Heterogeneous Hydraulic Conductivity Research Articles

Physics‐Informed Convolutional Decoder (PICD): A Novel Approach for Direct Inversion of Heterogeneous Subsurface Flow

AbstractWe propose a physics‐informed convolutional decoder (PICD) framework for inverse modeling of heterogenous groundwater flow. PICD stands out as a direct inversion method, eliminating the need for repeated forward model simulations. The framework combines data‐driven and physics‐driven approaches by integrating monitoring data and domain knowledge into the inversion process. PICD utilizes a convolutional decoder to effectively approximate the spatial distribution of hydraulic heads, while Karhunen–Loève expansion (KLE) is employed to parameterize hydraulic conductivities. During the training process, the stochastic vector in KLE and the parameters of the convolutional decoder are adjusted simultaneously to minimize the data‐mismatch and the physical violation. The final optimized stochastic vectors correspond to the estimation of hydraulic conductivities, and the trained convolutional decoder can predict the evolution and distribution of hydraulic heads. Various scenarios of groundwater flow are examined and results demonstrate the framework's capability to accurately estimate heterogeneous hydraulic conductivities and to deliver satisfactory predictions of hydraulic heads, even with sparse measurements.

Effects of hydraulic conductivity on simulating groundwater–land surface interactions over a typical endorheic river basin

Hydraulic conductivity is the key variable in subsurface configuration that is important in modeling groundwater and its interactions with land surface processes. However, an acquisition of quantitatively distributed hydraulic conductivity is challenging due to limitations on data availability, accuracy and resolution, which inhibits an advanced understanding of hydrologic and energy cycles at the basin scale. This study investigated the role of hydraulic conductivity in simulating groundwater-land surface interactions over a typical endorheic river basin in northwestern China. A 3D variably saturated groundwater model coupled to a land surface model was configured to account for interactions between groundwater dynamics and land energy fluxes. Four scenarios of hydraulic conductivity were tested and inter-compared, adopting both homogeneous and heterogeneous parameterization strategies based on Global Soil Dataset for use in Earth System Models (GSDE) and HYdrogeology MaPS (GLHYMPS). The overall objectives are to diagnose effects of hydraulic conductivity on simulating groundwater–land surface interactions, with a particular focus on differences in spatial distributions and statistical characteristics between scenarios, as well as the dynamics and feedback mechanisms between groundwater and land surface energy fluxes. Comprehensive simulations and analyses were conducted to discern spatial distributions and statistical characteristics of key hydrological variables (streamflow, water table depth, latent heat flux, sensible heat flux) under different scenarios. Results showed that the hydraulic conductivity of the saturated zone had a significant control over water table depth, while the combine hydraulic conductivity in the vadose zone and the saturated zone governed the evolution of streamflow. Moreover, the water table critical range—where the land energy budget is highly sensitive to groundwater table—was 0.1 m–5 m and the area might expand approximately 4.5 % after considering the heterogeneity of hydraulic conductivity. Compared to the homogeneous scenario, the spatial heterogeneity of hydraulic conductivity amplified water table depth variability while diminishing land energy fluxes variability.

A U-Net architecture as a surrogate model combined with a geostatistical spectral algorithm for transient groundwater flow inverse problems

Characterizing groundwater flow parameters is crucial for understanding complex aquifer systems, and inverse techniques play a fundamental role in modeling hydrogeological parameters and assessing their uncertainties. Nonetheless, the use of a forward model in these methods can be highly time-consuming, especially with an increasing number of model parameters. To address this issue, we propose a surrogate model based on a U-Net architecture that replaces the transient groundwater flow model, reducing runtime and enabling a fast quantification of uncertainties related to key parameters, including heterogeneous hydraulic conductivity, boundary conditions, specific storage, and pumping rate. The surrogate is trained using limited evaluations of the forward model to learn the physical relationship between hydraulic conductivity fields and transient hydraulic heads measured on-site. The physical principles of the studied problem, including boundary conditions, specific storage, and source terms, are also mapped and introduced as inputs to the model to enhance its understanding of the governing equation of transient groundwater flow. To speed up learning using image–image regression, the previously predicted transient hydraulic heads also serve as an input to predict the transient heads at the current time step. Once the model is trained, we use a spectral geostatistical method to solve the inverse problem, a pumping test of 12 h, using the surrogate model in place of the forward model. Our study demonstrates that the trained U-Net accurately reproduces the state variables corresponding to a specific parameter field, and in terms of computational demand, using U-Net as a surrogate model reduces the required computational time by approximately an order of magnitude for the defined problem. The proposed approach offers an efficient and accurate method for groundwater flow parameter characterization and uncertainty quantification in complex aquifer systems.

