Convection-dispersion Equation Research Articles

Study on non-Darcian flow and solute transport characteristics through a filled single fracture

ABSTRACT In this study, well-controlled laboratory experiments on flow and solute migration through a Filled-Single Fracture (FSF) of 3 mm aperture size were performed for different flow rates. Non-Darcian flow conditions prevailed throughout the experiment, which can be effectively explained using Forchheimer or Izbash’s equations. The constant mode was utilized to inject a conservative solute (NaCl) evenly through the FSF, following which the resulting Breakthrough curves (BTCs) were developed for three different average flow velocities. The computed BTCs were simulated by the conventional Convection-Dispersion Equation (CDE); and the single rate Mobile-Immobile (MIM) Model. A non-Fickian trend was observed for the attained BTCs, which cannot be interpreted by the Fickian CDE; however, the CDE model serves as the comparing base. MIM model was found much better than the CDE in fitting BTCs for three different average values of pore velocities. Statistical analysis on the goodness-of-fit further confirmed the suitability of MIM as much greater correlation coefficients were observed compared to the CDE model. Further, the BTCs were found to fit better at lower velocities. The greater computed dispersivity values are compatible with Taylor who suggested that much stronger dispersion occurs under non-Darcian flow conditions.

Approximate analytical solutions for multispecies convection-dispersion transport equation with variable parameters

Multispecies pollutant migration often occurs in polluted groundwater systems. Most of the multispecies problems that have been dealt in the literature assume constant transport parameters, primarily because analytical solutions for varying parameters become a challenge. The present study analytically solves a two-species convection-dispersion transport equation, considering spatially varying dispersion coefficient and seepage velocity, which corresponds to the steady migration in a steady flow domain. Indeed, the methodology can be adopted for other cases, such as the transient migration in a steady flow domain and transient migration in an unsteady flow domain, without any difficulty. Three kinds of homotopy-based methods, namely the homotopy perturbation method (HPM), homotopy analysis method (HAM), and optimal homotopy asymptotic method (OHAM), are used to derive approximate analytical solutions in the form of truncated series. In homotopy analysis method, the convergence-control parameter ℏ plays a key role in the convergence of the series solution. It is observed that for a specific case of this parameter, namely ℏ=−1, the HAM-based solution recovers the HPM-based solution. For the verification of homotopy-based solutions, we utilize the MATLAB routine pdepe, which efficiently solves a class of parabolic PDEs (single/system). An excellent agreement is found between the derived analytical solutions and the numerical solutions for all three methods. Further, a quantitative assessment is carried out for the derived solutions. Also, convergence theorems are proposed for the series solutions obtained using HAM and OHAM.

Decoupling Method for the Convective-Dominated Leaching Process of Ion-Adsorption-Type Rare-Earth Ores

Ion-adsorption-type rare-earth ores (IATREOs) that have experienced granite weathering have good permeability, and their leaching process involves the solute transport problem, which is dominated by convection. Because of the oscillation and dispersion errors of existing numerical methods for solving the convective-dominated solute transport equation, the results have low precision in the twin leaching process. In this paper, the convection–dispersion equation is decoupled into the dispersion equation, the convection equation, and the source-sink equation; the Crank–Nicolson and implicit difference methods are used to solve the dispersion equation and the source-sink equation, respectively. The solution of the convection equation is achieved on the basis of its physical interpretation. Therefore, a decoupling method for the convective-dominated solute transport equation is established. In comparison to the two examples with analytical solutions, the calculation errors of the method established in this paper are less than 2.00%, and it can solve the oscillation and dispersion problems. The rationality of the method is further demonstrated through the column leaching experiment of IATREOs. In comparison to the test results, the coefficients of determination of the breakthrough curves of rare-earth ions and ammonium ions calculated by the proposed method are all greater than 0.850, and the peak concentration error of rare-earth ions is less than 7.00%. This indicates that the proposed method can simulate the leaching process well. Furthermore, by combining the multiple/half method and the dichotomy method, an optimization method for determining the leaching agent amount was established to analyze the relationship between the leaching agent and the ratio of dispersion to pillar length. The results can provide a solution that can be used to mine IATREOs from experience to theory.

