Beyond Kozeny–Carman: Predicting the Permeability in Porous Media

Transport in Porous Media - Various processes such as heterogeneous reactions or biofilm growth alter a porous medium’s underlying geometric structure. This significantly affects its...     

An Analytical Model of Porosity–Permeability for Porous and Fractured Media

The classic Kozeny–Carman equation (KC) uses parameters that are empirically based or not readily measureable for predicting the permeability of unfractured porous media. Numerous published KC modifications share this disadvantage, which potentially limits the range of conditions under which the equations are applicable. It is not straightforward to formulate non-empirical general approaches due to the challenges of representing complex pore and fracture networks. Fractal-based expressions are increasingly popular in this regard, but have not yet been applied accurately and without empirical constants to estimating rock permeability. This study introduces a general non-empirical analytical KC-type expression for predicting matrix and fracture permeability during single-phase flow. It uses fractal methods to characterize geometric factors such as pore connectivity, non-uniform grain or crystal size distribution, pore arrangement, and fracture distribution in relation to pore distribution. Advances include (i) modification of the fractal approach used by Yu and coworkers for industrial applications to formulate KC-type expressions that are consistent with pore size observations on rocks. (ii) Consideration of cross-flow between pores that adhere to a fractal size distribution. (iii) Extension of the classic KC equation to fractured media absent empirical constants, a particular contribution of the study. Predictions based on the novel expression correspond well to measured matrix and fracture permeability data from natural sandstone and carbonate rocks, although the currently available dataset for fractures is sparse. The correspondence between model calculation results and matrix data is better than for existing models.     

Porosity–Permeability Relations for Evolving Pore Space: A Review with a Focus on (Bio-)geochemically Altered Porous Media

Reactive transport processes in a porous medium will often both cause changes to the pore structure, via precipitation and dissolution of biomass or minerals, and be affected by these changes, via changes to the material’s porosity and permeability. An understanding of the pore structure morphology and the changes to flow parameters during these processes is critical when modeling reactive transport. Commonly applied porosity–permeability relations in simulation models on the REV scale use a power-law relation, often with slight modifications, to describe such features; they are often used for modeling the effects of mineral precipitation and/or dissolution on permeability. To predict the reduction in permeability due to biomass growth, many different and often rather complex relations have been developed and published by a variety of authors. Some authors use exponential or simplified Kozeny–Carman relations. However, many of these relations do not lead to fundamentally different predictions of permeability alteration when compared to a simple power-law relation with a suitable exponent. Exceptions to this general trend are only few of the porosity–permeability relations developed for biomass clogging; these consider a residual permeability even when the pore space is completely filled with biomass. Other exceptions are relations that consider a critical porosity at which the porous medium becomes impermeable; this is often used when modeling the effect of mineral precipitation. This review first defines the scale on which porosity–permeability relations are typically used and aims at explaining why these relations are not unique. It shows the variety of existing approaches and concludes with their essential features.     

Simulation of Structure Change in Porous Media During Gas–Solid Reactions Using Cellular Automata Model

Transport in Porous Media - In some gas–solid reactions, a new solid substance is produced. The product acts as a shield and prevents the collision between gas and solid reactants which...     

A Three-Dimensional Model of Two-Phase Flows in a Porous Medium Accounting for Motion of the Liquid–Liquid Interface

A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation.     

Universal Frequency-Dependent Permeability of Heterogeneous Porous Media: Effective–Medium Approximation and Critical-Path Analysis

Transport in Porous Media - Experiments, numerical simulations, and analytical models for simple models of porous media, such as a single pore and spatially-periodic models, have provided evidence...     

Pore-Scale Imaging and Modelling of Reactive Flow in Evolving Porous Media: Tracking the Dynamics of the Fluid–Rock Interface

Transport in Porous Media - Fluid–mineral and fluid–rock interfaces are key parameters controlling the reactivity and fate of fluids in reservoir rocks and aquifers. The interface...     

