Confidence Limits on the Parameters and Predictions of Slightly Compressible, Single-Phase Reservoirs

Abstract

Methods of nonlinear regression theory were applied to the reservoir history-matching problem to determine the effect of erroneous problem to determine the effect of erroneous parameter estimates obtained from well testing parameter estimates obtained from well testing on the future prediction of reservoir pressures. Two examples were studied: well testing in a radial one-dimensional slightly compressible reservoir and in an undersaturated, two-dimensional, heterogeneous oil field. The reservoir parameters of permeability, porosity, external radius, and pore volume were considered, and the effects of pore volume were considered, and the effects of measurement error, test time, and flow rate on the confidence limits were computed. Introduction The operation of a reservoir simulator requires accurate estimates of the reservoir properties. However, the simulation parameters, such as permeability, porosity, and reservoir geometry, are permeability, porosity, and reservoir geometry, are usually unknown unless coring and physical property analysis have been undertaken. Because of the cost of these procedures, it is more desirable to use the pressures measured at the well during a well test pressures measured at the well during a well test and indirectly compute the important parameters of the system. By using history matching of the test data to obtain the system parameters, the future pressure behavior of the reservoir can be predicted pressure behavior of the reservoir can be predictedSeveral studies on history matching have indicated that the welltest approach for determining the reservoir parameters often suffers from incorrect and nonunique parameter estimates. The factors that affect the parameter estimation can be classified as model errors, observability, measurement errors or noise, history time, test procedure, and optimization procedure. Model errors arise from the inaccuracy of the model and the numerical integration. For example, a reservoir simulator is only a reasonable approximation for flow through porous media. Solution of a model equation by numerical means also introduces roundoff and discretization errors. Observability of the system plays an important role in estimating the reservoir parameters. Depending on the location of the well and the number of data points, it may not be possible to determine uniquely all reservoir parameters from the measurements made at that well. Observability is strictly a function of the reservoir model used. At a given well, pressure measurements may only reflect the values of the parameters in specific zones of the reservoir. If a specific zone away from the well does not affect the measured pressure, then the system is not observable at that particular location. A rigorous definition of observability can be found in other papers. Measurement errors in the pressures and flow rates are another source of unrealistic parameter estimates. Longer history times always give more information about the reservoir as long as the system remains in a dynamic state. The nature of the system input (well flow rate) also affects the accuracy of the estimates and predictions. The final source of incorrect parameter estimates arises because the history-matching problem, posed mathematically, is usually a nonlinear programming problem that must be solved computationally. Such problem that must be solved computationally. Such a problem yields multiple extrema that often can lead to a relative minimum (rather than a global minimum) in the numerical search for the smallest matching error. Also, the magnitude of the objective function can be quite insensitive to the parameters selected, thus causing the optimization procedure to terminate prematurely. The above factors control the history-matching process; with actual data, it is usually impossible process; with actual data, it is usually impossible to identify the exact contributions of each factor to the errors in the parameter estimates. Since a certain amount of error will be introduced into the estimated parameters from the history-matching process, it is parameters from the history-matching process, it is useful to study the magnitude of this error resulting from various sources under controlled simulation conditions. Also, it is important to determine how the errors in the parameters are reflected in the future predictions of the pressures. SPEJ P. 42

Lumped-parameter models for low-temperature geothermal fields and their application

“…The unknown parameters in nonlinear regression are constrained by using the so-called imaging method of Carvalho et al (1996). In addition, we compute 95% confidence intervals, standard deviation of fit, and correlation coefficients using standard definitions (Dogru et al, 1977). Calculation and inspection of these regression analysis statistics can help in identifying parameters that can be reliably determined from the available data set.…”

“…The statistical standard deviation of fit (or RMS) and confidence intervals are well-known tools for obtaining quantitative evaluations in model discrimination and uncertainty assessments in estimated parameters (Dogru et al, 1977;Anraku and Horne, 1995). The RMS value is an absolute measure of goodness of fit as well as of a statistical standard deviation of the fit, and thus gives an idea of model performance.…”

“…In this way we are able to characterize or appraise the uncertainty in the predicted performance, and all information about the reservoir model, reservoir parameters, and observed data are expressed in terms of probability distributions. Any uncertainties in estimated parameters and predictions are then assessed by sampling the overall probability distribution of the observed data as derived from history matching (Bard, 1974;Dogru et al, 1977). Note that this sampling provides multiple realizations of the predicted performance for a given production scenario.…”

Well reservoir model identification using translation and scale invariant higher order networks

“…r rD = --(3) fw (0.00o003 57ktd 1OgTD = logC2 + log(dT) (4) where r is is the distance of the point from the well bore (pipe) base, P is the pressure at that point, rw is the well bore radius, k = permeability, TD = dimensionless time, dT = time unit, dP = pressure change in time dT, q = flow rate, ~z = viscosity, + = porosity, ~ = compressibility and Bo = formation volume factor.…”

Estimation of Radial Permeability Distribution From Well-Test Data

“…Previous methods, however, have used finite-difference or finite-element methods to reduce the parameter dimension from infinitedimensional to a relatively small dimension that would give unique or stable results when an inverse calculation is attempted. [1][2][3] In this paper, I show that it is possible to retain the functional flavor of the permeability field by using the Backus and Gilbert 4 ,5 approach to solving the inverse problem.…”