Stability of Spectral Collocation Schemes with Explicit-Implicit-Null Time-Marching for Convection-Diffusion and Convection-Dispersion Equations

Stability of Spectral Collocation Schemes with Explicit-Implicit-Null Time-Marching for Convection-Diffusion and Convection-Dispersion Equations

Impact of Geometrical Features on Solute Transport Behavior through Rough-Walled Rock Fractures

The solute transport in the fractured rock is dominated by a single fracture. The geometric characteristics of single rough-walled fractures considerably influence their solute transport behavior. According to the self-affinity of the rough fractures, the fractal model of single fractures is established based on the fractional Brownian motion and the successive random accumulation method. The Navier–Stokes equation and solute transport convective-dispersion equation are employed to analyze the effect of fractal dimension and standard deviation of aperture on the solute transport characteristics. The results show that the concentration front and streamline distribution are inhomogeneous, and the residence time distribution (RTD) curves have obvious tailing. For the larger fractal dimension and the standard deviation of aperture, the fracture surface becomes rougher, aperture distribution becomes more scattered, and the average flow velocity becomes slower. As a result, the average time of solute transport is a power function of the fractal dimension, while the time variance and the time skewness present a negative linear correlation with the fractal dimension. For the standard deviation of aperture, the average time exhibits a linearly decreasing trend, the time variance is increased by a power function, and the skewness is increased logarithmically.

Sequential Extraction of 239+240Pu and the Vertical Distribution of 239+240Pu, 137Cs, and Heavy Metals in Chang-Bai Mountains’ Grassland Soil

To establish a 239+240Pu, 137Cs, and heavy metal reference soil site in Northeast China, three grassland soil cores were collected from the Lao-Ye-Ling Mountain, which is a branch of the Chang-Bai Mountains. The activities were in the range of 0.041–2.752 mBq/g for 239+240Pu and in the range of 0.2–57.6 mBq/g for 137Cs. The 240Pu/239Pu atomic ratios were 0.161–0.186, showing that 239+240Pu in the study area originated from the global fallout. Also, 239+240Pu, 137Cs, and excess 210Pb were enriched in the surface soil. The convective–dispersion equation model provided the lower apparent dispersion coefficients (D) and apparent convection velocities (v) for both 239+240Pu and 137Cs. The soil erosion rate in this study area was estimated by an erosion model, which was lower than that in the other area of Northeast China. 239+240Pu in the different soil fractions was analyzed by a sequential extraction method, and the result showed that 239+240Pu was mainly adsorbed by Fe/Mn oxides and their associated organic matter (NOMFe/Mn-oxides fraction). Except for Fe, all other heavy metals from the anthropogenic source were enriched in the surface soil. The pollution level of heavy metals in this study area was lower. In general, the alpine grassland soil of the Lao-Ye-Ling Mountain can be used as a reference soil site to study environmental pollution and soil erosion and provided some reference baselines for future multidisciplinary studies in Northeast Asia.

Chemotaxis along local chemical gradients enhanced bacteria dispersion and PAH bioavailability in a heterogenous porous medium

