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Pore-scale dilution of conservative solutes: An example
Water Resources Research
Abstract
We simulate flow and transport of a conservative nonsorbing tracer in an idealized periodic pore channel using finite element techniques. The concentration is computed; then the slowly varying concentration mean, variance, coefficient of variation, and reactor ratio are calculated through averaging over every cell. The coefficient of variation and reactor ratio are related and quantify the degree of dilution. Then a novel methodology is developed for the evaluation of macroscopic parameters (homogenization), including the variance decay coefficient, which measures the rate with which small-scale concentration fluctuations tend to diminish, and the large-time coefficient of proportionality between the concentration variance and the square of the mean concentration variance. The methodology is based on the solution of a steady advection-dispersion problem in a single cell (which acts as a representative elementary volume); the computed result is then integrated in order to compute the macroscopic parameters. These parameters are compared with the parameters computed through direct simulation on a cell-by-cell basis, and they are found to be in reasonably good agreement. When the macroscopic parameters are used in the macroscopic equations, they produce estimates of the concentration mean and variance that are in agreement with the results of direct simulation.
... Consideration of transport processes in the above context is more difficult. Various studies have led to the critical conclusion that continuum-scale formulations used to interpret (conservative and reactive) transport experiments are typically dependent on the type and strength of physical and chemical processes occurring at the level of individual pores [see, e.g., Rashidi et al., 1996;Cao and Kitanidis, 1998;Gramling et al., 2002;Tartakovsky et al., 2008Tartakovsky et al., , 2009Edery et al., 2010;Sanchez-Vila et al., 2010, and references therein]. An intrinsic weakness of continuum-scale interpretive formulations is an inadequate, averaged (or upscaled) description of pore-scale processes; under many conditions, such processes cannot simply be homogenized and transferred to a macroscale effective model. ...
... With particular reference to reactive transport settings, slow advection zones, where displacement is controlled mainly by diffusion, hinder instantaneous contacts among chemicals species. In this context, incomplete mixing has been observed in laboratory experiments [e.g., Gramling et al., 2002], where this effect is most relevant at the lowest concentration levels [e.g., Rashidi et al., 1996;Cao and Kitanidis, 1998;Sanchez-Vila et al., 2010]. Accounting for non-Fickian or anomalous transport observed at a macroscale [see, e.g., Dagan and Neuman, 1991;Cushman and Ginn, 1993;Berkowitz and Scher, 1997;Haggerty et al., 2000;Cirpka and Kitanidis, 2000;S anchez-Vila and Carrera, 2004;Berkowitz et al., 2006;Rubin et al., 2012;Edery et al., 2014] has been accomplished by various mathematical conceptualizations, including particle-based (or particle tracking, PT) methods [e.g., Gillespie, 1977;Lindenberg and Romero, 2007;Srinivasan et al., 2007;Palanichamy et al., 2007;Yuste et al., 2008;Edery et al., 2010]. ...
... Use of continuum-scale models and failure to properly account for these dynamics might lead to severe over-(or under-) estimation of solute concentrations, reaction rates, and reaction products. Typically, these effects are most significant for the lowest concentration levels [e.g., Rashidi et al., 1996;Cao and Kitanidis, 1998;Sanchez-Vila et al., 2010]. ...
We consider modeling approaches to characterize solute transport in porous media, integrating them into a unique theoretical and experimental framework for model evaluation and data interpretation. To date, development of (conservative and reactive chemical) transport models and formulation of model calibration methods grounded on sensitivity-based collection of measurements have been pursued in parallel. Key questions that remain include: For a given set of measurements, which conceptual picture of the transport processes, as embodied in a mathematical model or models, is most appropriate? What are the most valuable space-time locations for solute concentration measurements, depending on the model selected? How is model parameter uncertainty propagated to model output, and how does this propagation affect model calibration? We address these questions by merging parallel streams of research—model formulation, reduction, calibration, sensitivity analysis, and discrimination—offering our view on an emerging framework that guides (i) selection of an appropriate number and location of time-dependent concentration measurements given a transport model and (ii) assessment (through discrimination criteria) of the relative benefit of applying any particular model from a set of several models. Our strategy is to employ metrics to quantify the relative contribution of each uncertain model parameter to the variability of the model output. We evaluate these metrics through construction of a surrogate (or “meta”) transport model that has the additional benefit of enabling sensitivity analysis and model calibration at a highly reduced computational cost. We demonstrate the applicability of this framework, focusing on transport of reactive chemicals in laboratory-scale porous media.
... Under such assumption Kapoor and Kitanidis [11] demonstrated that at large times relative to a characteristic variance residence time ðt ) VRTÞ, the ''volume occupied by solute [i.e., dilution index] approaches that apparent from its spatial moments [i.e., spreading index]". On the other hand, the pore-scale investigation of Cao and Kitanidis [14] demonstrated that at the fringes of a tracer plume the solute is never fully mixed even for large diffusion times. The only other known study of dilution applied specifically to the pore-scale is reported by [15], who considered the effect of pore-scale geometry on dilution. ...
... Table 1. Cao and Kitanidis [14] argued that for t=s p D > 1 the homogenization theory can be applied. Based on this condition and results in Fig. 5 we can conclude that for Pe ¼ 24; EðtÞ=E max ðtÞ ¼ 0:92 is close to the asymptotic value. ...
... On the other hand, large ratios s D D =s p A for Pe ¼ 246 and 24,610 indicate that the solute may not completely dilute even an a domain that is significantly larger than the domain used in the simulations. Cao and Kitanidis [14] also showed that at the boundary of a plume solute never completely dilutes. Because of these we expect that even in significantly larger domains Darcy-scale model will overestimate the extent of the reaction. ...
Darcy-scale description of multicomponent reactive transport can significantly over-predict the extent of mixing-controlled reactions. Here we present a systematic pore- and Darcy-scale study of multicomponent reactive transport for various Peclet (Pe)(Pe) and Damkohler (Da)(Da) numbers. We use pore-scale simulations to parameterize and validate Darcy-scale model. Our results reveal that for small Pe, the Darcy-scale model of mixing-controlled reactions is accurate, but its accuracy deteriorates with increasing Pe. The dilution index is calculated based on the pore-scale simulations as a measure of the degree of true solute mixing (dilution) for different Pe. We find that true mixing decreases with increasing Pe. A strong correlation is found between the magnitude of the dilution index and the error in the Darcy-scale model prediction. The error (over-prediction of the mass of a product of the mixing-controlled reaction) increases with decreasing dilution index (reduced dilution).
... Under such assumption Kapoor and Kitanidis [11] demonstrated that at large times relative to a characteristic variance residence time ðt ) VRTÞ, the ''volume occupied by solute [i.e., dilution index] approaches that apparent from its spatial moments [i.e., spreading index]". On the other hand, the pore-scale investigation of Cao and Kitanidis [14] demonstrated that at the fringes of a tracer plume the solute is never fully mixed even for large diffusion times. The only other known study of dilution applied specifically to the pore-scale is reported by [15], who considered the effect of pore-scale geometry on dilution. ...
... Table 1. Cao and Kitanidis [14] argued that for t=s p D > 1 the homogenization theory can be applied. Based on this condition and results in Fig. 5 we can conclude that for Pe ¼ 24; EðtÞ=E max ðtÞ ¼ 0:92 is close to the asymptotic value. ...
... On the other hand, large ratios s D D =s p A for Pe ¼ 246 and 24,610 indicate that the solute may not completely dilute even an a domain that is significantly larger than the domain used in the simulations. Cao and Kitanidis [14] also showed that at the boundary of a plume solute never completely dilutes. Because of these we expect that even in significantly larger domains Darcy-scale model will overestimate the extent of the reaction. ...
Darcy-scale description of multicomponent reactive transport can significantly over-predict the extent of mixing-controlled reactions. Here we present a systematic pore-and Darcy-scale study of multicompo-nent reactive transport for various Peclet ðPeÞ and Damkohler ðDaÞ numbers. We use pore-scale simulations to parameterize and validate Darcy-scale model. Our results reveal that for small Pe, the Darcy-scale model of mixing-controlled reactions is accurate, but its accuracy deteriorates with increasing Pe. The dilution index is calculated based on the pore-scale simulations as a measure of the degree of true solute mixing (dilution) for different Pe. We find that true mixing decreases with increasing Pe. A strong correlation is found between the magnitude of the dilution index and the error in the Darcy-scale model prediction. The error (over-prediction of the mass of a product of the mixing-controlled reaction) increases with decreasing dilution index (reduced dilution).
... The main key in designing laboratory experiments for reactive transport is to ensure a deep knowledge and understanding of the conceptual model of the physical process. This involves studying the degree of heterogeneity in the porous media and how would state variables in the governing equations vary with the progression of the reactions [24,88]. A prime focus in this case would be to understand the physics of the flow at both the pore scale and the continuum (Darcy) scale. ...
... The resultant system signifies that each kinetic reaction rate only appears in one equation due to the linear (24) n e ×n c × n c ×n r = 0 n e ×n r (25) ...
The rapid advancements in the computational abilities of numerical simulations have attracted researchers to work on the area of reactive transport in porous media to improve the hydrocarbon production processes from mature reservoirs. In the hydrology community, reactive transport is well developed where the main research focuses on studying the movement of groundwater and contaminants in aquifers, and quantifying the effect of chemical reactions between the rocks and water. Recently, great efforts have been made to adapt similar models for petroleum applications where multiphase flow is experienced in the subsurface reservoirs. In such systems, thermodynamic and chemical equilibrium is key in establishing an accurate description of the states of the fluids existing in the reservoir. This paper presents a detailed and comprehensive review on the concepts of geochemical modeling, and how it can be mathematically adapted to petroleum multiphase flow problems in porous media. We introduce key physical concepts outlining the treatment of chemical reactions in experimental trials and then explain in detail how a network of chemical reactions can be modeled mathematically for numerical simulation applications. The steps of characterizing the physical behavior of the fluid flow—through a set of governing equations by either natural or molar variables formulations, and the methodology to simplify and incorporate the numerical algorithms into an existing reservoir simulation scheme are shown as well. We finally present two numerical cases from the literature to highlight the key variations between the different variable formulations and comment on the advantages and disadvantages of each approach.
... In this paper we focus on a particular unit cell geometry, a channel with sinusoidal boundaries, which in the context of porous media was first studied by Dykaar and Kitanidis (1996) and Cao and Kitanidis (1998) . While simple, it has been shown to capture some very rich dynamics, representative of complexities in real porous media. ...
... While simple, it has been shown to capture some very rich dynamics, representative of complexities in real porous media. Multiple studies have used this, or very similar, geometries to explore moderate to high Reynolds number flow and transport Bouquain et al., 2012;Chaudhary et al., 2013;Richmond et al., 2013 ) and to study the impact of geometry on macroscopic dispersion ( Bolster et al., 2009a;Cardenas, 20 08;20 09;LeBorgne et al., 2012 ) and dilution ( Cao and Kitanidis, 1998 ). The system is well studied and serves as an ideal benchmark against which to test our modeling approach. ...
