A Non-isothermal Pore Network Drying Model with Gravity Effect

The concept of immiscible displacement as an invasion percolation (IP) process driven by heat and mass transfer is used in a pore network model for convective drying of capillary porous media. The coupling between heat and mass transfer occurs at the liquid–gas interface through temperature-dependent equilibrium vapor pressure and surface tension as well as the phase change enthalpy (in evaporation and condensation). The interfacial effects due to capillary forces and gravity are combined in an invasion potential; viscous forces are neglected. Simulation results show stabilized invasion patterns and finite drying front width by the influence of gravity.     

A Triple Pore Network Model (T-PNM) for Gas Flow Simulation in Fractured,Micro-porous and Meso-porous Media

Transport in Porous Media - In this study, a novel triple pore network model (T-PNM) is introduced which is composed of a single pore network model (PNM) coupled to fractures and micro-porosities....     

Pore Network Analysis of Resistivity Index for Water-Wet Porous Media

Pore network analysis is used to investigate the effects of microscopic parameters of the pore structure such as pore geometry, pore-size distribution, pore space topology and fractal roughness porosity on resistivity index curves of strongly water-wet porous media. The pore structure is represented by a three-dimensional network of lamellar capillary tubes with fractal roughness features along their pore-walls. Oil-water drainage (conventional porous plate method) is simulated with a bond percolation-and-fractal roughness model without trapping of wetting fluid. The resistivity index, saturation exponent and capillary pressure are expressed as approximate functions of the pore network parameters by adopting some simplifying assumptions and using effective medium approximation, universal scaling laws of percolation theory and fractal geometry. Some new phenomenological models of resistivity index curves of porous media are derived. Finally, the eventual changes of resistivity index caused by the permanent entrapment of wetting fluid in the pore network are also studied.Resistivity index and saturation exponent are decreasing functions of the degree of correlation between pore volume and pore size as well as the width of the pore size distribution, whereas they are independent on the mean pore size. At low water saturations, the saturation exponent decreases or increases for pore systems of low or high fractal roughness porosity respectively, and obtains finite values only when the wetting fluid is not trapped in the pore network. The dependence of saturation exponent on water saturation weakens for strong correlation between pore volume and pore size, high network connectivity, medium pore-wall roughness porosity and medium width of the pore size distribution. The resistivity index can be described succesfully by generalized 3-parameter power functions of water saturation where the parameter values are related closely with the geometrical, topological and fractal properties of the pore structure.     

A Dynamic Network Model for Two-Phase Flow in Porous Media

We present a dynamic model of immiscible two-phase flow in a network representation of a porous medium. The model is based on the governing equations describing two-phase flow in porous media, and can handle both drainage, imbibition, and steady-state displacement. Dynamic wetting layers in corners of the pore space are incorporated, with focus on modeling resistivity measurements on saturated rocks at different capillary numbers. The flow simulations are performed on a realistic network of a sandpack which is perfectly water-wet. Our numerical results show saturation profiles for imbibition in agreement with experiments. For free spontaneous imbibition we find that the imbibition rate follows the Washburn relation, i.e., the water saturation increases proportionally to the square root of time. We also reproduce rate effects in the resistivity index for drainage and imbibition.     

A Dynamic Pore Network Model for Oil Displacement by Wettability-Altering Surfactant Solution

A dynamic pore network model, capable of predicting the displacement of oil from a porous medium by a wettability-altering and interfacial tension reducing surfactant solution, is presented. The key ingredients of the model are (1) a dynamic network model for the displacement of oil by aqueous phase taking account of capillary and viscous effects, (2) a simulation of the transport of surfactant through the network by advection and diffusion taking account of adsorption on the solid surface, and (3) the coupling of these two by linking the contact angle and interfacial tension appearing in the dynamic network simulation to the local concentration of surfactant computed in the transport simulation. The coupling is two-way: The flow field used to advect the surfactant concentration is that associated with the displacement of oil by the injected aqueous phase, and the surfactant concentration influences the flow field through its effect on the capillarity parameters. We present results obtained using the model to validate that it reproduces the displacement patterns observed by other authors in two-dimensional networks as capillary number and mobility ratio are varied, and to illustrate the effects of surfactant on displacement patterns. A mechanism is demonstrated whereby in an initially mixed-wet medium, surfactant-induced wettability alteration can lead to stabilization of displacement fronts.     

Smoothed Particle Hydrodynamics Model for Diffusion through Porous Media

A smoothed particle hydrodynamics (SPH) model is presented for the study of diffusion in spatially periodic porous media. The method of SPH is formulated to solve the convection–diffusion equation for tracer diffusion under steady state and transient conditions. Solutions obtained using SPH are compared with other available solutions and the model is used to calculate diffusion coefficients of spatially periodic porous media for the steady state diffusion problem. Diffusion coefficients are then used to calculate nondimensional diffusivities of the media. The effects of media properties on the values of nondimensional diffusivity are also presented.     

