An improved gray lattice Boltzmann model for simulating fluid flow in multi-scale porous media

A lattice Boltzmann (LB) model is proposed for simulating fluid flow in porous media by allowing the aggregates of finer-scale pores and solids to be treated as ‘equivalent media’. This model employs a partially bouncing-back scheme to mimic the resistance of each aggregate, represented as a gray node in the model, to the fluid flow. Like several other lattice Boltzmann models that take the same approach, which are collectively referred to as gray lattice Boltzmann (GLB) models in this paper, it introduces an extra model parameter, n_s, which represents a volume fraction of fluid particles to be bounced back by the solid phase rather than the volume fraction of the solid phase at each gray node. The proposed model is shown to conserve the mass even for heterogeneous media, while this model and that model of Walsh et al. (2009) [1], referred to the WBS model thereafter, are shown analytically to recover Darcy–Brinkman’s equations for homogenous and isotropic porous media where the effective viscosity and the permeability are related to n_s and the relaxation parameter of LB model. The key differences between these two models along with others are analyzed while their implications are highlighted. An attempt is made to rectify the misconception about the model parameter n_s being the volume fraction of the solid phase. Both models are then numerically verified against the analytical solutions for a set of homogenous porous models and compared each other for another two sets of heterogeneous porous models of practical importance. It is shown that the proposed model allows true no-slip boundary conditions to be incorporated with a significant effect on reducing errors that would otherwise heavily skew flow fields near solid walls. The proposed model is shown to be numerically more stable than the WBS model at solid walls and interfaces between two porous media. The causes to the instability in the latter case are examined. The link between these two GLB models and a generalized Navier–Stokes model [2] for heterogeneous but isotropic porous media are explored qualitatively. A procedure for estimating model parameter n_s is proposed.     

Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media

We introduce two new channeling indicators D_ic and D_cc based on the Lagrangian distribution of flow rates. On the basis of the participation ratio, these indicators characterize the extremes of both the flow-tube width distribution and the flow rate variation along flow lines. The participation ratio is an indicator biased toward the larger values of a distribution and is equal to the normalized ratio of the square of the first-order moment to the second-order moment. Compared with other existing indicators, they advantageously provide additional information on the flow channel geometry, are consistently applicable to both porous and fractured media, and are generally less variable for media generated using the same parameters than other indicators. Based on their computation for a broad range of porous and fracture permeability fields, we show that they consistently characterize two different geometric properties of channels. D_ic gives a characteristic scale of low-flow zones in porous media and a characteristic distance between effectively flowing structures in fractured cases. D_cc gives a characteristic scale of the extension of high-flow zones in porous media and a characteristic channel length in fractured media. D_ic is mostly determined by channel density and permeability variability. D_cc is, however, more affected by the nature of the correlation structure like the presence of permeability channels or fractures in porous media and the length distribution in fracture networks.     

Crossover from capillary fingering to compact invasion for two-phase drainage with stable viscosity ratios

Motivated by a wide range of applications from enhanced oil recovery to carbon dioxide sequestration, we have developed a two-dimensional, pore-level model of immiscible drainage, incorporating viscous, capillary, and gravitational effects. This model has been validated quantitatively, in the very different limits of zero viscosity ratio and zero capillary number; flow patterns from modeling agree well with experiment. For a range of stable viscosity ratios (μ_injected/μ_displaced ? 1), we have increased the capillary number, N_c, and studied the way in which the flows deviate from capillary fingering (the fractal flow of invasion percolation) and become compact for realistic capillary numbers. Results exhibiting this crossover from capillary fingering to compact invasion are presented for the average position of the injected fluid, the fluid–fluid interface, the saturation and fractional flow profiles, and the relative permeabilities. The agreement between our results and earlier theoretical predictions [Blunt M, King MJ, Scher H. Simulation and theory of two-phase flow in porous media. Phys Rev A 1992;46:7680–99; Lenormand R. Flow through porous media: limits of fractal patterns. Proc Roy Soc A 1989;423:159–68; Wilkinson D. Percolation effects in immiscible displacement. Phys Rev A 1986;34:1380–90; Xu B, Yortsos YC, Salin D. Invasion Percolation with viscous forces. Phys Rev E 1998;57:739–51] supports the validity of these general theoretical arguments, which were independent of the details of the porous media in both two and three dimensions.     