The Spatial Markov Model (SMM) is an upscaled model that has been used successfully to predict effective mean transport across a broad range of hydrologic settings. Here we propose a novel variant of the SMM, applicable to spatially periodic systems. This SMM is built using particle trajectories, rather than travel times. By applying the proposed SMM to a simple benchmark problem we demonstrate that it can predict mean effective transport, when compared to data from fully resolved direct numerical simulations. Next we propose a methodology for using this SMM framework to predict measures of mixing and dilution, that do not just depend on mean concentrations, but are strongly impacted by pore-scale concentration fluctuations. We use information from trajectories of particles to downscale and reconstruct pore-scale approximate concentration fields from which mixing and dilution measures are then calculated. The comparison between measurements from fully resolved simulations and predictions with the SMM agree very favorably.
... A very important issue is the determination of dispersion coefficient (Gelhar, 1993). Some studies have shown that it is not possible to understand mixing and dilution only under the approach of mean concentrations (Cao and Kitanidis, 1998;Raje and Kapoor, 2000). Rashidi et al. (1996) found experimentally the dependency of solute concentration with the pore size, while Cao and Kitanidis (1998) did the same numerically. ...
... Some studies have shown that it is not possible to understand mixing and dilution only under the approach of mean concentrations (Cao and Kitanidis, 1998;Raje and Kapoor, 2000). Rashidi et al. (1996) found experimentally the dependency of solute concentration with the pore size, while Cao and Kitanidis (1998) did the same numerically. Both coincide that concentration at poral scale is ruled by molecular diffusivity. ...
Most research on porous media is based on a continuum model that analyze transport
processes in a macroscale approximation, averaging on a large number of pores,
neglecting poral scale effects.
In reactive solute flow dynamics this approach can produce misleading results,
because reactants that are considered homogenized at Darcy scale are not actually
perfectly mixed. Numerical schemes based on this assumption are unable to reproduce
properly experimental results, because they are disregarding inhomogeneities at poral
scale, just where reactions are taking place.
This lack of homogeneity at poral scale (called segregation effect) can be addressed
in the transport equation using an effective reaction rate term, acting as a source. Within
this approach the main problem seems to be how to model the effective reaction rate. A
phenomenological model is proposed and its parameter is estimated by minimizing the
error between simulated results and experimental data. Correlation between the parameter
of the mathematical model and physical properties of the experimental set up is
discussed.
Given that this kind of processes are of advective - diffusive - reactive type, they
present three different characteristic time scales, depending of which is the dominant
process. Solving this problem numerically, attention must be paid to this fact, in order to
address the full complexity of this equation.
... Given the fact that in batch experiments, there are no spatial variations in the concentration field (as it is a well-mixed system), the reaction models and reaction parameters actually pertain to the pore scale. In an open system, where diffusion and/or hydrodynamic dispersion occur, gradients in concentration exist, as observed both in experiments (e.g., Rashidi et al., 1996;Kapoor et al., 1998;Taylor 1953;Aris, 2005) and through numerical simulations (e.g., Li et al., 2006;Cao and Kitanidis, 1998;Shapiro and Adler, 1997). As such, the reaction rates are an outcome of coupling between the reaction and hydrodynamic processes (Li et al., 2007); this coupling can be the reason for much faster laboratorymeasured reaction rates of many minerals than those observed in the field (White and Brantley, 2003;Maher et al., 2004). ...
... The effect of residence times on reactive transport was also addressed by Robinson and Viswanathan (2003), who showed the importance of pore-scale gradients, especially for nonlinear reactions; solute pulses of short duration; and systems with broad residence time distribution curves. Characteristic timescales of reaction processes pose constraints for transport models (Mo and Friedly, 2000;Cao and Kitanidis, 1998). ...
... Given the fact that in batch experiments, there are no spatial variations in the concentration field (as it is a well-mixed system), the reaction models and reaction parameters actually pertain to the pore scale. In an open system, where diffusion and/or hydrodynamic dispersion occur, gradients in concentration exist, as observed both in experiments (e.g., Rashidi et al., 1996;Kapoor et al., 1998;Taylor 1953;Aris, 2005) and through numerical simulations (e.g., Li et al., 2006;Cao and Kitanidis, 1998;Shapiro and Adler, 1997). As such, the reaction rates are an outcome of coupling between the reaction and hydrodynamic processes (Li et al., 2007); this coupling can be the reason for much faster laboratorymeasured reaction rates of many minerals than those observed in the field (White and Brantley, 2003;Maher et al., 2004). ...
... The effect of residence times on reactive transport was also addressed by Robinson and Viswanathan (2003), who showed the importance of pore-scale gradients, especially for nonlinear reactions; solute pulses of short duration; and systems with broad residence time distribution curves. Characteristic timescales of reaction processes pose constraints for transport models (Mo and Friedly, 2000;Cao and Kitanidis, 1998). ...
... Given the fact that in batch experiments, there are no spatial variations in the concentration field (as it is a well-mixed system), the reaction models and reaction parameters actually pertain to the pore scale. In an open system, where diffusion and/or hydrodynamic dispersion occur, gradients in concentration exist, as observed both in experiments (e.g., Rashidi et al., 1996;Kapoor et al., 1998;Taylor 1953;Aris, 2005) and through numerical simulations (e.g., Li et al., 2006;Cao and Kitanidis, 1998;Shapiro and Adler, 1997). As such, the reaction rates are an outcome of coupling between the reaction and hydrodynamic processes (Li et al., 2007); this coupling can be the reason for much faster laboratorymeasured reaction rates of many minerals than those observed in the field (White and Brantley, 2003;Maher et al., 2004). ...
... The effect of residence times on reactive transport was also addressed by Robinson and Viswanathan (2003), who showed the importance of pore-scale gradients, especially for nonlinear reactions; solute pulses of short duration; and systems with broad residence time distribution curves. Characteristic timescales of reaction processes pose constraints for transport models (Mo and Friedly, 2000;Cao and Kitanidis, 1998). ...
Adsorption of solutes in porous media is commonly modeled as an equilibrium process. Indeed, one may safely assume that within the pore space, the concentration of adsorbed solute at a point on the grain surface is algebraically related to the concentration in the fluid next to the grain. The same, however, cannot be said about average concentrations. In fact, during solute transport, concentration gradients develop within the pore space, and these could potentially give rise to a scale-dependent adsorption process. The main objective of this research is to develop relationship between pore-scale adsorption coefficient and corresponding upscaled adsorption parameters. Two approaches are used: Theoretical averaging and numerical upscaling. In the averaging approach, equilibrium adsorption is assumed at the pore-scale and solute transport equations are averaged over REV. This leads to explicit expressions for macro-scale adsorption rate constants as a function of micro-scale parameters. In the numerical approach, first we simulate solute transport within a single tube undergoing equilibrium adsorption at the pore wall, and then flux averaged concentration breakthrough curves are obtained. These are used to determine the upscaled adsorption rate constants as functions of pore-scale hydraulic and adsorption parameters. Results of the two approaches agree very well.
... Given the fact that in batch experiments, there are no spatial variations in the concentration field (as it is a well-mixed system), the reaction models and reaction parameters actually pertain to the pore scale. In an open system, where diffusion and/or hydrodynamic dispersion occur, gradients in concentration exist, as observed both in experiments (e.g., Rashidi et al. [1996], Kapoor et al. [1998], Taylor [1953], Aris [1999] and through numerical simulations (e.g., Li et al. [2006b], Cao and Kitanidis [1998], Shapiro and Adler [1997]). As such, the reaction rates are an outcome of coupling between the reaction and hydrodynamic processes [Li et al., 2008]; this coupling can be the reason for much faster laboratory measured reaction rates of many minerals than those observed in the field [White and Brantley, 2003, Maher et al., 2004. ...
... The effect of residence times on reactive transport was also addressed by Robinson and Viswanathan [2003], who showed the importance of pore-scale gradients, especially for nonlinear reactions; solute pulses of short duration; and systems with broad residence time distribution curves. Characteristic timescales of reaction processes pose constraints for transport models [Mo andFriedly, 2000, Cao andKitanidis, 1998]. Experimental studies (e.g., Guo and Thompson [2001]) as well as pore-scale numerical models [Knutson et al., 2001a] have shown the dependence of mass transfer coefficients (e.g., in dissolution process) on the hydrodynamics of porous media. ...
Pore-scale modeling provides opportunities to study transport phenomena in fundamental ways because detailed information is available at the microscopic pore scale. This offers the best hope for bridging the traditional gap that exists between pore scale and macro (lab) scale description of the process. As a result, consistent upscaling relations can be performed, based on physical processes defined at the appropriate scale. In the present study, we have used a Multi-Directional Pore Network (MDPN) for representing a porous medium. One of the main features of our network is that pore throats can be oriented not only in the three principal directions, but in 13 different directions, allowing a maximum coordination number of 26.
Using MDPN, flow and transport processes were simulated at the pore scale in detail by explicitly modeling the interfaces and mass exchange at surfaces. The solution of pore network model provides local concentrations and it allows relating concentrations and reaction rates at the macro scale to concentrations and reaction rates at the scale of individual pores, a scale at which reaction processes are well defined. Then, comparing the result of pore-scale simulations with the model representing the macro-scale behavior, we could study the relation between these two scales. We have considered mass transfer of reactive/adsorptive solutes though interfaces, under both saturated and partiality saturated conditions. While under saturated conditions the interfaces are only those of solid-water interfaces, under saturated conditions in addition to solid-water interfaces there will be mass transfer though air-water interfaces as well. Macroscopic quantities were obtained through averaging over the pore network domain. To meet our objectives we focus on both physical heterogeneity and topology (differently sized pores and coordination number distribution) and chemical processes.
There are many other novel and unique aspects to this book, through which, we develop more accurate and realistic schemes to study flow and transport processes. For this purpose we have developed an extensive FORTRAN 90 Modular Package which spans though generation of random structure networks, discretizing pore spaces on the basis of their saturation state, and solving flow and reactive transport under both saturated and unsaturated conditions using several complex algorithms. The governing equations are solved applying a fully implicit numerical scheme; however, efficient substitution methods have been applied which made the algorithm more computationally effective and appropriate for parallel computations.
Through coupling MDPN with multi-component reactive simulator, BRNS, we could simulate transport of species which may cause changes in porosity and permeability due to reaction with the solid phase. By averaging over a representative MDPN, we calculated upscaled parameters such as permeability, dispersion coefficient and measure of plume spreading, and upscaled adsorption parameters under saturated conditions. For partially saturated conditions the results were capillary pressured-saturation relation, relative permeability, unsaturated dispersivity-saturation relation, and upscale adsorption parameters. Whenever possible, we compared our results with the results of experimental observations and analytical equations. In this way, we could evaluate the limitations and sufficiency of the available analytical equations and macro scale models for prediction of transport behavior of (reactive) solutes.
... The calculated power-law coefficient was equal to 1.24, which is in good agreement with the experimental measurements for sandstones. Also, the advection-diffusion equation associated with the network of flow channels of porous media has been numerically developed to quantify the non-Fickian transport behaviour (Cao and Kitanidis, 1998;Garmeh et al., 2009;Jha et al., 2009). Accordingly, Garmeh et al. (2009) obtained the power-law coefficients in the range of 1.55 to 1.89 by simulating the performance of homogeneous unconsolidated porous media, which were higher than those of well-packed porous media. ...
