A Lagrangian Approach for Oil Recovery in Two-Phase Porous Media Flow
Bibliographic record
Abstract
Abstract This paper deals with a fully implicit finite difference scheme for the numerical solution of the Lagrangian form of the porous media fractional flow equation. It covers the theoretical background, description of mathematical formulations, basic assumptions in the models, normalization and transformation between Lagrangian and Eulerian formulations, specification of boundary conditions, discretization and solution method. Some numerical examples are described and the results compared favourably with co-current immiscible displacement data. The solution presented is more efficient than the finite element solution method in terms of requiring less matrix computation effort and it is much more stable than the explicit finite difference solution methods that have been presented in the literature. Due to the ease of formulation and use, the simulation algorithm presented can be used to formulate laboratory numerical simulators that can be used routinely for co-current flow numerical studies. Introduction An equation of the form: Equation (1) (Available In Full Paper) is one of the various Lagrangian forms of the fractional flow equation for two-phase, immiscible displacement in porous media. It is Lagrangian because this configuration allows the equation to be solved as a function of saturation and time. The Eulerian configuration enables the equation to be solved as a function of space and time. Bentsen(1) first derived Equation (1), also called the Bentsen equation(2,3). It was subsequently re-derived by Shen and Ruth(2,3) using a different derivation approach. The equation was derived by combining Darcy's equation for two-phase flow with the continuity equation and by transforming the resulting fractional flow Eulerian form to Lagrangian form. A summary of this derivation is presented later in the paper. This equation, together with normalized forms of the displacement (frontal advance) equation, can be solved to produce a numerical description of a two-phase, incompressible, immiscible displacement process in porous media. The main aim of using the Lagrangian approaches is to enable a more computationally efficient scheme and eliminate the need for space discretization as distance is eliminated from the resulting equation. The form presented in Equation (1) is particularly useful because it is very compact. Theoretical Background Douglas et al.(4) presented an Eulerian technique for solving linear waterflooding problems which involved using a transformed saturation variable and the use of a small mobile water saturation ahead of the flood front to enable the base of the flood front to move forward. Fayers and Sheldon(5) obtained solutions to the one-dimensional displacement equation using both the Lagrangian and the Eulerian approaches. McEwen(6) used the method of characteristics for a numerical solution of the linear displacement equation with capillary pressure. Hovanessian and Fayers(7) also solved numerically the equations describing waterflooding experiments that include both gravity and capillary effects. Their method is similar to that of Douglas et al.(4) because it involves the use of the Eulerian form of the fractional flow equation and the transformation of the resulting second-order non-linear partial differential equations. Hovanessian and Fayers(7) solution differs from those of Douglas et al.(4) because it includes the effect of gravity, allows for the calculation of pressure profiles and makes use of a tabular format for the input of relative permeability and capillary pressure values instead of the u
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".