Saturation–pressure relationships for two- and three-phase flow analogies for soft matter

Soft matter and porous media analogies are used to derive micro-scale informed pressure–saturation relationships for cell aggregates. These aggregates may consist of different cell types and an interstitial liquid. The extracellular matrix is the scaffold for these soft materials and is first taken as rigid. Extension to deformable material is also addressed.In tissue mechanics, micro-scale formulations are very often not sustainable from the computational point of view due to the complexity of the pore scale phase distributions and the large size of many problems of interest. Hence, models are formulated mainly at a larger scale, called the macro scale. We derive the relationships at that scale by exploiting theoretical results from two phase flow in porous media which incorporate information from the micro-scale. An example with an indirect validation of the obtained relationship is shown.     

Microscale Visual Study of End Effects at Permeability Discontinuities

The physical effect of multiphase fluid distribution and flow at permeability boundaries has not been fully investigated, particularly at the pore scale (1–100 μm), although such behaviour can significantly affect the overall scaled-up reservoir trapping capacity and production performance. In this article, microscale physical models have been used to qualitatively study the pore scale flow events at permeability boundaries, both high to low and vice versa, to gain a better understanding of the role of these boundaries and water saturation on multiphase displacement behaviour at the pore scale. We have used etched glass models of stripes of large and small (a factor of two) pores with circular matrix. Capillary pressure, which is the controlling parameter is itself dependant on pore size and its spatial distribution, the magnitude of the interfacial tensions and the wettability between the fluids and the solid surface of the models. Sometimes, the only way the non-wetting fluid can penetrate the boundary is through a fortuitous leakage, whereby the presence of an initial saturation reduces the controlling capillary pressure. Examples are demonstrated including mechanisms of end-effects and how capillary boundary resistance (due to capillary forces) can be broken down and fluid movement across the boundary can develop. These micromodel experiments show vividly that connate water can assist in these processes, particularly oil trapping and leakage of water across a permeability boundary.     

Multi-scale Optical Analyses of Dynamic Gas Saturation During Air Sparging into Glass Beads

A multi-scale optical imaging technique was developed allowing for the 2D observation of two phase flow in porous media at two different scales simultaneously: Using two coupled cameras, a 2D flow cell (0.5 × 0.5m²) is recorded entirely at the bench scale and at the pore scale with a spatial resolution of 0.5 and 0.01 mm, respectively. The technique is applied to study channelized gas flow in saturated glass beads. We analyze the phase distribution at the pore scale and derive a pixel-based method for the measurement of saturation at the larger scale. This method assumes linearity between the mean reflected light intensity and the local gas saturation if averaging is performed over representative areas (REV). The REV depends on the irregularity of the local pore structure and has a lower limit at the correlation length of the porous medium (somewhat above the size of the glass beads) and an upper limit which correlates with the width of gas channels. These limits could be quantified through optical analysis. The optical approach to estimate phase saturations was validated by gravimetric analysis where a characteristic ratio between the optically observed flow cell wall and the saturation within the bulk material was identified, which corresponds to the expectation based on geometrical considerations of the glass bead packing. Considering a transient flow experiment the optical method is demonstrated to be able to quantify the temporal evolution of the residual and the convective gas phase. We conclude that the new technique provides a valuable tool to improve our quantitative understanding of multiphase phenomena across different scales.     

A PDF propagation approach to model turbulent dispersion in swirling flows

A computationally efficient approach that solves for the spatial covariance matrix along the dense particle ensemble-averaged trajectory has been successfully applied to describe turbulent dispersion in swirling flows. The procedure to solve for the spatial covariance matrix is based on turbulence isotropy assumption, and it is analogous to Taylor's approach for turbulent dispersion. Unlike stochastic dispersion models, this approach does not involve computing a large number of individual particle trajectories in order to adequately represent the particle phase; a few representative particle ensembles are sufficient to describe turbulent dispersion. The particle Lagrangian properties required in this method are based on a previous study (Shirolkar and McQuay, 1998). The fluid phase information available from practical turbulence models is sufficient to estimate the time and length scales in the model. In this study, two different turbulence models are used to solve for the fluid phase – the standard k–ε model, and a multiple-time-scale (MTS) model. The models developed here are evaluated with the experiments of Sommerfeld and Qiu (1991). A direct comparison between the dispersion model developed in this study and a stochastic dispersion model based on the eddy lifetime concept is also provided. Estimates for the Reynolds stresses required in the stochastic model are obtained from a set of second-order algebraic relations. The results presented in the study demonstrate the computational efficiency of the present dispersion modeling approach. The results also show that the MTS model provides improved single-phase results in comparison to the k–ε model. The particle statistics, which are computed based on the fundamentals of the present approach, compare favorably with the experimental data. Furthermore, these statistics closely compare to those obtained using a stochastic dispersion model. Finally, the results indicate that the particle predictions are relatively unaffected by whether the Reynolds stresses are based on algebraic relations or on the turbulence isotropy assumption.     