Pumping‐induced Well Hydraulics and Groundwater Budget in a Leaky Aquifer System with Vertical Heterogeneity in Aquitard Hydraulic Properties

AbstractIn groundwater hydrology, aquitard heterogeneity is often less considered compared to aquifers, despite its significant impact on groundwater hydraulics and groundwater resources evaluation. A semi‐analytical solution is derived for pumping‐induced well hydraulics and groundwater budget with consideration of vertical heterogeneity in aquitard hydraulic conductivity (K) and specific storage (Ss). The proposed new solution is innovative in its partitioning of the aquitard into multiple homogeneous sub‐layers to enable consideration of various forms of vertically heterogeneous K or Ss. Two scenarios of analytical investigations are explored: one is the presence of aquitard interlayers with distinct K or Ss values, a common field‐scale occurrence; another is an exponentially depth‐decaying aquitard Ss, a regional‐scale phenomenon supported by statistical analysis. Analytical investigations reveal that a low‐K interlayer can significantly increase aquifer drawdown and enhance aquifer/aquitard depletion; a high‐Ss interlayer can noticeably reduce aquifer drawdown and increase aquitard depletion. Locations of low‐K or high‐Ss interlayers also significantly impact well hydraulics and groundwater budget. In the context of an exponentially depth‐decaying aquitard Ss, a larger decay exponent can enhance aquifer drawdown. When using current models with a vertically homogeneous aquitard, half the sum of the geometric and harmonic means of exponentially depth‐decaying aquitard Ss should be used to calculate aquitard depletion and unconfined aquifer leakage.

Enhancing predictions of remedial reagent transport via a vertical groundwater circulation well with high-resolution aquifer characterization

The vertical groundwater circulation well (GCW) is a commonly used technique in contaminated sites to remove secondary contaminants from low permeable zones. Early GCW studies often used simple subsurface hydraulic properties, such as anisotropic homogeneous aquifers or low conductivity lens/blocks, to mimic the complex subsurface heterogeneity. Although studies based on simplified representations of aquifer heterogeneity provide straightforward flow and transport information for engineering design of a GCW, they may over- or under-estimate contaminant fate and transport in the field. The objective of this study is to identify key heterogeneity factors that control the capture zone extension and to examine the extent to which the accuracy of estimated heterogeneity spatial distributions influences the prediction of remedial reagent transport. To achieve these objectives, we utilized Monte Carlo simulation to investigate the extension of the circulation zone in heterogeneous aquifers and to identify the key factors that contribute most to the variability of the circulation zone. Three commonly used geostatistical approaches (equivalent homogeneous, kriging, and highly parameterized methods) were employed to estimate the spatial distributions of key factors. The reliabilities of these estimated fields were evaluated through their remedial reagent transport predictability. The key factor analysis revealed that the mean porosity value, the variance of lnK, and the correlation length of lnK profoundly influence the lateral expansion of the capture zone. Neglecting the aquifer hydraulic conductivity heterogeneity underestimates the extension of the circulation zone and the spread of remedial reagent. Additionally, utilizing a highly parameterized approach to estimate the high-resolution K field can accurately reproduce the key remedial reagent distributions. The concentration arrival time and peak concentration are significantly improved compared to those predictions based on the equivalent homogeneous and kriged K fields.

