to com- pute sensitivity coefficients for two-dimensional single-phase flow problems. Carters procedure for single-phase flow has been extended to three-dimensional problems in a computationally efficient way by He et al. 3 For nonlinear problems, e.g., multiphase flow problems, the derivations of Refs. 2 and 3 do not apply. Thus, we are forced to seek other alternatives. One possible choice is the adjoint or optimal control method, introduced independently for the single- phase history-matching problem by Chen et al. 4 and by Chavent et al. 5 ~ For linear single-phase flow problems, Carter et al. 6 have shown that the adjoint method is equivalent to the Carter et al. 2 method. ! Although the adjoint method has been applied to multi- phase flow problems, primarily by Seinfeld and his students ~ see, for example, Lee and Seinfeld, 7 Yang et al. , 8 and Makhlouf et al. 9 ! these implementations have proved to be computationally inefficient. Specifically, the procedure has normally been applied only to compute the gradient of an objective function based on a sum of the squares of the production data mismatch term. Limited to this information, one is forced to minimize the objective func- tion using a steepest descent, variable-metric, or conjugate gradi- ent algorithm, which results in slow convergence. The sensitivity coefficients needed to form the Hessian for Newton or Gauss- Newton iteration are not available from the traditional adjoint sys- tems and, even if they were, the Hessian would typically be ill- conditioned because the history matching problem is generally ill-posed. Because of the computational cost of repeatedly solving the adjoint system for many iterations @ see, for example, Ma- khlouf et al....
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This note was uploaded on 01/24/2011 for the course ERE 284 taught by Professor . during the Spring '10 term at Stanford.
- Spring '10