Polycyclic aromatic hydrocarbons (PAHs) are ubiquitous, EPA-designated priority pollutants for soil and groundwater, remaining recalcitrant to bioremediation because of limited bioavailability. In this work, we used naphthalene as a model PAH and soil bacteria Pseudomonas putida G7 to investigate the potential role of chemotaxis to enhance access to PAHs in heterogenous porous media. To this aim, we conducted transport experiments and numerical simulations with chemotactic bacteria and naphthalene trapped within a non-aqueous phase liquid (NAPL) mainly in low permeable areas of a dual-permeability microfluidic device. Microscopic imaging showed higher accumulations of chemotactic bacteria, about eight times that of nonchemotactic bacteria, at the junctures between high and low permeability regions. Pore-scale simulations for fluid flow and naphthalene revealed that the junctures are stagnant areas of fluid flow, which generated strong and temporally persistent naphthalene gradients. The landscape and densities of bacterial accumulation at the junctures were strongly regulated by flow profiles and naphthalene gradients especially those transverse to flow. We conducted macroscale simulations using convective dispersion equations with an added chemotactic velocity to account for directed migration toward naphthalene. Simulated results showed good consistency with experiments and pore-scale simulation as normalized bacterial accumulation per mm of NAPL was 7.80, 7.84 and 7.71 mm−1 for experiments, pore-scale and macroscale simulations, respectively. Macroscale simulations indicated that in the absence of grain-boundary restrictions associated with the pore structure bacterial dispersion needed to be increased by 50 % to account for the interplay between chemotactic response and naphthalene gradients at the pore-scale level. Our work details the mechanism of pore-scale chemotaxis in enhancing bioavailability of PAHs and its impact on biomass retention at the system level, which provides a potential solution toward more efficient bioremediation for contaminants such as PAHs with limited bioavailability.

Transport behaviors of biochar particles in saturated porous media under DC electric field

The mobility of biochar in saturated quartz sand under a direct current (DC) electric field was investigated by column transport test. The effects of biochar preparation temperature (350 and 550 °C), solution chemistry (pH of 4, 7, and 10, and ion strength of 1, 10, 100 mM) and voltage gradient (0, 0.5 and 1.0 V cm−1) on the mobility of biochar were explored. It was found that DC electric field could significantly promote the migration of biochar, and the recovery rate of particles could be improved by 0.5–6.1 folds under 0.5 V cm−1. Higher voltage potential, solution pH and ionic strength were more favorable for biochar migration. The transport of biochar could be well interpreted by deterministic nonequilibrium convection-dispersion equation model. The enhanced mobility caused by DC electric field was attributed to the following reasons: enhanced electromigration following electrostatic attraction from the anode; increasing surface negative charges and functional groups on biochar surface as a result of electrochemical oxidization; reducing size blocking of biochar particles by decreasing particle size. Moreover, the interaction between biochar particles and electrode could alter solution chemistry, in particular, increasing solution pH, which in turn facilitated the transport of biochar. This study provided a perspective to modulate the transport behavior of biochar particle in the soil for the remediation of polluted sites.

Evaluation of machine learning methodologies using simple physics based conceptual models for flow in porous media

Machine learning (ML) techniques have drawn much attention in the engineering community due to recent advances in computational techniques and an enabling environment. However, often they are treated as black-box tools, which should be examined for their robustness and range of validity/applicability. This research presents an evaluation of their application to flow/transport in porous media, where exact solutions (obtained from physics-based models) are used to train ML algorithms to establish when and how these ML algorithms fail to predict the first order flow-physics. Exact solutions are used so as not to introduce artifacts from the numerical solutions.To test, validate, and predict the physics of flow in porous media using ML algorithms, one needs a reliable set of data that may not be readily available and/or the data might not be in suitable form (i.e. incomplete/missing reporting, metadata, or other relevant peripheral information). To overcome this, we first generated structured datasets for flow in porous media using simple representative building blocks of flow physics such as Buckley-Leverett, convection-dispersion equations, and viscous fingering. Then, the outcomes from those equations are fed into ML algorithms to examine their robustness and predictive strength of the key features, such as breakthrough time, and saturation and component profiles. In this research, we show that a physics-informed ML algorithm can capture the physical behavior and effects of various physical parameters (even when shocks and sharp gradients are present) on the features of the flow. Furthermore the ML approach can be utilized to solve inverse problems to estimate various physical parameters. In this study, we have focused on capturing the dominant flow physics on selected processes constituting the outcome of the physics of flow in porous media using ML algorithms. We expect that capturing the physics per selected processes can help solving more complex problems by providing correlative physical proxies.