Simultaneous identification of a non-point contaminant source with Gaussian spatially distributed release and heterogeneous hydraulic conductivity in an aquifer using the LES-MDA method

Space-temporal distribution of the contaminant plumes and aquifer properties is critical for groundwater management. However, most previous studies have focused on point source identification, barely exploring the identification of non-point sources. Xu et al. (2022) proposed to identify non-point sources but did not consider uncertainties in aquifer properties and release mass loading. In this work, we have implemented an application of the localized ensemble smoother with multiple data assimilation (LES-MDA) for the simultaneous identification of Gaussian hydraulic conductivities and non-point source parameters including Gaussian release mass-loading by assimilating both piezometric head and concentration observations in a synthetic confined aquifer. The results prove that the LES-MDA is not only capable of providing accurate identification of the spatial architecture of non-point contaminant sources and related release parameters (such as initial release time, and release duration) but also spatially heterogeneous release mass-loading and hydraulic conductivities.

ERT data assimilation to characterize aquifer hydraulic conductivity heterogeneity through a heat‐tracing experiment

AbstractGeothermal energy systems, such as heat pumps relying on aquifers, use renewable sources of energy that are accessible in urban areas. It is necessary to characterize the subsurface hydraulic properties prior to the installation of such systems. In this context, a heat‐tracing experiment is a typical field test that can help with the characterization of the subsurface. During a heat‐tracing experiment, monitoring with downhole temperature sensors, water‐level pressure transducers and electrical resistivity tomography (ERT) can be used to help characterize the hydrogeological properties. Previous monitoring tools have shortcomings, such as low‐resolution data and over‐smoothing; thus, they fail to reproduce the heterogeneity of hydrogeological properties. Ensemble Kalman filter (EnKF) is a promising tool that can overcome the over‐smoothing problem to replicate the hydrogeological property heterogeneity. In this work, we proposed a new procedure to assimilate time‐lapse cross‐borehole ERT data into a numerical model of groundwater flow and heat transfer, where the groundwater is extracted and heated water is reinjected into an unconfined sandy‐gravel aquifer. The finite element model (FEFLOW 7.3) of groundwater flow and heat transfer is integrated with petrophysical relationship and electrical forward modelling (ResIPy) to estimate cross‐borehole ERT measurements. Then, the estimated apparent resistivity is assimilated to update the hydraulic conductivity model using EnKF. The results of the application of the proposed approach to an experimental site located in Quebec City (Canada) demonstrate that the heterogeneity of K is correctly reproduced as the updated K model is reasonably consistent with the lithological log. In addition, the proposed approach was able to replicate the cross‐borehole ERT field and temperature measurements. The comparison between prior and posterior distribution of K with slug test results shows that the EnKF made the final (assimilated) distribution of K move towards K values inferred with slug tests.

Sequential Direct and Inverse Modeling of Underground Flows in the Upper Cheliff Alluvial Aquifer, Algeria

The Upper Cheliff plain lies 125 km south-west of Algiers, capital city of Algeria. It is characterized by its unconfined aquifer contained in a heterogeneous filling of Mio-Plio-Quaternary age. Nowadays, this aquifer is severely exploited to meet the domestic, industrial and agricultural needs in the region. Groundwater resources management by means of numerical modeling is highly required in order to improve our knowledge of the hydrogeological operation of this aquifer. In this work we implemented an approach based on a sequential solving of the direct and inverse problems. In the case of the direct problem, steadystate flow simulations revealed the persistence of large discrepancies between the measured and calculated piezometric levels, which reflect the heterogeneity of the aquifer. The zonation provided by the direct method, based on the conceptual model, is then fed in the inverse method. The latter, in addition to its speed and its convergence, made it possible to reduce the disparities with measurements as noted in the direct method. The main results highlight a heterogeneous hydraulic conductivity distribution which reaches its highest values in the central and north-western sectors of the aquifer, whereas the lowest values are found in the eastern, northern and southern sectors of the aquifer. Furthermore, calculation of the groundwater flow budget by the inverse model showed that rainfall recharge is the main input into the aquifer with 462.5x103 m3/day, whereas the most important outlet of the aquifer is the Cheliff Wadi with a flow rate of 627x103 m3/day.