Scale dependent dynamic capillary pressure effect for two-phase flow in porous media

Causes and effects of non-uniqueness in capillary pressure and saturation (P^c–S) relationship in porous media are of considerable concern to researchers of two-phase flow. In particular, a significant amounts of discussion have been generated regarding a parameter termed as dynamic coefficient (τ) which has been proposed for inclusion in the functional dependence of P^c–S relationship to quantify dynamic P^c and its relation with time derivative of saturation. While the dependence of the coefficient on fluid and porous media properties is less controversial, its relation to domain scale appears to be dependent on artefacts of experiments, mathematical models and the intra-domain averaging techniques. In an attempt to establish the reality of the scale dependency of the τ–S relationships, we carry out a series of well-defined laboratory experiments to determine τ–S relationships using three different sizes of cylindrical porous domains of silica sand. In this paper, we present our findings on the scale dependence of τ and its relation to high viscosity ratio (μ_r) silicone oil–water system, where μ_r is defined as the viscosity of non-wetting phase over that of the wetting phase. An order of magnitude increase in the value of τ was observed across various μ_r and domain scales. Also, an order of magnitude increase in τ is observed when τ at the top and the bottom sections in a domain are compared. Viscosity ratio and domain scales are found to have similar effects on the trend in τ–S relationship. We carry out a dimensional analysis of τ which shows how different variables, e.g., dimensionless τ and dimensionless domain volume (scale), may be correlated and provides a means to determine the influences of relevant variables on τ. A scaling relationship for τ was derived from the dimensionless analysis which was then validated against independent literature data. This showed that the τ–S relationships obtained from the literature and the scaling relationship match reasonably well.     

Dimensional analysis of two-phase flow including a rate-dependent capillary pressure–saturation relationship

The macroscopic modelling of two-phase flow processes in subsurface hydrosystems or industrial applications on the Darcy scale usually requires a constitutive relationship between capillary pressure and saturation, the P_c(S_w) relationship. Traditionally, it is assumed that a unique relation between P_c and S_w exists independently of the flow conditions as long as hysteretic effects can be neglected. Recently, this assumption has been questioned and alternative formulations have been suggested. For example, the extended P_c(S_w) relationship by Hassanizadeh and Gray [Hassanizadeh SM, Gray WG. Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resources 1990;13(4):169–86] proposes that the difference between the phase pressures to the equilibrium capillary pressure is a linear function of the rate of change of saturation, thereby introducing a constant of proportionality, the coefficient τ. It is desirable to identify cases where the extended relationship needs to be considered. Consequently, a dimensional analysis is performed on the basis of the two-phase balance equations. In addition to the well-known capillary and gravitational number, the dimensional analysis yields a new dimensionless number. The dynamic number Dy quantifies the ratio of dynamic capillary to viscous forces. Relating the dynamic to the capillary as well as the gravitational number gives the new numbers DyC and DyG, respectively. For given sets of fluid and porous medium parameters, the dimensionless numbers Dy and DyC are interpreted as functions of the characteristic length and flow velocity. The simulation of an imbibition process provides insight into the interpretation of the characteristic length scale. The most promising choice for this length scale seems to be the front width. We conclude that consideration of the extended P_c(S_w) relationship may be important for porous media with high permeability, small entry pressure and high coefficient τ when systems with a small characteristic length (e.g. steep front) and small characteristic time scale are under investigation.     

Field scale impacts of spatially correlated relative permeability in heterogeneous multiphase systems

Two-dimensional numerical simulations of two-phase (DNAPL-water) flow in spatially correlated random fields demonstrate the influence of nonwetting phase (NWP) relative permeability–saturation (k_r,N–S_W) relationships correlated to porous media intrinsic permeability (k). Both the volume of porous media invaded by the NWP and the length of time during which the NWP is migrating are under predicted if k_r,N–k correlation is not accounted for in the model formulation. Not accounting for the k_r,N–k correlation resulted in under predicting the volume of porous media invaded by up to approximately 10%, which is likely not significant for many practical applications. However, not accounting for the k_r,N–k correlation resulted in under predicting field scale migration times by up to a factor of 4, which is likely significant in that the migration times are on the order of years to several decades for the DNAPL (1,2-DCE) considered in this study. The under prediction of migration times was greater for lower permeability aquifers.     