... It is embedded as the standard in most popular reactive transport codes (e.g., Bea et al., 2009;Cederberg et al., 1985;De Simoni et al., 2005;Mangold & Tsang, 1991;Saaltink et al., 2004;Steefel & Lasaga, 1994;Steefel et al., 2015;Walter et al., 1994;Yeh & Tripathi, 1991, and references therein). However, field and laboratory observations, numerical simulations, and theoretical developments have demonstrated time and time again that the upscaled ADRE fails to adequately represent mixing and chemical reactions at all scales (Cao & Kitanidis, 1998;de Anna et al., 2014ade Anna et al., , 2014bEdery et al., 2009;Fernàndez-Garcia et al., 2008;Gramling et al., 2002;Palanichamy et al., 2009;Porta et al., 2016;Rashidi et al., 1996;Sanchez-Vila et al., 2010;Tartakovsky et al., 2008), because of its disregard for the local concentration fluctuations and the use of scale-averaged concentrations to compute reactions. In fact, there is the general consensus that reaction rates observed in the field tend to be much lower than those measured in laboratory experiments because of the presence of anticorrelated local fluctuations of reactant concentrations (e.g., Chiogna & Bellin, 2013;Ding et al., 2017). ...
The presence of solute concentration fluctuations at spatial scales much below the scale of resolution is a major challenge for modeling reactive transport in porous media. Overlooking small‐scale fluctuations, which is the usual procedure, often results in strong disagreements between field observations and model predictions, including, but not limited to, the overestimation of effective reaction rates. Existing innovative approaches that account for local reactant segregation do not provide a general mathematical formulation for the generation, transport, and decay of these fluctuations and their impact on chemical reactions. We propose a Lagrangian formulation based on the random motion of fluid particles carrying solute concentrations whose departure from the local mean is relaxed through multirate interaction by exchange with the mean (MRIEM). We derive and analyze the macroscopic descriptionof the local concentration covariance that emerges from the model, showing its potential to simulate the dynamics of mixing‐limited processes. The action of hydrodynamic dispersion on coarse‐scale concentration gradients is responsible for the production of local concentration covariance, whereas covariance destruction stems from the local mixing process represented by the MRIEM formulation. The temporal evolution of integrated mixing metrics in two simple scenarios shows the trends that characterize fully resolved physical systems, such as a late‐time power law decay of the relative importance of incomplete mixing with respect to the total mixing. Experimental observations of mixing‐limited reactive transport are successfully reproduced by the model.
... It is embedded as the standard in most popular reactive transport codes [e.g., Cederberg et al., 1985;Mangold and Tsang, 1991;Yeh and Tripathi, 1991;Steefel and Lasaga, 1994;Walter et al., 1994;Saaltink et al., 2004;De Simoni et al., 2005;Bea et al., 2009;Steefel et al., 2015, and references therein]. However, field and laboratory observations, numerical simulations and theoretical developments have demonstrated time and time again that the upscaled ADRE fails to adequately represent mixing and chemical reactions at all scales [Rashidi et al., 1996;Cao and Kitanidis, 1998;Gramling et al., 2002;Palanichamy et al., 2009;Tartakovsky et al., 2008;Fernàndez-Garcia et al., 2008;Edery et al., 2009;Sanchez-Vila et al., 2010;de Anna et al., 2014a,b;Porta et al., 2016], because of its disregard for the local concentration fluctuations and the use of scale-averaged concentrations to compute reactions. In fact, the main reason why reaction rates observed in the field tend to be much lower than those measured in laboratory experiments is the presence of anti-correlated local fluctuations of reactant concentrations [Chiogna and Bellin, 2013;Ding et al., 2017]. ...
... As a classical macroscopicscale approach, the advection-dispersion reaction equation (ADRE), based on a perfect mixing assumption, commonly overestimates the reaction rate and tends to generate a less reliable prediction. Many studies [3][4][5][6][7][8][9] have shown that the ADRE is generally incapable of describing the local mixing phenomenon and the neglected pore-scale incomplete mixing is a primary reason for the overestimated reaction rate. Therefore, the mixing-induced reaction transport through the fracture still needs to be better understood at the pore scale. ...
This pore-scale modeling study in single self-affine fractures showed that the heterogeneous flow field had a significant influence on the mixing-induced reaction transport. We generated the single self-affine fracture by the successive random additions (SRA) technique. The pore-scale model was developed by coupling the Navier-Stoke equation (NSE) and advection-diffusion equation with reaction (ADER). Eddies were captured in the self-affine fracture due to the increasing Reynolds number and the sudden expansion of aperture. The flux-weighted breakthrough curves (BTCs) of reaction product showed the typical non-Fickian characteristics (i.e., “early arrival” and “heavy tail”). It was found that the reactant was involved in eddies and then reacted inside the eddy-controlled domain. Consequently, eddies played a significant role in delaying the mass exchange process between the eddy-controlled domain and the main flow channel, which resulted in the “heavy tail” in BTCs. As the Reynolds number increased, the breakthrough time increased while the concentration peaks of BTCs decreased. Furthermore, the dilution index presenting the exponential of the Shannon entropy of a concentration probability distribution was used to quantify the degree of reactant mixing. The results showed that the quantification of dilution for nonreaction transport was in good agreement with the outcomes of mixing-induced reaction transport. The high Reynolds number and Peclet number had a negative influence on the mixing process at the early time whereas they led to the enhanced mixing process at the late time.
... Besides these experimental evidences for electrolyte transport, a number of pore-scale studies showed the importance of diffusion and incomplete mixing in pore throats and channels also at high flow velocities and Péclet numbers. Pore-scale investigations were based on both microfluidic experiments (Li et al., 2008;Zhang et al., 2010;de Anna et al., 2014;Scheven et al., 2013) and pore-scale simulations (e.g., Cao & Kitanidis, 1998;Tartakovsky et al., 2009;Molins et al., 2012;Rolle et al., 2012;Yoon et al., 2012;Rolle & Kitanidis, 2014;Crevacore et al., 2016). In 3 addition to the results of laboratory experiments, the outcomes of such pore-scale investigations have also been instrumental to improve the parameterization of key macroscopic quantities, such as hydrodynamic dispersion coefficients (e.g., Delgado, 2006;Bijelijc & Blunt, 2007;Chiogna et al., 2010;Rolle et al., 2012), necessary for the continuum-based description of solute transport in groundwater studies. ...
Transport of multicomponent electrolyte solutions in saturated porous media is affected by the electrostatic interactions between charged species. Such Coulombic interactions couple the displacement of the different ions in the pore water and remarkably impact mass transfer not only under diffusion‐ but also under advection‐dominated flow regimes. To accurately describe charge effects in flow‐through systems, we propose a multidimensional modeling approach based on the Nernst‐Planck formulation of diffusive/dispersive fluxes. The approach is implemented with a COMSOL‐PhreeqcRM coupling allowing us to solve multicomponent ionic conservative and reactive transport problems, in domains with different dimensionality (1‐D, 2‐D and 3‐D), and in homogeneous and heterogeneous media. The Nernst‐Planck based coupling has been benchmarked with analytical solutions, numerical simulations with another code, and high‐resolution experimental datasets. The latter include flow‐through experiments that have been carried out in this study to explore the effects of electrostatic interactions in fully three‐dimensional setups. The results of the simulations show excellent agreement for all the benchmarks problems, which were selected to illustrate the capabilities and the distinct features of the Nernst‐Planck based reactive transport code. The outcomes of this study illustrate the importance of Coulombic interactions during conservative and reactive transport of charged species in porous media and allow the quantification and visualization of the specific contributions to the diffusive/dispersive Nernst‐Planck fluxes, including the Fickian component, the term arising from the activity coefficient gradients, and the contribution due to electromigration.
... The Shannon entropy has been successfully applied to describe dilution during transport of conservative tracers and it is generally expressed in terms of dilution index, which essentially quantifies the effective volume of a solute slug [Kitanidis, 1994;Cao and Kitanidis, 1998;Tartakovsky et al., 2009;Dentz et al., 2011;Rolle et al., 2013;Schneider et al., 2013;Rolle and Kitanidis, 2014;Porta et al., 2015;Ben Neriah and Paster, 2016;Boon et al., 2016]. A modified version of the dilution index, the flux-related dilution index, has been proposed to quantify dilution as volumetric flux carrying the solute mass flux, and has been mostly used to quantify plume dilution in the case of continuous injection Chiogna et al., 2011;Muniruzzaman et al., 2014;Cirpka et al., 2015;Ye et al., 2015aYe et al., , 2015b. ...
... The Shannon entropy has been successfully applied to describe dilution during transport of conservative tracers and it is generally expressed in terms of dilution index, which essentially quantifies the effective volume of a solute slug [Kitanidis, 1994;Cao and Kitanidis, 1998;Tartakovsky et al., 2009;Dentz et al., 2011;Rolle et al., 2013;Schneider et al., 2013;Rolle and Kitanidis, 2014;Porta et al., 2015;Ben Neriah and Paster, 2016;Boon et al., 2016]. A modified version of the dilution index, the flux-related dilution index, has been proposed to quantify dilution as volumetric flux carrying the solute mass flux, and has been mostly used to quantify plume dilution in the case of continuous injection Chiogna et al., 2011;Muniruzzaman et al., 2014;Cirpka et al., 2015;Ye et al., 2015aYe et al., , 2015b. ...
Entropy-based metrics, such as the dilution index, have been proposed to quantify dilution and reactive mixing in solute transport problems. In this work, we derive the transient advection dispersion equation for the entropy density of a reactive plume. We restrict our analysis to the case where the concentration distribution of the transported species is Gaussian and we observe that, even in case of an instantaneous complete bimolecular reaction, dilution caused by dispersive processes dominates the entropy balance at early times and results in the net increase of the entropy density of a reactive species. Successively, the entropy of the reactant decreases until it vanishes. We show the existence of a unique critical value of dilution, which corresponds to the complete consumption of one of the reactants. This critical dilution index is independent of advective and dispersive processes, and depends only on the dimensionality of the problem, on the stoichiometry of the reaction and on the initial concentrations of the reactants. Furthermore, we provide simple analytical expressions to compute the critical reaction time, i.e., the time at which the critical dilution index is reached, for selected flow configurations. Our results show that, differently from the critical dilution index, the critical reaction time depends on solute transport processes such as advection and hydrodynamic dispersion.
... 23 24 Recent experiments and pore-scale simulations showed the key effects of incomplete 40 mixing in the pore channels on the observed macroscopic transport behavior [e.g., Raje 41 and Kapoor, 2000; Gramling et al., 2002; Tartakovsky et al., 2009] and have triggered 42 the development of new transport formulations and modeling approaches [e.g., Willman 43 Edery et al., 2009;Sanchez-Vila et al., 2010;Chiogna and Bellin, 2013; 44 Porta et al., 2015]. Incomplete mixing is due to the slow rate of diffusion, which allows 45 concentration gradients to be sustained at the small scale [Cao and Kitanidis, 1998; Rolle 46 et al., 2012]. In the study of transverse mixing, such effects were found to be responsible 47 Hochstelter et al., 2013]. ...
