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.     

Direct numerical simulation of pore-scale flow in a bead pack: Comparison with magnetic resonance imaging observations

A significant body of current research is aimed at developing methods for numerical simulation of flow and transport in porous media that explicitly resolve complex pore and solid geometries, and at utilizing such models to study the relationships between fundamental pore-scale processes and macroscopic manifestations at larger (i.e., Darcy) scales. A number of different numerical methods for pore-scale simulation have been developed, and have been extensively tested and validated for simplified geometries. However, validation of pore-scale simulations of fluid velocity for complex, three-dimensional (3D) pore geometries that are representative of natural porous media is challenging due to our limited ability to measure pore-scale velocity in such systems. Recent advances in magnetic resonance imaging (MRI) offer the opportunity to measure not only the pore geometry, but also local fluid velocities under steady-state flow conditions in 3D and with high spatial resolution. In this paper, we present a 3D velocity field measured at sub-pore resolution (tens of micrometers) over a centimeter-scale 3D domain using MRI methods. We have utilized the measured pore geometry to perform 3D simulations of Navier–Stokes flow over the same domain using direct numerical simulation techniques. We present a comparison of the numerical simulation results with the measured velocity field. It is shown that the numerical results match the observed velocity patterns well overall except for a variance and small systematic scaling which can be attributed to the known experimental uncertainty in the MRI measurements. The comparisons presented here provide strong validation of the pore-scale simulation methods and new insights for interpretation of uncertainty in MRI measurements of pore-scale velocity. This study also provides a potential benchmark for future comparison of other pore-scale simulation methods. © 2012 Elsevier Science. All rights reserved.     

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.     

Effective dispersion in a chemically heterogeneous medium under temporally fluctuating flow conditions

We investigate effective solute transport in a chemically heterogeneous medium subject to temporal fluctuations of the flow conditions. Focusing on spatial variations in the equilibrium adsorption properties, the corresponding fluctuating retardation factor is modeled as a stationary random space function. The temporal variability of the flow is represented by a stationary temporal random process. Solute spreading is quantified by effective dispersion coefficients, which are derived from the ensemble average of the second centered moments of the normalized solute distribution in a single disorder realization. Using first-order expansions in the variances of the respective random fields, we derive explicit compact expressions for the time behavior of the disorder induced contributions to the effective dispersion coefficients. Focusing on the contributions due to chemical heterogeneity and temporal fluctuations, we find enhanced transverse spreading characterized by a transverse effective dispersion coefficient that, in contrast to transport in steady flow fields, evolves to a disorder-induced macroscopic value (i.e., independent of local dispersion). At the same time, the asymptotic longitudinal dispersion coefficient can decrease. Under certain conditions the contribution to the longitudinal effective dispersion coefficient shows superdiffusive behavior, similar to that observed for transport in s stratified porous medium, before it decreases to its asymptotic value. The presented compact and easy to use expressions for the longitudinal and transverse effective dispersion coefficients can be used for the quantification of effective spreading and mixing in the context of the groundwater remediation based on hydraulic manipulation and for the effective modeling of reactive transport in heterogeneous media in general.     

Comparison of continuum and pore-scale models of nutrient biodegradation under transverse mixing conditions

Recent studies indicate that during in situ bioremediation of contaminated groundwater, degradation occurs primarily along transverse mixing zones. Classical reactive-transport models overpredict the amount of degradation because solute spreading and mixing are not distinguished. Efforts to correct this have focused on modifying both dispersion and reaction terms, but no consensus on the best approach has emerged. In this work, a pore-scale model was used to simulate degradation along a transverse mixing zone between two required nutrients, and a continuum model with fitted parameters was used to match degradation rates from the pore-scale model. The pore-scale model solves for the flow field, concentration field, and biomass development within pore spaces of porous medium. For the continuum model, the flow field and biomass distributions are assumed to be homogeneous, and the fitting parameters are the transverse dispersion coefficient (D_T) and maximum substrate utilization rate (k_S,c). Results from the pore-scale model show that degradation rates near the system inlet are limited by the reaction rate, while degradation rates downgradient are limited by transverse mixing. For the continuum model, the value of D_T may be adjusted so that the degradation rate with distance matches that from the pore-scale model in the mixing-limited region. However, adjusting the value of k_S only improves the fit to pore-scale results within the reaction-limited region. Comparison with field and laboratory experiments suggests that the length of the reaction rate-limited region is small compared to the length scale over which degradation occurs. This indicates that along transverse mixing zones in the field, values of k_S are unimportant and only the value of D_T must be accurately fit.     

Effects of compound-specific dilution on transient transport and solute breakthrough: A pore-scale analysis

This pore-scale modeling study in saturated porous media shows that compound-specific effects are important not only at steady-state and for the lateral displacement of solutes with different diffusivities but also for transient transport and solute breakthrough. We performed flow and transport simulations in two-dimensional pore-scale domains with different arrangement of the solid grains leading to distinct characteristics of flow variability and connectivity, representing mildly and highly heterogeneous porous media, respectively. The results obtained for a range of average velocities representative of groundwater flow (0.1–10 m/day), show significant effects of aqueous diffusion on solute breakthrough curves. However, the magnitude of such effects can be masked by the flux-averaging approach used to measure solute breakthrough and can hinder the correct interpretation of the true dilution of different solutes. We propose, as a metric of mixing, a transient flux-related dilution index that allows quantifying the evolution of solute dilution at a given position along the main flow direction. For the different solute transport scenarios we obtained dilution breakthrough curves that complement and add important information to traditional solute breakthrough curves. Such dilution breakthrough curves allow capturing the compound-specific mixing of the different solutes and provide useful insights on the interplay between advective and diffusive processes, mass transfer limitations, and incomplete mixing in the heterogeneous pore-scale domains. The quantification of dilution for conservative solutes is in good agreement with the outcomes of mixing-controlled reactive transport simulations, in which the mass and concentration breakthrough curves of the product of an instantaneous transformation of two initially segregated reactants were used as measures of reactive mixing.     