The impact of heterogeneity at various spatial locations on dune-induced hyporheic exchange

The hyporheic exchange is not evenly influenced by heterogeneous hydraulic conductivity (K) values throughout the streambed. Few studies have investigated the impact of K heterogeneity in different regions of the streambed on the hyporheic exchange. This study employed a first-order stochastic analysis and a dislocation superposition method to examine the cross-correlation between hyporheic exchange flux (HEF) and K anywhere in synthetic, heterogeneous, 2-D cross-sectional streambed models. Results show that hyporheic exchange flux positively correlates with K heterogeneity in every part of the streambeds. This relationship provides a mechanistic explanation for the impact of high/low K zones on the HEF in the hyporheic zone. The effect of K impact on the HEF intensifies in streambeds when 1) upwelling groundwater flux increases, 2) the dunes are steep and asymmetric, or 3) the streambed is stratified. This study develops an empirical linear formula based on dune height and stoss length for estimating the high cross-correlation zone (HCZ) locations, potentially guiding the sampling scheme of hydraulic conductivities for evaluating the HEF in practical fieldwork.

Drug transport modeling in solid tumors: A computational exploration of spatial heterogeneity of biophysical properties

Inadequate uptake of therapeutic agents by tumor cells is still a major barrier in clinical cancer therapy. Mathematical modeling is a powerful tool to describe and investigate the transport phenomena involved. However, current models for interstitial flow and drug delivery in solid tumors have not yet embedded the existing heterogeneity of tumor biomechanical properties. The purpose of this study is to introduce a novel and more realistic methodology for computational models of solid tumor perfusion and drug delivery accounting for these regional heterogeneities as well as lymphatic drainage effects.Several tumor geometries were studied using an advanced computational fluid dynamics (CFD) modeling approach of intratumor interstitial fluid flow and drug transport. Hereby, the following novelties were implemented: (i) the heterogeneity of tumor-specific hydraulic conductivity and capillary permeability; (ii) the effect of lymphatic drainage on interstitial fluid flow and drug penetration.Tumor size and shape both have a crucial role on the interstitial fluid flow regime as well as drug transport illustrating a direct correlation with interstitial fluid pressure (IFP) and an inverse correlation with drug penetration, except for large tumors having a diameter larger than 50 mm. The results also suggest that the interstitial fluid flow and drug penetration in small tumors depend on tumor shape. A parameter study on the necrotic core size illustrated that the core effect (i.e. fluid flow and drug penetration alteration) was only profound in small tumors. Interestingly, the impact of a necrotic core on drug penetration differs depending on the tumor shape from having no effect in ideally spherical tumors to a clear effect in elliptical tumors with a necrotic core. A realistic presence of lymphatic vessels only slightly affected tumor perfusion, having no substantial effect on drug delivery.In conclusion, our findings illustrated that our novel parametric CFD modeling strategy in combination with accurate profiling of heterogeneous tumor biophysical properties can provide a powerful tool for better insights into tumor perfusion and drug transport, enabling effective therapy planning.

Effects of aquitard windows on groundwater fluctuations within a coastal leaky aquifer system: An analytical and experimental study

Low-permeability aquitards are important sedimentary units that protect the underlying confined aquifers from contamination by seawater and anthropogenic sources in coastal leaky aquifer systems. Therefore, the integrity of the aquitard is of crucial importance. However, the aquitard integrity may be vulnerable to horizontal heterogeneities in the form of aquitard windows. In this study, an innovative analytical model was developed to describe groundwater flow in a coastal leaky aquifer system with horizontal heterogeneities in hydraulic properties of either the aquitard or the confined aquifer. Consequently, the effects of aquitard windows, which manifest as horizontally local heterogeneities in hydraulic conductivity and/or specific storage, on the tidal wave propagation can be considered in the analysis. Analytical solutions were derived for tidal-head or constant-flux inland boundary conditions. The derivation process avoided previous methodologies that involved the assembly of large-size matrices. The newly developed analytical model can cover multiple existing analytical models that are associated with the assumptions of homogeneity in hydraulic properties of the aquifer and the aquitard, infinite horizontal extent of the coastal leaky aquifer system, impermeability of the aquitard, or ignorance of aquitard storativity. Analytical investigations revealed that high-permeability aquitard windows acting as preferential-flow conduits tended to dampen the tide-induced head propagation within the underlying confined aquifer. Moreover, an aquitard with a higher storativity (but the same permeability) and of considerable size can result in an apparent propagation bias. The newly developed analytical model in this study was further corroborated through sandbox experiments. Both piezometer measurements and parameter estimates indicated the non-negligible effects of high-permeability aquitard windows on the tidal wave propagation. The sandbox investigation also highlighted the importance of geostatistical inverse modeling in characterizing local heterogeneities within the confined aquifer and in detecting the location and size of aquitard windows, when measurement data of tide-induced aquifer head fluctuations were provided.