Soil mineral nitrogen dynamics in fallow periods in a rainfed semiarid Mediterranean agricultural system

Rainfed agricultural systems in semiarid Mediterranean environments are subject to erratic but often heavy rainfall events. As an agronomic practice, fallow periods can be included even within the existing EU common policy for crop diversification. This research aimed to quantify the effects of a previous mineral fertilization on the soil mineral nitrogen (N min ) content and on potential nitrate leaching during no-tilled fallow periods of a crop rotation. Water and N min soil measurements obtained during three fallow seasons were used. Data were also used to check if the Leaching Estimation and Chemistry Model (LEACHM) can be used for the soil N min prediction after fallow. During these periods, the N min measured in the soil profile increased on average by 125 kg N ha -1 , while the model averaged an increase of 95 kg N ha -1 . The estimate of N leached ranged from 11 to 38 kg N ha -1 . The N balance simulated using LEACHM might differ from the actual processes. After a drought period followed by a soil water replenishment, the calibrated LEACHM underestimated soil N min probably due to the “Birch effect”. After occasional rainy spells when soil quickly became saturated, LEACHM overestimated soil N min probably due to occasional N 2 O emissions not being fully accounted by the model and to specific preferential water flow, which might produce a greater nitrate leaching than that simulated by LEACHM using the convection-dispersion equation . The results show that soil N min measurements by sampling after the fallow period cannot be replaced by LEACHM simulations. These samplings are of interest in fallow established inside a rotation and to avoid over-fertilization in the following cropping season, while reducing N environmental impacts.

Anomalous transport for multispecies reactive system with first order decay: time-fractional model

The prediction of pollutant migration and its concentration variation in the subsurface hydrology is vitally requisite for the assessment and restorative treatment of polluted groundwater system. Pollutant migration for the multispecies reactive system cannot be reliably investigated by classical form of convection-dispersion equation (CDE), due to the presence of more than one reactive species. This paper establishes a time-fractional model for multispecies reactive system under the first order sequential reaction network to understand the anomalous or non-Fickian migration phenomenon for reactive pollutants. At present, most of the fractional models are presented for the classical CDE to investigate migration phenomenon for single species system, not for the multispecies reactive system due to the complexity of the modelled framework. The impact of fractional derivative model is analysed for variable dependent migration parameters and constant parameters, both for the multispecies reactive migration phenomenon. The fractional derivative is expressed in the Caputo sense and to handle the non-linearity of problem, Homotopy perturbation method (HPM) is adopted. The advantage of this method, to get the solutions, is that the HPM is independent of small parameters required for the deformation process as used in other existing perturbation techniques, which make it much more convenient to use for non-linear systems. The impact of the fractional derivative index and other migration parameters is graphically depicted for the reactive species and significant influence of fractional term is observed. The derived solutions are then validated by using the corresponding solutions obtained by other existing well-established methods to ensure the convergence of the HPM solutions. As there do not exist any solutions for such fractional models for multispecies reactive system, this study may be advantageous to convey better understanding for the anomalous or non-Fickian migration for reactive pollutants and their remediation planning in the groundwater resources.

