Intercomparison of 3D pore-scale flow and solute transport simulation methods

Multiple numerical approaches have been developed to simulate porous media fluid flow and solute transport at the pore scale. These include 1) methods that explicitly model the three-dimensional geometry of pore spaces and 2) methods that conceptualize the pore space as a topologically consistent set of stylized pore bodies and pore throats. In previous work we validated a model of the first type, using computational fluid dynamics (CFD) codes employing a standard finite volume method (FVM), against magnetic resonance velocimetry (MRV) measurements of pore-scale velocities. Here we expand that validation to include additional models of the first type based on the lattice Boltzmann method (LBM) and smoothed particle hydrodynamics (SPH), as well as a model of the second type, a pore-network model (PNM). The PNM approach used in the current study was recently improved and demonstrated to accurately simulate solute transport in a two-dimensional experiment. While the PNM approach is computationally much less demanding than direct numerical simulation methods, the effect of conceptualizing complex three-dimensional pore geometries on solute transport in the manner of PNMs has not been fully determined. We apply all four approaches (FVM-based CFD, LBM, SPH and PNM) to simulate pore-scale velocity distributions and (for capable codes) nonreactive solute transport, and intercompare the model results. Comparisons are drawn both in terms of macroscopic variables (e.g., permeability, solute breakthrough curves) and microscopic variables (e.g., local velocities and concentrations). Generally good agreement was achieved among the various approaches, but some differences were observed depending on the model context. The intercomparison work was challenging because of variable capabilities of the codes, and inspired some code enhancements to allow consistent comparison of flow and transport simulations across the full suite of methods. This study provides support for confidence in a variety of pore-scale modeling methods and motivates further development and application of pore-scale simulation methods.     

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.     

Micro-computed tomography pore-scale study of flow in porous media: Effect of voxel resolution

A fundamental understanding of flow in porous media at the pore-scale is necessary to be able to upscale average displacement processes from core to reservoir scale. The study of fluid flow in porous media at the pore-scale consists of two key procedures: Imaging - reconstruction of three-dimensional (3D) pore space images; and modelling such as with single and two-phase flow simulations with Lattice-Boltzmann (LB) or Pore-Network (PN) Modelling. Here we analyse pore-scale results to predict petrophysical properties such as porosity, single-phase permeability and multi-phase properties at different length scales. The fundamental issue is to understand the image resolution dependency of transport properties, in order to up-scale the flow physics from pore to core scale. In this work, we use a high resolution micro-computed tomography (micro-CT) scanner to image and reconstruct three dimensional pore-scale images of five sandstones (Bentheimer, Berea, Clashach, Doddington and Stainton) and five complex carbonates (Ketton, Estaillades, Middle Eastern sample 3, Middle Eastern sample 5 and Indiana Limestone 1) at four different voxel resolutions (4.4 µm, 6.2 µm, 8.3 µm and 10.2 µm), scanning the same physical field of view. Implementing three phase segmentation (macro-pore phase, intermediate phase and grain phase) on pore-scale images helps to understand the importance of connected macro-porosity in the fluid flow for the samples studied. We then compute the petrophysical properties for all the samples using PN and LB simulations in order to study the influence of voxel resolution on petrophysical properties. We then introduce a numerical coarsening scheme which is used to coarsen a high voxel resolution image (4.4 µm) to lower resolutions (6.2 µm, 8.3 µm and 10.2 µm) and study the impact of coarsening data on macroscopic and multi-phase properties. Numerical coarsening of high resolution data is found to be superior to using a lower resolution scan because it avoids the problem of partial volume effects and reduces the scaling effect by preserving the pore-space properties influencing the transport properties. This is evidently compared in this study by predicting several pore network properties such as number of pores and throats, average pore and throat radius and coordination number for both scan based analysis and numerical coarsened data.     