Computing the Longtime Behaviour of NMR Propagators in Porous Media Using a Pore Network Random Walk Model

We present a pore network model combined with a random walk algorithm allowing the simulation of molecular displacement distributions in porous media as measured by NMR. A particular feature of this technique is the ability to probe the time evolution of these distributions. The objective is to predict the displacement behaviour for time intervals larger than the experimental observation time and explore the asymptotic dispersion regime at long times. Starting from 3D micro-CT images, we computed the variance of displacement distributions of water molecules in a Fontainebleau sand and found very good agreement of the time evolution of the variance with experimental data, without fitting parameter. The model confirms a weak superdispersion in the asymptotic regime. In addition, we conclude that, since pore network models do not take into account small scale features of the porous medium (e.g., surface roughness and grain shape), the origin of the observed superdispersion is mainly due to the topology and geometry of the porous medium.     

A Process-Based Pore Network Model Construction for Granular Packings Under Large Plastic Deformations

Transport in Porous Media - We propose a process-based method for constructing a pore network model of granular packings under large deformations. The method uses the radical Voronoi tessellation...     

Predicting Vertical Flow Barriers Using Tracer Diffusion in Partially Saturated,Layered Porous Media

Sudden changes in isotopic tracer concentration in pore waters have been interpreted as indicating barriers to vertical advective flow through porous rocks in the subsurface, e.g. step changes in \(^{87}\hbox {Sr}/^{86}\) Sr ratio are often used in the oil and gas industry as a signature of reservoir compartmentalisation. This study shows that this is not necessarily the case. It can take millions of years for such step changes to equilibrate by diffusion if there is no flow resulting from pressure or density gradients even in high permeability, high porosity rocks, particularly if the water saturation is low. Changes in tracer concentration gradients can be good indicators of changes in porosity (or water saturation) between layers. In contrast changes in sorption without a change in porosity are almost impossible to identify. The time taken for concentration gradients to equilibrate is affected by the layer properties but can be quickly estimated from the harmonic average of the effective diffusion coefficient for each layer and a simple analytical expression for a homogeneous system. This was achieved by performing a sensitivity analysis on different layer properties (porosity contrast, saturation contrast, sorption contrast, thickness ratio) using existing analytical solutions for diffusion in layered systems.     

A (Dual) Network Model for Heat Transfer in Porous Media

Transport in Porous Media - We present a dual network model to simulate coupled single-phase flow and energy transport in porous media including conditions under which local thermal equilibrium...     

A Geometrical Pore Model for Estimating the Microscopical Pore Geometry of Soil with Infiltration Measurements

This paper presents a simple geometrical pore model designed to relate characteristic pore radii of the porous network of soils with macroscopic infiltration parameters. The model composed of a stack of spherical hollow elements is described with two radii values: the pore access radius and the actual pore radius. The model was compared to cylindrical pore models and its mathematical consistency was assessed. Soil sorptivity S and the second parameter A of the Philip infiltration equation (1957), have been determined by numerically simulated infiltration. A diagram and an empirical relation have been set in order to relate the pore access and pore radii to the infiltration parameters S and A. The consistency of the model was validated by comparing the predicted sorptivity and hydraulic conductivity values, with the widely used unsaturated soil hydraulic functions (van Genuchten, 1980). The model showed good agreement with experimental infiltration data, and it is therefore concluded that the use of a model with two radii improves the relation between microscopic pore size and macroscopic infiltration parameters.     

Experimental Investigation of Transitional Flow in Porous Media with Usage of a Pore Doublet Model

The transition from laminar to turbulent flow in porous media is studied with a new method. To mimic interconnected pores, a simplified geometry consisting of a pipe with a relatively large diameter that is split into two parallel pipes with different diameters is studied. This is a pore-doublet setup and the pressure drops over the parallel pipes are recorded by pressure transducers for different flow rates. Results show that the flow in the parallel pipes is redistributed when turbulent slugs pass through one of them. The presence of the slugs is revealed by positive skewness in the pressure signals as well as an increase of the standard deviation of the pressure drops and correlation between the pressure drops of the pipes. A frequency analysis of the pressure drops show that lower band frequency pressure variations in one pipe are communicated to the other pipe while higher band frequencies are filtered out.     

