Sensitivity Coefficients for Three-Phase Flow History Matching
Authors: Li R. , Reynolds A.C. , Oliver D.S. 
Published: March 2010

Abstract
Changing the value of the porosity or the horizontal or vertical permeability in any grid cell in a reservoir simulator by a small amount often results in a small change in the value of a property predicted by the simulator. The ratio of change in prediction to change in reservoir property is called the sensitivity coefficient. In this paper, we describe the use of the adjoint system of equations to compute the sensitivity of wellbore pressure, water-oil ratio, and gas-oil ratio to changes in gridblock permeability and porosity. Unlike some other methods of computing sensitivity coefficients, this method is applicable for problems with large numbers of model parameters and for problems in which cross-flow and compressibility are significant.
Although the adjoint system has previously been used to compute the gradient of an objective function for three-phase flow data with respect to model parameters, the sensitivity coefficients are significantly more powerful as they enable the use of Newton-like methods with quadratic convergence, i.e., estimation of the covariance of model estimates based on the inverse of the Hessian, and they provide insight into the information content of data. We use several three-phase flow examples with solution-gas drive, gas injection, and gravity segregation to illustrate these ideas.

Introduction
In this work, we develop a computationally efficient adjoint procedure for computing sensitivity coefficients for general threedimensional, three-phase flow problems. Although originally introduced for linear problems by Chen et al.(1) and Chavent et al.,(2) the adjoint method has also been applied to multiphase flow history matching problems; see, for example, Wasserman et al.,(3) Lee and Seinfeld(4), Yang and Watson(5), and Makhlouf et al.(6) These authors applied the adjoint method to compute the gradient of the sum of the squares of the production data mismatch terms and minimized the data mismatch function using steepest descent or conjugate gradient algorithms. The steepest descent and conjugate gradient methods tend to converge slowly, however, so Newton-like methods are advantageous when the number of data is small. For example, Wu et al.(7) applied the adjoint method to calculate sensitivity coefficients of wellbore pressure and WOR with respect to model parameters in two-dimensional, two-phase (oil and water) problems and used a Gauss-Newton algorithm to minimize the data mismatch function. Approximately ten iterations were required to obtain convergence for a problem with 1,250 model parameters.
A more common choice for computing sensitivity coefficients for multiphase flow problems is the sensitivity equation(8) or gradient simulator method(9). Tan and Kalogerakis(10) used the sensitivity equations to compute the sensitivity of production data to a relatively few model parameters. Zonation was used in their work as a method for reducing the number of parameters, and correlation between parameters was used to determine which parameters should be combined. They discussed the importance of computation of sensitivities for estimation of covariance of model parameters, and for obtaining the most rapid convergence possible with the Gauss-Newton method.
Tan(11) used the sensitivity equation approach to compute sensitivities for a three-dimensional, three-phase flow problem.