On the Influence of Pore-Scale Dispersion in Nonergodic Transport in Heterogeneous Formations

Flow of an inert solute in an heterogeneous aquifer is usually considered as dominated by large-scale advection. As a consequence, the pore-scale dispersion, i.e. the pore scale mechanism acting at scales lower than that characteristic of the heterogeneous field, is usually neglected in the computation of global quantities like the solute plume spatial moments. Here the effect of pore-scale dispersion is taken into account in order to find its influence on the longitudinal asymptotic dispersivity D₁₁we examine both the two-dimensional and the three-dimensional flow cases. In the calculations, we consider the finite size of the solute initial plume, i.e. we analyze both the ergodic and the nonergodic cases. With Pe the Péclat number, defined as Pe=U/D, where U, , D are the mean fluid velocity, the heterogeneity characteristic length and the pore-scale dispersion coefficient respectively, we show that the infinite Péclat approximation is in most cases quite adequate, at least in the range of Péclat number usually encountered in practice (Pe > 10²). A noteworthy exception is when the formation log-conductivity field is highly anisotropic. In this case, pore-scale may have a significant impact on D₁₁, especially when the solute plume initial dimensions are not much larger than the heterogeneities' lengthscale. In all cases, D₁₁ appears to be more sensitive to the pore-scale dispersive mechanisms under nonergodic conditions, i.e. for plume initial size less than about 10 log-conductivity integral scales.     

Predictive Pore-Scale Modeling of Single and Multiphase Flow

We show how to predict flow properties for a variety of rocks using pore-scale modeling with geologically realistic networks. The pore space is represented by a topologically disordered lattice of pores connected by throats that have angular cross-sections. We successfully predict single-phase non-Newtonian rheology, and two and three-phase relative permeability for water-wet media. The pore size distribution of the network can be tuned to match capillary pressure data when a network representation of the system of interest is unavailable. The aim of this work is not simply to match experiments, but to use easily acquired data to estimate difficult to measure properties and to predict trends in data for different rock types or displacement sequences.     

Microtomography and Pore-Scale Modeling of Two-Phase Fluid Distribution

Synchrotron-based X-ray microtomography (micro CT) at the Advanced Light Source (ALS) line 8.3.2 at the Lawrence Berkeley National Laboratory produces three-dimensional micron-scale-resolution digital images of the pore space of the reservoir rock along with the spacial distribution of the fluids. Pore-scale visualization of carbon dioxide flooding experiments performed at a reservoir pressure demonstrates that the injected gas fills some pores and pore clusters, and entirely bypasses the others. Using 3D digital images of the pore space as input data, the method of maximal inscribed spheres (MIS) predicts two-phase fluid distribution in capillary equilibrium. Verification against the tomography images shows a good agreement between the computed fluid distribution in the pores and the experimental data. The model-predicted capillary pressure curves and tomography-based porosimetry distributions compared favorably with the mercury injection data. Thus, micro CT in combination with modeling based on the MIS is a viable approach to study the pore-scale mechanisms of CO₂ injection into an aquifer, as well as more general multi-phase flows.     

An Orthorhombic Lattice Boltzmann Model for Pore-Scale Simulation of Fluid Flow in Porous Media

One application of the lattice Boltzmann equation (LBE) models is in combination with tomography to simulate pore-scale flow and transport processes in porous media. Most LBE models in the literature are based on cubic lattice, and if the voxels in a tomography image are not cubic or cannot be divided into cubes due to computational limitations, these models will lose most of their advantages. How to deal with such images is, hence, an interest in use of the LBE model to simulate pore-scale processes. In this paper, we present an orthorhombic LBE model based on the single-relaxation time approach with the relaxation parameter varying with lattice directions. The equilibrium distribution functions in the standard LBE model were modified to correct the anisotropy induced by the non-cubic lattice, and the calculations of the fluid density and momentum were also redefined in order to maintain the conservation of mass and momentum during the collision. We tested the model against analytical solution for fluid flow in a tube, and against the standard cubic-based LBE model for fluid flow in a duct with an island inside. The model was then applied to simulate fluid flow in a 3D image in attempts to analyse the errors if the voxels in the image are not cubic but are assumed to be cubic.     

