A Lagrangian Approach for Oil Recovery in Two-Phase Porous Media Flow
Notice bibliographique
Résumé
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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Prédiction machine sur la base complète
Imitation des enseignantsNi prévalence calibrée, ni vérité terrain. Validation humaine à venir. Le volet Gemma est une étiquette directe du modèle pour chaque travail de la base, lue sur la notice réduite au titre. Le volet Codex est un classifieur appris des 10 348 étiquettes directes de Codex et calibré sur les taux pondérés de l'échantillon; les champs sans appui suffisant ne portent aucun appel Codex. Le mode candidate est l'union des deux volets; le consensus est leur intersection. Ces sorties portent le statut machine_predicted_unvalidated et ne sont pas des étiquettes humaines.
Scores du classifieur distillé par catégorie (deux têtes)
| Catégorie | Codex | Gemma |
|---|---|---|
| Métarecherche | 0,000 | 0,001 |
| Méta-épidémiologie (sens strict) | 0,000 | 0,000 |
| Méta-épidémiologie (sens large) | 0,000 | 0,000 |
| Bibliométrie | 0,000 | 0,000 |
| Études des sciences et des technologies | 0,000 | 0,001 |
| Communication savante | 0,000 | 0,001 |
| Science ouverte | 0,001 | 0,001 |
| Intégrité de la recherche | 0,001 | 0,001 |
| Charge utile insuffisante (le modèle a refusé de juger) | 0,002 | 0,000 |
Scores machine (provisoires)
Les deux têtes enseignantes du modèle étudiant, lues sur ce travail. Un score ordonne la base pour la relecture; il n'affirme jamais une catégorie, et le statut de validation accompagne chaque rangée tel quel.
Scores de référence d'un modèle non mature (critères de maturité non atteints, 7 itérations). Un score ordonne; il n'affirme jamais une catégorie.
score_only:v0-immature-baseline · tel quel depuis la passe de notation : score_only signifie que le nombre peut ordonner les travaux, et qu'aucune étiquette de catégorie n'en découleClassification
machine, non validéePrédiction automatique; un appel candidat d’une seule source (Gemma direct ou Codex distillé), pas un consensus.
Le détail, modèle par modèle et score par score, se trouve en fin de page sous « Comment cette classification a été obtenue ».