Elliptic Regions and Stable Solutions for Three-Phase flow in Porous Media

In the limit of zero capillary pressure, solutions to the equations governing three-phase flow, obtained using common empirical relative permeability models, exhibit complex wavespeeds for certain saturation values (elliptic regions) that result in unstable and non-unique solutions. We analyze a simple but physically realizable pore-scale model: a bundle of cylindrical capillary tubes, to investigate whether the presence of these elliptic regions is an artifact of using unphysical relative permeabilities. Without gravity, the model does not yield elliptic regions unless the most non-wetting phase is the most viscous and the most wetting phase is the least viscous. With gravity, the model yields elliptic regions for any combination of viscosities, and these regions occupy a significant fraction of the saturation space. We then present converged, stable numerical solutions for one-dimensional flow, which include capillary pressure. These demonstrate that, even when capillary forces are small relative to viscous forces, they have a significant effect on solutions which cross or enter the elliptic region. We conclude that elliptic regions can occur for a physically realizable model of a porous medium, and that capillary pressure should be included explicitly in three-phase numerical simulators to obtain stable, physically meaningful solutions which reproduce the correct sequence of saturation changes.     

Mechanistic Model of Steady-State Two-Phase Flow in Porous Media Based on Ganglion Dynamics

Recent experimental work has shown that the pore-scale flow mechanism during steady-state two-phase flow in porous media is ganglion dynamics (GD) over a broad and practically significant range of the system parameters. This observation suggests that our conception and theoretical treatment of fractional flow in porous media need careful reconsideration. Here is proposed a mechanistic model of steady-state two-phase flow in those cases where the dominant flow regime is ganglion dynamics. The approach is based on the ganglion population balance equations in combination with a microflow network simulator. The fundamental information on the cooperative flow behavior of the two fluids at the scale of a few hundred pores is expressed through the system factors, which are functions of the system parameters and are calculated using the simulator. These system factors are utilized by the population balance equations to predict the macroscopic behavior of the process. The dependence of the conventional relative permeability coefficients not only on the wetting fluid saturation S_wbut also on the capillary number, Ca, the viscosity ratio the wettability (⁰ a, ⁰ r), the coalescence factor, Co, as well as the porous medium geometry and topology is explained and predicted on a mechanistic basis. Sample calculations have been performed for steady-state fully developed (SSFD) and steady-state nonfully developed (SSnonFD) flow conditions. The number distributions of the moving and the stranded ganglia, the mean ganglion size, the fraction of the nonwetting fluid in the form of mobile ganglia, the ratio of the conventional relative permeability coefficients and the fractional flows are studied as functions of the system parameters and are correlated with the flow phenomena at pore level and the system factors.     

Critical Review of Foamy Oil Flow

Foamy oil flow is a term coined to describe a form of twophase oilgas flow that appears to occur during solution gas drive in some heavy oil reservoirs and does not fit the classical models of twophase flow. Most of the evidence supporting the presence of this unusual flow mechanism is circumstantial and comes from attempts to explain much higher than expected well productivity and primary recovery factors in several heavy oil reservoirs. This paper is a review of the available literature on foamy oil flow in primary production of heavy oils under solution gas drive.The mechanisms operating in solution gas drive in heavy oil reservoirs are briefly discussed. The issues related to supersaturation in oil phase, bubble nucleation, critical gas saturation, and relative permeability are discussed. The possible role of rate processes related to the release of solution gas and the formation of a segregated gas phase is reviewed. The porescale mechanisms involved in creation and propagation of dispersed gas flow are discussed. Several published mathematical models of foamy solution gas drive are reviewed with focus on their limitations.The review shows that the theoretical and experimental investigations of foamy oil flow are still in early stages. Although the occurrence of foamy oil flow has been verified in laboratory experiments, its existence at the reservoir scale has not been confirmed. The theoretical understanding of the mechanisms underlying foamy oil flow remains poor.     

Co-history Matching: A Way Forward for Estimating Representative Saturation Functions

Core-scale experiments and analyses would often lead to estimation of saturation functions (relative permeability and capillary pressure). However, despite previous attempts on developing analytical and numerical methods, the estimated flow functions may not be representative of coreflood experiments when it comes to predicting similar experiments due to non-uniqueness issues of inverse problems. In this work, a novel approach was developed for estimation of relative permeability and capillary pressure simultaneously using the results of “multiple” corefloods together, which is called “co-history matching.” To examine this methodology, a synthetic (numerical) model was considered using core properties obtained from pore network model. The outcome was satisfactorily similar to original saturation functions. Also, two real coreflood experiments were performed where water at high and low rates were injected under reservoir conditions (live fluid systems) using a carbonate reservoir core. The results indicated that the profiles of oil recovery and differential pressure (dP) would be significantly affected by injection rate scenarios in non-water wet systems. The outcome of co-history matching could indicate that, one set of relative permeability and capillary pressure curves can reproduce the experimental data for all corefloods.     