Comparison of travel-time and geostatistical inversion approaches for hydraulic tomography: Synthetic modeling study on data density and well configuration issues

Hydraulic tomography (HT) has been proven to be an effective approach in mapping the heterogeneity of hydraulic parameters. The travel-time based inversion (TTI) and geostatistical inversion (GI) approaches are two of several interpretation methods for HT data. In particular, GI is used to estimate heterogeneous hydraulic conductivity (K) and specific storage (Ss) tomograms, while TTI yields the diffusivity (D = K/Ss) tomogram. The main objective of this paper is to compare the performances of these two methods through a synthetic study. Two cases are designed based on different monitoring configurations. In each case, two independent scenarios are designed as: (1) providing the same dataset, and (2) providing datasets based on recommended strategies for each inversion approach. Then, the estimated tomograms are evaluated by direct comparison of estimated parameter distributions and assessment of model validation results. Comparison results show that the advantages of TTI are: (1) imaging of structural features representing high-D zones; and (2) requirement of less data for inverse modeling. In contrast, the advantages of GI are: (1) estimation of parameters throughout the simulation domain; (2) better characterization of low-D zones; and (3) generating the best estimated tomogram leading to best drawdown predictions. Collectively, our study suggests that TTI is suitable for rapid, coarse characterization of heterogeneity that could be utilized for providing hydrogeological structures for GI as an initial model. GI is more robust and preferable to applications that require higher accuracy in parameter estimation over a larger volume.

Spatio-temporal monitoring of leachates dispersion beneath a solid wastes dump in basement complex of southwestern Nigeria

For real-time monitoring and modeling of leachates dispersion beneath Arada wastes dump, Ogbomoso geologically located within Southwestern Nigeria basement complex, an integrated technique involving time-lapse Electrical Resistivity Tomography (ERT), hydrogeology, and geotechnic was used. ERT was performed using Wenner electrodes configuration in a Northeast-Southwest azimuth on the wastes dump monthly from March to September, and twice in March and September, at the control site, during the wet season. The Total Dissolved Solids (TDS), Electrical Conductivity (EC) and static water level of two monitoring wells on the dumpsite were measured during every ERT surveying. Twenty four soil samples were taken at 0.5 m intervals from 2.0 to 3.0 m deep bores on the anomalously low resistivity spots discovered on the subsurface resistivity distribution of the wastes dump, and sieve analysis and permeability tests were performed in the laboratory on these samples. Using the approach of grain-size analysis, the hydraulic conductivity and porosity were calculated empirically. On the wastes dump and control site, the resistivities of spatially distributed lenses of material ranged within 1.45–37.3 Ωm and 22.5–52.6 Ωm, respectively, suggesting leachates and clay anomalies. The wastes dump subsurface might be regarded as a porous medium since its porosity ranged between 38.2 and 44.2%, and the permeability range of (5.4–9.6) x 10−5 m/s showed that water could freely drain from it. Water samples taken from the monitoring wells on the waste dump had TDS and EC that ranged from 520 to 970 ppm and 1.05 to 2.37 mS/cm, respectively, but decreased in the southwest direction of groundwater flow, indicating leachates dispersion. The hydraulic conductivity ranged within 0.00285–0.176 m/s but varied anomalously and spatially indicating heterogeneous hydraulic conductivity field capable of initiating local variation in velocity that caused leachate to disperse longitudinally and transversely. The rate of longitudinal and transverse leachates dispersion as revealed by subsurface resistivity distribution images ranged within 0.000–0.250 m/day and 0.001–0.045 m/day, respectively. The former rate was 5.6 folds the latter rate and hence fell within the stipulated 5–100 for dispersion process that conformed to natural consequence of dispersivity. The conductivity-time curves (CTCs) of the leachates anomalies' steepness and doubly-peaked properties indicated Fickian dispersion, while different proportion times; t-1.588 and t-2.478, which are not equal to t-0.5, showed that the CTCs described non-Fickian dispersion at a significantly later time.