Modeling graphene oxide transport and retention in biochar

Experimental data from fixed-bed column studies and a numerical model based on convection-dispersion equations were used to describe transport and retention of Graphene Oxide (GO) in sand, biochar (BC), and BC modified with nanoscale zero-valent iron (BC-nZVI). Three blocking functions, namely no blocking, site-blocking, and depth-dependent blocking, were used to analyze GO transport and retention behavior in each media as a function of Ionic Strength (IS). An inverse modeling approach was implemented to determine the attachment coefficient (Ka) and maximum solid-phase retention capacity (Smax). The Langmuirian attachment model with site-blocking function effectively described experimental GO breakthrough curves (R2 ~ 0.70–0.99) compared to other models, indicating the importance of introducing a limit on the attachment capacity of the media. The Ka values for BC and BC-nZVI were significantly higher than sand, attributable to high porosity, roughness, and surface chemical properties. The models predicted an increasing trend in Ka (0.065 to 0.615 min−1) in BC with increasing IS (0.1 to 10 mM), while Ka values decreased (2.26 to 0.349 min−1) for BC-nZVI. A consistent increase in Smax was observed for both BC and BC-nZVI with increasing IS. Scenario analysis was conducted to further understand the effect of influent IS, GO concentration, and treatment depth. BC-nZVI exhibited a higher Ka and Smax and as a result, higher GO retention than BC at lower IS (0.1 and 1.0 mM). BC-nZVI had a relatively lower Ka (0.349 min−1) at 10 mM IS, however, it outperformed BC when GO retention capacities are compared over a longer period attributable to a higher Smax (6.47). Complete GO breakthrough occurred in a 5 cm media after 350 and 465 days for BC and BC-nZVI, respectively at 10 mM IS and influent concentration of 0.1 mg·L−1. GO breakthrough time increased with increasing treatment depth, however, the relation was non-linear.

Modeling the transport behavior of Pb(II), Ni(II) and Cd(II) in the complex heavy metal pollution site under the influence of coexisting ions

Complex metal contaminated sites are generally more harmful than single metal contaminated sites. Therefore, this study combined experiment and mathematical model to investigate the migration of Pb(II), Ni(II) and Cd(II) in single and multi-metal systems. The data from batch and column experiments were fitted well by Langmuir and nonequilibrium convection-dispersion equation. In the single metal system, the adsorption capacity of metals in soil was Pb> >Cd>Ni. In multi-metal systems, Pb(II), Ni(II) and Cd(II) retention in column decreased by 42.8%, 56.2% and 92.8% respectively due to the competition effect. The competitive effect enhanced the migration of three metals in the column. Ni(II) and Cd(II) showed irregular breakthrough curves, called “overshooting”, under the effect of Pb(II). The phenomenon of overshooting made the outlet concentration C of the column greater than the inlet concentration C0 (C/C0>1), leading to serious potential risks to groundwater. In addition, results from speciation of metals showed that the competition effect and overshooting mainly occurred in exchangeable and carbonate fractions. Understanding the transport of metals in complex contaminated sites is important for risk assessment.

Spatial distribution and migration of 239+240Pu in Chinese soils

The migration of radionuclides is a critical threat to the soil and groundwater environment. This study investigates highly radiological toxic 239+240Pu in 647 surface soils and 66 soil cores in China. First, the spatial distributions of 239+240Pu activities and 240Pu/239Pu ratios are presented in Chinese surface soils. Second, four different types of vertical distribution of 239+240Pu, namely 45.4%, 40.9%, 9.1% and 4.5% of Chinese soil cores proportions are integrated using statistical analysis. Furthermore, different soil types are accompanied by different 239+240Pu vertical distributions, which relate closely to the 239+240Pu migration. Finally, based on the Convection Dispersion Equation (CDE) model, the present work find that the apparent convection velocities of 239+240Pu are ranging from 0.00032 ± 0.00031 cm a−1 to 0.473 ± 0.083 cm a−1. As shown by the four typical vertical activity distribution of 239+240Pu in soil cores, the deepened activity maximum value position of 239+240Pu implies a fast migration rate or apparent convection velocity. This study, for the first time, suggests a significant linear correlation (R2 = 0.847) between the depth of 239+240Pu maximum value (cm) and the apparent convection velocity (v, cm a−1). We hypothesize that v usually does not exceed 0.5 cm a−1 in the CDE model. A significant linear correlation is also identified between apparent dispersion coefficient (D, cm2 a−1) and v2 in Chinese soil cores from the CDE model. It will provide an important reference for harmful heavy metal migration research in the future.

