On the aquifer's integrated balance equations

A general methodology is presented for describing transport phenomena in porous media at a macroscopic level. Then, these macroscopic balance equations are integrated (or averaged) along the vertical for confined, leaky and phreatic aquifers.The results are employed to derive (averaged) aquifer equations for the flow of water and of a solute (hydrodynamic dispersion). It is shown that in all cases, the resulting equation is identical to that derived on the basis of an assumption of horizontal flow (the Dupuit assumption).Macrodispersion, occurring at the aquifer level, is discussed and appropriate coefficients are proposed.     

A derivation of the equations for transport of liquid and heat in three dimensions in a fractured porous medium

Governing equations can be derived for transient, three-dimensional transport of heat and mass in compressible, liquid saturated fractured porous media. Recently developed mathematical techniques can be used which relate local space averages of derivatives of medium properties to the derivatives of those averages. Using these techniques, well established thermomechanical transport equations which apply at microscopic points may be transformed into equations in macroscopic variables; i.e. in variables which pertain to the scale of observation. In the absence of chemical reactions, transfer between source entities and the medium may be taken care of in a consistent, physically realistic way, such that macroscopic source terms arise naturally in the course of the macroscopization procedures.     

Seismic wave equations in tight oil/gas sandstone media

Tight oil/gas medium is a special porous medium, which plays a significant role in oil and gas exploration. This paper is devoted to the derivation of wave equations in such a media, which take a much simpler form compared to the general equations in the poroelasticity theory and can be employed for parameter inversion from seismic data. We start with the fluid and solid motion equations at a pore scale, and deduce the complete Biot's equations by applying the volume averaging technique.The underlying assumptions are carefully clarified. Moreover, time dependence of the permeability in tight oil/gas media is discussed based on available results from rock physical experiments. Leveraging the Kozeny-Carman equation, time dependence of the porosity is theoretically investigated. We derive the wave equations in tight oil/gas media based on the complete Biot's equations under some reasonable assumptions on the media. The derived wave equations have the similar form as the diffusiveviscous wave equations. A comparison of the two sets of wave equations reveals explicit relations between the coefficients in diffusive-viscous wave equations and the measurable parameters for the tight oil/gas media. The derived equations are validated by numerical results. Based on the derived equations, reflection and transmission properties for a single tight interlayer are investigated. The numerical results demonstrate that the reflection and transmission of the seismic waves are affected by the thickness and attenuation of the interlayer, which is of great significance for the exploration of oil and gas.     

The dispersive flux in transport phenomena in porous media

Employing the principles of continuum mechanics and a volumetric averaging approach to the derivation of the macroscopic balance equation of an extensive quantity of a fluid phase in a porous medium, the paper derives a macroscopic expression for the dispersive flux that appears in the latter as a result of averaging. It is shown that the dispersive flux obeys a Fickian type law, i.e., it is proportional to the macroscopic density gradient of the considered extensive quantity. The nature of the coefficient of dispersion that appears in the expression of the dispersive flux is analyzed and interpreted.     

Efficient flow and transport simulations in reconstructed 3D pore geometries

Upscaling pore-scale processes into macroscopic quantities such as hydrodynamic dispersion is still not a straightforward matter for porous media with complex pore space geometries. Recently it has become possible to obtain very realistic 3D geometries for the pore system of real rocks using either numerical reconstruction or micro-CT measurements. In this work, we present a finite element–finite volume simulation method for modeling single-phase fluid flow and solute transport in experimentally obtained 3D pore geometries. Algebraic multigrid techniques and parallelization allow us to solve the Stokes and advection–diffusion equations on large meshes with several millions of elements. We apply this method in a proof-of-concept study of a digitized Fontainebleau sandstone sample. We use the calculated velocity to simulate pore-scale solute transport and diffusion. From this, we are able to calculate the a priori emergent macroscopic hydrodynamic dispersion coefficient of the porous medium for a given molecular diffusion D_m of the solute species. By performing this calculation at a range of flow rates, we can correctly predict all of the observed flow regimes from diffusion dominated to convection dominated.     

Immiscible two-phase fluid flows in deformable porous media

Macroscopic differential equations of mass and momentum balance for two immiscible fluids in a deformable porous medium are derived in an Eulerian framework using the continuum theory of mixtures. After inclusion of constitutive relationships, the resulting momentum balance equations feature terms characterizing the coupling among the fluid phases and the solid matrix caused by their relative accelerations. These terms, which imply a number of interesting phenomena, do not appear in current hydrologic models of subsurface multiphase flow. Our equations of momentum balance are shown to reduce to the Berryman–Thigpen–Chen model of bulk elastic wave propagation through unsaturated porous media after simplification (e.g., isothermal conditions, neglect of gravity, etc.) and under the assumption of constant volume fractions and material densities. When specialized to the case of a porous medium containing a single fluid and an elastic solid, our momentum balance equations reduce to the well-known Biot model of poroelasticity. We also show that mass balance alone is sufficient to derive the Biot model stress–strain relations, provided that a closure condition for porosity change suggested by de la Cruz and Spanos is invoked. Finally, a relation between elastic parameters and inertial coupling coefficients is derived that permits the partial differential equations of the Biot model to be decoupled into a telegraph equation and a wave equation whose respective dependent variables are two different linear combinations of the dilatations of the solid and the fluid.     