Chaotic advection at the pore scale: Mechanisms,upscaling and implications for macroscopic transport

The macroscopic spreading and mixing of solute plumes in saturated porous media is ultimately controlled by processes operating at the pore scale. Whilst the conventional picture of pore-scale mechanical dispersion and molecular diffusion leading to persistent hydrodynamic dispersion is well accepted, this paradigm is inherently two-dimensional (2D) in nature and neglects important three-dimensional (3D) phenomena. We discuss how the kinematics of steady 3D flow at the pore scale generate chaotic advection—involving exponential stretching and folding of fluid elements—the mechanisms by which it arises and implications of microscopic chaos for macroscopic dispersion and mixing. Prohibited in steady 2D flow due to topological constraints, these phenomena are ubiquitous due to the topological complexity inherent to all 3D porous media. Consequently 3D porous media flows generate profoundly different fluid deformation and mixing processes to those of 2D flow. The interplay of chaotic advection and broad transit time distributions can be incorporated into a continuous-time random walk (CTRW) framework to predict macroscopic solute mixing and spreading. We show how these results may be generalised to real porous architectures via a CTRW model of fluid deformation, leading to stochastic models of macroscopic dispersion and mixing which both honour the pore-scale kinematics and are directly conditioned on the pore-scale architecture.     

From the pore scale to the lab scale: 3-D lab experiment and numerical simulation of drainage in heterogeneous porous media

A well-controlled 3-D experiment with pre-defined block heterogeneities is conducted, where neutron tomography is used to map 3-D water distribution after two successive drainage steps. The material and hydraulic properties of the two sands are first measured in the laboratory with multistep outflow experiments. Additionally, the pore structure of the sands is acquired by means of image analysis of synchrotron tomography data and the structure is used for pore-scale simulation of one- and two-phase flow with Lattice-Boltzmann methods. This gives us another set of material and hydraulic parameters of the sands. The two sets of hydraulic properties (from the lab scale and from the pore scale) are then used in numerical simulations of the 3-D experiment.     

Using Resin‐Based 3D Printing to Build Geometrically Accurate Proxies of Porous Sedimentary Rocks

Three‐dimensional (3D) printing is capable of transforming intricate digital models into tangible objects, allowing geoscientists to replicate the geometry of 3D pore networks of sedimentary rocks. We provide a refined method for building scalable pore‐network models (“proxies”) using stereolithography 3D printing that can be used in repeated flow experiments (e.g., core flooding, permeametry, porosimetry). Typically, this workflow involves two steps, model design and 3D printing. In this study, we explore how the addition of post‐processing and validation can reduce uncertainty in the 3D‐printed proxy accuracy (difference of proxy geometry from the digital model). Post‐processing is a multi‐step cleaning of porous proxies involving pressurized ethanol flushing and oven drying. Proxies are validated by: (1) helium porosimetry and (2) digital measurements of porosity from thin‐section images of 3D‐printed proxies. 3D printer resolution was determined by measuring the smallest open channel in 3D‐printed “gap test” wafers. This resolution (400 µm) was insufficient to build porosity of Fontainebleau sandstone (～13%) from computed tomography data at the sample's natural scale, so proxies were printed at 15‐, 23‐, and 30‐fold magnifications to validate the workflow. Helium porosities of the 3D‐printed proxies differed from digital calculations by up to 7% points. Results improved after pressurized flushing with ethanol (e.g., porosity difference reduced to ～1% point), though uncertainties remain regarding the nature of sub‐micron “artifact” pores imparted by the 3D printing process. This study shows the benefits of including post‐processing and validation in any workflow to produce porous rock proxies.     

Dissolution of residual non-aqueous phase liquids in porous media: pore-scale mechanisms and mass transfer rates