Elucidating the mass transfer mechanism of CrVI adsorption by encapsulated chitosan-carbon nanotubes-iron beads in packed-bed columns

This study evaluates the CrVI breakthrough behaviors of a continuous-flow column packed with composite chitosan-MWCNTs‑iron beads under various operating conditions. Under the tested range of experimental parameters, a maximum of 54% of CrVI removal was achieved at: water flow rate, 1 mL/min, feed CrVI concentration, 30 mg/L; and packed bed height 8 cm. A homogeneous surface diffusion model (HSDM) involving convection-dispersion and diffusion equations was formulated, numerically solved, and experimentally validated. The high degree of conformity between the calculated and experimental breakthrough curves facilitated the determination of mass transfer parameters including axial dispersion DL, (1.30 × 10−8 to 1.56 × 10−7 m2/s) and surface diffusion Ds, (7.28 × 10−11 to 1.80 × 10−10 m2/s) via an error minimizing approach. The DL value was a function of molecular diffusion, which varied with the flow velocity, mass loading, and bed height. The Ds values were slightly higher than those previously reported, owing to the heterogeneous nature of the adsorbent. The external film diffusion coefficient (kf) was also determined by using Wilson-Geankoplis empirical correlation, which was recognized as a rate-controlling factor during CrVI mass transfer to the adsorbent beads. Furthermore, sensitivity analysis showed that a transition from diffuse- to shock-front occurred with a decreasing DL, while a decrease in the slope of breakthrough curves was observed by decreasing Ds and bed porosity, and increasing kf and Langmuir parameters. Lastly, the study demonstrated a high correlation between utilization of fractional bed capacity and axial Peclet number. This method is suitable for optimizing operating conditions to maximize the utilization of a fixed-bed columns.

Barrier Longevity of NaCl-Laden Soil against Subterranean Termites in an Earth Embankment

Subterranean termite-induced damage to earth embankments in agricultural systems occurs globally. NaCl-laden soil barriers (NLSBs) are an environmentally sustainable termite control method, and have exhibited good potential in preventing termite-related tunneling damage in Zhejiang Province, China. The persistence of the NaCl concentration in NLSBs is a key characteristic for the long-term prevention of subterranean termite infestations. This study is a scientific attempt to estimate the field efficacy and barrier longevity of NLSBs in reservoir embankments based on the Richards equation and the convection–dispersion equation using HYDRUS (2D/3D). The observed and simulated NaCl concentrations at the end of a 1915-day simulation were compared. The results indicated that the proposed model performed well and can effectively characterize the water flow and salt transport in NLSBs. The salt desalination rate of the NLSB in the upstream slope was higher than that in the downstream slope, both of which were significantly higher than that at the embankment axis. Regardless of the type of embankment (homogeneous or core-wall), the barrier longevity of NaCl-laden soil against subterranean termites can reach 50 years with an optimized NaCl/soil ratio in different parts of the embankment.

Transport of chemotactic bacteria in granular media with randomly distributed chemoattractant-containing NAPL ganglia: Modeling and simulation

Chemotaxis is the biased movement of bacteria upon sensing chemical gradients, which can facilitate bioremediation of nonaqueous phase liquids (NAPLs) by transporting oil-degrading bacteria more efficiently to contaminants in groundwater aquifers. Transport phenomena of chemotactic bacteria in porous media can be described within the context of a convection-dispersion equation (CDE) that includes an additional convection-like chemotactic velocity. In this paper, we present a novel modification of a previously reported CDE by including a single first-order kinetic term which can differentiate bacterial sorption on porous matrix and chemoattractant-containing NAPL ganglia. Our model was solved numerically using a finite-element scheme to compute bacterial transport in granular media at a continuum level and also captured chemotaxis to discrete, randomly distributed NAPL ganglia of chemoattractant sources. Simulation results revealed the influence of oil ganglia on bacterial transport. It was found that chemotactic bacteria exhibited localized hotspots near NAPL ganglia because of chemotaxis toward and sorption on oil surfaces. The presence of oil ganglia reduced recovery of chemotactic bacteria by 71% and nonchemotactic bacteria by 44%, which accompanied enhanced retention inside the column. Model results suggested that chemotactic bacteria can locate NAPL and thereby sorb on surfaces more efficiently, which may overcome the limited bioavailability of contaminants dissolved in NAPLs. Simulated outputs demonstrated good agreement with experimental data in literature. Our model provides insight into the impact of chemotaxis and interaction with oil-phase chemoattractant sources on bacterial transport in porous media.

