Abstract

An automatic history-matching algorithm has been developed to determine relative permeabilities and capillary pressure curves simultaneously frog the production history of a displacement test of oil and water through a core. Relative permeability and capillary pressure curves are represented by power functions. Those representations contain sole parameters which are found by minimizing an objective function. The objective function is ford by the sum of the square of the differences between experimentally measured and numerically simulated production data. The numerical simulator is an IMPES finite-difference program which models the one dimensional two-phase flow. The program which models the one dimensional two-phase flow. The analyzation is performed by a non-linear regression algorithm: the Quasi-Newton Approximation for the Least-Squares Problem Broyden- Fletcher-Goldfarb-Shanno formulae).The automatic history matching algorithm is applied to analyze the influence of capillary pressure on the determination of relative permeability curves. The algorithm is tested with simulated data and with actual experimental data. Database consisting of simulated experiments with random errors are tested to demonstrate the feasibility of the method. Besides, they provide exact relative permeability curves for comparison purposes. In effect, results obtained by using simulated data show the convergence and uniqueness of the algorithm. But the main observation is that relative permeabilities estimates can be in error when capillary pressure terms are neglected. Several sets of actual experimental data at inlet constant pressure are also tested. convergent solutions are always found. pressure are also tested. convergent solutions are always found. Three different objective functions are applied. They are formed by oil rates, total fluid rates or a combination of both. The solution is dependent on the chosen objective function. Best adjustments are found by applying a combination of oil and total fluid rates, and including capillary pressure.

Introduction

Them measurement of relative permeabilities fall into two basic categories: the steady-state (Penn State) technique or the unsteady state (Welge) technique. The steady-state technique is the east direct, but it Is expensive and time-consuming. On the other hand, the unsteady-state experiment involves the displacement of oil by water through a sample of reservoir-rock initially saturated with oil and connate water. This technique is much quicker and less expensive than the steady-state procedure and in consequence is customarily performed - but the relative-permeability curves have to be inferred indirectly frog pressure drop and fluid production data measured during the test. In fact, these curves are calculated by matching the production data with results of a mathematical model of the two-phase flow. Traditionally, the matching is performed with, graphical methods based on the Buckley-Leverett model - i.e., JBN or JR methods. These procedures are not adequate for heterogeneous cores, or for low flow rate experiments in which capillary pressure terms cannot be disregarded. Furthermore they require the graphical or numerical differentiation of measured data, and this process of differentiation amplifies measurement errors. Automatic history matching techniques have been put forward to overcome these limitations. An automatic adjustment algorithm is composed of a numerical simulator of the unsteady one dimensional biphasic flow of oil and water through the core functional representation of the relative permeability curves in terms of a set of adjustable parameters; and a least squares criterion to minimize the differences between measured and calculated production history by varying the adjustable parameters. parameters. In those regression based techniques relative permeabilities have been estimated taking into account capillary pressure effects or neglecting them.

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