Electrical conduction and formation factor in isotropic porous media

A macroscopic form of Ohm's law is obtained for isotropic porous media saturated with an electrically conductive fluid by using volumetric averaging concepts. Closure of the macroscopic charge transport equation is aided by approximative modelling of the average geometric structures of three different types of isotropic porous media, namely foamlike materials, granular media and crossflow over prismatic bundles. Modelling of the microscopic charge transport necessitated the introduction of a representative interstitial flux of charge carriers and required quantification of the geometric tortuosity applicable to transport phenomena in general. Deterministic expressions for the formation factor are obtained and compare favourably with experimental results.     

Non-Fickian mass transport in fractured porous media

The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection-dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.     

Reactive transport in disordered media: Role of fluctuations in interpretation of laboratory experiments

We review the analysis of the dynamics of reactive transport in disordered media, emphasizing the nature of the chemical reactions and the role of small-scale fluctuations induced by the structure of the porous medium. We are motivated by results and interpretations of laboratory-scale experiments, for which detailed characterization of the system is possible. Modeling approaches based on continuum and particle tracking (PT) schemes are examined critically, highlighting how fluctuations are incorporated. The continuum approach spans a large literature. Traditional formats of reactive transport equations, such as the advection–dispersion–reaction equation (ADRE), are based on a series of assumptions related mainly to scale separation and relative magnitude of time scales involved in the reactive transport setting. These assumptions as well as further developments are assessed in depth. PT methods offer an alternative means of accounting for pore-scale dynamics, wherein space–time transitions are drawn from appropriate probability distributions that have been tested to account for anomalous transport. While PT methods have been employed for many years to describe conservative transport, their application to laboratory-scale reactive transport problems in the context of both Fickian and non-Fickian regimes is relatively recent. We concentrate on experimental observations of different types of reactions in disordered media: (1) the dynamics of a bimolecular reactive transport (A + B → C) in passive (non-reactive) media, and (2) a multi-step chemical reaction, as exemplified in the process of dedolomitization involving both dissolution and precipitation. The fluctuations in a number of the key variables controlling the processes prove to have a dominant role; elucidation of this role forms the basis of the present study and the comparison of methods.     

Upscaling geochemical reaction rates using pore-scale network modeling

Geochemical reaction rate laws are often measured using crushed minerals in well-mixed laboratory systems that are designed to eliminate mass transport limitations. Such rate laws are often used directly in reactive transport models to predict the reaction and transport of chemical species in consolidated porous media found in subsurface environments. Due to the inherent heterogeneities of porous media, such use of lab-measured rate laws may introduce errors, leading to a need to develop methods for upscaling reaction rates. In this work, we present a methodology for using pore-scale network modeling to investigate scaling effects in geochemical reaction rates. The reactive transport processes are simulated at the pore scale, accounting for heterogeneities of both physical and mineral properties. Mass balance principles are then used to calculate reaction rates at the continuum scale. To examine the scaling behavior of reaction kinetics, these continuum-scale rates from the network model are compared to the rates calculated by directly using laboratory-measured reaction rate laws and ignoring pore-scale heterogeneities. In this work, this methodology is demonstrated by upscaling anorthite and kaolinite reaction rates under simulation conditions relevant to geological CO₂ sequestration. Simulation results show that under conditions with CO₂ present at high concentrations, pore-scale concentrations of reactive species and reaction rates vary spatially by orders of magnitude, and the scaling effect is significant. With a much smaller CO₂ concentration, the scaling effect is relatively small. These results indicate that the increased acidity associated with geological sequestration can generate conditions for which proper scaling tools are yet to be developed. This work demonstrates the use of pore-scale network modeling as a valuable research tool for examining upscaling of geochemical kinetics. The pore-scale model allows the effects of pore-scale heterogeneities to be integrated into system behavior at multiple scales, thereby identifying important factors that contribute to the scaling effect.     

General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow

Equations which describe single phase fluid flow and transport through an elastic porous media are obtained by applying constitutive theory to a set of general multiphase mass, momentum, energy, and entropy equations. Linearization of these equations yields a set of equations solvable upon specification of the material coefficients which arise. Further restriction of the flow to small velocities proves that Darcy's law is a special case of the general momentum balance.     

