Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling

To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size, network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces fitted to capillary pressure, saturation, and interfacial area (P _c –S _w –a _nw ) points as well as to relative permeability, saturation, and interfacial area (k _r –S _w –a _nw ) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different surfaces to P _c –S _w –a _nw points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation (P _c –S _w ) and the relative permeability–saturation (k _r –S _w ) curves.     

Capillary hyperdisperson of wetting liquids in fractal porous media

Recent displacement experiments show anomalously rapid spreading of water during imbibition into a prewet porous medium. We explain this phenomenon, calledhyperdispersion, as viscous flow along fractal pore walls in thin films of thicknessh governed by disjoining forces and capillarity. At high capillary pressure, total wetting phase saturation is the sum of thin-film and pendular stucture inventories:S _w =S _tf +S _ps . In many cases, disjoining pressure is inversely proportional to a powerm of film thicknessh, i.e. h _–m , so thatS _tf P _c ^–1/m. The contribution of fractal pendular structures to wetting phase saturation often obeys a power lawS _ps P _c ^(3–D), whereD is the Hausdorff or fractal dimension of pore wall roughness. Hence, if wetting phase inventory is primarily pendular structures, and if thin films control the hydraulic resistance of wetting phase, the capillary dispersion coefficient obeysD _c S _w ^v , where v=[3–m(4–D) ]/m(3–D). The spreading ishyperdispersive, i.e.D _c (S _w ) rises as wetting phase saturation approaches zero, ifm>3/(4–D),hypodispersive, i.e.D _c (S ₂) falls as wetting phase saturation tends to zero, ifm<3>D), anddiffusion-like ifm=3/(4–D). Asymptotic analysis of the capillary diffusion equation is presented.     

Uniqueness of Specific Interfacial Area–Capillary Pressure–Saturation Relationship Under Non-Equilibrium Conditions in Two-Phase Porous Media Flow

The capillary pressure?Csaturation (P ^c?CS ^w) relationship is one of the central constitutive relationships used in two-phase flow simulations. There are two major concerns regarding this relation. These concerns are partially studied in a hypothetical porous medium using a dynamic pore-network model called DYPOSIT, which has been employed and extended for this study: (a) P ^c?CS ^w relationship is measured empirically under equilibrium conditions. It is then used in Darcy-based simulations for all dynamic conditions. This is only valid if there is a guarantee that this relationship is unique for a given flow process (drainage or imbibition) independent of dynamic conditions; (b) It is also known that P ^c?CS ^w relationship is flow process dependent. Depending on drainage and imbibition, different curves can be achieved, which are referred to as ??hysteresis??. A thermodynamically derived theory (Hassanizadeh and Gray, Water Resour Res 29: 3389?C3904, 1993a) suggests that, by introducing a new state variable, called the specific interfacial area (a ^nw, defined as the ratio of fluid?Cfluid interfacial area to the total volume of the domain), it is possible to define a unique relation between capillary pressure, saturation, and interfacial area. This study investigates these two aspects of capillary pressure?Csaturation relationship using a dynamic pore-network model. The simulation results imply that P ^c?CS ^w relation not only depends on flow process (drainage and imbibition) but also on dynamic conditions for a given flow process. Moreover, this study attempts to obtain the first preliminary insights into the global functionality of capillary pressure?Csaturation?Cinterfacial area relationship under equilibrium and non-equilibrium conditions and the uniqueness of P ^c?CS ^w?Ca ^nw relationship.     

