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Title: The finite element analysis of multiphase flow through porous media
Author: Johnson, K. H.
Awarding Body: University College of Swansea
Current Institution: Swansea University
Date of Award: 1982
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This thesis reports the development of a finite-element simulator, 7DeN, for multiphase flows of compressible fluids through arbitrarily-shaped porous media. The equations express mass continuity in each phase and incorporate Darcy's law; the variables are pressure in one phase and saturations of the others. Up to three phases may be specified but this is extendible. Spatial discretization is by the Galerkin procedure, general enough to admit a wide variety of elements: one-, two-, and three-dimensional, of almost any order, and integrated by any Gaussian-type rule. The shape functions, calculated only at the integration points, are read off a data file and combined through an isoparametric transformation into a matrix integration; the derived coefficients are stored to avoid recalculation of the Jacobian. The capacity (C) matrix may be partially or completely lumped onto its diagonal. Time-stepping uses a Crank-Nicolson procedure whereby the nonlinear matrices are evaluated with variables interpolated to some midtime level: this may be varied between a fully-forward and a fully-backward difference procedure. Nonlinearities are handled by substitution in a predictor-corrector iteration scheme; an option is included for extrapolation of the zeroth iterative estimate. The resulting asymmetric system is assembled and solved using a front solver. Boundary conditions include set values for nodal unknowns (Dirichlet), nodal flares (Neumann), and a non-linear mobility condition where the total nodal volumetric flux is apportioned according to the fraction of total mobility that phase contributes. Flows required to satisfy Dirichlet conditions are evaluated. Parametric functions (e.g. relative permeabilities, capillary pressure, densities, and viscosities) are input as coefficients for straight-line segments. Standard FORTRAN is used to allow easy conversion to any computer system: to date the program has been implemented on a CDC7600 and an ICL 1904S. Much of the work included involves an extensive series of tests on the Buckley-Leverett problem-a 2-phase, 1D case with zero capillary pressure. The celerity of the saturation shock front which develops in this problem was measured as it passed through the mesh and it was found that this converges quadratically, as At/Ax is reduced, to a value different from that predicted by the Buckley-Leverett analysis. It is conjectured that this is due to there being a near-discontinuity which is preferred by the numerical model since it in some "best" way satisfies the partial differential equations among the continuous solutions available to the model. This waveform depends on the mobilities within the shock and does not in general satisfy the entropy condition which characterizes the Buckley-Leverett solution. Methods used previously to overcome this difficulty, namely the addition of diffusion by upstream differencing or by an artificial capillary pressure gradient and a modification of the mobilities, are considered and their potential uses and drawbacks evaluated. Oscillations which occur near the shock with a consistent (unlumped) C are explained as a feature of the least-squares best fit to the shock. A five-spot problem is also attempted to show the applicability of the program and these interpretations to two-dimensional problems. The results of other tests are briefly described demonstrating features such as compressibility, three-phase flows, three-dimensionality, and axisymmetry.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available