A Fractal Model for Gas–Water Relative Permeability in Inorganic Shale with Nanoscale Pores

A reliable gas–water relative permeability model in shale is extremely important for the accurate numerical simulation of gas–water two-phase flow (e.g., fracturing fluid flowback) in gas-shale reservoirs, which has important implication for the economic development of gas-shale reservoir. A gas–water relative permeability model in inorganic shale with nanoscale pores at laboratory condition and reservoir condition was proposed based on the fractal scaling theory and modified non-slip boundary of continuity equation in the nanotube. The model not only considers the gas slippage in the entire Knudsen regime, multilayer sticking (near-wall high-viscosity water) and the quantified thickness of water film, but also combines the real gas effect and stress dependence effect. The presented model has been validated by various experiments data of sandstone with microscale pores and bulk shale with nanoscale pores. The results show that: (1) The Knudsen diffusion and slippage effects enhance the gas relative permeability dramatically; however, it is not obviously affected at high pressure. (2) The multilayer sticking effect and water film should not be neglected: the multilayer sticking would reduce the water relative permeability as well as slightly decrease gas relative permeability, and the film flow has a negative impact on both of the gas and water relative permeability. (3) The increased fractal dimension for pore size distribution or tortuosity would increase gas relative permeability but decrease the water relative permeability for a given saturation; however, the effect on relative permeability is not that notable. (4) The real gas effect is beneficial for the gas relative permeability, and the influence is considerable when the pressure is high enough and when the nanopores of bulk shale are mostly with smaller size. For the stress dependence, not like the intrinsic permeability, none of the gas or water relative permeability is sensitive to the net pressure and it can be ignored completely.     

Homogenization of a Hele–Shaw Problem in Periodic and Random Media

We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary velocity, and we establish the uniform convergence of the free boundaries.     

Double-Diffusive Natural Convection Flows with Thermosolutal Symmetry in Porous Media in the Presence of the Soret–Dufour Effects

The double-diffusive natural convection past a vertical plate embedded in a fluid-saturated porous medium is considered in the boundary-layer and Boussinesq approximations. It is assumed that the Soret–Dufour cross-diffusion effects are significant. The heat and mass fluxes on the plate are prescribed as functions of the surface coordinate x. The general similarity reduction of the problem for power-law and exponential variation of the wall fluxes is given. In the case of thermosolutal symmetry, when the similar temperature and concentration fields become coincident, exact analytical as well as numerical solutions are reported and discussed in some detail. For the flows without thermosolutal symmetry, the final similarity equations have been solved numerically, by paying attention to the influence of the Soret and Dufour numbers on the departure from thermosolutal symmetry. The reported results focus on the wall temperatures and concentrations, whose reciprocals are Nusselt and Sherwood numbers, respectively.     

Influence of Viscous Fingering on Dynamic Saturation–Pressure Curves in Porous Media

We report on results from primary drainage experiments on quasi-two-dimensional porous models. The models are transparent, allowing the displacement process and structure to be monitored in space and time during primary drainage experiments carried out at various speeds. By combining detailed information on the displacement structure with global measurements of pressure, saturation and the capillary number Ca, we obtain a scaling relation relating pressure, saturation, system size and capillary number. This scaling relation allows pressure–saturation curves for a wide range of capillary numbers to be collapsed on the same master curve. We also show that in the case of primary drainage, the dynamic effect in the capillary pressure–saturation relationship observed on partially water saturated soil samples might be explained by the combined effect of capillary pressure along the invasion front of the gaseous phase, and pressure changes caused by viscous effects in the wetting fluid phase.     

Infiltration of Salt Solute in Homogeneous and Saturated Porous Media – An Analytical Solution Evaluated by Numerical Simulations

In many cases various land disposal activities (e.g. infiltration, injection wells) constitute an important potential source of groundwater contamination. Using a 2D physical model, the behaviour of the infiltration of a salt solute, locally injected in a homogeneous and saturated porous medium, has been analysed. Under various experimental conditions (density effects, injection flow rate) the salt solute penetrates the porous media and leads to a steady-state regime inside the mixing zone. By using experimental observations, the basic equations describing the flow and transport phenomena can be simplified and an analytical solution obtained. Its validity is subject to numerical verification. The numerical model, based on the development of the mass balance equation expressed by its conservative form, uses a combination of the mixed hybrid finite element (MHFE) and discontinuous finite element (DFE) methods. The efficiency of this numerical model was previously verified on standard benchmarks, for example Elder's problem and Henry's problem. In the first step, the qualitative good agreement between the experimental and numerical results enabled us to use the numerical model in order to verify some hypotheses resulting from visual observations. Thus, the numerical results reveal the existence of a steady-state regime inside the mixing zones. Nevertheless, both its vertical and longitudinal extensions are less than those observed in the physical model. In the second step, the numerical results enable to establish the validity domain as well as the accuracy of the proposed analytical solution.     