A Multiscale Network Model for Simulating Moisture Transfer Properties of Porous Media

A multiscale network model is presented to model unsaturated moisture transfer in hygroscopic capillary-porous materials showing a broad pore-size distribution. Both capillary effects and water sorption phenomena, water vapour and liquid water transfer are considered. The multiscale approach is based on the concept of examining the porous space at different levels of magnification. The conservation of the water vapour permeability of dry material is used as scaling criterion to link the different pore scales. A macroscopic permeability is deduced from the permeabilities calculated at the different levels of magnification. Each level of magnification is modelled using an isotropic nonplanar 2D cross-squared network. The multiscale network simulates the enhancement of water vapour permeability due to capillary condensation, the hysteresis phenomenon between wetting and drying, and the steep increase of moisture permeability at the critical moisture saturation level. The calculated network permeabilities are compared with experimental data for calcium silicate and ceramic brick and a good agreement is observed.     

Cross Effects in Solid Media with Diffusion

Based on thermodynamics of irreversible processes, possible cross effects in isothermal diffusion in solids with internal surfaces are analyzed. Expressions for the fluxes of components and for the fluxes of a newly introduced thermodynamic parameter, specific area of internal surfaces per unit volume or mass of the medium, are obtained. An analysis of these relations and relations between the stresses and strains for various cases is given. In derivation of the governing relations, it was assumed that diffusion proceeds through interstitial sites.     

Capillary Hysteresis in Neutrally Wettable Fibrous Media: A Pore Network Study of a Fuel Cell Electrode

Hysteresis in the saturation versus capillary pressure curves of neutrally wettable fibrous media was simulated with a random pore network model using a Voronoi diagram approach. The network was calibrated to fit experimental air-water capillary pressure data collected for carbon fibre paper commonly used as a gas diffusion layer in fuel cells. These materials exhibit unusually strong capillary hysteresis, to the extent that water injection and withdrawal occur at positive and negative capillary pressures, respectively. Without the need to invoke contact angle hysteresis, this capillary behaviour is re-produced when using a pore-scale model based on the curvature of a meniscus passing through the centre of a toroid. The classic Washburn relation was shown to produce erroneous results, and its use is not recommended when modelling fibrous media. The important effect of saturation distribution on the effective diffusivity of the medium was also investigated for both water injection and withdrawal cases. The findings have bearing on the understanding of both capillarity in fibrous media and fuel cell design.     

Reliability of Algorithms Interpreting Topological and Geometric Properties of Porous Media for Pore Network Modelling

Pore network models (PNMs) offer a computationally efficient way to analyse transport in porous media. Their effectiveness depends on how well they represent the topology and geometry of real pore systems, for example as imaged by X-ray CT. The performance of two popular algorithms, maximum ball and watershed, is evaluated for three porous systems: an idealised medium with known pore throat properties and two rocks with different morphogenesis—carbonate and sandstone. It is demonstrated that while the extracted PNM simulates simple flow (permeability) with acceptable accuracy, their topological and geometric properties are significantly different. This suggests that such PNM may not serve more complex studies, such as reactive/convective transport of contaminants or bacteria, and further research is necessary to improve the interpretation of real pore spaces with networks. Linear topology–geometry relations are derived and presented to stimulate development of more realistic PNM.

     

Macroscopic Two-phase Flow in Porous Media Assuming the Diffuse-interface Model at Pore Level

The paper presents a model for two-phase flow, where liquid and gas are treated as one fluid with variable density. A one-component fluid and the diffuse-interface model for two-phase flow are assumed at pore level. The wetting properties of the fluid are described by the Cahn theory. Macroscopic equations are deduced in the framework of the Marle formalism. It is shown that two-phase flow in porous media can be described by the Cahn–Hilliard equation for the mass density. The concept of relative permeability is not needed. For non-neutral wetting, it is shown that a capillary pressure exists but that it is not a function of state. Two numerical illustrations are presented, one of them showing that the model is, at least in a simple steady-state situation, compatible with the generalized two-continuum model.     

Pore Network Modelling of Electrical Resistivity and Capillary Pressure Characteristics

In order to model petrophysical properties of hydrocarbon reservoir rocks, the underlying physics occurring in realistic rock pore structures must be captured. Experimental evidence showing variations of wetting occurring within a pore, and existence of the so-called 'non-Archie' behaviour, has led to numerical models using pore shapes with crevices (for example, square, elliptic, star-like shapes, etc.). This paper presents theoretical derivations and simulation results of a new pore space network model for the prediction of petrophysical properties of reservoir rocks. The effects of key pore geometrical factors such as pore shape, pore size distribution and pore co-ordination number (pore connectivity) have been incorporated into the theoretical model. In particular, the model is used to investigate the effects of wettability and saturation history on electrical resistivity and capillary pressure characteristics. The petrophysical characteristics were simulated for reservoir rock samples. The use of the more realistic grain boundary pore (GBP) shape allows simulation of the generic behaviour of sandstone rocks, with various wetting scenarios. The predictions are in close agreement with electrical resistivity and capillary pressure characteristics observed in experiments.