A Multiscale Method Coupling Network and Continuum Models in Porous Media I: Steady-State Single Phase Flow

2012 ◽  
Vol 10 (2) ◽  
pp. 515-549 ◽  
Author(s):  
Jay Chu ◽  
Björn Engquist ◽  
Maša Prodanović ◽  
Richard Tsai
Author(s):  
Guang Dong ◽  
Yulan Song

The topology optimization method is extended to solve a single phase flow in porous media optimization problem based on the Two Point Flux Approximation model. In particular, this paper discusses both strong form and matrix form equations for the flow in porous media. The design variables and design objective are well defined for this topology optimization problem, which is based on the Solid Isotropic Material with Penalization approach. The optimization problem is solved by the Generalized Sequential Approximate Optimization algorithm iteratively. To show the effectiveness of the topology optimization in solving the single phase flow in porous media, the examples of two-dimensional grid cell TPFA model with impermeable regions as constrains are presented in the numerical example section.


Author(s):  
William G. Gray ◽  
Michael A. Celia

The mathematical study of flow in porous media is typically based on the 1856 empirical result of Henri Darcy. This result, known as Darcy’s law, states that the velocity of a single-phase flow through a porous medium is proportional to the hydraulic gradient. The publication of Darcy’s work has been referred to as “the birth of groundwater hydrology as a quantitative science” (Freeze and Cherry, 1979). Although Darcy’s original equation was found to be valid for slow, steady, one-dimensional, single-phase flow through a homogeneous and isotropic sand, it has been applied in the succeeding 140 years to complex transient flows that involve multiple phases in heterogeneous media. To attain this generality, a modification has been made to the original formula, such that the constant of proportionality between flow and hydraulic gradient is allowed to be a spatially varying function of the system properties. The extended version of Darcy’s law is expressed in the following form: qα=-Kα . Jα (2.1) where qα is the volumetric flow rate per unit area vector of the α-phase fluid, Kα is the hydraulic conductivity tensor of the α-phase and is a function of the viscosity and saturation of the α-phase and of the solid matrix, and Jα is the vector hydraulic gradient that drives the flow. The quantities Jα and Kα account for pressure and gravitational effects as well as the interactions that occur between adjacent phases. Although this generalization is occasionally criticized for its shortcomings, equation (2.1) is considered today to be a fundamental principle in analysis of porous media flows (e.g., McWhorter and Sunada, 1977). If, indeed, Darcy’s experimental result is the birth of quantitative hydrology, a need still remains to build quantitative analysis of porous media flow on a strong theoretical foundation. The problem of unsaturated flow of water has been attacked using experimental and theoretical tools since the early part of this century. Sposito (1986) attributes the beginnings of the study of soil water flow as a subdiscipline of physics to the fundamental work of Buckingham (1907), which uses a saturation-dependent hydraulic conductivity and a capillary potential for the hydraulic gradient.