Experiments designed to elucidate the pore-scale mechanisms of the dissolution of a residual non-aqueous phase liquid (NAPL), trapped in the form of ganglia within a porous medium, are discussed. These experiments were conducted using transparent glass micromodels with controlled pore geometry, so that the evolution of the size and shape of individual NAPL ganglia and, hence, the pore-scale mass transfer rates and mass transfer coefficients could be determined by image analysis. The micromodel design permitted reasonably accurate control of the pore water velocity, so that the mass transfer coefficients could be correlated in terms of a local (pore-scale) Peclet number. A simple mathematical model, incorporating convection and diffusion in a slit geometry was developed and used successfully to predict the observed mass transfer rates. For the case of non-wetting NAPL ganglia, water flow through the corners in the pore walls was seen to control the rate of NAPL dissolution, as recently postulated by Dillard and Blunt [Water Resour. Res. 36 (2000) 439–454]. Break-up of doublet non-wetting phase ganglia into singlet ganglia by snap-off in pore throats was also observed, confirming the interplay between capillarity and mass transfer. Additionally, the effect of wettability on dissolution mass transfer was demonstrated. Under conditions of preferential NAPL wettability, mass transfer from NAPL films covering the solid surfaces was seen to control the dissolution process. Supply of NAPL from the trapped ganglia to these films by capillary flow along pore corners was observed to result in a sequence of pore drainage events that increase the interfacial area for mass transfer. These observations provide new experimental evidence for the role of capillarity, wettability and corner flow on NAPL ganglia dissolution.     

Does pore water velocity affect the reaction rates of adsorptive solute transport in soils? Demonstration with pore-scale modelling

Modelling adsorptive solute transport in soils needs a number of parameters to describe its reaction kinetics and the values of these parameters are usually determined from batch and displacement experiments. Some experimental results reveal that when describing the adsorption as first-order kinetics, its associated reaction rates are not constants but vary with pore water velocity. Explanation of this varies but an independent verification of each explanation is difficult because simultaneously measuring the spatiotemporal distributions of dissolved and adsorbed solutes in soils is formidable. Pore-scale modelling could play an important role to address this gap and has received increased attention over the past few years. This paper investigated the transport of adsorptive solute in a simple porous medium using pore-scale modelling. Fluid flow through the void space of the medium was assumed to be laminar and in saturated condition, and solute transport consisted of advection and molecular diffusion; the sorption and desorption occurring at the fluid–solid interface were modelled as linear first-order kinetics. Based on the simulated spatiotemporal distribution of dissolved and adsorbed solutes at pore scale, volumetric-average reaction kinetics at macroscopic scale and its associated reactive parameters were measured. Both homogeneous adsorption where the reaction rates at microscopic scale are constant, and heterogeneous adsorption where the reaction rates vary from site to site, were investigated. The results indicate that, in contrast to previously thought, the macroscopic reaction rates directly measured from the pore-scale simulations do not change with pore velocity under both homogeneous and heterogeneous adsorptions. In particular, we found that for the homogeneous adsorption, the macroscopic adsorption remains first-order kinetic and can be described by constant reaction rates, regardless of flow rate; whilst for the heterogeneous adsorption, the macroscopic adsorption kinetics continues not to be affected by flow rate but is no longer first-order kinetics that can be described by constant reaction rates. We discuss how these findings could help explain some contrary literature reports over the dependence of reaction rates on pore water velocity.     

Experimental analysis of pore-scale flow and transport in porous media

A novel, non-intrusive fluorescence imaging technique has been used to quantitatively measure the pore geometry, fluid velocity, and solute concentration within a saturated, three-dimensional porous medium. Discrete numerical averages of these quantities have been made over a representative volume of the medium and used to estimate macroscopic quantities that appear in conventional continuum models of flow and transport. The approach is meant to illustrate how microscopic information can be measured, averaged, and used to characterize medium-scale processes that are typically approximated constitutively. The experimental system consisted of a clear, cylindrical column packed with clear spherical beads and a refractive index-matched fluid seeded with fluorescent tracer particles and solute dye. By illuminating the fluid within the column with a scanning planar laser beam, details of flow and concentration within the pore spaces can be quantitatively observed, allowing for three-dimensional, dimensional, time dependent information to be obtained at good resolution. In time dependent information to be obtained at good resolution. In the current experiment, volumetrically averaged velocities and void-to-volume ratios are first compared with bulk measurements of fluid flux and medium porosity. Microscopic measurements of concentration are then used to construct cross-sectionally averaged profiles, mean breakthrough curves, and direct measurements of the dispersive flux, velocity variance, and concentration variance. In turn, the dispersive flux measurements are compared with mean concentration gradients to provide a basis for confirming the Fickian dispersion model and estimating dispersion coefficients for the medium. Coefficients determined in this manner are compared with others based upon traditional length-scale arguments, mean breakthrough analyses, and curve fits with numerical simulations.     

