Abstract
The estimation of reservoir properties is inherently an underdetermined problem (one having a nonunique solution) because of the large number of unknown parameters relative to the available data. parameters relative to the available data. The common zonation approach to reducing the number of parameters introduces considerable modeling error by insisting that reservoir properties are uniform within each zone and by assigning the boundaries of these zones more or less arbitrarily. In this paper, Bayesian estimation theory is applied to history matching as an alternative to zonation. By using a priori statistical information on the unknown parameters, the problem becomes statistically better determined. Bayesian estimation and zonation are applied to the problem of porosity and permeability estimation in a one-dimensional, permeability estimation in a one-dimensional, one-phase reservoir.
Introduction
The estimation of parameters such as porosity and permeability in a reservoir model using well production and pressure data is commonly referred production and pressure data is commonly referred to as history matching. Although an inhomogeneous reservoir is in principle specified by an infinite number of parameters, a computational reservoir model can only contain a finite number. The most detailed description is obtained by allowing porosity and permeability to vary independently at each block of the spatial grid used in the finite-difference solution. While minimizing the modeling error, this approach entails a great deal of uncertainty because of the large number of unknowns compared with the limited data available. Thus, in a given problem, many different sets of property estimates problem, many different sets of property estimates may provide satisfactory and essentially indistinquishable data fits. Some of these parameter estimates can be grossly in error with respect to the actual properties, and as a result can lead to erroneous prediction of future reservoir behavior. To reduce the statistical uncertainty one must either decrease the number of unknowns or utilize additional information.
A commonly used procedure for reducing the number of unknown parameters is zonation; the reservoir is divided into a small number of zones, in each of which the properties are treated as uniform. A modeling error is thus introduced through the assumption of uniform properties within each zone and through the more or less arbitrary assignment of the zone boundaries. As the number of zones is decreased, the error due to statistical uncertainty decreases while the modeling error increases. The total error passes through a minimum at some intermediate number of zones. The specification of this optimum level of description, which has been briefly considered in past work, will be treated in detail in a future report.
An alternative to decreasing the statistical uncertainty by reducing the number of unknown parameters is the utilization of additional parameters is the utilization of additional information. This information need not be limited to measurements on the reservoir under study, but can be based on prior geological information about property variability in reservoirs of the same type. property variability in reservoirs of the same type. This paper examines this alternative method of reducing statistical uncertainty. The prior geological information is utilized by a formulation akin to classical Bayesian estimation. The Bayesian estimation is illustrated and is compared with the zonation approach for the case of a hypothetical, one-dimensional reservoir with variable porosity and permeability.
The numerical simulations are used to investigate questions such as the optimum number of parameters in zonation and the effect of erroneous prior statistics in Bayesian estimation, and to compare the two methods. Considerable attention is also given to computational aspects such as convergence rate and computer time required by two of the most commonly used minimization algorithms, Marquardt's and the conjugate gradient.
NATURE OF PRIOR GEOLOGICAL INFORMATION
The application of probabilistic models in geology is the subject of a recent review.
SPEJ
P. 337
A Study on the Bayesian Estimation of Unknown Parameters of a Nonlinear Controlled System