Numerical modeling of formation damage by two-phase particulate transport processes during CO2 injection in deep heterogeneous porous media

Prediction of CO₂ injection performance in deep subsurface porous media relies on the ability of the well to maintain high flow rates of carbon dioxide during several decades typically without fracturing the host formation or damaging the well. Dynamics of solid particulate suspensions in permeable media are recognized as one major factor leading to injection well plugging in sandstones. The invading supercritical liquid-like fluid can contain exogenous fine suspensions or endogenous particles generated in situ by physical and chemical interactions or hydrodynamic release mechanisms. Suspended solids can plug the pores possibly leading to formation damage and permeability reduction in the vicinity of the injector. In this study we developed a finite volume simulator to predict the injectivity decline near CO₂ injection wells and also for production wells in the context of enhanced oil recovery. The numerical model solves a system of two coupled sets of finite volume equations corresponding to the pressure-saturation two-phase flow, and a second subsystem of solute and particle convection-diffusion equations. Particle transport equations are subject to mechanistic rate laws of colloidal, hydrodynamic release from pore surfaces, blocking in pore bodies and pore throats, and interphase particle transfer. The model was validated against available laboratory experiments at the core scale. Example results reveal that lower CO₂ residual saturation and formation porosity enhance CO₂-wet particle mobility and clogging around sinks and production wells. We conclude from more realistic simulations with heterogeneous permeability spanning several orders of magnitude that the control mode of mobilization, capture of particles, and permeability reduction processes strongly depends on the type of permeability distribution and connectivity between injection and production wells.     

Visualization experiments of biodegradation in porous media and calculation of the biodegradation rate

Biodegradation in porous media is studied with carefully controlled and well-characterized experiments in model porous media constructed of etched glass. Porous media of this type allow visual observation of the phenomena that take place at pore scale. An aqueous solution of five organic pollutants (toluene, phenol, o-cresol, naphthalene and 1,2,3-trimethylbenzene) was used as a model NAPL (representing creosote). The bacteria used were Pseudomonas fluorescens, which are indigenous (even predominant) in many contaminated soils. The maximum aqueous concentrations of the specific organic substances, below which biodegradation becomes possible, were determined as a function of temperature from toxicity experiments. Visualization experiments were made under various flow velocities and organic loadings to study the morphology and thickness of the biofilm as a function of the pore size and the distance from the entrance, and the efficiency of biodegradation. The efficiency of biodegradation decreased as the aqueous concentration of NAPL at the inlet increased and/or as the flow velocity increased. The thickness of biofilm decreased as the distance from the inlet increased and/or the pore diameter decreased. A quasi-steady-state theoretical model of biodegradation was used to calculate the values of the mesoscopic biochemical rates and to predict the profile of NAPL concentration in the porous medium and the thickness of biofilm in pores. The agreement between experimental data and model predictions is quite satisfactory.     

Stochastic modeling of the permeability of randomly generated porous media

Permeability of porous media in subsurface environments is subject to potentially large uncertainties due to the heterogeneity of natural systems. In this study, a first-order reliability method (FORM) is combined with a lattice Boltzmann method (LBM) to estimate the permeability of randomly generated porous media. The proposed procedure provides an increased ease of addressing complex pore structures by employing LBM to model fluid flow, while inheriting the computational efficiency from FORM. Macroscale-equivalent permeability can thus be estimated with significantly reduced computational efforts, while maintaining a connection to the complex microscale fluid dynamics within a pore structure environment. Implemented on several randomly generated porous media domains, the proposed method provides 13–120 times the efficiency compared to Monte Carlo methods.     

Uncertainty and equifinality in calibrating distributed roughness coefficients in a flood propagation model with limited data

Monte-Carlo simulations of a two-dimensional finite element model of a flood in the southern part of Sicily were used to explore the parameter space of distributed bed-roughness coefficients. For many real-world events specific data are extremely limited so that there is not only fuzziness in the information available to calibrate the model, but fuzziness in the degree of acceptability of model predictions based upon the different parameter values, owing to model structural errors. Here the GLUE procedure is used to compare model predictions and observations for a certain event, coupled with both a fuzzy-rule-based calibration, and a calibration technique based upon normal and heteroscedastic distributions of the predicted residuals. The fuzzy-rule-based calibration is suited to an event of this kind, where the information about the flood is highly uncertain and arises from several different types of observation. The likelihood (relative possibility) distributions predicted by the two calibration techniques are similar, although the fuzzy approach enabled us to constrain the parameter distributions more usefully, to lie within a range which was consistent with the modellers' a priori knowledge of the system.     