Study of fracture permeability using Lattice Gas Automata

We study the problem of flow permeability of fracture joints using Lattice-Gas Automata simulations. We model the fracture as a rough channel bounded by a self-affine surface. Changing the surface roughness exponent, rough walls having different microstructures are obtained. Different relative roughnesses — defined as the height of the largest surface asperity divided by the mean aperture — are obtained pulling apart the two surfaces that constitute the rough walls of the channel. We calculate the macroscopic variables volume flow rate and pressure difference using microscopic balances. In the low Reynolds number regime the pressure difference and the flow rate are linearly related (the behavior is described by Darcy's law). In this regime, we study the effect of geometry on the permeability. We have found that permeability is independent of the surface roughness exponentH and it is fully determined in terms of the relative roughness and mean aperture of the fracture joint. For larger Reynolds numbers a transition to a regime in which pressure difference and flow rate are not longer linearly related is observed. This transition is observed in a domain of Reynolds numbers for which the behavior in a smooth channel remains linear. We discuss this transition.     

The Effect of Wettability on Three-Phase Relative Permeability

We study three-phase flow in water-wet, oil-wet, and fractionally-wet sandpacks. We use CT scanning to measure directly the oil and water relative permeabilites for three-phase gravity drainage. In an analogue experiment, we measure pressure gradients in the gas phase to determine the gas relative permeability. Thus we find all three relative permeabilities as a function of saturation. We find that the gas relative permeability is approximately half as much in a oil-wet medium than in an water-wet medium at the same gas saturation. The water relative permeability in the water-wet medium and the oil relative permeability in the oil-wet medium are similar. In the water-wet medium the oil relative permeability scales as k _roS _o ⁴ for S _o>S _or, where S _or is the waterflood residual oil saturation. With octane as the oil phase, k _roS _o ² for S _o<S _or, while with decane as the oil phase, k _ro falls sharply for S _o<S _or. The water relative permeability in the oil-wet medium resembles the oil relative permeability in the water-wet medium for a non-spreading oil such as decane. These observations can be explained in terms of wetting, spreading, and the pore scale configurations of fluid.     

Quadratic systems of conservation laws with generic behavior at infinity

We identify quadratic systems of conservation laws with generic behavior at infinity, where the genericity conditions derive naturally when studying weak solutions of conservation laws. Namely, we identify quadratic models for which the vector field associated with the viscosity admissibility criterion has properties at infinity that are true for an open and dense subset of the set of all planar quadratic vector fields in the metric associated with the Euclidian space of coefficients. We determine the boundaries of the regions containing, generic models in the parameter space of coefficients of quadratic models. We show that when crossing the boundaries of nongeneric models transversally, the Poincaré compactification of the corresponding vector field undergoes either a saddle node or a transcritical bifurcation at infinity. For quadratic models with a bounded elliptic region we calculate the loci of nongeneric models assuming the viscosity matrix to be the identity. We obtain a two-parameter normal form for such models and show that the boundaries that determine generic models in the two-dimensional parameter space correspond to the Schaeffer-Shearer classification of models with an isolated umbilic point. Since the loci of nongeneric models are invariant under the equivalence transformations that preserve weak solutions of conservation laws, understanding their behavior at infinity promises to provide an insight into a general classification of quadratic conservation laws.     

Behavior of Nonaqueous Phase Liquids in Fractured Porous Media under Two‐Phase Flow Conditions

After dense nonaqueous phase liquids (DNAPLs) travel downward through the subsurface, they typically come to rest on fractured bedrock or tight clay layers, which become additional pathways for DNAPL migration. DNAPLs trapped in fractures are continuous sources of groundwater contamination. To decide whether they can be left in place to dissolve or volatilize, or must be removed with active treatment, the movement of DNAPLs in fractured media must be understood at a fundamental level. This work presents numerical simulations of the movements of DNAPLs in naturally fractured media under twophase flow conditions. The flow is modeled using a multiphase network flow model, used to develop predictive capabilities for DNAPL flow in fractures. Capillary pressure–saturation–relative permeability curves are developed for twophase flow in fractures. Comparisons are made between the behavior in crystalline, almost impermeable rocks (e.g. granite) and more permeable rocks like sandstone, to understand the effects of the rock matrix on the displacement of the DNAPLs in the fracture. For capillarydominated flow, displacements occur as a sequence of jumps, as the invading phase overcomes the capillary pressure at downgradient apertures. Preferential channels for the displacement of nonaqueous phase are formed due to high fracture aperture in some regions.     