A Numerical Study of Micro-Heterogeneity Effects on Upscaled Properties of Two-Phase Flow in Porous Media

Commonly, capillary pressure–saturation–relative permeability (P ^c–S–K _r) relationships are obtained by means of laboratory experiments carried out on soil samples that are up to 10–12 cm long. In obtaining these relationships, it is implicitly assumed that the soil sample is homogeneous. However, it is well known that even at such scales, some micro-heterogeneities may exist. These heterogeneous regions will have distinct multiphase flow properties and will affect saturation and distribution of wetting and non-wetting phases within the soil sample. This, in turn, may affect the measured two-phase flow relationships. In the present work, numerical simulations have been carried out to investigate how the variations in nature, amount, and distribution of sub-sample scale heterogeneities affect P ^c–S–K _r relationships for dense non-aqueous phase liquid (DNAPL) and water flow. Fourteen combinations of sand types and heterogeneous patterns have been defined. These include binary combinations of coarse sand imbedded in fine sand and vice versa. The domains size is chosen so that it represents typical laboratory samples used in the measurements of P ^c–S–K _r curves. Upscaled drainage and imbibition P ^c–S–K _r relationships for various heterogeneity patterns have been obtained and compared in order to determine the relative significance of the heterogeneity patterns. Our results show that for micro-heterogeneities of the type shown here, the upscaled P ^c–S curve mainly follows the corresponding curve for the background sand. Only irreducible water saturation (in drainage) and residual DNAPL saturation (in imbibition) are affected by the presence and intensity of heterogeneities.     

Modeling Thermal-Hydrologic Processes for a Heated Fractured Rock System: Impact of a Capillary-Pressure Maximum

Various thermal-hydrologic models have been developed to simulate thermal-hydrologic conditions in emplacement drifts and surrounding host rock for the proposed high-level nuclear waste repository at Yucca Mountain, Nevada. The modeling involves two-phase (liquid and gas) and two-component (water and air) transport in a fractured-rock system, which is conceptualized as a dual-permeability medium. Simulated hydrologic processes depend upon calibrated system parameters, such as the van Genuchten α and m, which quantify the capillary properties of the fractures and rock matrix. Typically, these parameters are not calibrated for strongly heat-driven conditions, i.e., conditions for which boiling and rock dryout occur. The objective of this study is to modify the relationship between capillary pressure and saturation, P _c(S), for strongly heated conditions that drive saturation below the residual saturation (S → 0). We offer various extensions to the van Genuchten capillary-pressure function and compare results from a thermal-hydrologic model with data collected during the Drift-Scale Test, an in situ thermal test at Yucca Mountain, to investigate the suitability of these various P _c extension methods. The study suggests that the use of extension methods and the imposition of a capillary-pressure cap (or maximum) improve the agreement between Drift-Scale Test data and model results for strongly heat-driven conditions. However, for thermal-hydrologic models of the Yucca Mountain nuclear waste repository, temperature and relative humidity are insensitive to the choice of extension method for the capillary-pressure function. Therefore, the choice of extension method applied to models of drift-scale thermal-hydrologic behavior at Yucca Mountain can be made on the basis of numerical performance.     

Determination of Capillary Pressure Function from Resistivity Data

A model has been derived theoretically to correlate capillary pressure and resistivity index based on the fractal scaling theory. The model is simple and predicts a power law relationship between capillary pressure and resistivity index (P _c = p _e · I ^β) in a specific range of low water saturation. To verify the model, gas-water capillary pressure and resistivity were measured simultaneously at a room temperature in 14 core samples from two formations in an oil reservoir. The permeability of the core samples ranged from 0.028 to over 3000 md. The porosity ranged from less than 8 to over 30. Capillary pressure curves were measured using a semi-permeable porous-plate technique. The model was tested against the experimental data obtained in this study. The results demonstrated that the model could match the experimental data in a specific range of low water saturation. The experimental results also support the fractal scaling theory in a low water saturation range. The new model developed in this study may be deployed to determine capillary pressure from resistivity data both in laboratories and reservoirs, especially in the case in which permeability is low or it is difficult to measure capillary pressure.     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P ^c–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P ^c–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P ^c–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.     