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.     

Robustness of flood-model calibration using single and multiple events

ABSTRACT

Lack of discharge data for model calibration is challenging for flood prediction in ungauged basins. Since establishment and maintenance of a permanent discharge station is resource demanding, a possible remedy could be to measure discharge only for a few events. We tested the hypothesis that a few flood-event hydrographs in a tropical basin would be sufficient to calibrate a bucket-type rainfall–runoff model, namely the HBV model, and proposed a new event-based calibration method to adequately predict floods. Parameter sets were chosen based on calibration of different scenarios of data availability, and their ability to predict floods was assessed. Compared to not having any discharge data, flood predictions improved already when one event was used for calibration. The results further suggest that two to four events for calibration may considerably improve flood predictions with regard to accuracy and uncertainty reduction, whereas adding more events beyond this resulted in small performance gains.     

Pore geometry of sandstone derived from pulsed field gradient NMR

Several parameters of pore geometry are needed for estimating permeability which is a key parameter for the characterization of reservoir sandstones. Powerful techniques for probing the pore space are the self-diffusion and the relaxation time NMR methods. However, the quality of results depends on the petrophysical model which underlies the interpretation of measurements. We applied the pulsed field gradient nuclear magnetic resonance (PFG-NMR) technique and measured time-dependent self-diffusion coefficients, D(Δ), of water in anhydrite cemented sandstones with low porosity and high tortuosity. The conventional method of fitting data with a function (Padé approximation) of the surface-to-volume ratio and the tortuosity yielded uncertain results. As part of a novel approach, we developed a numerical simulation code based on physical principles and a fractal pore space model. We compared our method with the Padé approximation and tested with data from the literature. For porous media with low tortuosity values and simple geometry such as randomly packed glass beads, both methods are in good agreement and give similar results. In sedimentary rocks, however, the new method of calculation is able to determine more accurate details of the pore geometry. However, its main advantage occurs in porous media with fractal geometry or with constricted pores, where the D(Δ)/D₀ curves deviate from the characteristic shape that is found for randomly packed glass beads.     

Effect of single and multi-site calibration techniques on hydrological model performance,parameter estimation and predictive uncertainty: a case study in the Logone catchment,Lake Chad basin

Understanding hydrological processes at catchment scale through the use of hydrological model parameters is essential for enhancing water resource management. Given the difficulty of using lump parameters to calibrate distributed catchment hydrological models in spatially heterogeneous catchments, a multiple calibration technique was adopted to enhance model calibration in this study. Different calibration techniques were used to calibrate the Soil and Water Assessment Tool (SWAT) model at different locations along the Logone river channel. These were: single-site calibration (SSC); sequential calibration (SC); and simultaneous multi-site calibration (SMSC). Results indicate that it is possible to reveal differences in hydrological behavior between the upstream and downstream parts of the catchment using different parameter values. Using all calibration techniques, model performance indicators were mostly above the minimum threshold of 0.60 and 0.65 for Nash Sutcliff Efficiency (NSE) and coefficient of determination (R ²) respectively, at both daily and monthly time-steps. Model uncertainty analysis showed that more than 60% of observed streamflow values were bracketed within the 95% prediction uncertainty (95PPU) band after calibration and validation. Furthermore, results indicated that the SC technique out-performed the other two methods (SSC and SMSC). It was also observed that although the SMSC technique uses streamflow data from all gauging stations during calibration and validation, thereby taking into account the catchment spatial variability, the choice of each calibration method will depend on the application and spatial scale of implementation of the modelling results in the catchment.     

