| A numerical algorithm is developed to estimate absolute permeability in multi-phase, multilayered petroleum reservoirs based upon noisy observation data, such as pressure, water cut, gas-oil ratio and rates of liquid and gas production from individual layers. The industrial Black-Oil code, CLASS, (Chevron Limited Application Simulation System), is used as the basic reservoir simulator in conjunction with this history matching algorithm. Since the history matching inverse problem is ill-posed due to its large dimensionality and the insensitivity of the permeability to measured well data, regularization and spline approximation of the spatially varying absolute permeability are employed to render the problem computationally well behaved. A stabilizing functional with a gradient operator is used to measure the non-smoothness of the parameter estimates in the regularization approach, and the regularization parameter is determined automatically during the computation. The numerical minimization algorithm is based on the partial conjugate gradient method of Nazareth. Numerical examples are considered in two- and three-phase reservoirs with three layers. The effects of the degree of regularization, spline approximation versus zonation, and differing true areal permeability distributions on the performance of the method are considered. |