An equivalent medium model for wave simulation in fractured porous rocks

Seismic wave propagation in reservoir rocks is often strongly affected by fractures and micropores. Elastic properties of fractured reservoirs are studied using a fractured porous rock model, in which fractures are considered to be embedded in a homogeneous porous background. The paper presents an equivalent media model for fractured porous rocks. Fractures are described in a stress‐strain relationship in terms of fracture‐induced anisotropy. The equations of poroelasticity are used to describe the background porous matrix and the contents of the fractures are inserted into a matrix. Based on the fractured equivalent‐medium theory and Biot's equations of poroelasticity, two sets of porosity are considered in a constitutive equation. The porous matrix permeability and fracture permeability are analysed by using the continuum media seepage theory in equations of motion. We then design a fractured porous equivalent medium and derive the modified effective constants for low‐frequency elastic constants due to the presence of fractures. The expressions of elastic constants are concise and are directly related to the properties of the main porous matrix, the inserted fractures and the pore fluid. The phase velocity and attenuation of the fractured porous equivalent media are investigated based on this model. Numerical simulations are performed. We show that the fractures and pores strongly influence wave propagation, induce anisotropy and cause poroelastic behaviour in the wavefields. We observe that the presence of fractures gives rise to changes in phase velocity and attenuation, especially for the slow P‐wave in the direction parallel to the fracture plane.     

Variable-density flow and transport in porous media: approaches and challenges

We review the state of the art in modeling of variable-density flow and transport in porous media, including conceptual models for convection systems, governing balance equations, phenomenological laws, constitutive relations for fluid density and viscosity, and numerical methods for solving the resulting nonlinear multifield problems. The discussion of numerical methods addresses strategies for solving the coupled spatio-temporal convection process, consistent velocity approximation, and error-based mesh adaptation techniques. As numerical models for those nonlinear systems must be carefully verified in appropriate tests, we discuss weaknesses and inconsistencies of current model-verification methods as well as benchmark solutions. We give examples of field-related applications to illustrate specific challenges of further research, where heterogeneities and large scales are important.     

Direct numerical simulations of interface dynamics to link capillary pressure and total surface energy

The flow of two immiscible fluids through a porous medium depends on the complex interplay between gravity, capillarity, and viscous forces. The interaction between these forces and the geometry of the medium gives rise to a variety of complex flow regimes that are difficult to describe using continuum models. Although a number of pore-scale models have been employed, a careful investigation of the macroscopic effects of pore-scale processes requires methods based on conservation principles in order to reduce the number of modeling assumptions. In this work we perform direct numerical simulations of drainage by solving Navier–Stokes equations in the pore space and employing the Volume Of Fluid (VOF) method to track the evolution of the fluid–fluid interface. After demonstrating that the method is able to deal with large viscosity contrasts and model the transition from stable flow to viscous fingering, we focus on the macroscopic capillary pressure and we compare different definitions of this quantity under quasi-static and dynamic conditions. We show that the difference between the intrinsic phase-average pressures, which is commonly used as definition of Darcy-scale capillary pressure, is subject to several limitations and it is not accurate in presence of viscous effects or trapping. In contrast, a definition based on the variation of the total surface energy provides an accurate estimate of the macroscopic capillary pressure. This definition, which links the capillary pressure to its physical origin, allows a better separation of viscous effects and does not depend on the presence of trapped fluid clusters.     

Scaling of capillary,gravity and viscous forces affecting flow morphology in unsaturated porous media

Interplay between capillary, gravity and viscous forces in unsaturated porous media gives rise to a range of complex flow phenomena affecting morphology, stability and dynamics of wetting and drainage fronts. Similar average phase contents may result in significantly different fluid distribution and patterns affecting macroscopic transport properties of the unsaturated medium. The formulation of general force balance within simplified pore spaces yields scaling relationships for motion of liquid elements in which gravitational force in excess of capillary pinning force scales linearly with viscous force. Displacement fluid front morphology is described using dimensionless force ratios expressed as Bond and Capillary numbers. The concise representations of a wide range of flow regimes with scaling relations, and predictive capabilities of front morphology based on dimensionless numbers lend support to certain generalizations. Considering available experimental data, we are able to define conditions for onset of unstable and intermittent flows leading to enhanced liquid and gas entrapment. These results provide a basis for delineation of a tentative value of Bo ∼ 0.05 as an upper limit of applicability of the Richards equation (at pore to sample scales) and related continuum-based flow models.     

Experimental and numerical study of the validity of Hele–Shaw cell as analogue model for variable-density flow in homogeneous porous media

Several laboratory experiments were conducted to identify the validity domain under which a Hele–Shaw cell may serve as a suitable analogue for variable-density flow in homogeneous porous media. These experiments are concerned with the injection into a Hele–Shaw cell of a salt solute at different concentrations and flow rates. The experimental data analysis highlighted two types of mixing zone shape: with and without ‘fingers’. A semi-empirical criterion based on the ratio between gravitational and injected velocities was used to forecast the change from one shape to another. The experimental data were then analysed using numerical solutions of the classical Hele–Shaw equations by taking into account an anisotropic dispersion tensor whose components depend on fluid density gradients. The good agreement between experimental and numerical results clearly shows that the validity of the concentration-dependent dispersion tensor strongly depends on the local Péclet number variation. For Péclet numbers lower 50, the Hele–Shaw cell can be considered as an analogous model of a homogeneous and isotropic 2D porous medium. It can be successfully used to study, at the laboratory scale, the gravitational instability effects induced by flow and transport phenomena into a porous medium.