This study investigates the effects of compound-specific diffusion/dispersion and electrochemical migration on transient solute transport in saturated porous media. We conducted laboratory bench-scale experiments, under advection-dominated regimes (seepage velocity: 0.5, 5, 25 m/d), in a quasi two-dimensional flow-through setup using pulse injection of multiple tracers (both uncharged and ionic species). Extensive sampling and measurement of solutes' concentrations (∼1500 samples; >3000 measurements) were performed at the outlet of the flow-through setup, at high spatial and temporal resolution. The experimental results show that compound-specific effects and charge-induced Coulombic interactions are important not only at low velocities and/or for steady state plumes but also for transient transport under high flow velocities. Such effects can lead to a remarkably different behavior of measured breakthrough curves also at very high Péclet numbers. To quantitatively interpret the experimental results, we used four modeling approaches: classical advection-dispersion equation (ADE), continuous time random walk (CTRW), dual-domain mass transfer model (DDMT), and a multicomponent ionic dispersion model. The latter is based on the multicomponent formulation of coupled diffusive/dispersive fluxes and was used to describe and explain the electrostatic effects of charged species. Furthermore, we determined experimentally the temporal profiles of the flux-related dilution index. This metric of mixing, used in connection with the traditional solute breakthrough curves, proved to be useful to correctly distinguish between plume spreading and mixing, particularly for the cases in which the sole analysis of integrated concentration breakthrough curves may lead to erroneous interpretation of plume dilution.
... However, pore-scale mixing is governed by the gradients in reactant concentrations, especially at low concentrations, which can impact the largerscale reactive transport behavior. As shown by, e.g., Rashidi et al. [1996] and Cao and Kitanidis [1998], the actual amount of mixing between reactants (and hence the amount c 2016 American Geophysical Union. All Rights Reserved. ...
Reactive chemical transport plays a key role in geological media, across scales from pores to an aquifer. Systems can be altered by changes in solution chemistry and a wide variety of chemical transformations, including precipitation/dissolution reactions that cause feedbacks that directly affect the flow and transport regime. The combination of these processes with advective-dispersive-diffusive transport in heterogeneous media leads to a rich spectrum of complex dynamics. The principal challenge in modeling reactive transport is to account for the subtle effects of fluctuations in the flow field and species concentrations; spatial or temporal averaging generally suppresses these effects. Moreover, it is critical to ground model conceptualizations and test model outputs against laboratory experiments and field measurements. This review emphasizes the integration of these aspects, considering carefully-designed and controlled experiments at both laboratory and field scales, in the context of development and solution of reactive transport models based on continuum-scale and particle tracking approaches. We first discuss laboratory experiments and field measurements that define the scope of the phenomena and provide data for model comparison. We continue by surveying models involving advection-dispersion-reaction equation and continuous time random walk formulations. The integration of measurements and models is then examined, considering a series of case studies in different frameworks. We delineate the underlying assumptions, and strengths and weaknesses, of these analyses, and the role of probabilistic effects. We also show the key importance of quantifying the spreading and mixing of reactive species, recognizing the role of small-scale physical and chemical fluctuations that control the initiation of reactions.
... Pore-scale concentration variations are important in remediation reactions [Cao and Kitanidis, 1998]. Dykaar and Kitanidis [1996] found that the slow rate of molecular diffusion across pores toward prokaryote cells attached to the rock matrix can result in steep concentration gradients if reaction rates or pore diameters are large. ...
Toxic organic contaminants may enter the subsurface as slightly soluble and volatile non-aqueous-phase-liquids (NAPLs) or as dissolved solutes resulting in contaminant plumes emanating from the source zone. A large body of research published in Water Resources Research has been devoted to characterizing and understanding processes controlling the transport and fate of these organic contaminants and the effectiveness of natural attenuation, bioremediation and other remedial technologies. These contributions include studies of NAPL flow, entrapment, and interphase mass transfer that have advanced from the analysis of simple systems with uniform properties and equilibrium contaminant phase partitioning to complex systems with pore- and macro-scale heterogeneity and rate-limited interphase mass transfer. Understanding of the fate of dissolved organic plumes has advanced from when biodegradation was thought to require oxygen, to recognition of the importance of anaerobic biodegradation, multiple redox zones, microbial enzyme kinetics, and mixing of organic contaminants and electron acceptors at plume fringes. Challenges remain in understanding the impacts of physical, chemical, biological and hydrogeological heterogeneity, pore-scale interactions, and mixing on the fate of organic contaminants. Further effort is needed to successfully incorporate these processes into field-scale predictions of transport and fate. Regulations have greatly reduced the frequency of new point source contamination problems; however, remediation at many legacy plumes remains challenging. A number of fields of current relevance are benefiting from research advances from point-source contaminant research. These include geologic carbon sequestration, nonpoint source contamination, aquifer storage and recovery, the fate of contaminants from oil and gas development, and enhanced bioremediation. This article is protected by copyright. All rights reserved.
... However, there remains a strong need for detailed experiments that provide high-resolution data sets to test models, and for improved models that can account for the complex patterns of reactive transport. Modeling, design of appropriate experiments, and determination of representative measurements must address the key role of fluctuations in governing state variables (e.g., concentrations, velocities) caused by medium heterogeneities1234 . For reactive transport, experiments indicate that chemical reactions are localized to pore-scale sites, with medium heterogeneity affecting the nature of reactions and transport of reactants; the high degree of variability in pore-scale mixing impacts the larger-scale reactive transport behavior. ...
We analyze dynamic behavior of chemically reactive species in a porous medium, subject to anomalous transport. In this context, we present transport experiments in a refraction-index-matched, three-dimensional, water-saturated porous medium. A pH indicator (Congo red) was used as either a conservative or a reactive tracer, depending on the tracer solution pH relative to that of the background solution. The porous medium consisted of an acrylic polymer material formed as spherical beads that have pH-buffering capacity. The magnitude of reaction during transport through the porous medium was related to the color change of the Congo red, via image analysis. Here, we focused on point injection of the tracer into a macroscopically uniform flow field containing water at a pH different from that of the injected tracer. The setup yielded measurements of the temporally evolving spatial (local-in-space) concentration field. Parallel experiments with the same tracer, but without reactions (no changes in pH), enabled identification of the transport itself to be anomalous (non-Fickian); this was quantified by a continuous time random walk (CTRW) formulation. A CTRW particle tracking model was then used to quantify the spatial and temporal migration of both the conservative and reactive tracer plumes. Model parameters related to the anomalous transport were determined from the conservative tracer experiments. An additional term accounting for chemical reaction was established solely from analysis of the reactant concentrations, and significantly, no other fitting parameters were required. The measurements and analysis emphasized the localized nature of reaction, caused by small-scale concentration fluctuations and preferential pathways. In addition, a threshold radius for pH-controlled reactive transport processes was defined under buffering conditions, which delineated the region in which reactions occurred rapidly.
... Further, they transferred results of conservative transport to reactive transport in cases where mixing is governed by local scale dispersion, Their analysis was based on the spatially integrated breakthrough curve as a result of advective-dispersive transport in independent stream tubes with identical Peclet number but different seepage velocity. Cao and Kitanidis (1998) studied flow and transport of a nonsorbing tracer in an idealized pore channel. They developed a methodology for the evaluation of macroscopic parameters based on homogenization and illustrated it with an example of dilution at a small scale to determine dilution. ...
... They observed an increased influence of Reynolds number on longitudinal dispersion for Re > 20 and also noted that this influence was less pronounced for disordered unit cells than for ordered unit cells. Cao and Kitanidis [20] studied dispersion and dilution in a 2D sinusoidal-walled channel in the Stokes regime, and compared derived macroscopic solutions with results of direct numerical simulation. Bijeljic and colleagues [21,22] used a pore-network modeling method to study dispersion in porous media, and identified inertial effects as a possible explanation for changes in effective dispersion at high Re. ...
... Indeed, pore-scale mixing is governed by the gradients in reactant concentrations, especially at low concentrations, which can impact the larger-scale reactive transport behavior. As shown by, e.g., Rashidi et al. [1996] and Cao and Kitanidis [1998], the actual amount of mixing between reactants (and hence the amount of product) may be less than would be otherwise predicted by continuum models, due to averaging of the concentrations and the mixing zone. ...
We use a particle tracking approach to analyze the dynamics that control bimolecular reactive transport (A + B → C) in porous media. Particle transitions are governed by spatial and temporal distributions to account for the transport within a continuous time random walk framework. Particle tracking simulations are compared to measurements from a laboratory experiment of bimolecular reactive transport in a constant flow field. The simulations capture the experimental sequence of evolving C particle profiles using a marginally Fickian temporal distribution to quantify the particle transitions. The first profile is a fit with the model parameters, and subsequent ones are predictions. The rate of production of reaction product C over time is found to follow a power law. At early times after the injection of A particles into a uniform distribution of B particles, the strong contact and reaction between A and B particles induces the formation of a spatial void between the reactants. At longer times, the production of C is nearly constant and depends on the fluctuations of velocities of reactant particles that can surmount the void. We probe the behavioral dependence of the A, B, and C spatial profiles on the spectra of velocity fluctuations of the reactants. The latter are generated by different temporal distributions, namely, a decaying exponential distribution, which is equivalent to advective-dispersive (Fickian) transport, and the truncated power law with degrees of non-Fickian behavior, which is characteristic of transport in heterogeneous media. We demonstrate that the C profile exhibits subtle dynamics because of competition between the dispersion (spreading of the plumes) of A and B and the (power law) production rate.
... This kind of measurement was applied to determine mixing of conservative tracers at the laboratory [e.g., Ursino et al., 2001;Moroni et al., 2007;Rossi et al., 2007Rossi et al., , 2008 and at the field scale [e.g., Thierrin and Kitanidis, 1994]. The dilution index was also used in pore-scale modeling [e.g., Cao and Kitanidis, 1998], in modeling studies in heterogeneous porous media Kitanidis, 1996, 1998;Beckie, 1998;Pannone and Kitanidis, 1999;McLaughlin and Ruan, 2001], and even to determine numerical dispersion of transport codes [Chilakapati and Yabusaki, 1999]. Recently, this measure was applied in pore-scale multicomponent reactive transport simulations [Tartakovsky et al., 2009]. ...
The correct quantification of mixing is of utmost importance for modeling reactive transport in porous media and for assessing the fate and transport of contaminants in the subsurface. An appropriate measure of mixing in heterogeneous porous formations should correctly capture the effects on mixing intensity of various processes at different scales, such as local dispersion and the mixing enhancement due to heterogeneities. In this work, we use the concept of flux-related dilution index as a measure of transverse mixing. This quantity expresses the dilution of the mass flux of a conservative tracer solution over the total discharge of the system, and is particularly suited to address problems where a compound is continuously injected into the domain. We focus our attention on two-dimensional systems under steady state flow conditions and investigate both conservative and reactive transport in homogeneous and heterogeneous porous media at different scales. For mixing-controlled reactive systems, we introduce and illustrate the concept of critical dilution index, which represents the amount of mixing required for complete degradation of a continuously emitted plume undergoing decay upon mixing with ambient water. We perform two-dimensional numerical experiments at bench and field scales in homogeneous and heterogeneous conductivity fields. These numerical simulations show that the flux-related dilution index quantifies mixing and that the concept of critical dilution index is a useful measure to relate the mixing of conservative tracers to mixing-controlled degradation of reactive compounds.
... Flow-through bench-scale experiments were performed to study transverse mixing of conservative and (bio)reactive solutes in homogeneous (e.g., Delgado and Carvalho 2001;Klenk and Grathwohl 2002;Huang et al. 2003;Gaganis et al. 2005;Tartakovsky et al. 2008;Rolle et al. 2010) and heterogeneous (e.g., Bauer et al. 2009a, Bauer et al. 2009bRolle et al. 2009) porous media. At the smaller pore scale, recent advances include numerical modeling studies (Cao and Kitanidis 1998;Knutson et al. 2007;Acharya et al. 2007;Bijeljic and Blunt 2007), theoretical investigation of solute dispersion (Porter et al. 2010) and microfluidic experiments (Willingham et al. 2008;Zhang et al. 2010). These detailed studies at the pore scale have contributed to improve the understanding of transverse mixing and mixing-controlled reactions by directly taking into account the physical mechanisms governing solute transport. ...
