Estimation of properties in petroleum reservoirs

The determination of parameters in a dynamical system, on the basis of noisy observations of its state is variously known as parameter estimation, identification or the inverse problem. In this work, the determination of porous rock property distribution in a petroleum reservoir using the production rate records and observed pressures (the history matching problem) is considered. The history matching problem is inherently underdetermined because of the large number of unknown parameters relative to the available data. The number of unknowns can be reduced by representing the distributions by a small number of parameters (parameterization). The commonly used zonation approach involves a parameterization, but introduces a considerable modeling error. In chapter 1, Bayesian estimation theory is extended to history matching as an alternative to zonation; it is sought to alleviate the underdeterminacy through specification of a priori statistical information about the unknown parameters. Application of Bayesian estimation and zonation to the problem of porosity and permeability estimation in a one-dimensional single-phase reservoir indicates that the former yields superior estimates; this holds true even when the prior statistics involve large errors. The application of the conjugate gradient and the Gauss-Newton (or Marquardt's) algorithms for history matching is investigated, and the numerical effort for zonation and Bayesian estimation in one- and two-dimensional reservoirs is estimated in detail. In chapter 2, analytic expressions are derived for the sensitivities of an observed oil pressure to small, arbitrary changes in the porosity and permeability distributions in a one-dimensional reservoir. The results indicate that highly oscillatory components of either have very small influence on the pressure and thus cannot be determined by history matching. Further, the dependence of all the observed pressures on the unknown parameters is linearized, for small deviation, about two reference property distributions. The linear relation is analyzed to yield quantitative information concerning the statistical properties of the problem. Iterative corrections in the history matching algorithms are identified with various pseudo-inverses of the linear relation, thus explaining the properties of the resulting estimates. The nature of the linear relation is found to be not strongly dependent on the reference property distributions used for linearization; thus such analysis can be performed prior to estimation. It is discussed how the linearized analysis can be used to determine the determinacy of any given parameterization. The information derived from the linearized analysis and that in the a priori statistics is synthesized in chapter 3 to predict covariances for the zonation and Bayesian estimates. Since the results of the linearized analysis depend only weakly on the reference distribution, the predicted covariances are valid for a class of reservoirs having "true" property distributions with identical prior statistics. A good agreement is found when the predicted variances are compared with actual mean square estimate errors in simulations with four distributions with given prior statistics. The sensitivity of the estimates and their covariance to changes and errors in the specification of the prior statistics are investigated in considerable detail. The determination of zonation with smallest trace of estimate covariance for a given problem is considered. The design of Marquardt's algorithm to yield the smallest expected total estimate error for a given zonation is discussed