Approximate Semi-analytical Solution for Injection–Falloff–Production Well Test: An Analytical Tool for the In Situ Estimation of Relative Permeability Curves

An injection–falloff–production test (IFPT) was originally proposed in Chen et al. (in: SPE conference paper, 2006. doi: 10.2118/103271-MS, SPE Reserv Eval Eng 11(1):95–107, 2008) as a well test for the in situ estimation of two-phase relative permeability curves to be used for simulating multiphase flows in porous media. Hence, we develop an approximate semi-analytical solution for the two-phase saturation distribution in an oil–water system during the flowback period of an IFPT according to the mathematical theory of waves. In fact, we show that the weak solution we construct for the saturation equation for the flowback period satisfies the Oleinik entropy condition and hence is unique. In addition, we allow the governing relative permeabilities during the flowback period to be different from the relative permeabilities during injection. Using the saturation solution with the steady-state pressure theory of Thompson and Reynolds, we obtain a solution for the wellbore pressure during the flowback period. By comparing results from our solution with those from a commercial numerical simulator, we show that our approximate semi-analytical solution yields accurate saturation profiles and bottom hole pressures history. The use of very small time steps and a highly refined radial grid is necessary to generate a good solution from a reservoir simulator. The approximate analytical pressure solution developed is used as a forward model to match pressure and water flow rate data from an IFPT in order to estimate reservoir rock absolute permeability and skin factor in conjunction with in situ imbibition and drainage water–oil relative permeabilities.     

On the Validation of a Compositional Model for the Simulation of $$\text {CO}_2$$ Injection into Saline Aquifers

In this work we apply a recently proposed Bayesian Markov chain Monte Carlo framework (Akbarabadi et al. in Comput Geosci 19(6):1231–1250, 2015) to quantify uncertainty in the three-dimensional permeability field of a rock core. This process establishes the credibility of a compositional two-phase flow model to describe the displacement of brine by \(\text {CO}_2\) and \(\text {CO}_2\) storage in saline aquifers. We investigate the predictive capabilities of the compositional model in the context of an unsteady-state \(\text {CO}_2\)-brine drainage experiment at the laboratory scale, performed at field-scale aquifer conditions. We employ forward models consisting of a system of discretized partial differential equations along with relative permeability curves obtained by a curve fitting of experimental measurements. We consider a forward model to be validated when: (1) numerical simulations reveal that the Bayesian framework has accurately characterized the core’s permeability and (2) Monte Carlo predictions show excellent agreement between measured and simulated data. A large set of numerical studies with an accurate compositional simulator shows that forward models have been successfully validated. For such models, our numerical results show that we are able to capture all the dominant features and general trends of the \(\text {CO}_2\) saturation fields observed in the core. Our study is consistent with the design and findings of real experiments. Fluid properties, relative permeability data, measured porosity field, physical dimensions, and thermodynamic conditions are the same as those reported in Akbarabadi and Piri (Adv Water Resour 52:190–206, 2013). However, the measured saturation data are from flow experiments different from those reported in Akbarabadi and Piri (2013), and will be presented here.     

Macro-Scale Dynamic Effects in Homogeneous and Heterogeneous Porous Media

It is known that the classical capillary pressure-saturation relationship may be deficient under non-equilibrium conditions when large saturation changes may occur. An extended relationship has been proposed in the literature which correlates the rate of change of saturation to the difference between the phase pressures and the equilibrium capillary pressure. This linear relationship contains a damping coefficient, \tau, that may be a function of saturation. The extended relationship is examined at the macro-scale through simulations using the two-phase simulator MUFTE-UG. In these simulations, it is assumed that the traditional equilibrium relationship between the water saturation and the difference in fluid pressures holds locally. Steady-state and dynamic numerical experiments are performed where a non-wetting phase displaces a wetting phase in homogeneous and heterogeneous domains with varying boundary conditions, domain size, and soil parameters. From these simulations the damping coefficient can be identified as a (non-linear) function of the water saturation. It is shown that the value of increases with an increased domain size and/or with decreased intrinsic permeability. Also, the value of for a domain with a spatially correlated random distribution of intrinsic permeability is compared to a homogeneous domain with equivalent permeability; they are shown to be almost equal.     

A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.     