The equations of miscible flow with negligible molecular diffusion

A set of equations with generalized permeability functions has been proposed by de la Cruz and Spanos, Whitaker, and Kalaydjian to describe three-dimensional immiscible two-phase flow. We have employed the zero interfacial tension limit of these equations to model two phase miscible flow with negligible molecular diffusion. A solution to these equations is found; we find the generalized permeabilities to depend upon two empirically determined functions of saturation which we denote asA andB. This solution is also used to analyze how dispersion arises in miscible flow; in particular we show that the dispersion evolves at a constant rate. In turn this permits us to predict and understand the asymmetry and long tailing in breakthrough curves, and the scale and fluid velocity dependence of the longitudinal dispersion coefficient. Finally, we illustrate how an experimental breakthrough curve can be used to infer the saturation dependence of the underlying functionsA andB.Roman Letters A a surface area; cross-sectional area of a slim tube or core - A _1s pore scale area of interface between solid and fluid 1 - A ₁₂ pore scale area of interface between fluid 1 and fluid 2 - A(S ₁) fluid flow weighting function defined by Equation (3.21) - a _i ,b _a ,c _a ,d _i macro scale parameters,i=1...2 (Section 3); polynomial coefficients,i=1...N (Section 7) - B(S ₁) fluid flow weighting function defined by Equation (3.16) - c _e effluent concentration - c _i mass concentration fluidi=1...2 - c _fi fractional mass concentration of fluidi=1...2 - D dispersion tensor - D _m mechanical dispersion tensor - D ₀ molecular dispersion tensor - D _L longitudinal dispersion coefficient - D _T transverse dispersion coefficient - D _L ⁰ defined by Equation (6.21) - F(c _f2) defined by Equation (5.17) - f ₁(S ₁) fractional flow - g acceleration of gravity - j ₂ deviation mass flux of fluid 2 - K permeability of porous medium - K _ij generalized relative permeability function,i=1...2,j=1...2 - K _ri relative permeability functions,i=1...2 - L length of a slim tube or core - M _i total mass of fluidi=1...2 in volumeV - N number of points used to generate numerical curves - n unit normal to a surface - P pressure - P _i pressure in fluidi=1...2 - P _c capillary pressure - P ₁₂ macroscopic capillary pressure parameter - P(x) normal distribution function - q Darcy velocity of total fluid - q _i Darcy velocity of fluidi=1...2 - S _i saturation of fluidi=1...2 - S _L a low saturation value forS ₁ - S _H a high saturation value forS ₁ - u average intersitial fluid velocity - u _S isosaturation velocity - V volume used for volume averaging - V(c _f2) function defined by Equation (6.28) - V _e effluent volume - V _f fluid volume - V _i volume of fluidi=1...2 (Section 2); injected fluid volume - V _p pore volume of a slim tube or core - v macro scale fluid velocity - v _i macro scale velocity of fluidi=1...2 - _q (S ₁) isosaturation speed - _g (S ₁) component of isosaturation velocity due to gravity - w(S _L,S _H,t) width of a displacement front - w(t) overall width of a displacement front Greek Letters static interfacial tension - _ME macroscopic dispersivity - divergence operator - porosity - _i fraction of pore space occupied by fluidi=1...2 - (S ₁) effective viscosity of the fluid - _i viscosity of fluidi=1...2 - ₁₂ macroscopic fluid viscosity coupling parameter - macro scale fluid density - _i density of fluid i=1...2 - _q effective gravitational fluid density     