In this study, we performed multitracer laboratory bench-scale experiments and pore-scale simulations in different homogeneous saturated porous media (i.e., different grain sizes) with the objective of (i) obtaining a generalized parameterization of transverse hydrodynamic dispersion at the continuum Darcy scale; (ii) gaining an improved understanding of the role of basic transport processes (i.e., advection and molecular diffusion) at the subcontinuum scale and their effect on the macroscopic description of transverse mixing in porous media; (iii) quantifying the importance of compound-specific properties such as aqueous diffusivities for transport of different solutes. The results show that a non-linear compound-dependent parameterization of transverse hydrodynamic dispersion is required to capture the observed lateral displacement over a wide range of seepage velocities (0.1–35 m/day). With pore-scale simulations, we can prove the hypothesis that the interplay between advective and diffusive mass transfer results in vertical concentration gradients leading to incomplete mixing in the pore channels. We quantify mixing in the pore throats using the concept of flux-related dilution index and show that different solutes undergoing transport in a flow-through system with a given average velocity can show different degrees of incomplete mixing. Furthermore, it is this compound-specific incomplete mixing within pores that causes different local transverse (mechanical) dispersion to result at the Darcy scale for high flow velocities. We conclude that physical processes at the microscopic level significantly determine the observed macroscopic behavior and, therefore, should be properly reflected in up-scaled parameterizations of transport processes such as local hydrodynamic dispersion coefficients.
... In this work we focus on the dilution index, which, for a conservative species i with concentration C i [ML À3 ], is defined as: [Kitanidis, 1994]. This metric has been successfully applied in many contexts to measure the degree of dilution and to effectively distinguish between spreading and dilution [e.g., Cao and Kitanidis, 1998;Beckie, 1998;Tartakovsky et al., 2009;Bolster et al., 2011b]. The dilution index represents the exponential of the Shannon entropy of a concentration probability distribution p [L À3 ] defined over a volume V ...
Mixing processes significantly affect reactive solute transport in
fluids. For example, contaminant degradation in environmental aquatic
systems can be limited either by the availability of one or more
reactants, brought into contact by physical mixing, or by the kinetics
of the (bio)chemical transformations. Appropriate metrics are needed to
accurately quantify the interplay between mixing and reactive processes.
The exponential of the Shannon entropy of the concentration probability
distribution has been proposed and applied to quantify the dilution of
conservative solutes either in a given volume (dilution index) or in a
given water flux (flux-related dilution index). In this work we derive
the transport equation for the entropy of a reactive solute. Adopting a
flux-related framework, we show that the degree of uniformity of the
solute mass flux distribution for a reactive species and its rate of
change are informative measures of physical and (bio)chemical processes
and their complex interaction.
... [3] In order to model and understand the physics behind mixing in porous media, it is necessary to define the scale of interest at which we aim to address a problem [Bear, 1972;Dentz et al., 2011]. Porous media structures can be analyzed at the pore scale [e.g., Cao and Kitanidis, 1998;Li et al., 2006;Tartakovsky et al., 2009;Willingham et al., 2008Willingham et al., , 2010, at the Darcy scale [e.g., Berkowitz et al., 2000;Ginn et al., 2001] or at the larger field scale [e.g., Thierrin and Kitanidis, 1994;Kapoor and Kitanidis, 1998;Heinz et al., 2003;Englert et al., 2009;Li et al., 2010]. Depending on the system we are interested in, we may focus just on one of these scales. ...
Different measures of dilution have been proposed to describe solute
mixing in heterogeneous porous media. Most of these approaches lead to
the definition of effective dispersion coefficients. In order to
quantify mixing, these up-scaled parameters should account for both
local-scale dispersion and effects of flow variability in heterogeneous
formations (e.g., flow focusing in high-conductivity and defocusing in
low-conductivity inclusions). The correct quantification of mixing is
particularly important for transport of compounds undergoing reactions.
Recent results of multitracer laboratory experiments showed a dependency
of local transverse dispersion on molecular diffusion over a wide range
of flow velocities, implying compound-specific transverse mixing even at
intermediate and high Péclet numbers. The goal of this study is
to assess the relevance of a compound-specific local-scale transverse
dispersion on conservative and reactive mixing in heterogeneous domains
at the field scale. We restrict our analysis to steady state
two-dimensional flow and transport with continuous injection from a line
source. We present numerical simulations in heterogeneous domains with
different characteristics of variability in the conductivity field, and
apply as measures of solute mixing: (1) the effective transverse
dispersion coefficient derived from second central spatial moments, (2)
a dispersion coefficient derived from flux-related second central
spatial moments, (3) the scalar dissipation rate, and (4) a dispersion
coefficient derived from the flux-related dilution index. The results
indicate compound-specific transverse mixing behavior also at the field
scale which is particularly significant in case of low to moderately
heterogeneous porous media. Moreover, we show that measures of dilution
calculated in a flux-related framework result in an improved
quantification of mixing processes and allow to define up-scaled
parameters (i.e., effective transverse dispersion coefficients) affected
by a low degree of uncertainty. For mixing-controlled reactive transport
we illustrate the importance of compound-dependent local effects on the
length of reactive solute plumes.
This chapter describes the Langevin advection‐diffusion‐reaction (LADR) model, an effective stochastic model defined on the scale L. For reactive transport, results show that the error in the Darcy advection‐dispersion‐reaction model increases with the decreasing dilution index. The error is defined as the ratio between masses of the reaction product obtained from the pore‐ and Darcy‐scale simulations. The chapter then demonstrates that separate treatment of advective and diffusion mixing is more accurate for modeling multicomponent reactive transport. Pore‐scale flow and transport equations are solved numerically using the smoothed particle hydrodynamics (SPH) method. Multicomponent reactive transport in porous media is a challenging problem because of the complex interplay between diffusive and advective mixing and reactions. In the case of the concentration‐gradient‐driven Rayleigh‐Taylor instability, author's results show that the advection‐dispersion model underestimates the concentration gradient across the front separating two miscible fluids and the rate of the Rayleigh‐Taylor instability growth.
The limiting of reaction rates in porous media due to the imperfect mixing of reactants is a well studied phenomenon. It has been observed on both the field and laboratory scale and studied using a range of numerical techniques including continuum approaches and Lagrangian particle tracking methods. A new method is presented here for the simulation of mixing-limited reactions using continuum methods, based on the idea of separating each reactant into a mixed and an unmixed fraction, of which only the mixed fraction can react. The method is shown to be capable of producing mixing-limited reaction rates associated with one-, two-, and three-dimensional systems, and of producing outputs well fitted to the experimental data presented in Gramling et al. [1].
The process of dispersion in porous media is the effect of combined variability in fluid velocity and concentration at scales smaller than the ones resolved that contributes to spreading and mixing. It is usually introduced in textbooks and taught in classes through the Fick-Scheidegger parameterization, which is introduced as a scientific law of universal validity. This parameterization is based on observations in bench-scale laboratory experiments using homogeneous media. Fickian means that dispersive flux is proportional to the gradient of the resolved concentration while the Scheidegger parameterization is a particular way to compute the dispersion coefficients. The unresolved scales are thus associated with the pore-grain geometry that is ignored when the composite pore-grain medium is replaced by a homogeneous continuum. However, the challenge faced in practice is how to account for dispersion in numerical models that discretize the domain into blocks, often cubic meters in size, that contain multiple geologic facies. Although the Fick-Scheidegger parameterization is by far the one most commonly used, its validity has been questioned. This work presents a method of teaching dispersion that emphasizes the physical basis of dispersion and highlights the conditions under which a Fickian dispersion model is justified. In particular, we show that Fickian dispersion has a solid physical basis provided that an equilibrium condition is met. The issue of the Scheidegger parameterization is more complex but it is shown that the approximation that the dispersion coefficients should scale linearly with the mean velocity is often reasonable, at least as a practical approximation, but may not necessarily be always appropriate. Generally in Hydrogeology, the Scheidegger feature of constant dispersivity is considered as a physical law and inseparable from the Fickian model, but both perceptions are wrong. We also explain why Fickian dispersion fails under certain conditions, such as dispersion inside and directly upstream of a contaminant source. Other issues discussed are the relevance of column tests and confusion regarding the meaning of terms dispersion and Fickian.
This chapter is concerned with dispersion of nonreactive solutes in saturated porous media. It reviews velocity‐based solute transport models, and defines their parameters. The chapter presents the different mechanisms contributing to the spreading of a conservative nonreactive solute in saturated soil. It also defines pore characteristics relevant to solute transport and discusses methods of measuring them. The chapter also reviews experimental studies relating pore characteristics to solute dispersion. It is then devoted to theoretical models for solute transport based on various pore characteristics. These models are presented in order of increasing complexity in their representation of the pore space geometry. Emerging areas of research are identified. Further research into the feasibility of predicting dispersion at the reservoir scale by renormalizing core‐scale dispersivity measurements may prove fruitful. Finally, the chapter summarizes relationships between solute spreading and pore characteristics, and discusses the predictive potential of pore‐based transport models.
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.
Dispersive transport in porous media is usually described through a Fickian model, in which the flux is the product of a dispersion tensor times the concentration gradient. This model is based on certain implicit assumptions, including slowly varying conditions. About fifty years ago, it was first suggested that the parameterization of the second-order dispersion tensor for anisotropic porous media involves a fourth-order dispersivity tensor. However, the properties of the dispersivity tensor have not been adequately studied. This work contributes to achieving a better grasp of dispersion in anisotropic porous media through a number of ways. First, with clearly stated assumptions and from first principles, we use the method of moments to derive a mathematical formula for the fourth-order dispersivity tensor, and show that it is a function of pore geometry, fluid velocity, and pore diffusion. Second, by using pore-scale flow and transport simulations through orderly and randomly packed 2-D and 3-D porous media, we evaluate the effects of the three factors on dispersivity. Different relationships with the Peclét number are observed for the longitudinal and transverse dispersivities and for orderly and randomly packed media. Third, we discuss the limitations of 2-D periodic media with simple structures in computing transverse dispersivity, which is more accurately predicted in the 3-D periodic media and 2-D randomly packed media. Fourth, we exhibit through numerical simulations that the method of moments can, computational limitations notwithstanding, be extended to stationary porous media.
We review the analysis of the dynamics of reactive transport in
disordered media, emphasizing the nature of the chemical reactions and
the role of small-scale fluctuations induced by the structure of the
porous medium. We are motivated by results and interpretations of
laboratory-scale experiments, for which detailed characterization of the
system is possible. Modeling approaches based on continuum and particle
tracking (PT) schemes are examined critically, highlighting how
fluctuations are incorporated. The continuum approach spans a large
literature. Traditional formats of reactive transport equations, such as
the advection-dispersion-reaction equation (ADRE), are based on a series
of assumptions related mainly to scale separation and relative magnitude
of time scales involved in the reactive transport setting. These
assumptions as well as further developments are assessed in depth. PT
methods offer an alternative means of accounting for pore-scale
dynamics, wherein space-time transitions are drawn from appropriate
probability distributions that have been tested to account for anomalous
transport. While PT methods have been employed for many years to
describe conservative transport, their application to laboratory-scale
reactive transport problems in the context of both Fickian and
non-Fickian regimes is relatively recent. We concentrate on experimental
observations of different types of reactions in disordered media: (1)
the dynamics of a bimolecular reactive transport (A + B → C) in
passive (non-reactive) media, and (2) a multi-step chemical reaction, as
exemplified in the process of dedolomitization involving both
dissolution and precipitation. The fluctuations in a number of the key
variables controlling the processes prove to have a dominant role;
elucidation of this role forms the basis of the present study and the
comparison of methods.
