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Unformatted text preview: Conditioning Geostatistical Models to TwoPhase 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 twophase flow production data is derived. The proce dure is applied to calculate the sensitivity of wellbore pressure and wateroil ratio to reservoir simulator gridblock permeabilities and porosities. Using these sensitivity coefficients, an efficient form of the GaussNewton 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 wateroil ratio data obtained under twophase ~ 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 twodimensional 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 twodimensional singlephase flow problems. Carters procedure for singlephase flow has been extended to threedimensional 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 historymatching problem by Chen et al. 4 and by Chavent et al. 5 ~ For linear singlephase 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, variablemetric, 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 illposed. 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.
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