Effect of Mean Network Coordination Number on Dispersivity Characteristics

In this study, we investigate the role of topology on the macroscopic (centimeter scale) dispersion characteristics derived from pore-network models. We consider 3D random porous networks extracted from a regular cubic lattice with coordination number distributed in accordance with real porous structures. We use physically consistent rules including ideal mixing in pore bodies, molecular diffusion, and Taylor dispersion in pore throats to simulate transport at the pore-scale level. Fundamental properties of porous networks are based on statistical distributions of basic parameters. Theoretical calculations demonstrate strong correspondence with data obtained from numerical experiments. For low coordination numbers, we observe normal transport in porous networks. Anomalous effects expressed by tailing in concentration evolution are seen for higher coordination numbers. We find that the mean network coordination number has significant influence on averaged characteristics of porous networks such as geometric and hydraulic dispersivity, while other topological properties are of less significance. We give an explicit formula that describes the decrease of geometric dispersivity with growing mean coordination number. The results demonstrate the importance of network topology for models for flow and transport in porous media.     

A Novel Framework for Integration of Random-Walk Particle-Tracking Simulation in Subsurface Multi-phase Flow Modeling

Coarse-scale models are generally preferred in the numerical simulation of multi-phase flow due to computational constraints. However, capturing the effects of fine-scale heterogeneity on flow and isolating the impacts of numerical (artificial) dispersion, which increases with scale, are not trivial. In this paper, a particle-tracking method is devised and integrated in a scale-up workflow to estimate the conditional probability distributions of multi-phase flow functions, which can be considered as inputs in coarse-scale simulations with existing commercial packages. First, a novel particle-tracking method is developed to solve the saturation transport equation. The transport calculation is coupled with a velocity update, following the implicit pressure, explicit saturation framework, to solve the governing equations of two-phase immiscible flow. Each phase particle is advanced in a deterministic convection step according to the phase velocity, as well as in a stochastic dispersion step based on the random Brownian motion. A kernel-based formulation is proposed for computation of fluid saturation in accordance with the phase particle distribution. A novel aspect is that this method employs the kernel approach to construct saturation from phase particle distribution, which is an important improvement to the conventional box method that necessitates a large number of particles per grid cell for consistent saturation interpolation. The model is validated against various analytical solutions. Finally, the validated model is integrated in a statistical scale-up procedure to calibrate effective, or “pseudo,” multi-phase flow functions (e.g., relative permeability functions). The proposed scale-up framework does not impose any length scale requirement regarding the distribution of sub-grid heterogeneities.     

Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage

The aim of the paper is to model at a large scale, the formation damage in porous media by erosion and deposition of solid particles. We start from the equations governing the pore-scale processes of erosion, deposition, convection and diffusion. The macroscopic equivalent behaviour is investigated by using a homogenization method. Four characteristic models with different dominating phenomena at the pore scale are determined. The main results are twofold: first dispersion-deposition and dispersion-erosion phenomena are shown at the macroscopic scale for peculiar values of the dimensionless numbers; furthermore, and contrarily to phenomenological models, erosion and deposition generally occur in regions of intense and slow flow, respectively.     

Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach

A pore scale analysis is implemented in this numerical study to investigate the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure. The macroscopic characteristics of the transport phenomena are evaluated with an averaging technique of the controlling variables at a pore scale level in an elementary cell of the porous structure. The Darcy–Forchheimer model describes the fluid motion through the porous medium while the continuity and Navier–Stokes equations are applied within the unit cell. An average energy equation is employed for the thermal part of the porous medium. The macroscopic pressure loss is computed in order to evaluate the dominant microscopic inertial effects. Local fluctuations of velocity and temperature at the pore scale are instrumental in the quantification of the thermal dispersion through the total effective thermal diffusivity. The numerical results demonstrate that microscopic inertia contributes significantly to the magnitude of the macroscopic pressure loss, in some instances with as much as 70%. Depending on the nature of the porous medium, the thermal dispersion may have a marked bearing on the heat transfer, particularly in the streamwise direction for a highly conducting fluid and certain values of the Peclet number.     