Mixing and a nonlinear bimolecular chemical reaction (reactant A + reactant B → product; reaction rate r = κc
1c
2) in laminar shear flow are investigated. It is found that asymptotically the dominant balance between the rates of production and dissipation of the mean-squared concentration fluctuations \((\sigma_{c_1 }^2 ,\sigma_{c_2 }^2)\) and cross-covariance of concentration fluctuations \((\overline {c_1 c_2 })\) occurs under nonreactive and reactive conditions. Longitudinal dispersion of the cross-sectional averages (C
1, C
2), and variances and the cross-covariance of reactant concentrations can be asymptotically quantified by the classic Taylor dispersion coefficient (D) even under reactive conditions. The characteristic time-scale (τ) over which molecular diffusion dissipates concentration variance and the cross-covariance of reactant concentrations is also shown to be the same under nonreactive and reactive conditions. A variational estimate of τ is shown to be close to the values inferred from detailed numerical simulation. The production-dissipation balance implies that the cross-sectional averaged reaction rate follows \(\overline r =\kappa_{eff} C_1 C_2 \) and \(\kappa _{eff} \approx \kappa \left[ {1+2D\tau \left( {{\partial \ln C_1 } \mathord{\left/ {\vphantom {{\partial \ln C_1 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)\left( {{\partial \ln C_2 } \mathord{\left/ {\vphantom {{\partial \ln C_2 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)} \right]\). The effective reaction rate parameter (κ
eff
) is higher than that of well-mixed batch test reaction rate constant (κ) for initially overlapping species and κ
eff
is smaller than κ for initially non-overlapping species.
We provide a quantitative interpretation of the column experiment reported by Gramling et al. (2002). The experiment involves advection-dominated transport in porous media of three dissolved species, i.e., two reactants undergoing a fast irreversible reaction and the resulting product. The authors found that their observations could not be properly fitted with a model based on an advection-dispersion-reaction equation (ADRE) assuming the reaction was instantaneous, the actual measured total reaction product being lower than predictions for all times. The data have been recently well reproduced by Edery et al. (2009, 2010) by means of a particle tracking approach in a continuous time random walk framework. These and other authors have questioned the use of partial differential equation (PDE)–based approaches to quantify reactive transport because of the difficulty in capturing local-scale mixing and reaction. We take precisely this approach and interpret the experiments mentioned by means of a continuum-scale model based on the ADRE. Our approach differs from previous modeling attempts in that we imbue effects of incomplete mixing at the pore scale in a time-dependent kinetic reaction term and show that this model allows quantitative interpretation of the experiments in terms of both reaction product profiles and time-dependent global production rate. The time dependence of the kinetic term presented accounts for the progressive effects of incomplete mixing due to pore-scale rate-limited mass transfer, and follows a power law, which is consistent with the compilation of existing experiments reported by Haggerty et al. (2004). Our interpretation can form the basis for further research to assess the potential use of PDE approaches for the interpretation of reactive transport problems in moderately heterogeneous media.
We quantitatively account for the measured concentration profile from a laboratory experiment of bimolecular (A + B -> C) reactive transport in a porous medium with a particle tracking (PT) model. The PT results are in contrast to the analytical solution of the continuum scale advection-dispersion-reaction equation, which results in an excess quantity of reaction product (C). The approaches differ in the treatment of the mixing zone, the fluctuations due to the low reactant concentrations, and the localized nature of the reaction. The PT can accommodate a range of transport modes with different temporal distributions. An exponential temporal distribution is equivalent to Fickian transport, which we use for the comparison to the laboratory data; a truncated power-law (TPL) temporal distribution yields a non-Fickian transport characteristic of heterogeneous media. We study the influence of disorder on the mixing zone and the product concentration profiles via these contrasting transport modes.
Consider the advection and dispersion of a conservative nonsorbing solute in a spatially variable but statistically homogeneous velocity field. A Lagrangian approach leads to expressions for the large-time mean and variance of concentration. The expressions require only the macroscopic velocity vector Um, the macrodispersion tensor Dm, and an additional tensor Theta. The macroscopic velocities and macrodispersivities are well known from numerous previous studies, but Theta is introduced here for the first time. The tensor Theta is needed to describe the kinetics of dilution of a plume: Two cases with the same Um and Dm have different dilution characteristics depending on Theta. The characteristic times of dilution are given by tensor ThetaDm-1. It is demonstrated, for the first time through a Lagrangian approach, that the coefficient of variation of concentration at the center of the plume becomes proportionate to 1/t, as was previously shown in the Eulerian theory of Kapoor and Gelhar [1994a, b]. Expressions for the geometric mean of the dilution index and the reactor ratio are derived. Numerical simulations support the validity of the approach.
The physical problem of advection, dispersion, and reaction of a suite of chemicals is formulated mathematically as a set of coupled time-dependent nonlinear partial differential equations. Partially based on the Gear scheme for time discretization, the algorithm developed in this paper decouples equations into separate linearized problems. Then, using finite element approximation, these resulting linear advection-dispersion equations are further transformed, via a re-assembling of variational formulation, into an elliptic Stokes-like problem, which can be solved by an existing Stokes solver previously employed when solving the flow problem to produce a velocity background. The methodology is implemented in the simulation of transport and transformations of an electron donor, an electron acceptor, and active biomass. This application provides insights into biofilm evolution and pore clogging, and demonstrates mesh self-adjustment in the process of biofilm growth.
1] In this work we develop the macroscale transport equation for dispersion of a nonreactive chemical species, with a particular focus on the influence of inertial contributions at moderate Reynolds numbers. Our starting point is the continuum level description of transport written at the subpore scale. Volume averaging is used to upscale these equations to develop the macroscale solute balance that applies at the Darcy scale. We develop a fully transient version of the ancillary closure problem that predicts the total dispersion tensor, and we solve the closure using finite Fourier transforms. The result of this effort is a nonlocal macroscale transport equation, where the nonlocal dispersion depends upon the microscale geometry of the pore space and the physical characteristics of the fluid. Both the longitudinal and transverse components of the total dispersion tensor are computed for a simple three-dimensional unit cell. The computational results indicate that a simple three-dimensional periodic unit cell is able to capture the correct behavior for the longitudinal dispersion in the range 10 1 < Pe p < 2.5 Â 10 5 , although the magnitude of the longitudinal dispersion coefficient is underpredicted by up to a factor of about 4. For the transverse dispersion coefficient, the simple unit cell provides much less satisfactory results when compared with experimental data. The inertial effects for the longitudinal dispersion coefficient were relatively small, but for the transverse dispersion coefficient, inertial effects were predicted to increase the transverse dispersion coefficient up to 40 times that which would be predicted for Stokes flow.
The Stokes problem describes flow of an incompressible constant-viscosity fluid when the Reynolds number is small so that inertial and transient-time effects are negligible. The numerical solution of the Stokes problem requires special care, since classical finite element discretization schemes, such as piecewise linear interpolation for both the velocity and the pressure, fail to perform. Even when an appropriate scheme is adopted, the grid must be selected so that the error is as small as possible. Much of the challenge in solving Stokes problems is how to account for complex geometry and to capture important features such as flow separation. This paper applies adaptive mesh techniques, using a posteriori error estimates, in the finite element solution of the Stokes equations that model flow at pore scales. Different selected numerical test cases associated with various porous geometrics are presented and discussed to demonstrate the accuracy and efficiency of our methodology.
A key challenge for predictive modeling of transverse mixing and reaction of solutes in groundwater is to determine values of transverse dispersivity (αT) in heterogeneous flow fields that accurately describe mixing and reaction at the pore scale. We evaluated the effects of flow focusing in high permeability zones on mixing enhancement using experimental micromodel flow cells and pore-scale lattice-Boltzmann-finite-volume model (LB-FVM) simulations. Micromodel results were directly compared to LB-FVM simulations using two different pore structures, and excellent agreement was obtained. Six different flow focusing pore structures were then systematically tested using LB-FVM, and both analytical solutions and a two-dimensional (2D) continuum-scale model were used to fit αT values to pore-scale results. Pore-scale results indicate that the overall rate of mixing-limited reaction increased by up to 40% when flow focusing occurred, and it was greater in pore structures with longer flow focusing regions and greater porosity contrast. For each pore structure, αT values from analytical solutions of transverse concentration profiles or total product at a given longitudinal location showed good agreement for nonreactive and reactive solutes, and values determined in flow focusing zones were always smaller than those downgradient after the flow focusing zone. Transverse dispersivity values from the 2D continuum model were between values within and downgradient from the flow focusing zone determined from analytical solutions. Also, total product and transverse concentration profiles along the entire pore structure from the 2D continuum model matched pore scale results. These results indicate that accurate quantification of pore-scale flow focusing with transverse dispersion coefficients is possible only when the entire flow and concentration fields are considered.
The objectives of this work were to determine if a pore-scale model could accurately capture the physical and chemical processes that control transverse mixing and reaction in microfluidic pore structures (i.e., micromodels), and to directly evaluate the effects of porous media geometry on a transverse mixing-limited chemical reaction. We directly compare pore-scale numerical simulations using a lattice-Boltzmann finite volume model (LB-FVM) with micromodel experiments using identical pore structures and flow rates, and we examine the effects of grain size, grain orientation, and intraparticle porosity upon the extent of a fast bimolecular reaction. For both the micromodel experiments and LB-FVM simulations, two reactive substrates are introduced into a network of pores via two separate and parallel fluid streams. The substrates mix within the porous media transverse to flow and undergo instantaneous reaction. Results indicate that (i) the LB-FVM simulations accurately captured the physical and chemical process in the micromodel experiments, (ii) grain size alone is not sufficient to quantify mixing at the pore scale, (iii) interfacial contact area between reactive species plumes is a controlling factor for mixing and extent of chemical reaction, (iv) at steady state, mixing and chemical reaction can occur within aggregates due to interconnected intra-aggregate porosity, (v) grain orientation significantly affects mixing and extent of reaction, and (vi) flow focusing enhances transverse mixing by bringing stream lines which were initially distal into close proximity thereby enhancing transverse concentration gradients. This study suggests that subcontinuum effects can play an important role in the overall extent of mixing and reaction in groundwater, and hence may need to be considered when evaluating reactive transport.