Theory-guided full convolutional neural network: An efficient surrogate model for inverse problems in subsurface contaminant transport

Identification of the location and strength of a contaminant source in an aquifer is a challenging but crucial task. Efficient surrogate models can be constructed to replace traditional time-consuming simulators while solving this inverse problem. In recent years, with the rapid development of machine learning (ML) algorithms, the artificial neural network (ANN) has been proven to be an efficient way for surrogate modeling. However, it may be difficult for ANN-based algorithms to learn the convection-dispersion equation and predict the contaminant concentration field due to their point-to-point learning scheme. Because of their strong localized features, the concentration fields can be seen as images. In contrast, the convolutional neural network (CNN) can extract spatial information better due to its convolutional structure. Herein, a theory-guided full convolutional neural network (TgFCNN) model is proposed to solve inverse problems in subsurface contaminant transport. TgFCNN can construct robust and reliable surrogate models with limited training realizations, and be further utilized for inverse modeling tasks. The loss function of TgFCNN comprises the residual of governing equations of contaminant transport, as well as data mismatch. Moreover, the iterative ensemble smoother (IES) method is employed to update the parameters while solving the inverse problems. The proposed TgFCNN model is evaluated in four scenarios. The results demonstrate that the TgFCNN model possesses strong generalization and extrapolation abilities, and satisfactory accuracy when estimating unknown contaminant source parameters, as well as the permeability field. The time consumption of the TgFCNN surrogate model for inverse tasks is also greatly reduced compared to using traditional simulators directly.

Macroscopic Model for Passive Mass Dispersion in Porous Media Including Knudsen and Diffusive Slip Effects

In this work, a macroscopic model for incompressible and Newtonian gas flow coupled to Fickian and advective transport of a passive solute in rigid and homogeneous porous media is derived. At the pore-scale, both momentum and mass transport phenomena are coupled, not only by the convective mechanism in the mass transport equation, but also in the solid-fluid interfacial boundary condition. This boundary condition is a generalization of the Kramers-Kistemaker slip condition that includes the Knudsen effects. The resulting upscaled model, applicable in the bulk of the porous medium, corresponds to: 1) A Darcy-type model that involves an apparent permeability tensor, complemented by a dispersive term and 2) A macroscopic convection-dispersion equation for the solute, in which both the macroscopic velocity and the total dispersion tensor are influenced by the slip effects taking place at the pore-scale. The use of the model is restricted by the starting assumptions imposed in the governing equations at the pore scale and by the (spatial and temporal) constraints involved in the upscaling process. The different regimes of application of the model, in terms of the Péclet number values, are discussed as well as its extents and limitations. This new model generalizes previous attempts that only include either Knudsen or diffusive slip effects in porous media.

Variational principle of the one-dimensional convection–dispersion equation with fractal derivatives

The convection–dispersion equation has always been a classic equation for studying pollutant migration models. There are certain deviations in scientific research because of the existence of the impurity of the medium and the nonsmooth boundary. In this paper, we introduced the one-dimensional convection–dispersion equation with fractal derivatives in fractal space, and established the fractal variational formula of the equation through the semi-inverse method. The fractal variational formula we have obtained can provide the conservation laws in an energy form in the fractal space and possible solution structures of the given equation. An analytical solution is obtained through the two-scale transform method and Laplace transform.