Evolution of Pore-Scale Morphology of Oil Shale During Pyrolysis: A Quantitative Analysis

This article describes experimental results and the numerical validation for multiphase, multicomponent evaporation in porous media 1d flow. We apply the model of Lindner et al. (Transp. Porous Media 112(2):313–332, 2016. doi: 10.1007/s11242-016-0646-6). The permeability of the porous medium is measured in an additional setup with a constant head permeameter to verify the validity of Darcy flow. The heat losses are considered in an analytical approach of correlating measured temperatures and heat inputs with enthalpies. A method of interpreting the experimental results is discussed to determine the phase state. We can show good qualitative agreement of the shift and position of the evaporation region when varying boundary conditions such as mass flux, concentration and heat input.     

Pore-Scale Simulation of Shear Thinning Fluid Flow Using Lattice Boltzmann Method

The present work attempts to identify the roles of flow and geometric variables on the scaling factor which is a necessary parameter for modeling the apparent viscosity of non-Newtonian fluid in porous media. While idealizing the porous media microstructure as arrays of circular and square cylinders, the present study uses multi-relaxation time lattice Boltzmann method to conduct pore-scale simulation of shear thinning non-Newtonian fluid flow. Variation in the size and inclusion ratio of the solid cylinders generates wide range of porous media with varying porosity and permeability. The present study also used stochastic reconstruction technique to generate realistic, random porous microstructures. For each case, pore-scale fluid flow simulation enables the calculation of equivalent viscosity based on the computed shear rate within the pores. It is observed that the scaling factor has strong dependence on porosity, permeability, tortuosity and the percolation threshold, while approaching the maximum value at the percolation threshold porosity. The present investigation quantifies and proposes meaningful correlations between the scaling factor and the macroscopic properties of the porous media.     

Multiblock Pore-Scale Modeling and Upscaling of Reactive Transport: Application to Carbon Sequestration

In order to safely store CO₂ in depleted reservoirs and deep saline aquifers, a better understanding of the storage mechanisms of CO₂ is needed. Reaction of CO₂ with minerals to form precipitate in the subsurface helps to securely store CO₂ over geologic time periods, but a concern is the formation of localized channels through which CO₂ could travel at large, localized rates. Pore-scale network modeling is an attractive option for modeling and understanding this inherently pore-level process, but the relatively small domains of pore-scale network models may prevent accurate upscaling. Here, we develop a transient, single-phase, reactive pore-network model that includes reduction of throat conductivity as a result of precipitation. The novelty of this study is the implementation of a new mortar/transport method for coupling pore networks together at model interfaces that ensure continuity of pressures, species concentrations, and fluxes. The coupling allows for modeling at larger scales which may lead to more accurate upscaling approaches. Here, we couple pore-scale models with large variation in permeability and porosity which result in initial preferential pathways for flow. Our simulation results suggest that the preferential pathways close due to precipitation, but are not redirected at late times.     

Pore-Scale Lattice Boltzmann Modeling and 4D X-ray Computed Microtomography Imaging of Fracture-Matrix Fluid Transfer

We present sequential X-ray computed microtomography (CMT) images of matrix drainage in a fractured, sintered glass-granule-pack. Sequential (4D) CMT imaging captured the capillary-dominated displacement of the oil-occupied matrix by the surfactant-brine-occupied fracture at the pore scale. The sintered glass-granule-pack was designed to have minimal pore space beyond the resolution of CMT imaging, ensuring that the pore space of the matrix connected to the fracture could be captured in its entirety. This provided an opportunity to validate the increasingly common lattice Boltzmann modeling technique against experimental images at the pore scale. Although the surfactant was found to alter the wettability of the originally weakly oil-wet glass to water-wet, the fracture-matrix fluid transfer is found to be a drainage process, showing minimal counter-current migration of the initial wetting phase (decane). The LB simulations were found to closely match experimental rates of fracture-matrix fluid transfer, and trends in the saturation profiles, but not the irreducible wetting-phase saturation behind the flooding front. The underestimation of the irreducible wetting phase saturation suggests that finer image and lattice resolutions than those reported here may be required for accurate prediction of some macroscale multiphase flow properties, at a sizable computational cost.     