Micromodel Observation of the Role of Oil Layers in Three-Phase Flow

We have studied the flow of a non-aqueous phase liquid (NAPL, or oil), water and air at the pore scale using a micromodel. The pore space pattern from a photomicrograph of a two-dimensional section through a Berea sandstone was etched onto a silicon wafer. The sizes of the pores in the micromodel are in the range 3–30,m and are the same as observed in the rock from which the image was taken. We conducted three-phase displacement experiments at low capillary numbers (in the order of 10^-7) to observe the presence of predicted displacement mechanisms at the pore scale. We observed stable oil layers between the wetting phase (water) and the non-wetting phase (gas) for the water–decane–air system, which has a negative equilibrium spreading coefficient, as well as four different types of double displacements where one fluid displaces another that displaces a third. Double imbibition and double drainage are readily observed, but the existence of an oil layer surrounding the gas phase makes the other double displacement combinations very unlikely.     

Numerical Calculation of Effective Diffusion in Unsaturated Porous Media by the TRT Lattice Boltzmann Method

Numerical models that solve transport of pollutants at the macroscopic scale in unsaturated porous media need the effective diffusion dependence on saturation as an input. We conducted numerical computations at the pore scale in order to obtain the effective diffusion curve as a function of saturation for an academic sphere packing porous medium and for a real porous medium where pore structure knowledge was obtained through X-ray tomography. The computations were performed using a combination of lattice Boltzmann models based on two relaxation time (TRT) scheme. The first stage of the calculations consisted in recovering the water spatial distribution into the pore structure for several fixed saturations using a phase separation TRT lattice Boltzmann model. Then, we performed diffusion computation of a non-reactive solute in the connected water structure using a diffusion TRT lattice Boltzmann model. Finally, the effective diffusion for each selected saturation value was estimated through inversion of a macroscopic classical analytical solution.     

Dynamic Network Model of Mobility Control in Emulsion Flow Through Porous Media

Modeling the flow of emulsion in porous media is extremely challenging due to the complex nature of the associated flows and multiscale phenomena. At the pore scale, the dispersed phase size can be of the same order of magnitude of the pore length scale and therefore effective viscosity models do not apply. A physically meaningful macroscopic flow model must incorporate the transport of the dispersed phase through the porous material and the changes on flow resistance due to drop deformation as it flows through pore throats. In this work, we present a dynamic capillary network model that uses experimentally determined pore-level constitutive relationships between flow rate and pressure drop in constricted capillaries to obtain representative transient macroscopic flow behavior emerging from microscopic emulsion flow at the pore level. A parametric analysis is conducted to study the effect of dispersed phase droplet size and capillary number on the flow response to both emulsion and alternating water/emulsion flooding in porous media. The results clearly show that emulsion flooding changes the continuous-phase mobility and consequently flow paths through the porous media, and how the intensity of mobility control can be tuned by the emulsion characteristics.     

Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes

This paper investigates the transverse and torsional wave in single- and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36×10⁻⁹ and 0.95×10⁻⁹ m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e₀ for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.     

Experimental investigation of characteristic length scale in periodic heterogeneous porous media

An experimental investigation of scale-dependent dispersion in periodic heterogeneous porous media was conducted. Models with two-, three- and four-layer periodic heterogeneities were constructed to investigate the effect of heterogeneity size on the scale-dependence of dispersion. Longitudinal dispersion coefficients were determined as a function of column length by measuring the breakthrough of a continuous injection of potassium chloride tracer solution. Chloride ion concentration was monitored by recording the millivolt potential of silver/silver chloride electrodes placed at intervals along the length of the column. In all three models, dispersion appeared to be scale dependent up to a distance of approximately 20–30 times the size of the repeated heterogeneity group (hydraulic unit). Because all three models suggested a similar dependence, it was concluded that a medium with periodic heterogeneity may likely be characterized by the scale of its hydraulic unit.     

