Petroleum Reservoir Simulation

The purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem. Parameter identification methods are classified under the error criterion used in the formulation of the inverse problem. The problem of ill-posedness in connection with the inverse problem is addressed. Typical inverse solution techniques are highlighted. The review also includes the evaluation of methods used for computing the sensitivity matrix. Statistics which can be used to estimate the parameter uncertainty are outlined. Attempts have been made to compare and contrast representative inverse procedures, and direction for future research is suggested.

Review of Parameter Identification Procedures in Groundwater Hydrology: The Inverse Problem

History-matching problems, in which reservoir parameters arc to be estimated from well pressure data, are formulated as optimal control problems. The necessary conditions for optimality lead naturally to gradient optimization methods for determining the optimal parameter estimates. The key feature of the approach is that reservoir properties are considered as continuous functions
of position rather than as uniform in a certain number of zones. The optimal control approach is illustrated on a hypothetical reservoir and on an actual Saudi Arabian reservoir, both characterized by single-phase flow. A significant saving in computing time over conventional constant-zone gradient optimization methods is demonstrated.

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A New Algorithm for Automatic History Matching

Owing to the deterministic nature of most groundwater flow models there has been a tendency in the past to overlook the strong element of uncertainty that invariably enters into the problem of parameter identification. It is shown that because of this uncertainty an approach based on the minimization of a single error functional does not in general lead to satisfactory results. A multiple-objective decision process is postulated taking into account all the available information on the aquifer flow system as well as the range of environmental conditions under which the system is expected to operate in the future. According to this new approach a continuous or discrete set of alternative solutions to the identification problem is generated with the aid of mathematical programing techniques, and the decision maker is asked to apply his own value judgment in selecting a particular model structure. The method is illustrated by applying parametric linear programing to a finite element model of steady state flow in a locally anisotropic aquifer. The reliability of each parameter estimate is ascertained with the aid of a postoptimal sensitivity analysis.

Calibration of distributed parameter groundwater flow models viewed as a multiple‐objective decision process under uncertainty

A new statistically based approach to the problem of estimating spatially varying aquifer transmissivities on the basis of steady state water level data is presented. The method involves solving a family of generalized nonlinear regression problems and then selecting one particular solution from this family by means of a comparative analysis of residuals. A linearized error analysis of the solution is included. This analysis allows one to estimate the covariance of the transmissivity estimates as well as the square error of the estimates of hydraulic head. In addition to the explicitly statistical orientation of the method, it has an additional feature of permitting the user to incorporate a priori information about the transmissivities. This information may be based on actual field data such as pumping tests, or on statistical data accumulated from similar aquifers elsewhere in the world. A highly efficient explicit numerical scheme for solving the inverse problem in an approximate manner when errors in water level data are sufficiently small is also described. When these errors are large, the explicit scheme may still be useful for obtaining a rapid initial idea about the approximate location of the optimum solution. Paper 1 presents the theory and illustrates it by a theoretical example. The purpose of this example is to demonstrate the effectiveness of our method in dealing with noisy data obtained from a known model. Application of the method to real data will be described in paper 2.

A statistical approach to the inverse problem of aquifer hydrology: 1. Theory

A New Technique for Determining Reservoir Description from Field Performance Data

An Efficient Model for Evaluating Gas Field Gathering System Design

The recovery of cushion gas upon ultimate depletion of an aquifer storage reservoir is dependent upon reservoir heterogeneity, aquifer strength, production rate and fluid and rock properties such as density, viscosity, relative permeability, capillary pressure, and residual gas saturation. This study illustrates the use of multi-dimensional, 2-phase, compressible fluid flow calculations to simulate the depletion. Results which illustrate the non-exhaustive examination of the effects of heterogeneity, aquifer strength, and gas production rate are presented. The study indicates a strong dependence of recovery upon reservoir heterogeneity. The multi- dimensional type of calculation employed appears necessary to reliably estimate recoverable cushion gas for any particular reservoir.

Calculation of Gas Recovery Upon Ultimate Depletion of Aquifer Storage

Use of Numerical Models To Develop and Operate Gas Storage Reservoirs

INTRODUCTION This paper describes a numerical Many gas wells exhibit pressure model for analyzing gas-well tests test behavior which is difficult if and predicting long-term deliverability. not impossible to interpret using Field applications presented include conventional methods of analysis. an interpretation of a gas well test Difficulty of interpretation is frein a tight sand leading to an accurate quently encountered in low permeability long-term deliverability projection. reservoirs and in layered reservoirs The model presented numerically simuwith limited or incomplete crossflow. lates two-dimensional (r-z) gas flow In these cases, assumptions in conand accounts for effects of turbulence, ventional analysis methods, such as skin, afterflow, partial penetration, complete (or no) crossflow and pressure-dependent permeability and negligible effects of afterflow or any degree of crossflow ranging from interlayer recirculation through the complete to none. Through a novel wellbore, are frequently invalid. treatment of the equations describing reservoir flow, skin and afterflow, This paper describes a numerical the model simulates shutin at the wellmodel which accounts for many factors head and then calculates afterflow and which are neglected in conventional any subsequent circulation of gas methods of analysis. The model through the wellbore from some layers numerically simulates two-dimensional to others. (r-z), transient gas fiow in a cyiinder References and illustrations at end representing the drainage volume of a of paper. single well. The calculations account 2 ANALYSIS AND PREDICTION for effects of turbulence, skin, afterflow, partial penetration, pressuredependent permeability and any degree of crossflow ranging from complete to none. Equations describing gas flow in the reservoir, skin effect and afterflow are combined in a manner which allows simulation of shutin at wellhead rather than bottomhole; the model calculates afterflow and any recirculation of gas through the wellbore from some layers to others. Thus, the calculated results show the effects of afterflow and recirculation on shape of the pressure buildup curve. Field applications presented illustrate use of the model to predict the long-term flow characteristics of gas “w”ells prim to ~~hnse+inn tQ Z3 ““....-----pipeline. The wells selected for illustration have been tested with both short and long-term tests to indicate the reliability of the method. An additional field application shows use of the model to explain and reproduce long-term (up to 600 days) gas well buildups. The method presented is equally applicable to simulation of oil well tests and performance and the slightly modified equations for that case are given in the Appendix. BRIEF DESCRIPTION OF THE MODEL Equations comprising the model are described in detail in the Appendix. Only a brief outline of the method is presented here. The basic equation of the model is eq. (1) describing transient, two-dimensional (r-z) gas flow in a cylindrical drainage volume*:

J.R. Dempsey

Papers

A New Technique for Determining Reservoir Description from Field Performance Data

An Efficient Model for Evaluating Gas Field Gathering System Design

Calculation of Gas Recovery Upon Ultimate Depletion of Aquifer Storage

Use of Numerical Models To Develop and Operate Gas Storage Reservoirs

Analysis and Prediction of Gas Well Performance