| Abstract: | Splines have been applied to a variety of engineering applications. They used to be defined via piecewise polynomials. It is now recognized that a spline function can be more efficiently represented as a linear combination of a finite number of linearly independent spline elements called B-splines. They are numerically stable and versatile to use in engineering computations. In this study, B-splines were used to construct approximating functions of inexact measurements and to develop a reservoir simulator for the immiscible displacement. Physical properties of engineering importance are frequently estimated from experimental measurements. These measurements, as usual, are scattered and subject to measurement error. For purposes of communications and calculations, it is desirable to have a functional representation of these inexact discrete measurements. Spline is a natural choice for function approximation. With the use of B -splines, the accuracy of spline approximation depends mainly on the selection of knots. We have developed a systematic procedure for the selection of knots. For a given set of measurements, this algorithm automatically chooses the number of knots as well as their locations used to calculate spline estimates. These knots are chosen to minimize a nonlinear least-squares objective function. This new algorithm is robust; trial and error is no longer needed. It always determines the (least-squares) optimum fits. If the number of knots is specified and fixed during calculations, a global minimum fit is expected. In many data-fitting exercises, it is desired that the fitted curve retains certain features that may be attributed to the measured response function. Two situations com m only encountered are that the response function may be known to be monotonic increasing (or decreasing) or convex upwards (or downwards). Our algorithm is capable of preserving these desired shape features. It is carried out by solving a constrained optimization problem. At times people may need to predict or simulate the response of a physical process under assumed conditions. Reservoir simulation, for example, has long been used to predict and evaluate reservoir performance under alternative recovery schemes. Here, we considered the problem of simulating the immiscible displacement occurring in a petroleum reservoir. B-splines were used as the unifying bases for this new simulator. The versatile properties of B-splines makes it possible to specify a variety of basis functions frequently applied in reservoir simulations with the same spline routines. This feature allows our simulator to represent a broad class of problem s. A new collocation approximation was used to develop this simulator; it is accurate and always maintains a material balance. For convenience, the following discussions will be divided into two parts. In the first part, the problem of fitting inexact measurements will be addressed. In the second part, the problem of modeling the immiscible displacement process in an oil reservoir will be discussed. |
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