Displacement of trapped oil from water-wet reservoir rock

Displacement of oil trapped in water-wet reservoirs was analyzed using percolation theory. The critical capillary number of the CDC (Capillary Desaturation Curve) was be predicted based on the pore structure of the medium. The mobilization and stability theories proposed by Stegemeier were used to correlate oil cluster length to the capillary number needed to mobilize the trapped oil. Under the assumption that all pore chambers have the same size, a procedure was developed using the drainage capillary pressure curve and effective accessibility function to predict the CDC curve for a given medium. The prediction of critical capillary numbers was compared with the experimental data from 32 sandstone samples by Chatzis and Morrow. Also, the CDC curve of one sandstone sample was calculated using the procedure developed in this work and compared with the measured data. Very good agreements were obtained.Nomenclature a average radius of a liquid filament [m] - c constant - D pore throat diameter [m] - D _a advancing diameter of an oil cluster [m] - D _af average flowing diameter of the medium [m] - D _da controlling diameter of the medium [m] - D _r receding diameter of an oil cluster [m] - D _X difficulty index - f ratio of length to average radius of an oil cluster - F _i interfacial forces [N] - F _p force from pressure gradient [N] - g wettability function - k absolute permeability [m²] - l length of an oil cluster [m] - l _m mobile oil cluster length [m] - l _s stable oil cluster length [m] - l _w wavelength [m] - n* relative length of an oil cluster - N _c 1 capillary number defined by Equation (1) - N _c 2 capillary number defined by Equation (2) - P _b probability of oil filling a pore - P _c percolation threshold value - p _c capillary pressure [N/m²] - r radius of a pore [m] - r _e average pore radius [m] - S _n the nonwetting phase saturation - S _or residual oil saturation - S _orn normalized oil saturation - v Darcy flow rate [m/s] - X _t total fraction of pores - X _t ^a accessibility - X _e ^a effective accessibility - (D) pore throat size distribution function - _a advancing contact angle - _r receding contact angle - porosity - density of the liquid [kg/m³] - constant in Equation (4) - dynamic length of an oil cluster [m] - interfacial tension [N/m] - viscosity [N/(m s)] - p pressure gradient [N/m³]     

Modeling the Formation of Fluid Banks During Counter-Current Flow in Porous Media

Fluid banks sometimes form during gravity-driven counter-current flow in certain natural reservoir processes. Prediction of flow performance in such systems depends on our understanding of the bank-formation process. Traditional modeling methods using a single capillary pressure curve based on a final saturation distribution have successfully simulated counter-current flow without fluid banks. However, it has been difficult to simulate counter-current flow with fluid banks. In this paper, we describe the successful saturation-history-dependent modeling of counter-current flow experiments that result in fluid banks. The method used to simulate the experiments takes into account hysteresis in capillary pressure and relative permeabilities. Each spatial element in the model follows a distinct trajectory on the capillary pressure versus saturation map, which consists of the capillary hysteresis loop and the associated capillary pressure scanning curves. The new modeling method successfully captured the formation of the fluid banks observed in the experiments, including their development with time. Results show that bank formation is favored where the p_c-versus-saturation slope is low. Experiments documented in the literature that exhibited formation of fluid banks were also successfully simulated.     

Two-Phase Flow Properties Prediction from Small-Scale Data Using Pore-Network Modeling

The objective of this work is to evaluate the prediction accuracy of network modeling to calculate transport properties of porous media based on the interpretation of mercury invasion capillary pressure curves only. A pore-scale modeling approach is used to model the multi-phase flow and calculate gas/oil relative permeability curves. The characteristics of the 3-D pore-network are defined with the requirement that the network model satisfactorily reproduces the capillary pressure curve (P_c curve), the porosity and the permeability. A sensitivity study on the effect of the input parameters on the prediction of capillary pressure and gas/oil relative permeability curves is presented. The simulations show that different input parameters can lead to similarly good reproductions of the experimental P_c, although the predicted relative permeabilities K_r are somewhat widespread. This means that the information derived from a mercury invasion P_c curve is not sufficient to characterize transport properties of a porous medium. The simulations indicate that more quantitative information on the wall roughness and the node/bond aspect ratio would be necessary to better constrain the problem. There is also evidence that in narrow pore size distributions pore body volume and pore throat radius are correlated while in broad pore size distributions they would be uncorrelated.     

The laminar hole pressure for Newtonian fluids

For flows with wall turbulence the hole pressure, P _H , was shown empirically by Franklin and Wallace (J Fluid Mech, 42, 33–48, 1970) to depend solely on R ₊, the Reynolds number constructed from the friction velocity and the hole diameter b. Here this dependence is extended to the laminar regime by numerical simulation of a Newtonian fluid flowing in a plane channel (gap H) with a deep tap hole on one wall. Calculated hole pressures are in good agreement with experimental values, and for two hole sizes are well represented by: (P _H −P _HS )/τ _w = √(k ² + c ² R ₊²)−k, where the Stokes hole pressure P _HS /τ _w = s (b/H)³, k, c, s are fitted constants, and τ _w is the wall shear stress.     