Calibration of the Wetland Hydrodynamic Model to the Everglades Nutrient Removal Project

The objective of this study was to calibrate the Everglades Wetland Hydrodynamic Model (EWHM) to the Everglades Nutrient Removal (ENR) Project, from April 1995 through July of 1996. Model predictions were evaluated graphically and statistically against field observations to quantify the accuracy of model predictions and evaluate the success of model calibration. Comparisons between model predictions and field observations of water surface elevations at interior stations indicated that the model was successfully calibrated and model predictions were highly correlated with observed water surface elevations (r² ranged from 0.79 to 0.84). Model-predicted chloride (Cl) concentrations fell within the observed range of field observations, further confirming the success of model calibration. Good agreement found in these comparisons between observed and predicted results warrants the use of the model in a predictive mode. This is further supported by noting that the model contains no adjustable constants and requires no computational fitting of parameters to experimental data as is necessary in many previous obstructed flow studies.     

Multilayer shallow water flow using lattice Boltzmann method with high performance computing

A multilayer lattice Boltzmann (LB) model is introduced to solve three-dimensional wind-driven shallow water flow problems. The multilayer LB model avoids the expensive Navier–Stokes equations and obtains stratified horizontal flow velocities as vertical velocities are relatively small and the flow is still within the shallow water regime. A single relaxation time BGK method is used to solve each layer coupled by the vertical viscosity forcing term. To increase solution stability, an implicit step is suggested to obtain flow velocities. The main advantage of using the LBM is that after selecting appropriate equilibrium distribution functions, the LB algorithm is only slightly modified for each layer and retains all the simplicities of the LBM within the high performance computing (HPC) environment. The performance of the parallel LB model for the multilayer shallow water equations is investigated on CPU-based HPC environments using OpenMP. We found that the explicit loop control with cache optimization in LBM gives better performance on execution time, speedup and efficiency than the implicit loop control as the number of processors increases. Numerical examples are presented to verify the multilayer LB model against analytical solutions. We demonstrate the model’s capability of calculating lateral and vertical distributions of velocities for wind-driven circulation over non-uniform bathymetry.     

Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces

Pore-scale forces have a significant effect on the macroscopic behaviour of multiphase flow through porous media. This paper studies the effect of these forces using a new volume-of-fluid based finite volume method developed for simulating two-phase flow directly on micro-CT images of porous media. An analytical analysis of the relationship between the pore-scale forces and the Darcy-scale pressure drops is presented. We use this analysis to propose unambiguous definitions of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each phase, and then use them to obtain the relative permeability curves. We show that this definition is consistent with conventional laboratory/field measurements by comparing our predictions with experimental relative permeability. We present single and two-phase flow simulations for primary oil injection followed by water injection on a sandpack and a Berea sandstone. The two-phase flow simulations are presented at different capillary numbers which cover the transition from capillary fingering at low capillary numbers to a more viscous fingering displacement pattern at higher capillary numbers, and the effect of capillary number on the relative permeability curves is investigated. Overall, this paper presents a new finite volume-based methodology for the detailed analysis of two-phase flow directly on micro-CT images of porous media and upscaling of the results to the Darcy scale.     

Stochastic analysis of immiscible displacement of the fluids with arbitrary viscosities and its dependence on support scale of hydrological data

Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow.     

Flow in unsaturated random porous media,nonlinear numerical analysis and comparison to analytical stochastic models

This work presents a rigorous numerical validation of analytical stochastic models of steady state unsaturated flow in heterogeneous porous media. It also provides a crucial link between stochastic theory based on simplifying assumptions and empirical field and simulation evidence of variably saturated flow in actual or realistic hypothetical heterogeneous porous media. Statistical properties of unsaturated hydraulic conductivity, soil water tension, and soil water flux in heterogeneous soils are investigated through high resolution Monte Carlo simulations of a wide range of steady state flow problems in a quasi-unbounded domain. In agreement with assumptions in analytical stochastic models of unsaturated flow, hydraulic conductivity and soil water tension are found to be lognormally and normally distributed, respectively. In contrast, simulations indicate that in moderate to strong variable conductivity fields, longitudinal flux is highly skewed. Transverse flux distributions are leptokurtic. the moments of the probability distributions obtained from Monte Carlo simulations are compared to modified first-order analytical models. Under moderate to strong heterogeneous soil flux conditions (σ²_y≥1), analytical solutions overestimate variability in soil water tension by up to 40% as soil heterogeneity increases, and underestimate variability of both flux components by up to a factor 5. Theoretically predicted model (cross-)covariance agree well with the numerical sample (cross-)covarianaces. Statistical moments are shown to be consistent with observed physical characteristics of unsaturated flow in heterogeneous soils.©1998 Elsevier Science Limited. All rights reserved