History-matching problems, in which reservoir
An algorithm is developed to estimate absolute permeability in multi-phase, multilayered petroleum reservoirs. Summary A numerical algorithm is developed to estimate absolute permeability in multi-phase, multilayered petroleum reservoirs 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. A Chevron Black-Oil code 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 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 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. Introduction Estimation of the properties of multilayered, multiphase reservoirs remains an important problem in reservoir analysis. The knowledge of these properties, such as absolute permeability, forms the basis for determining an optimal strategy of oil recovery. The process of estimating unknown properties in a mathematical process of estimating unknown properties in a mathematical reservoir model to give the best fit to measure well data is called history matching. The history matching problem is a notoriously difficult one for several reasons:(1)In a reservoir. properties vary with location; thus, conceptually an infinite number of parameters are required for a full description of the reservoir. Computationally, a reservoir simulator contains a finite number of parameters corresponding to the number of grid blocks, and in field-scale simulations, a simulator may contain on the order of 10,000 grid blocks.(2)The history matching problem is theoretically imposed, which means that small instabilities in the data can lead to large perturbations in the estimated parameters.(3)Many actual perturbations in the estimated parameters.(3)Many actual reservoirs involve significant vertical as well as horizontal property variations, requiring the estimation of property property variations, requiring the estimation of property distributions in both directions.(3)History matching situations may involve full threephase (oil, water and gas) behavior, wherein traditional observations at wells, such as pressures and water-oil ratio, may not be adequate to enable property estimation; the sorts of observation data necessary have not been addressed previously in the literature. There exist a number of prior papers on automatic history matching, which have been organized according to reservoir dimensionality and number of phases considered in Table 1. The goal of the present work is to develop a general history matching code applicable for three-dimensional, three-phase reservoirs. To alleviate the ill-conditioning in the history matching problem, several approaches have been tried, such as decreasing problem, several approaches have been tried, such as decreasing the number of parameters to be estimated and, in addition, utilizing any available information to constrain the choice of the unknown parameters. One way of reducing the number of parameters is to parameters. One way of reducing the number of parameters is to divide the reservoir into a relatively small number of zones and to assume that the properties are uniform within each zone. While this approach is effective in reducing the number of unknowns, sufficient a priori information is not usually available to enable specification of the zones on any physical basis. A modification to zonation is to use prior information expressed as an assumed probability distribution for the zonal reservoir properties. If probability distribution for the zonal reservoir properties. If certain a priori knowledge is assumed about the mean values and correlations of the parameters, the history matching performance index can be modified to include a term that penalizes the weighted deviations of the parameters from their assumed mean values. Although it has been shown that better-conditioned estimates may be obtained when a priori statistical information is used, sufficient knowledge of the nature of the unknown parameters is not generally available to specify the parameters needed to carry out such a Bayesian estimation.
This paper applies material presented by Chen et al. and by Chavent et al to practical reservoir problems. The pressure history-matching algorithm used is initially based on a discretized single-phase reservoir model. Multiphase effects are approximately treated in the single-phase model by multiplying the transmissibility and storage terms by saturation-dependent terms that are obtained from a multiphase simulator run. Thus, all the history matching is performed by a "pseduo" single-phase model. The multiplicative factors for transmissibility and storage are updated when necessary. The matching technique can change any model permeability thickness or porosity thickness value. Three field examples are given. Introduction History matching using optimal-control theory was introduced by two sets of authors. Their contributions were a major breakthrough in attacking the long-standing goal of automatic history matching. This paper extends the work presented by Chen et al. and Chavent et al. Specifically, we focus on three areas.We derive the optimal-control algorithm using a discrete formulation. Our reservoir simulator, which is a set of ordinary differential equations, is adjoined to the function to be minimized. The first variation is taken to yield equations for computing Lagrange multipliers. These Lagrange multipliers are then used for computing a gradient vector. The discrete formulation keeps the adjoint equations consistent with the reservoir simulator.We include the effects of saturation change in history-matching pressures. We do this in a fashion that circumvents the need for developing a full multiphase optimal-control code.We show detailed results of the application of the optimal-control algorithm to three field examples. DERIVATION OF ADJOINT EQUATIONS Most implicit-pressure/explicit-saturation-type, finite-difference reservoir simulators perform two calculation stages for each time step. The first stage involves solving an "expansivity equation" for pressure. The expansivity equation is obtained by summing the material-balance equations for oil, gas, and water flow. Once the pressures are implicitly obtained from the expansivity equation, the phase saturations can be updated using their respective balance equations. A typical expansivity equation is shown in Appendix B, Eq. B-1. When we write the reservoir simulation equations as partial differential equations, we assume that the parameters to be estimated are continuous functions of position. The partial-differential-equation formulation is partial-differential-equation formulation is generally termed a distributed-parameter system. However, upon solving these partial differential equations, the model is discretized so that the partial differential equations are replaced by partial differential equations are replaced by sets of ordinary differential equations, and the parameters that were continuous functions of parameters that were continuous functions of position become specific values. Eq. B-1 is a position become specific values. Eq. B-1 is a set of ordinary differential equations that reflects lumping of parameters. Each cell has three associated parameters: a right-side permeability thickness, a bottom permeability thickness, and a pore volume. pore volume.Once the discretized model is written and we have one or more ordinary differential equations per cell, we can then adjoin these differential equations to the integral to be minimized by using one Lagrange multiplier per differential equation. The ordinary differential equations for the Lagrange multipliers are now derived as part of the necessary conditions for stationariness of the augmented objective function. These ordinary differential equations are termed the adjoint system of equations. A detailed example of the procedure discussed in this paragraph is given in Appendix A. The ordinary-differential-equation formulation of the optimal-control algorithm is more appropriate for use with reservoir simulators than the partial-differential-equation derivation found in partial-differential-equation derivation found in Refs. 1 and 2. SPEJ P. 347
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