Analytical Modeling of Flow Behavior for Wormholes in Naturally Fractured–Vuggy Porous Media

Acidizing technology has been widely applied when developing naturally fractured–vuggy reservoirs. So testing and evaluating acidizing wells’ pressure behavior become necessary for further improving the wells’ performance. Analyzing transient pressure data can estimate some key reservoir parameters. Generally speaking, carbonate minerals are usually composed of dolomite and calcite which are easy to be dissolved by hydrochloric acid which is often used to react with the rock to create a high conductivity channel, namely wormhole. Pressure transient behavior in fractured–vuggy reservoirs has been studied for many years; however, the models of acidizing wells with wormholes were not reported in previous studies. This article presented an analytical model for wormholes in naturally fractured–vuggy carbonate reservoirs, and wormholes solutions were obtained through point sink integral method. The results were validated accurately by comparing with previous results and numerical simulation. Then in this paper, type curves were established to recognize the flow characteristics, and flow was divided into six flow regimes comprehensively. The calculative results showed that the characteristics of type curves were influenced by inter-porosity flow factor, wormhole number, fluids capacitance coefficient. We also showed that the pressure behavior was affected by the angles between wormholes, and the pressure depletion increased as the angle decreased, because the wormholes were closer, their interaction became stronger. At the end, a reservoir example was showed to demonstrate the methodology of new type curve analysis.     

The Use of Chemical Potential to Describe Water Transfer in Complex Media with Strong Solid–Liquid Bonding

We consider a complex medium composed of finely intertwined micro-skeletons and micro-compartments where water transfer can occur. In these media, at low moisture content, water pressure measurement is not longer possible. Mass transfer is then expressed in terms of chemical potential gradient. The assumption of local thermodynamic equilibrium, resulting in the uniformity of water chemical potential in all microstructures, is essential to define a sorption isotherm reflecting the relationship between water activity and average moisture content. In this case, it is also possible to describe water transfer by using the chemical potential gradient. Radial water transfer in wood is examined using a destructive method for calculating water flux and chemical potential gradient at the same position and at the same time. We deduce the variation of transport coefficient as a function of moisture content.     

Transport of Polymer Particles in Oil–Water Flow in Porous Media: Enhancing Oil Recovery

Transport in Porous Media - 3D printing with powders offers the most analogous method to the natural way in which clastic reservoir rocks are formed, resulting in pore network textures and...     

Impact of the Buoyancy–Viscous Force Balance on Two-Phase Flow in Layered Porous Media

Motivated by geological carbon storage and hydrocarbon recovery, the effect of buoyancy and viscous forces on the displacement of one fluid by a second immiscible fluid, along parallel and dipping layers of contrasting permeability, is characterized using five independent dimensionless numbers and a dimensionless storage or recovery efficiency. Application of simple dimensionless models shows that increased longitudinal buoyancy effects increase storage efficiency by reducing the distance between the leading edges of the injected phase in each layer and decreasing the residual displaced phase saturation behind the leading edge of the displacing phase. Increased transverse buoyancy crossflow increases storage efficiency if it competes with permeability layering effects, but reduces storage efficiency otherwise. When both longitudinal and transverse buoyancy effects are varied simultaneously, a purely geometrical dip angle group defines whether changes in storage efficiency are dominated by changes in the longitudinal or transverse buoyancy effects. In the limit of buoyancy-segregated flow, we report an equivalent, unidimensional flow model which allows rapid prediction of storage efficiency. The model presented accounts for both dip and layering, thereby generalizing earlier work which accounted for each of these but not both together. We suggest that the predicted storage efficiency can be used to compare and rank geostatistical realizations, and complements earlier heterogeneity measures which are applicable in the viscous limit.