Microbial degradation of contaminants in the subsurface requires the availability of nutrients; this is impacted by porous media heterogeneity and the degree of transverse mixing. Two types of microfluidic pore structures etched into silicon wafers (i.e., micromodels), (i) a homogeneous distribution of cylindrical posts and (ii) aggregates of large and small cylindrical posts, were used to evaluate the impact of heterogeneity on growth of a pure culture (Delftia acidovorans) that degrades (R)-2-(2,4-dichlorophenoxy)propionate (R-2,4-DP). Following inoculation, dissolved O2 and R-2,4-DP were introduced as two parallel streams that mixed transverse to the direction of flow. In the homogeneous micromodel, biomass growth was uniform in pore bodies along the center mixing line, while in the aggregate micromodel, preferential growth occurred between aggregates and slower less dense growth occurred throughout aggregates along the center mixing line. The homogeneous micromodel had more rapid growth overall (2 times) and more R-2,4-DP degradation (9.5%) than the aggregate pore structure (5.7%). Simulation results from a pore-scale reactive transport model indicate mass transfer limitations within aggregates along the center mixing line decreased overall reaction; hence, slower biomass growth rates relative to the homogeneous micromodel are expected. Results from this study contribute to a better understanding of the coupling between mass transfer, reaction rates, and biomass growth in complex porous media and suggest successful implementation and analysis of bioremediation systems requires knowledge of subsurface heterogeneity.
Groundwater transport models that accurately describe spreading of nonreactive solutes in an aquifer can poorly predict concentrations of reactive solutes. The dispersive term in the advection-dispersion equation can overpredict pore-scale mixing, and thereby overpredict homogeneous chemical reaction. We quantified this experimentally by imaging instantaneous colorimetric reactions between solutions of aqueous CuSO4 and EDTA4- within a 30-cm long translucent chamber packed with cryolite sand that closely matched the optical index of refraction of water. A charge-coupled device camera was used to quantify concentrations of blue CuEDTA2- within the chamber as it was produced by mixing of the two reactants at different flow rates. We compared these experimental results with a new analytic solution for instantaneous bimolecular reaction coupled with advection and dispersion of the product and reactants. For all flow rates, the concentrations of CuEDTA2- recorded in the experiments were about 20% less than predicted by the analytic solution, thereby demonstrating that models assuming complete mixing at the pore scale can overpredict reaction during transport.
Neglecting local solute dispersion, we prove that under realistic conditions the ensemble moments of the local concentration satisfy a mean convection dispersion equation (CDE) for both conservative and instantaneously adsorbing solutes, provided that the solute velocity field is stochastically stationary. Thus the same algorithm can be used to model the ensemble mean concentration and its uncertainty. For uniform initial conditions the solution of the mean CDE contains all the information about the local concentration probability distribution function, which is not Gaussian. The governing equation for k-point ensemble moments of the concentration is similar in form to the mean CDE, with the same effective dispersion coefficient and convective velocity. The special case of an impulse initial condition is used to illustrate that volume averaging can significantly change the concentration coefficient of variation except at the plume front. The persistence in the longitudinal direction of concentration moments and concentration k-point moments is caused by large longitudinal megadispersion. Finally, it is suggested that ``ergodicity'' should be interpreted operationally in terms of an acceptably small coefficient of variation of the concentration field so that this concept can be applied to experimental field data.
The problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two- and three-dimensional flows. Approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity. The solute concentration, regarded as a random variable, is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty. The uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale. In any case the concentration expectation does not obey a diffusion type equation in the case of two-dimensional flows, unless the solute body has traveled a distance larger than a few tens transmissivity integral scales. This distance may be exceedingly large in many conceivable applications.
The Stokes problem describes flow of an incompressible constant-viscosity fluid when the Reynolds number is small so that inertial and transient-time effects are negligible. The numerical solution of the Stokes problem requires special care, since classical finite element discretization schemes, such as piecewise linear interpolation for both the velocity and the pressure, fail to perform. Even when an appropriate scheme is adopted, the grid must be selected so that the error is as small as possible. Much of the challenge in solving Stokes problems is how to account for complex geometry and to capture important features such as flow separation. This paper applies adaptive mesh techniques, using a posteriori error estimates, in the finite element solution of the Stokes equations that model flow at pore scales. Different selected numerical test cases associated with various porous geometrics are presented and discussed to demonstrate the accuracy and efficiency of our methodology.
The pore system of a packed bed is represented by a system of canals in order to permit probability computations for a foreign particle carried by the pore liquid movement to arrive at a certain place in a certain time. The computations lead to explicit values for the coefficient of longitudinal and transversal diffusion. A test device is described which permits the determination of the longitudinal diffusivity. The relationship between test result and theory is discussed.
Low Reynolds number flow theory finds wide application in such diverse fields as sedimentation, fluidization, particle-size classification, dust and mist collection, filtration, centrifugation, polymer and suspension rheology, flow through porous media, colloid science, aerosol and hydrosal technology, lubrication theory, blood flow, Brownian motion, geophysics, meteorology, and a host of other disciplines. This text provides a comprehensive and detailed account of the physical and mathematical principles underlying such phenomena, heretofore available only in the original literature.
The main objective of this work is to introduce a new macroscopic measure of dilution, the dilution index E. Examples serve to demonstrate the usefulness of the measure. A general expression for the rate of dilution of a tracer plume is derived. The exact rate of increase of the dilution index under the idealized conditions of constant dispersion coefficients and a Gaussian plume is computed, and a lower bound is found to the same quantity for non-Gaussian plumes. For the general heterogeneous case the analysis demonstrates that the instantaneous rate of increase of ln E is proportional to the small-scale dispersion coefficients, everthing else being the same. The rate of increase of ln E depends also on the degree of irregularity in the shape of the plume. Thus, in the long term, geological heterogeneity should increase the rate of dilution because spatial variability in the flow velocity tends to deform plumes and make them less regular. -from Author
Transport of inert solutes in natural porous formations is dominated by convection and by the large-scale heterogeneity of permeability. A solute body inserted in the formation spreads because of the variation of velocity among and along the stream tubes which cross the plume. With neglect of the slow effect of pore-scale dispersion the solute particles preserve their initial concentration, but the body as a whole spreads in an irregular manner (Figures 1, 2, and ). The transport theory, based on representation of permeability and velocity as random space functions, can predict the expected value and variance of concentration, but under the above conditions, the coefficient of variation may be large. In contrast, the spatial moments of the solute body are less susceptible to uncertainty, depending on the transverse dimensions of the plume and on the travel time. The first and second spatial moments are regarded as random functions of time, and their expected value and variance are derived in terms of the velocity field. The moments are assumed to satisfy the ergodic hypothesis if their coefficients of variation are negligible. The conditions which ensure the fulfillment of this requirement are examined. The “effective dispersion coefficients” are defined with the aid of the spatial moments and are shown to depend generally on the initial size of the solute body and on travel time. The results are illustrated by an analytical solution of transport in a stratified formation with the average velocity parallel to the bedding.
This paper applies stochastic methods to the analysis and prediction of
solute transport in heterogeneous saturated porous media. Partial
differential equations for three unconditional ensemble moments (the
concentration mean, concentration covariance, and velocity concentration
cross covariance) are derived by applying perturbation techniques to the
governing transport equation for a conservative solute. Concentration
uncertainty is assumed to be the result of unmodeled small-scale
fluctuations in a steady state velocity field. The moment expressions,
which describe how each moment evolves over time and space, resemble the
classic deterministic advection-dispersion equation and can be solved
using similar methods. A solution procedure based on a Galerkin finite
element algorithm is illustrated with a hypothetical two-dimensional
example. For this example the required steady state velocity statistics
are obtained from an infinite domain spectral solution of the stochastic
groundwater flow equation. The perturbation solution is shown to
reproduce the statistics obtained from a Monte Carlo simulation quite
well for a natural log conductivity standard deviation of 0.5 and
moderately well for a natural log conductivity standard deviation of
1.0. The computational effort required for a perturbation solution is
significantly less than that required for a Monte Carlo solution of
acceptable accuracy. Sensitivity analyses conducted with the
perturbation approach provide qualitative confirmation of a number of
results obtained by other investigators for more restrictive special
cases.
In the mid-seventies, a new area of research has emerged in subsurface hydrology, namely sto chastic modeling of flow and transport. This development has been motivated by the recognition of the ubiquitous presence of heterogeneities in natural formations and of their effect upon transport and flow, on the one hand, and by the vast expansion of computational capability provided by elec tronic machines, on the other. Apart from this, one of the areas in which spatial variability of for mation properties plays a cardinal role is of contaminant transport, a subject of growing interest and concern. I have been quite fortunate to be engaged in research in this area from its inception and to wit ness the rapid growth of the community and of the literature on spatial variability and its impact upon subsurface hydrology. In view of this increasing interest, I decided a few years ago that it would be useful to present the subject in a systematic and comprehensive manner in order to help those who wish to engage themselves in research or application of this new field. I viewed as my primary task to analyze the large scale heterogeneity of aquifers and its effect, presuming that the reader already possesses a background in traditional hydrology. This is achieved in Parts 3, 4 and 5 of the text which incorporate the pertinent material."
This book presents a coherent introduction to boundary integral, boundary element and singularity methods for steady and unsteady flow at zero Reynolds number. The focus of the discussion is not only on the theoretical foundation, but also on the practical application and computer implementation. The text is supplemented with a number of examples and unsolved problems, many drawn from the field of particulate creeping flows. The material is selected so that the book may serve both as a reference monograph and as a textbook in a graduate course on fluid mechanics or computational fluid mechanics.
A unified Eulerian-Lagrangian theory is presented for the transport of a
conservative solute in a random velocity field that satisfies locally
∇ · v(x, t) = f(x, t), where f(x, t) is a random function
including sources and/or the time derivative of head. Solute
concentration satisfies locally the Eulerian equation ∂c(x,
t)/∂t + ∇ · J(x, t) = g(x, t), where J(x, t) is
advective solute flux and g(x, t) is a random source independent of f(x,
t). We consider the prediction of c(x, t) and J(x, t) by means of their
unbiased ensemble moments <c(x, t)>ν and <J(x,
t)>ν conditioned (as implied by the subscript) on local
hydraulic measurements through the use of the latter in obtaining a
relatively smooth unbiased estimate ν(x, t) of v(x, t). These
predictors satisfy ∂<c(x, t)>v/∂t + ∇
· <J(x, t)>ν = <g(x,
t)>ν, where <J(x, t)>ν = ν(x,
t)<c(x, t)>ν + Qν(x, t) and
Qν(x, t) is a dispersive flux. We show that
Qν, is given exactly by three space-time convolution
integrals of conditional Lagrangian kernels αν with
∇·Qν, βν with
∇ ν, and γν with
ν for a broad class of v(x, t) fields, including
fractals. This implies that Qν(x, t) is generally nonlocal
and non-Fickian, rendering <c(x, t)>ν non-Gaussian.
The direct contribution of random variations in f to Qν
depends on ν rather than on
∇ ν,. We elucidate the nature of the above
kernels; discuss conditions under which the convolution of
βν and ∇ becomes pseudo-Fickian, with
a Lagrangian dispersion tensor similar to that derived in 1921 by
Taylor; recall a 1952 result by Batchelor which yields an exact
expression for ν at early time; use the latter to
conclude that linearizations which predict that ν
bifurcates at early time when the probability density function of v is
unimodal cannot be correct; propose instead a weak approximation which
leads to a nonlinear integro-differential equation for
ν due to an instantaneous point source and which
improves with the quantity and quality of hydraulic data; demonstrate
that the weak approximation is analogous to the "direct interaction"
closure of turbulence theory; offer non-Fickian and pseudo-Fickian weak
approximations for the second conditional moment of the concentration
prediction error; demonstrate that it yields the so-called "two-particle
covariance" as a special case; conclude that the (conditional) variance
of c does not become infinite merely as a consequence of disregarding
local dispersion; and discuss how to estimate explicitly the cumulative
release of a contaminant across a "compliance surface" together with the
associated estimation error.