Pore-Scale Flow in Surfactant Flooding

Recovery of oil from the blocks of an initially oil-wet, naturally fractured, reservoir as a result of counter-current flow following introduction of aqueous wettability-altering surfactant into the fracture system is considered, as an example of a practical process in which phenomena acting at the single pore-scale are vital to the economic displacement of oil at the macroscopic scale. A Darcy model for the process is set up, and solutions computed illustrating the recovery rate controlling role of the bulk diffusion of surfactant. A central ingredient of this model is the capillary pressure relation, linking the local values of the pressure difference between the oleic and aqueous phases, the aqueous saturation and the surfactant concentration. Using ideas from single capillary models of oil displacement from oil-wet tubes by wettability-altering surfactant, we speculate that the use of a capillary pressure function, with dependences as assumed, may not adequately represent the Darcy scale consequences of processes acting at the single pore-scale. Multi-scale simulation, resolving both sub-pore and multi-pore flow processes may be necessary to resolve this point. Some general comments are made concerning the issues faced when modelling complex displacement processes in porous media starting from the pore-scale and working upwards.     

Pore-Scale Modeling of Electrokinetics in Geomaterials

Pore-scale finite-volume continuum models of electrokinetic processes are used to predict the Debye lengths, velocity, and potential profiles for two-dimensional arrays of circles, ellipses and squares with different orientations. The pore-scale continuum model solves the coupled Navier–Stokes, Poisson, and Nernst–Planck equations to characterize the electro-osmotic pressure and streaming potentials developed on the application of an external voltage and pressure difference, respectively. This model is used to predict the macroscale permeabilities of geomaterials via the widely used Carmen–Kozeny equation and through the electrokinetic coupling coefficients. The permeability results for a two-dimensional X-ray tomography-derived sand microstructure are within the same order of magnitude as the experimentally calculated values. The effect of the particle aspect ratio and orientation on the electrokinetic coupling coefficients and subsequently the electrical and hydraulic tortuosity of the porous media has been determined. These calculations suggest a highly tortuous geomaterial can be efficient for applications like decontamination and desalination.

     

Comprehensive Seepage Simulation of Fluid Flow in Multi-scaled Shale Gas Reservoirs

Shale gas seepage behaviour is a multi-field/-scale problem and makes transient pressure analysis a very challenging task. Non-Darcy flow in nanopores is prominent due to the broken of continuity hypothesis. Slippage effect and Knudsen diffusion are two important seepage mechanisms in nanopores, while recent studies show surface diffusion is another important transporting mechanism on surface of nanopores. Porous kerogen system contains large amounts of dissolved gas, which should not be overlooked. In this study, a comprehensive mathematical model was established by pseudo-quadruple porosity medium conception, coupling the effects of slippage flow, Knudsen diffusion, surface diffusion, ad-/desorption and gas transferring from kerogen to nanopore system, while fluid flow in fractures/macropores is described by Darcy’s law. Transient pressure behaviours of a multiple fractured horizontal well in box-shaped shale gas reservoir were studied, with nine possible flow regimes divided and parameters sensitivity analysed. Adsorbed constant and dissolved constant were defined to reflect the amount of adsorbed gas and dissolved gas, respectively. Research shows that adsorbed gas and dissolved gas are two important gas storage forms, neither of which should be neglected. The study can not only help us understand fluid flow mechanisms in nanopores from microscopic perspective, but enable us to analyse production performance and determine key operational parameters from macroscopic perspective.     

Quasi-Hydrodynamic Model of Multiphase Fluid Flows Taking into Account Phase Interaction

A quasi-hydrodynamic system of equations describing flows of a heat-conducting viscous compressible multiphase multicomponent fluid is constructed taking into account surface effects. The system was obtained by generalizing the methods of obtaining a single-phase quasi-hydrodynamic system and a multicomponent flow model with surface effects based on the concept of microforces and microstresses. The equations are derived using the Coleman–Noll procedure. The results of the simulations show that the constructed model is applicable for modeling multiphase multicomponent flows with allowance for surface effects on the interfaces.     