Influence of Microbial Growth on Hydraulic Properties of Pore Networks

From laboratory experiments it is known that bacterial biomass is able to influence the hydraulic properties of saturated porous media, an effect called bioclogging. To interpret the observations of these experiments and to predict possible bioclogging effects on the field scale it is necessary to use transport models, which are able to include bioclogging. For these models it is necessary to know the relation between the amount of biomass and the hydraulic conductivity of the porous medium. Usually these relations were determined using bundles of parallel pore channels and do not account for interconnections between the pores in more than one dimension. The present study uses two-dimensional pore network models to study the effects of bioclogging on the pore scale. Numerical simulations were done for two different scenarios of the growth of biomass in the pores. Scenario 1 assumes microbial growth in discrete colonies clogging particular pores completely. Scenario 2 assumes microbial growth as a biofilm growing on the wall of each pore. In both scenarios the hydraulic conductivity was reduced by at least two orders of magnitude, but for the colony scenario much less biomass was needed to get a maximal clogging effect and a better agreement with previously published experimental data could be found. For both scenarios it was shown that heterogeneous pore networks could be clogged with less biomass than more homogeneous ones.     

Percolation theory approach to transport phenomena in porous media

The percolation theory approach to static and dynamic properties of the single- and two-phase fluid flow in porous media is described. Using percolation cluster scaling laws, one can obtain functional relations between the saturation fraction of a given phase and the capillary pressure, the relative permeability, and the dispersion coefficient, in drainage and imbibition processes. In addition, the scale dependency of the transport coefficient is shown to be an outcome of the fractal nature of pore space and of the random flow pattern of the fluids or contaminant.     

A theory of macrodispersion for the scale-up problem

Dispersion is the result, observable on large length scales, of events which are random on small length scales. When the length scale on which the randomness operates is not small, relative to the observations, then classical dispersion theory fails. The scale up problem refers to situations in which randomness occurs on all length scales, and for which classical dispersion theory necessarily fails. The purpose of this article is to present non-Fickian, theories of dispersion, which do not assume a scale separation between the randomness and the observed consequences, and which do not assume a single length scale.Porous media flow properties are heterogeneous on all length scales. The geological variation on length scales below the observational length scale can be regarded as unknown and unknowable, and thus as a random variable.We develop a systematic theory relating scaling behavior of the geological heterogeneity to the scaling behavior of the fluid dispersivity. Three qualitatively distinct regimes (Fickian, non-Fickian and nonrenormalizable) are found. The theory gives consistent answers within several distinct analytic approximations, and with numerical simulation of the equations of porous media flow.Comparison to field data is made. The use of Kriging to generate constrained ensembles for conditional simulation is discussed.     

Determination of Nanoparticle Macrotransport Coefficients from Pore Scale Processes

Nanoparticle transport in porous media is modeled using a hierarchical set of differential equations corresponding to pore scale and macroscale. At the pore scale, movement and interaction of a single particle with a solid matrix is modeled using the advection–dispersion–sorption equation. A single nanoparticle entering the space encounters viscous, diffusion and surface forces. Surface forces (electrostatic and van der Waals forces) between nanoparticles and mineral grains appear as sorption propensity on solid matrix boundary condition. These local events are then transformed into a macroscale continuum by imposing periodic boundary conditions for contiguous unit cells representing porous media and using a scheme of moment analysis. At the macroscale, propagation and retention of particles are characterized by three position-independent coefficients: mean nanoparticle velocity vector \({\bar{\mathbf{U}}}^*\), macroscopic dispersion coefficient \({\bar{\mathbf{D}}}^*\), and mean nanoparticle retention rate constant \({\bar{K}}^*\). The modeling results are validated with a set of nanoparticle transport tests in porous microchips. We also present simulations of realistic porous media, where an actual image of sandstone samples is processed into binary tones. The representative unit cells are constructed from the resulting binary image by searching for areas within the sample with maximum similarities to the whole sample in terms of porosity and specific surface area, which are found to show strong correlations with the resulting \({\bar{\mathbf{U}}}^*\) and \({\bar{K}}^*\), respectively.     

Wave propagation and dispersion in elasto-plastic microstructured materials

A Mindlin continuum model that incorporates both a dependence upon the microstructure and inelastic (nonlinear) behavior is used to study dispersive effects in elasto-plastic microstructured materials. A one-dimensional equation of motion of such material systems is derived based on a combination of the Mindlin microcontinuum model and a hardening model both at the macroscopic and microscopic level. The dispersion relation of propagating waves is established and compared to the classical linear elastic and gradient-dependent solutions. It is shown that the observed wave dispersion is the result of introducing microstructural effects and material inelasticity. The introduction of an internal characteristic length scale regularizes the ill-posedness of the set of partial differential equations governing the wave propagation. The phase speed does not necessarily become imaginary at the onset of plastic softening, as it is the case in classical continuum models and the dispersive character of such models constrains strain softening regions to localize.