A multiple-point statistics algorithm for 3D pore space reconstruction from 2D images

Fluid flow behavior in a porous medium is a function of the geometry and topology of its pore space. The construction of a three dimensional pore space model of a porous medium is therefore an important first step in characterizing the medium and predicting its flow properties. A stochastic technique for reconstruction of the 3D pore structure of unstructured random porous media from a 2D thin section training image is presented. The proposed technique relies on successive 2D multiple point statistics simulations coupled to a multi-scale conditioning data extraction procedure. The Single Normal Equation Simulation Algorithm (SNESIM), originally developed as a tool for reproduction of long-range, curvilinear features of geological structures, serves as the simulation engine. Various validating criteria such as marginal distributions of pore and grain, directional variograms, multiple-point connectivity curves, single phase effective permeability and two phase relative permeability calculations are used to analyze the results. The method is tested on a sample of Berea sandstone for which a 3D micro-CT scanning image is available. The results confirm that the equi-probable 3D realizations obtained preserve the typical patterns of the pore space that exist in thin sections, reproduce the long-range connectivities, capture the characteristics of anisotropy in both horizontal and vertical directions and have single and two phase flow characteristics consistent with those of the measured 3D micro-CT image.     

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.     

Effective macroscopic transport parameters between parallel plates with constant concentration boundaries

A macroscopic transport model is developed, following the Taylor shear dispersion analysis procedure, for a 2D laminar shear flow between parallel plates possessing a constant specified concentration. This idealized geometry models flow with contaminant dissolution at pore-scale in a contaminant source zone and flow in a rock fracture with dissolving walls. We upscale a macroscopic transient transport model with effective transport coefficients of mean velocity, macroscopic dispersion, and first-order mass transfer rate. To validate the macroscopic model the mean concentration, covariance, and wall concentration gradient are compared to the results of numerical simulations of the advection–diffusion equation and the Graetz solution. Results indicate that in the presence of local-scale variations and constant concentration boundaries, the upscaled mean velocity and macrodispersion coefficient differ from those of the Taylor–Aris dispersion, and the mass transfer flux described by the first-order mass transfer model is larger than the diffusive mass flux from the constant wall. In addition, the upscaled first-order mass transfer coefficient in the macroscopic model depends only on the plate gap and diffusion coefficient. Therefore, the upscaled first-order mass transfer coefficient is independent of the mean velocity and travel distance, leading to a constant pore-scale Sherwood number of 12. By contrast, the effective Sherwood number determined by the diffusive mass flux is a function of the Peclet number for small Peclet number, and approaches a constant of 10.3 for large Peclet number.     

Micro-scale experimental investigation of the effect of flow rate on trapping in sandstone and carbonate rock samples

We present the results of a pore-scale experimental study of residual trapping in consolidated sandstone and carbonate rock samples under confining stress. We investigate how the changes in wetting phase flow rate impacts pore-scale distribution of fluids during imbibition in natural, water-wet porous media. We systematically study pore-scale trapping of the nonwetting phase as well as size and distribution of its disconnected globules. Seven sets of drainage-imbibition experiments were performed with brine and oil as the wetting and nonwetting phases, respectively. We utilized a two-phase miniature core-flooding apparatus integrated with an X-ray microtomography system to examine pore-scale fluid distributions in small Bentheimer sandstone (D = 4.9 mm and L = 13 mm) and Gambier limestone (D = 4.4 mm and L = 75 mm) core samples. The results show that with increase in capillary number, the residual oil saturation at the end of the imbibition reduces from 0.46 to 0.20 in Bemtheimer sandstone and from 0.46 to 0.28 in Gambier limestone. We use pore-scale displacement mechanisms, in-situ wettability characteristics, and pore size distribution information to explain the observed capillary desaturation trends. The reduction was believed to be caused by alteration of the order in which pore-scale displacements took place during imbibition. Furthermore, increase in capillary number produced significantly different pore-scale fluid distributions during imbibition. We explored the pore fluid occupancies and studied size and distribution of the trapped oil clusters during different imbibition experiments. The results clearly show that as the capillary number increases, imbibition produces smaller trapped oil globules. In other words, the volume of individual trapped oil globules decreased at higher brine flow rates. Finally, we observed that the pore space in the limestone sample was considerably altered through matrix dissolution at extremely high brine flow rates. This increased the sample porosity from 44% to 62% and permeability from 7.3 D to 80 D. Imbibition in the altered pore space produced lower residual oil saturation (from 0.28 to 0.22) and significantly different distribution of trapped oil globules.     