Pore-Scale Modelling of Rate Effects in Waterflooding

We develop a rate-dependent network model that accounts for viscous forces by solving for the wetting and non-wetting phase pressure and which allows wetting layer swelling near an advancing flood front. The model incorporates a new time-dependent algorithm by accounting for partial filling of elements. We use the model to study the effects of capillary number, mobility ratio and contact angle distribution on waterflood displacement patterns, saturation and velocity profiles. By using large networks, generated from a new stochastic network algorithm, we reproduce Buckley–Leverett profiles directly from pore-scale modelling thereby providing a bridge between pore-scale and macro-scale transport.     

Analysis of Pore Pressure Distribution in Shale Formations under Hydraulic,Chemical, Thermal and Electrical Interactions

Change in pore pressure in chemically active rocks such as shale is caused by several mechanisms and numerous studies have been carried out to investigate these mechanisms. However, some important coupling terms or driving forces have been neglected in these studies due to simplifying assumptions. In this study, a hydro-chemo-thermo-electrical model based on finite element method is presented to investigate the change in pore pressure in shale formations resulted from thermal, hydraulic, chemical and electric potential gradients. The change in pore pressure is induced by hydraulic conduction, chemical, electrical and thermal osmotic flow. In order to solve the problem of ion transfer under the influence of an electrical field, the Nernst–Planck equation is used. In addition, ion advection is considered to investigate its possible effect on ion transfer for the range of shale permeability. All equations are derived based on the thermodynamics of irreversible processes in a discontinuous system. The numerical results are compared against existing and derived uncoupled analytical solutions and good agreement is observed. The numerical results showed that the ion transfer and pore pressure are considerably affected by the electric field in the vicinity of the wellbore. It was also found that advection can play a remarkable role in ion transfer in shale formations. It was further shown that the change in pore pressure in shale formation is characterized by the combined effect of hydraulic, chemical, thermal and electro osmotic flow.     

A Resistance Model for Newtonian and Power-Law Non-Newtonian Fluid Transport in Porous Media

A theoretical model based on the force balance between pressure, viscous force, and inertia force is proposed to predict the flow resistance of Newtonian and power-law non-Newtonian fluids through porous packed beds. The present model takes inertia effect into consideration, and the flow regime can be extended from Darcy flow to non-Darcy flow. It is demonstrated that the present model can predict most available experimental data well. The present results are also compared to the Ergun equation and other drag correlations.     

Advective Transport in Heterogeneous Formations: The Impact of Spatial Anisotropy on the Breakthrough Curve

Water flow and solute transport take place in formations of spatially variable conductivity K. The logconductivity Y?= ln K is modeled as a random stationary space function, of normal univariate pdf (of mean In K _G and variance ${\sigma_{Y}^{2}}$ ) and of axisymmetric autocorrelation of integral scales I _h,I _v (anisotropy ratio f?=?I _v/I _h?<?1). The head gradient and the velocity are uniform in the mean, parallel to bedding, and of constant and given as J and U, respectively. Transport is ruled by advection, which typically overwhelms pore scale dispersion in the breakthrough curve (BTC) determination. In the present study we analyze the impact of anisotropy f on the BTC of a passive solute, which is related to the mass flux??? (t, x) at a control plane at x. While a considerable body of literature dealt with weakly heterogeneous formations ( ${\sigma _{Y}^{2} <1 }$ ), the present study addresses the case of ${\sigma _{Y}^{2} >1 }$ , which is of interest for many aquifers and is more difficult to solve either numerically or by approximations. We approach the three dimensional problem by modeling the structure as an ensemble of densely packed oblate spheroids of semi-major and semi-minor axis R and f R, respectively, and independent lognormal K, submerged in a matrix of uniform conductivity K _ef, the effective conductivity of the ensemble. The detailed numerical simulations of transport show that the BTC is insensitive to the value of the anisotropy ratio f, i.e.,??? (t, x) I _h/U depends only on ${\sigma _{Y}^{2}}$ (except for small differences in the tail). This important result implies that transport, as quantified by BTCs or spatial longitudinal mass distributions, can be modeled accurately by the much simpler solutions developed in the past for isotropic media, like e.g., the semi-analytical self-consistent approximation.     

Dispersion in Heterogeneous Geological Formations: Preface (Transport in Porous Media – Special Issue)

Transport in Porous Media -