Citation
Shah, Piyush Chimanlal (1977) Estimation of properties in petroleum reservoirs. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd12042006081055
Abstract
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 onedimensional singlephase 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 GaussNewton (or Marquardt's) algorithms for history matching is investigated, and the numerical effort for zonation and Bayesian estimation in one and twodimensional 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 onedimensional 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 pseudoinverses 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.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Aeronautics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1 October 1976 
Record Number:  CaltechETD:etd12042006081055 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd12042006081055 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  4774 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  20 Dec 2006 
Last Modified:  26 Dec 2012 03:11 
Thesis Files

PDF (Shah_pc_1977.pdf)
 Final Version
See Usage Policy. 10Mb 
Repository Staff Only: item control page