Numerical estimation of carbonate rock properties using multiscale images

Characterizing the pore space of rock samples using three‐dimensional (3D) X‐ray computed tomography images is a crucial step in digital rock physics. Indeed, the quality of the pore network extracted has a high impact on the prediction of rock properties such as porosity, permeability and elastic moduli. In carbonate rocks, it is usually very difficult to find a single image resolution which fully captures the sample pore network because of the heterogeneities existing at different scales. Hence, to overcome this limitation a multiscale analysis of the pore space may be needed. In this paper, we present a method to estimate porosity and elastic properties of clean carbonate (without clay content) samples from 3D X‐ray microtomography images at multiple resolutions. We perform a three‐phase segmentation to separate grains, pores and unresolved porous phase using 19 μm resolution images of each core plug. Then, we use images with higher resolution (between 0.3 and 2 μm) of microplugs extracted from the core plug samples. These subsets of images are assumed to be representative of the unresolved phase. We estimate the porosity and elastic properties of each sample by extrapolating the microplug properties to the whole unresolved phase. In addition, we compute the absolute permeability using the lattice Boltzmann method on the microplug images due to the low resolution of the core plug images. In order to validate the results of the numerical simulations, we compare our results with available laboratory measurements at the core plug scale. Porosity average simulations for the eight samples agree within 13%. Permeability numerical predictions provide realistic values in the range of experimental data but with a higher relative error. Finally, elastic moduli show the highest disagreements, with simulation error values exceeding 150% for three samples.     

Simulating dispersion in porous media and the influence of segmentation on stagnancy in carbonates

Understanding the transport of chemical components in porous media is fundamentally important to many reservoir processes such as contaminant transport and reactive flows involved in CO₂ sequestration. Carbonate rocks in particular present difficulties for pore-scale simulations because they contain large amounts of sub-micron porosity. In this work, we introduce a new hybrid simulation model to calculate hydrodynamic dispersion in pore-scale images of real porous media and use this to elucidate the origins and behaviour of stagnant zones arising in transport simulations using micro-CT images of carbonates. For this purpose a stochastic particle model for simulating the transport of a solute is coupled to a Lattice-Boltzmann algorithm to calculate the flow field. The particle method incorporates second order spatial and temporal resolution to resolve finer features of the domain. We demonstrate how dispersion coefficients can be accurately obtained in capillaries, where corresponding analytical solutions are available, even when these are resolved to just a few lattice units. Then we compute molecular displacement distributions for pore-spaces of varying complexity: a pack of beads; a Bentheimer sandstone; and a Portland carbonate. Our calculated propagator distributions are compared directly with recent experimental PFG-NMR propagator distributions (Scheven et al., 2005; Mitchell et al., 2008), the latter excluding spin relaxation mechanisms. We observe that the calculated transport propagators can be quantitatively compared with the experimental distribution, provided that spin relaxations in the experiment are excluded, and good agreement is found for both the sandstone and the carbonate. However, due to the absence of explicit micro-porosity from the carbonate pore space image used for flow field simulations we note that there are fundamental differences in the physical origins of the stagnant zones for micro-porous rocks between simulation and experiment. We show that for a given micro-CT image of a carbonate, small variations in the parameters chosen for the segmentation process lead to different amounts of stagnancy which diffuse away at different rates. Finally, we use a filtering method to show that this is due to the presence of spurious isolated pores which arise from the segmentation process and suggest an approach to overcome this limitation.