Numerical study of non-uniqueness of the factors influencing relative permeability in heterogeneous porous media by lattice Boltzmann method

The effects of capillary number Ca and viscosity ratio M on non-uniqueness of the relative permeability (RP) – wetting saturation (S_w) relationships of steady-state immiscible two-phase flow in the heterogeneous porous media are studied by a developed lattice Boltzmann method (LBM). In this work, it is suggested that the non-uniqueness of capillary number Ca and viscosity ratio M influencing RP–S_w curves is due to the presence of the heterogeneity of porous media. For the different Ca and M, the RP–S_w curves in the homogenous and heterogeneous porous media with the same porosity are discussed in detail. It is found that (1) the pore-size distributions in porous media have significant effect on the RPs of both wetting phase (WP) and non-wetting phase (NWP); (2) the heterogeneity of porous media has negative effect on the increment of the RP of NWP caused by the large capillary number. The lubricating effect is strongly dependent on pore-size distributions in porous media; and (3) the large viscosity ratio increases the RP of NWP, while it has little effect on the RP of WP. The effect on the RP of NWP may be magnified by the heterogeneity of porous media.     

A Numerical Model of Tracer Transport in a Non-isothermal Two-Phase Flow System for {\text{ CO}}_2 Geological Storage Characterization

For the purpose of characterizing geologically stored $\text{ CO}_{2}Air sparging is an in situ soil/groundwater remediation technology, which involves the injection of pressurized air through air sparging well below the zone of contamination. To investigate the rate-dependent flow properties during multistep air sparging, a rule-based dynamic two-phase flow model was developed and applied to a 3D pore network which is employed to characterize the void structure of porous media. The simulated dynamic two-phase flow at the pore scale or microscale was translated into functional relationships at the continuum-scale of capillary pressure?Csaturation (P ^c?CS) and relative permeability??saturation (K _r?CS) relationships. A significant contribution from the air injection pressure step and duration time of each air injection pressure on both of the above relationships was observed during the multistep air sparging tests. It is observed from the simulation that at a given matric potential, larger amount of water is retained during transient flow than that during steady flow. Shorter the duration of each air injection pressure step, there is higher fraction of retained water. The relative air/water permeability values are also greatly affected by the pressure step. With large air injection pressure step, the air/water relative permeability is much higher than that with a smaller air injection pressure step at the same water saturation level. However, the impact of pressure step on relative permeability is not consistent for flows with different capillary numbers (N _ca). When compared with relative air permeability, relative water permeability has a higher scatter. It was further observed that the dynamic effects on the relative permeability curve are more apparent for networks with larger pore sizes than that with smaller pore sizes. In addition, the effect of pore size on relative water permeability is higher than that on relative air permeability.     