Numerical Investigations of the Steady State Relative Permeability of a Simplified Porous Medium

The purpose of this paper is to investigate, by flow simulations in a uniform pore-space geometry, how the co and countercurrent steady state relative permeabilities depend on the following parameters: phase saturation, wettability, driving force and viscosity ratio. The main results are as follows: (i) with few exceptions, relative permeabilities are convex functions of saturation; (ii) the cocurrent relative permeabilities increase while the countercurrent ones decrease with the driving force; (iii) with one exception, phase 2 relative permeabilities decrease and phase 1 relative permeabilities increase with the viscosity ratio M=₁/₂; (iv) the countercurrent relative permeabilities are always less than the cocurrent ones, the difference being partly due to the opposing effect of the viscous coupling, and partly to different levels of capillary forces; (v) the pore-level saturation distribution, and hence the size of the viscous coupling, can be very different between the cocurrent and the countercurrent cases so that it is in general incorrect to estimate the full mobility tensor from cocurrent and countercurrent steady state experiments, as suggested by Bentsen and Manai (1993).(Now at AS Norske Shell, Norway.) e-mail:     

Gas Condensate Relative Permeability for Well Calculations

This paper addresses several issues related to the modeling and experimental design of relative permeabilities used for simulating gas condensate well deliverability. Based on the properties of compositional flow equations, we make use of the fact that relative permeability ratio k _rg/k _ro is a purely thermodynamic variable, replacing saturation, when flow is steady-state. The key relation defining steady-state flow in gas condensate wells is relative permeability k _rg as a function of k _rg/k _ro. Consequently, determination of saturation and k _r as a function of saturation is not important for this specific calculation. Once the k _rg=f(k _rg/k _ro) relationship is experimentally established and correlated with capillary number (N _c), accurate modeling of condensate blockage is possible. A generalized model is developed for relative permeability as the function of k _rg/k _ro and capillary number. This model enables us to link the immiscible or rock curves with miscible or 'straight-line curves by a transition function dependent on the capillary number. This model is also extended to the case of high-rate, inertial gas flow within the steady-state condensate blockage regionand locally at the wellbore. We have paid particular attention to the effect of hysteresis on the relation k _rg=f(k _rg/k _ro), based on our observation that many repeated cycles of partial/complete imbibition and drainage occur in the near-well region during the life of a gas condensate well. Finally, the composite effect of condensate blockage is handled using a Muskat pseudopressure model, where relative permeabilities are corrected for the positive effect of capillary number dependence and the negative effect of inertial high velocity flow. Special steady-state experimental procedures have been developed to measure k _rg as a function of k _rg/k _ro and N _c. Saturations, though they can be measured, are not necessary. An approach for fitting steady-state gas condensate relative permeability data has been developed and used for modeling relative permeability curves.     

Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems

A simple process-based model of three-phase displacement cycles for both spreading and non-spreading oils in a mixed-wet capillary bundle model is presented. All possible pore filling sequences are determined analytically and it is found that the number of pore occupancies that are permitted on physical grounds is actually quite restricted. For typical non-spreading gas/oil/water systems, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils. These two cases correspond to whether water or gas is the intermediate-wetting phase in oil-wet pores as determined by the corresponding contact angles, that is, cos ^o _gw > 0 or cos ^o _gw < 0, respectively. Analysis of the derived pore occupancies leads to the establishment of a number of relationships showing the phase dependencies of three-phase capillary pressures and relative permeabilities in mixed-wet systems. It is shown that different relationships hold in different regions of the ternary diagram and the morphology of these regions is discussed in terms of various rock/fluid properties. Up to three distinct phase-dependency regions may appear for a non-spreading oil and this reduces to two for a spreading oil. In each region, we find that only one phase may be specified as being the intermediate-wetting phase and it is only the relative permeability of this phase and the capillary pressure between the two remaining phases that depend upon more than one saturation. Given the simplicity of the model, a remarkable variety of behaviour is predicted. Moreover, the emergent three-phase saturation-dependency regions developed in this paper should prove useful in: (a) guiding improved empirical approaches of how two-phase data should be combined to obtain the corresponding three-phase capillary pressures and relative permeabilities; and (b) determining particular displacement sequences that require additional investigation using a more complete process-based 3D pore-scale network model.     

Pressure transient response of stochastically heterogeneous fractured reservoirs

A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblattet al.'s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonuniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inhomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Roots's solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parameters and, using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parameters and, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoir and, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.     