1985 ◽  
Vol 25 (01) ◽  
pp. 101-112 ◽  
Author(s):  
Stanley C. Jones

Jones, Stanley C., SPE, Marathon Oil Co. Abstract Displacements were conducted in Berea cores to gain insight into the mechanism of tertiary oil displacement and propagation by a micellar slug. Contrary to expectation, propagation by a micellar slug. Contrary to expectation, the first oil mobilized by micellar fluid was among the first oil (instead of the last oil) to be produced, giving the appearance of either viscous fingering or of unusually large dispersion. To eliminate the possibility of unfavorable mobility ratios caused by oil/water/surfactant interaction, we conducted several runs in which an injected hydrocarbon displaced another hydrocarbon, initially at residual saturation. In other experiments, water (the wetting phase) at irreducible saturation was displaced by a distinguishable injected aqueous phase. Injected hydrocarbon appeared in the produced fluids immediately after oil breakthrough, yielding behavior similar to the micellar-slug experiments. Even with a favorable viscosity ratio of less than 0.01, the apparent dispersion was huge. However, mixing zones in the wetting-phase displacements were quite normal and similar to those observed for single-phase flow. Nonwetting-phase fronts (injected hydrocarbon displacing resident hydrocarbon) are smeared much more than wetting-phase fronts because the entrance of hydrocarbon into smaller water-filled pore throats is delayed until the capillary entrance pressure is overcome by differences in the flowing oil and water pressure gradients. Oil might not be displaced from the smaller pores until long after oil breakthrough. Nonwetting-phase dispersion, which occurs in many EOR processes, can be expected to be one or two orders of magnitude greater than dispersion measured in single-phase-flow experiments. Entrance of the wetting phase, however, is not delayed; hence, wetting-phase Mixing zones are short. Introduction Experiments for this study were inspired by the question: How is residual oil, which has been mobilized by a micellar slug, transported? More specifically, does the first oil mobilized by a slug (near the injection end of a core) contact and mobilize oil downstream from it, which displaces more oil even farther downstream? If this were the case, the first oil to be produced would be the most-downstream oil (i.e., oil nearest the outlet). The last oil produced would be the first oil mobilized from the produced would be the first oil mobilized from the injection end of the core. This scheme is somewhat analogous to pushing a broom across a floor covered with a heavy layer of dust. The first dust encountered by the broom stays next to the broom. As the accumulated layer of dust in front of the broom becomes adequately compacted, it pushes dust ahead of it to from an ever-widening band or "dust bank" ahead of the broom. The dust farthest ahead of the broom is the first to be pushed into the dustpan, and the dust first encountered by the broom is the last to be pushed in. Or is this concept all wrong? Another model postulates that the oil first contacted by a micellar slug is mobilized and quickly travels away from the slug so that the downstream oil is contacted and mobilized by the slug, not by the first-mobilized oil. If this process were to proceed to its logical conclusion, the first-produced oil would proceed to its logical conclusion, the first-produced oil would be from the inlet end of the core, and the last-produced from the outlet end. Either of these two extremes would be modified by dispersion, which smears sharp fronts by mixing displaced and displacing fluids. Dispersion in porous media has been investigated extensively. Perkins and Johnston have reviewed several studies, mostly involving single-phase flow. The simultaneous injection of the water with light hydrocarbon solvents is a technique used to reduce solvent mobility and viscous fingering. Raimondi et al. performed steady-state experiments in which flowing performed steady-state experiments in which flowing water and oil were miscibly displaced by the simultaneous injection of water and solvent. They found that the longitudinal mixing coefficient for the hydrocarbon phase increased sharply with increasing water above the irreducible saturation. The displacement of the wetting phase was not greatly affected by the presence of the nonwetting phase. However, a large amount of oil that initially phase. However, a large amount of oil that initially seemed to be trapped by water was eventually recovered by continued solvent injection. Raimondi and Torcaso later found that some oil, particularly at high water-to-solvent injection ratios, was particularly at high water-to-solvent injection ratios, was trapped permanently, provided that injection rates, ratios, and pressure drops were unchanged in switching from water/oil to water/solvent injection. Fitzgerald and Nielsen also found that only part of the in-place crude was recovered by solvent injection. Moreover, solvent appeared in the effluent shortly after oil breakthrough. Oil recovery was further decreased when solvent and water were injected simultaneously. Thomas et al. reported slightly increased wetting-phase longitudinal mixing during simultaneous water/oil injection as the wetting-phase saturation decreased. Non-wetting-phase mixing increased substantially as the nonwetting-phase saturation decreased from 100%. SPEJ p. 101


1984 ◽  
Vol 106 (2) ◽  
pp. 193-200 ◽  
Author(s):  
V. Arp ◽  
J. M. Persichetti ◽  
Guo-bang Chen

The Gru¨neisen parameter has long been used in equations of state for solids to relate thermodynamic properties to lattice vibrational spectra [1]. A few papers have extended the concept to studies of liquid structure. Knopoff and Shapiro [2] have evaluated a Gru¨neisen parameter for water and for mercury, attempting to relate its temperature dependence in a limited range to atomic clustering within the liquid. Sharma [3], in a series of papers, has evaluated a pseudo-Gru¨neisen parameter in mercury and liquefied gases and related it to internal pressures, a solubility parameter, and clustering phenomena. In this paper we evaluate the Gru¨neisen parameter for a variety of fluids, and show how it occurs in many problems in compressible fluid hydrodynamics, without reference to concepts of liquid structure. The work extends that reported in an earlier paper for the special case of steady state, single phase flow [4].


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