Estimation of three-phase relative permeabilities

Starting from the statistical structural model of Alemánet al. (1988), we have developed an alternative to Stone's (1970, 1973; Aziz and Settari, 1979) methods for estimating steady-state, three-phase relative permeabilities from two sets of steady-state, two-phase relative permeabilities. Our result reduces to Stone's (1970; Aziz and Settari, 1979) first method, when the steady-state, two-phase relative permeability of the intermediate-wetting phase with respect to either the wetting phase or the nonwetting phase is a linear function of the saturation of the intermediate-wetting phase. As the curvature of either of these relative permeability functions increases, the deviation of our result from Stone's (1970; Aziz and Settari, 1979) first method increases. Currently, there are no data available that are sufficiently complete to form the basis of a comparison between our result and either of the methods of Stone (1970, 1973; Aziz and Settari, 1979).Notation a free parameter in Equation (19) - B(m, n) Beta function defined by Equation (17) - F ^(w), F^(nw) defined by Equations (31) and (27), respectively - G ⁽ⁱ⁾ defined by Equations (37) and (39) - H ⁽ⁱ⁾ defined by Equations (38) and (40) - k ⁽ⁱ⁾ three-phase relative permeability fo phasei - k ^(i)* defined by Equations (34) through (36) - k ^(i,j) relative permeability to phasei during a two-phase flow with phasej, possibly in the presence of an immobile phase - k ^(i,j)* defined by analogy with Equations (41) and (42) - k ^(i,j)** defined by Equations (49), (50), (53), and (54) - k _max ⁽ⁱ⁾ defined by Equation (11) - k ₁₉₇₀ ^(iw) defined by Equation (10) - k ₁₉₇₃ ^(iw)* defined by Equation (58) - k ₁₉₇₃ ^(iw) defined by Equation (13) - L length and diameter of cylindrical averaging surfaceS - L _t length of an individual capillary tube enclosed byS - L _t ^* defined by Equation (19) - L _t,min length of pore whose radius isR _max - N total number of pores contained within the averaging surfaceS - p ₁ ⁽ⁱ⁾ ,p ₂ ⁽ⁱ⁾ pressure of phasei at entrance and exit of averaging surfaceS, respectively - p defined by Equation (21) - p _c ^(i,j) capillary pressure function - p _c ^(i,j)* defined by Equations (23), (29), and (32) - p ⁽ⁱ⁾ intrinsic average of pressure within phasei defined by Alemánet al. (1988) - R pore radius - R ^* defined by Equation (18) - R _max maximum pore radius that occurs withinS - s ⁽ⁱ⁾ local saturation of phasei - s ^(i)* defined by Equation (7) - s _min ⁽ⁱ⁾ minimum or immobile saturation of phasei - S averaging surface introduced in local volume averaging - V ⁽ⁱ⁾ volume of phasei occupying the pore space enclosed byS Greek Letters , parameters in the Beta distribution defined by Equation (16) - ^(w), ^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (6) - ^(i,j) interfacial tension between phasesi andj - (x) Gamma function - defined by Equation (57) - , spherical coordinates in system centered upon the axis of the averaging surfaceS - _max maximum value of , 45 °, in view of assumption (9) - ^(i,j) contact angle between phasesi andj measured through the displacing phase - ^(w),^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (12) Other gradient operator Amoco Production Company, PO Box 591 Tulsa, OK 74102, U.S.A.     

Steady-state solutions to Bentsen's equation

Based on experimentally observed phenomena and the physical requirement of a unique value of saturation at any location within a porous medium, a restrictive condition for a valid solution to Bentsen's equation is derived: ?² f/?S ²≤0. The steady-state solution to Bentsen's equation is shown to be identical to the Buckley-Leverett solution to the displacement equation, and the steady-state solution for the fractional flow is shown to be independent of the capillary number. It is proved that under steady-state conditions, the capillary term of the fractional flow equation in the frontal region does not depend on the capillary number. Therefore, the unrealistic triple-valued saturation profile of the original Buckley-Leverett solution resulted because the capillary term was in-appropriately neglected. The break-through recovery efficiency,Τ _bt , is shown to be a function of the capillary number. As the capillary number decreases, the break-through recovery efficiency increases and the maximum value ofΤ _bt can be obtained asN _c → 0. The Buckley-Leverett solution is the limiting solution asN _c → 0.     

Reynolds' analogy for the thermal convection driven by nonisothermal stretching surfaces

In the present paper the steady boundary-layer flows induced by permeable stretching surfaces with variable temperature distribution are investigated under the aspect of Reynolds' analogy r = St _x /C _f(x). It is shown that for certain stretching velocities and wall temperature distributions, “Reynolds' function”r, i.e. the ratio of the local Stanton number St _x and the skin friction coefficient C _f(x) equals −1/2 for any value of the Prandtl number Pr and of the dimensionless suction/injection velocity f _w. In all of these cases, the dimensionless temperature field ϑ is connected to the dimensionless downstream velocity f ^′ by the simple relationship ϑ=(f ^′)^Pr. It is also shown that in the general case, Reynolds' function r may possess several singularities in f _w. The largest of them represents a critical value, so that for f _w<f _w,crit the solutions of the energy equation (although they still satisfy all the boundary conditions) become nonphysical.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.