Topological groundwater hydrodynamics

Topological groundwater hydrodynamics is an emerging subdiscipline in the mechanics of fluids in porous media whose objective is to investigate the invariant geometric properties of subsurface flow and transport processes. In this paper, the topological characteristics of groundwater flows governed by the Darcy law are studied. It is demonstrated that: (i) the topological constraint of zero helicity density during flow is equivalent to the Darcy law; (ii) both steady and unsteady groundwater flows through aquifers whose hydraulic conductivity is an arbitrary scalar function of position and time are confined to surfaces on which the streamlines of the flow are geodesic curves; (iii) the surfaces to which the flows are confined either are flat or are tori; and (iv) chaotic streamlines are not possible for these flows, implying that they are inherently poorly mixing in advective solute transport.     

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.     

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.     

On the aquifer's integrated balance equations

A general methodology is presented for describing transport phenomena in porous media at a macroscopic level. Then, these macroscopic balance equations are integrated (or averaged) along the vertical for confined, leaky and phreatic aquifers.The results are employed to derive (averaged) aquifer equations for the flow of water and of a solute (hydrodynamic dispersion). It is shown that in all cases, the resulting equation is identical to that derived on the basis of an assumption of horizontal flow (the Dupuit assumption).Macrodispersion, occurring at the aquifer level, is discussed and appropriate coefficients are proposed.     

Experimental and numerical study of the validity of Hele–Shaw cell as analogue model for variable-density flow in homogeneous porous media

Several laboratory experiments were conducted to identify the validity domain under which a Hele–Shaw cell may serve as a suitable analogue for variable-density flow in homogeneous porous media. These experiments are concerned with the injection into a Hele–Shaw cell of a salt solute at different concentrations and flow rates. The experimental data analysis highlighted two types of mixing zone shape: with and without ‘fingers’. A semi-empirical criterion based on the ratio between gravitational and injected velocities was used to forecast the change from one shape to another. The experimental data were then analysed using numerical solutions of the classical Hele–Shaw equations by taking into account an anisotropic dispersion tensor whose components depend on fluid density gradients. The good agreement between experimental and numerical results clearly shows that the validity of the concentration-dependent dispersion tensor strongly depends on the local Péclet number variation. For Péclet numbers lower 50, the Hele–Shaw cell can be considered as an analogous model of a homogeneous and isotropic 2D porous medium. It can be successfully used to study, at the laboratory scale, the gravitational instability effects induced by flow and transport phenomena into a porous medium.     

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.     

基于数字岩心动态应力应变模拟的非均匀孔隙介质波致流固相对运动刻画

地震波传播激发的不同尺度的流固相对运动(宏观、中观和微观)是许多沉积岩地层中地震波频散和衰减的主要原因,然而野外观测和试验测量都难以对非均匀多孔介质孔隙压力弛豫物理过程进行精细刻画.通过数字岩石物理技术,本文建立了三个典型的数字岩心分别用于表征孔隙结构、岩石骨架和斑状饱和流体引起的非均质性,利用动态应力应变模拟技术计算数字岩心的位移和孔隙流体增量图像.通过分析和比较三个数字岩心的位移和孔隙压力增量图像,细致刻画了发生于非均匀含流体多孔介质内的宏观、中观和微观尺度的流固相对运动:1)宏观尺度的波致孔隙流体流动导致波长尺度上数字岩心不同区域的孔隙压力和位移差异;2)中观尺度的流体流动发生在软层与硬层之间、气层与液层之间;3)微观尺度的流体流动发生在孔隙内部或相邻孔隙之间.数值模拟试验也证明基于数字岩心的动态应力应变模拟技术可以从微观尺度上更好的理解波致孔隙流体流动发生的物理机理,从而为建立岩石骨架、孔隙流体、孔隙结构非均质性和弹性波频散-衰减特征的映射关系奠定基础.     

An exact solution of solute transport by one-dimensional random velocity fields

The problem of one-dimensional transport of passive solute by a random steady velocity field is investigated. This problem is representative of solute movement in porous media, for example, in vertical flow through a horizontally stratified formation of variable porosity with a constant flux at the soil surface. Relating moments of particle travel time and displacement, exact expressions for the advection and dispersion coefficients in the Focker-Planck equation are compared with the perturbation results for large distances. The first- and second-order approximations for the dispersion coefficient are robust for a lognormal velocity field. The mean Lagrangian velocity is the harmonic mean of the Eulerian velocity for large distances. This is an artifact of one-dimensional flow where the continuity equation provides for a divergence free fluid flux, rather than a divergence free fluid velocity.     

An exact solution of solute transport by one-dimensional random velocity fields

The problem of one-dimensional transport of passive solute by a random steady velocity field is investigated. This problem is representative of solute movement in porous media, for example, in vertical flow through a horizontally stratified formation of variable porosity with a constant flux at the soil surface. Relating moments of particle travel time and displacement, exact expressions for the advection and dispersion coefficients in the Focker-Planck equation are compared with the perturbation results for large distances. The first- and second-order approximations for the dispersion coefficient are robust for a lognormal velocity field. The mean Lagrangian velocity is the harmonic mean of the Eulerian velocity for large distances. This is an artifact of one-dimensional flow where the continuity equation provides for a divergence free fluid flux, rather than a divergence free fluid velocity.