By Myron B. III. Allen
This research is an outgrowth of my doctoral dissertation at Princeton college. i'm quite thankful to Professors George F. Pinder and William G. grey of Princeton for his or her suggestion in the course of either my learn and my writing. i think that finite-element collocation holds promise as a numer ical scheme for modeling complex flows in porous media. besides the fact that, there seems a "conventional knowledge" holding that collocation is hopelessly beset by way of oscillations and is, not directly, essentially beside the point for multiphase flows. i am hoping to dispel those objections, figuring out that others will stay for additional paintings. The U. S. nationwide technological know-how origin funded a lot of this research via supply quantity NSF-CEE-8111240. desk OF CONTENTS summary ;; FOREWORD ;; ; bankruptcy ONE. THE actual method. 1.1 advent. 1 1.2 The reservoir and its contents. five 1.3 Reservoir mechanics. nine 1.4 Supplementary constraints. 18 1.5 Governing equations. 26 bankruptcy . REPRESENTING FLUID-PHASE habit. 39 2.1 Thermodynamics of the fluid procedure. forty 2.2 usual equation-of-state equipment. forty five 2.3 Maxwell-set interpolation.
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Additional resources for Collocation Techniques for Modeling Compositional Flows in Oil Reservoirs
WN• p ) are unknown. 2-9a) for i = 1, ... 2-9b) Solving this system numerically requires an iterative method. The Newton-Raphson method is attractive except for the unwieldy computations needed to evaluate the derivatives of f~ appearing in the Jacobian 1 matrix. 2-9) using a quasi-Newton or secant method based on the use of finite differences. 2-9) by R(t) = 0, where R: RN+ 1 ~ RN+ 1 is the nonlinear function whose roots we seek. · ~. , ~ k J .. (t ) 1J '" e. being the j-th unit basis vector.
5-17) are independent. 5-10). ), 1 1 i = 1, ... 5-18) and discontinuities in w. travel at the speed 1 U r. 1 = qp-1 [ fi ]/[ Wi ], i = 1, ... 5-19) 35 By definition, a given composition (w 1 ' ... ,w N_1 ) propagates as a coherent wave provided all of the variables wi at a given time and place (x,t) in the reservoir move with the same speed; in symbols, i = 1, ... 5-20a) Similarly, for a discontinuity to be coherent, we must have = A, i = 1, ... 5-20b) Consider the case when N = 3. A dw. 5-20) become q pdf.
Thus, while solving the compositional equations numerically is a problem of considerable practical importance, it is also a task lacking somewhat in mathematically rigorous support. 5-3) to heuristic remarks based on analogies with the simpler cases. The formation and persistence of steep composition gradients occur in problems where the dissipative influences of species dispersion and capillary pressure gradients are dominated by convection owing to applied pressure gradients. practice. Such conditions are common in oilfield When dispersion and capillarity are absent the governing equations take the form of hyperbolic conservation laws, and, as the simplified analyses show, we should expect discontinuities to form.