A rigorous theory of dispersion in both granular and sintered spatially-periodic porous media is presented, utilizing concepts originating from Brownian motion theory. A precise prescription is derived for calculating both the Darcy-scale interstitial velocity vector {v}* and dispersivity dyadic {D}* of a tracer particle. These are expressed in terms of the local fluid velocity vector field v at each point within the interstices of a unit cell of the spatially periodic array and, for the dispersivity, the molecular diffusivity D of the tracer particle through the fluid. Though the theory is complete, numerical results are not yet available owing to the complex structure of the local interstitial velocity field v. However, as an illustrative exercise, the theory is shown to correctly reduce in an appropriate limiting case to the well-known Taylor-Aris results for dispersion in circular capillaries.
The concentration variance, i.e., mean squared concentration fluctuations, undergoes mean advection, a local dispersive flux, and a macrodispersive flux due to a correlation between squared concentration perturbations and velocity perturbations. The products of the macrodispersion coefficient and the squared gradient of the mean concentration field determine the rate of production of concentration variance. The rate of dissipation of concentration variance is determined by the product of the local dispersion coefficient and the mean squared gradient of the concentration perturbation field. Variance dissipation is represented as a first-order decay with the decay coefficient equal to twice the sum of the local dispersion coefficient divided by the squared concentration microscale. The concentration microscale, estimated for an advection-dominated log hydraulic conductivity microscale, is an increasing function of the log conductivity microscale. Thus the larger the log conductivity microscale is, the slower is the rate of dissipation of concentration fluctuations by local dispersion and vice versa. The wave number squared dependence of fluctuation dissipation requires intensive sampling to realistically model the log conductivity spectrum and its microscale, which determines the rate of dissipation of concentration fluctuations by the action of local dispersion. There is no mechanism of destroying concentration fluctuations without the action of local dispersion.
For a multidimensional finite-size impulse input, analytical solutions to the conservation equation for concentration variance σc2 are presented. Due to the dissipating action of local dispersion, at large times, σc is a decreasing fraction of the mean concentration. The Cape Cod bromide tracer exhibits this decrease. The larger the log conductivity microscale is, the slower is the action of local dispersion, and the slower is the predicted rate of decrease of the ratio of σc and the mean concentration (i.e., the coefficient of variation) with time, at large times, and vice versa. The coefficient of variation increases with distance from the center. A balance between the rates of production and dissipation of σc2 relates it linearly to the squared gradients of the mean concentration field, away from the center of mass. For the zero local dispersion case, σc is an unboundedly growing multiple with time of the mean concentration. The longitudinal spatial second moment and macrodispersivities are insensitive to the inclusion/exclusion of local dispersion and therefore do not differentiate between the concentration fields for the two different cases. In contrast, the spatial-temporal evolution of σc2 is singularly determined by the dissipating action of local dispersion. Measurements of local dispersivities need to be made along with a characterization of hydraulic conductivity variations to assess contaminant concentrations in aquifers.
J. A. Cole (Department of Geodesy and Geo-physics, University of Cambridge, Cambridge, England)-The writer finds the paper most valuable,
because he [Cole, 1957] has made experiments with columns of granular material, basically similar to those described by the author, in order
to determine the relative importance of: Effect (a) radial molecular diffusion in each pore combined with a velocity gradient
across the pore, and Effect (b) the geometry of the pore system in producing a longitudinal dispersion of an injected substance.
A spectrally based perturbation approach is used to evaluate the
concentration variability for a steady concentration field in a
three-dimensional statistically homogeneous and anisotropic aquifer. The
analysis assumes small, locally stationary concentration perturbations,
and consequently, is valid only after mean displacements that are large
compared to the scale of aquifer heterogeneity. It is shown that the
concentration variance is proportional to the square of the local mean
concentration gradient and the variance and the correlation scales of
log-hydraulic conductivity (ln K) and is inversely proportional to the
local dispersivity. The concentration covariance function is highly
anisotropic, with the largest correlation length aligned to the mean
flow direction. Another important finding is the sensitivity of the
longitudinal persistence of the concentration field to the high wave
number behavior of the input ln K spectrum. The commonly employed
exponential-type covariance function, which corresponds to a
nondifferentiable random field, results in extremely large concentration
correlation lengths along the flow direction. Input spectra
corresponding to differentiable fields with more rapid high wave number
cutoffs produce a significant reduction of the longitudinal correlation
length.
The physical problem of advection, dispersion, and reaction of a suite of chemicals is formulated mathematically as a set of coupled time-dependent nonlinear partial differential equations. Partially based on the Gear scheme for time discretization, the algorithm developed in this paper decouples equations into separate linearized problems. Then, using finite element approximation, these resulting linear advection-dispersion equations are further transformed, via a re-assembling of variational formulation, into an elliptic Stokes-like problem, which can be solved by an existing Stokes solver previously employed when solving the flow problem to produce a velocity background. The methodology is implemented in the simulation of transport and transformations of an electron donor, an electron acceptor, and active biomass. This application provides insights into biofilm evolution and pore clogging, and demonstrates mesh self-adjustment in the process of biofilm growth.
A physically based model is developed to study the transport of a solute utilized by microorganisms forming a biofilm coating on soil grains in a porous medium. A wavy-walled channel is used as a geometrical model of a porous medium and a biofilm is attached to the channel wall. Within the biofilm the solute is consumed according to a first-order volumetric rate. A numerical study is performed to obtain the dependence of the macrotransport coefficients on the Peclet number and Damkohler number. It is found that in some cases of practical importance the pore fluid is not well mixed, and mass transport limitations can control macroreaction rates. For diffusion-limited cases (large Damkohler numbers) increased solvent velocity can enhance the macroreaction rate by a factor of almost 3. Mean solute and mean solvent velocities are, in general, not equal, and mean solute velocities can exceed mean solvent velocities by 60% at high Damkohler numbers. These results agree qualitatively with those of a previous numerical study by Edwards et al. [1993]. The results also suggest that due to the spatially variable pore geometry, the biomass nearest the pore throat is more effective at consuming the solute than biomass in the pore chamber. A comparison is made between mass transfer correlations and the results determined for the macroreaction rate coefficient. We find that over a limited range of Peclet numbers a macroscale Sherwood number follows the Pe l/3 behavior determined from experimental mass transfer correlations and predicted by boundary layer theory.
The theory of homogenization which is a rigorous method of averaging by multiple scale expansions, is applied here to the transport of a solute in a porous medium. The main assumption is that the matrix has a periodic pore structure on the local scale. Starting from the pores with the Navier-Stokes equations for the fluid motion and the usual convective-diffusion equation for the solute, we give an alternative derivation of the three-dimensional macroscale dispersion tensor for solute concentration. The original result was first found by Brenner by extending Brownian motion theory. The method of homogenization is an expedient approach based on conventional continuum equations and the technique of multiple-scale expansions, and can be extended to more complex media involving three or more contrasting scales with periodicity in every but the largest scale.
Macrodispersion is spreading of a substance induced by spatial variations in local advective velocity at field scales. Consider the case that the steady-state seepage velocity and the local dispersion coefficients in a heterogeneous formation may be modeled as periodic in all directions in an unbounded domain. The equations satisfied by the first two spatial moments of the concentration are derived for the case of a conservative non-reacting solute. It is shown that the moments can be calculated from the solution of well-defined deterministic boundary value problems. Then, it is described how the rate of increase of the first two moments can be calculated at large times using a Taylor-Aris analysis as generalized by Brenner. It is demonstrated that the second-order tensor of macrodispersion (or effective dispersion) can be computed through the solution of steady-state boundary-value problems followed by the determination of volume averages. The analysis is based solely on volume averaging and is not limited by the assumption that the fluctuations are small. The large-time results are valid when the system is in a form of equilibrium in which a tagged particle samples all locations in an appropriately defined phase space with equal probability.
Dilution of solute in two-dimensionally periodic heterogeneous porous media is assessed by numerically simulating advection-dispersion. The concentration fluctuations, created by advective heterogeneity, are destroyed by local dispersion, over a characteristic variance residence time (VRT). For an impulse introduction of solute, initially, plumes become increasingly irregular with time—the coefficient of variation (CV) of concentration grows with time. A priori, the spatial second moment and mean concentrations are insufficient measures of dilution, because concentration fluctuations can be large. At large times (t > VRT) the relative concentration fluctuations weaken—the concentration CV was observed to slowly decrease with time. At the center of mass the general trend of the decreasing CV follows VRT/t (predicted by Kapoor and Gelhar). The VRT is found to be an increasing function of the log hydraulic conductivity microscale. In employing effective dispersion coefficents to model the mean concentration, it needs to be recognized that excursions of concentrations around the mean are singularly determined by local dispersion.
Transport in heterogeneous formations: Spatial moments, ergodicity, and effective dispersion Longitudinal and transverse diffusion in granular deposits (abstract) Macrotransport of a biologically reacting solute through porous media
- G Dagan
- B B Dykaar
- P K Kitanidis
Dagan, G., Transport in heterogeneous formations: Spatial moments, ergodicity, and effective dispersion, Water Resour. Res., 26(6), 1281-1290, 1990. De Josselin De Jong, G., Longitudinal and transverse diffusion in granular deposits (abstract), Eos Trans. AGU, 39(1), 67-74, 1958. Dykaar, B. B., and P. K. Kitanidis, Macrotransport of a biologically reacting solute through porous media, Water Resour. Res., 32(2), 307-320, 1996.
Asymptotic Analysis for Periodic Structures Dispersion resulting from flow through spatially periodic porous media Adaptive finite-element simulation of Stokes flow in porous media An algorithm for solving reactive advec-tion-dispersion problems, interim report
- J Bear
- A Bensoussan
- J L Lions
- G Papanicolaou
- H Brenner
- H Brenner
- D A Edwards
Bear, J., Dynamics of Fluids in Porous Media, Elsevier, New York, 1972. Bensoussan, A., J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, New York, 1978. Brenner, H., Dispersion resulting from flow through spatially periodic porous media, Philos. Trans. R. Soc. London, Ser. A, 297, 81-133, 1980. Brenner, H., and D. A. Edwards, Macrotransport Processes, Butter-worth-Heinemann, Newton, Mass., 1993. Cao, J., and P. K. Kitanidis, Adaptive finite-element simulation of Stokes flow in porous media, Adv. Water Resour., in press, 1998a. Cao, J., and P. K. Kitanidis, An algorithm for solving reactive advec-tion-dispersion problems, interim report, Civ. and Environ. Eng., Stanford Univ., Stanford, Calif., 1998b.
Groundwater Pollution Stochastic Subsurface Hydrology, Prentice-Hall, Engle-wood Cliffs Stochastic analysis of nonstation-ary subsurface solute transport, 1, Unconditional moments
- J J Fried
- L W Gelhar
- W D Graham
- D Mclaughlin
Fried, J. J., Groundwater Pollution, Elsevier, New York, 1975. Gelhar, L. W., Stochastic Subsurface Hydrology, Prentice-Hall, Engle-wood Cliffs, N.J., 1993. Graham, W. D., and D. McLaughlin, Stochastic analysis of nonstation-ary subsurface solute transport, 1, Unconditional moments, Water Resour. Res., 25(2), 215-232, 1989.
Groundwater Pollution
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