New concepts in relative permeabilities at high capillary numbers for surfactant flooding

Flooding oil reservoirs with surfactant solutions can increase the amount of oil that can be recovered. Macroscopic modelling of the process requires relative permeabilities to be functions of saturation and capillary number. With only limited experimental data, relative permeabilities have usually been assumed to be linear functions of saturation at high capillary numbers. The experimental data is reviewed, some of which suggest that this assumption is not necessarily correct. The basis for the assumption is therefore reviewed and it is concluded that the linear model corresponds to microscopically segregated flow in the porous medium. Based on new but equally plausible complementary assumptions about the flow pattern, a mixed flow model is derived. These models are then shown to be limiting cases of a droplet model which represents the mixing scale within the porous medium and gives a physical basis for interpolating between the models. The models are based on physical concepts of flow in a porous medium and so the approach described here represents a significant improvement in the understanding of high capillary number flow. This is shown by the fact that fewer parameters are needed to describe experimental data.Notation A total cross-sectional area assigned to capillary bundle - A ⁽ⁱ⁾ physical cross-sectional area of tube i - c ⁽ⁱ⁾ ordered configurational label for droplets in tube i - c configuration label for tube i (order not considered) - D defined by Equation (26) - E(...) expectation value with respect to the trinomial distribution - S _r ⁽⁾ fractional flow of phase - k absolute permeability - k _r relative permeability of phase - k _r ⁰ endpoint relative permeability of phase - L capillary tube length in bundle model - m ⁽ⁱ⁾ number of droplets of phase a occupying tube i - n exponent for phase a in Equation (2) - N number of droplets in bundle model - N _c capillary number - p pressure - p(c') probability of configuration c - Q ⁽ⁱ⁾ total volume flow rate in tube i - S saturation of phase - S flowing saturation of phase - S _r residual saturation of phase - S _r ⁽⁾ saturations when fractional flow of phase is 1 in the case of varying residual saturations for three-phase flow ( ) - t _c residence time for droplet configuration c - v ⁽ⁱ⁾ total fluid velocity in bundle tube i - , phase label - p pressure differential across capillary bundle - ⁽ⁱ⁾ tube conductivity defined by Equation (7) - viscosity of phase - interfacial tension - gradient operator - ... average over tube droplet configurations     

Nonstationary filtration in partially saturated porous media

From the mathematical formulation of a one-dimensional flow through a partially saturated porous medium, we arrive at a nonlinear free boundary problem, the boundary being between the saturated and the unsaturated regions in the medium. In particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part.Existence, uniqueness, a maximum principle and regularity properties are proved for weak solutions of a Cauchy-Dirichlet problem in the cylinder {(x,t): 0x1, t0} and the nature, in particular the regularity, of the free boundary is discussed.Finally, it is shown that solutions of a large class of Cauchy-Dirichlet problems converge towards a stationary solution as t and estimates are given for the rate of convergence.     

Numerical Homogenization of the Absolute Permeability Using the Conformal-Nodal and Mixed-Hybrid Finite Element Method

The modeling of hydrocarbon reservoirs and of aquifer-aquitard systems can be separated into two activities: geological modeling and fluid flow modeling. The geological model focuses on the geometry and the dimensions of the subsurface layers and faults, and on its rock types. The fluid flow model focuses on quantities like pressure, flux and dissipation, which are related to each other by rock parameters like permeability, storage coefficient, porosity and capillary pressure. The absolute permeability, which is the relevant parameter for steady single-phase flow of a fluid with constant viscosity and density, is studied here. When trying to match the geological model with the fluid flow model, it generally turns out that the spatial scale of the fluid flow model is built from units that are at least a hundred times larger in volume than the units of the geological model. To counter this mismatch in scales, the fine-scale permeabilities of the geological data model have to be upscaled' to coarse-scale permeabilities that relate the spatially averaged pressure, flux and dissipation to each other. The upscaled permeabilities may be considered as complicated averages, which are derived from the spatially averaged flow quantities in such a way that the continuity equation, Darcy's law and the dissipation equation remain valid on the coarse scale. In this paper the theory of upscaling will be presented from a physical point of view aiming at understanding, rather than mathematical rigorousness. Under the simplifying assumption of spatial periodicity of the fine-scale permeability distributions, homogenization theory can be applied. However, even then the spatial distribution of the permeability is generally so intricate that exact solutions of the homogenized permeability cannot be found. Therefore, numerical approximation methods have to be applied. To be able to estimate the approximation error, two numerical methods have been developed: one based on the conventional nodal finite element method (CN-FEM) and the other based on the mixed-hybrid finite element method (MH-FEM). CN-FEM gives an upper bound for the sum of the diagonal components of the homogenized mobility matrix, while MH-FEM gives a lower bound. Three numerical examples are presented.     

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.