Physics-embedded inverse analysis with algorithmic differentiation for the earth’s subsurface

Inverse analysis has been utilized to understand unknown underground geological properties by matching the observational data with simulators. To overcome the underconstrained nature of inverse problems and achieve good performance, an approach is presented with embedded physics and a technique known as algorithmic differentiation. We use a physics-embedded generative model, which takes statistically simple parameters as input and outputs subsurface properties (e.g., permeability or P-wave velocity), that embeds physical knowledge of the subsurface properties into inverse analysis and improves its performance. We tested the application of this approach on four geologic problems: two heterogeneous hydraulic conductivity fields, a hydraulic fracture network, and a seismic inversion for P-wave velocity. This physics-embedded inverse analysis approach consistently characterizes these geological problems accurately. Furthermore, the excellent performance in matching the observational data demonstrates the reliability of the proposed method. Moreover, the application of algorithmic differentiation makes this an easy and fast approach to inverse analysis when dealing with complicated geological structures.

Richards Equation at the Hillslope Scale: Can We Resolve the Heterogeneity of Soil Hydraulic Material Properties?

AbstractProcess‐based modeling of soil water movement with the Richards equation requires the description of soil hydraulic material properties, which are highly uncertain and heterogeneous at all scales. This limits the applicability of the Richards equation at larger scales beyond the patch scale. The experimental capabilities of the three hillslopes of the Landscape Evolution Observatory (LEO) at Biosphere 2 provide a unique opportunity to observe the heterogeneity of hydraulic material properties at the hillslope scale. We performed a gravity flow experiment where through constant irrigation the water content increases until the hydraulic conductivity matches the irrigation flux above. The dense water content sensor network at LEO then allows mapping of the heterogeneity of hydraulic conductivity at a meter scale resolution. The experiment revealed spatial structures within the hillslopes, mainly a vertical trend with the lowest hydraulic conductivity close to the surface. However, the variation between neighboring sensors is high, showing that the heterogeneity cannot be fully resolved even at LEO. By representing the heterogeneity in models through Miller scaling we showed the impact on hillslope discharge. For the hillslope with the smallest heterogeneity, representing the dominant structures was sufficient. However, for the two hillslopes with the larger overall heterogeneity, adding further details of the local heterogeneity did impact the discharge further. This highlights the limitations of the Richards equation, which requires the heterogeneous field of material properties, at the hillslope scale and shows the relevance to improving our understanding of effective parameters to be able to apply the process‐based model to larger scales.

PFOS Mass Flux Reduction/Mass Removal: Impacts of a Lower-Permeability Sand Lens within Otherwise Homogeneous Systems.

Perfluorooctane sulfonic acid (PFOS) is one of the most common per- and polyfluoroalkyl substances (PFAS) and is a significant risk driver for these emerging contaminants of concern. A series of two-dimensional flow cell experiments was conducted to investigate the impact of flow field heterogeneity on the transport, attenuation, and mass removal of PFOS. A simplified model heterogeneous system was employed consisting of a lower-permeability fine sand lens placed within a higher-permeability coarse sand matrix. Three nonreactive tracers with different aqueous diffusion coefficients, sodium chloride, pentafluorobenzoic acid, and β-cyclodextrin, were used to characterize the influence of diffusive mass transfer on transport and for comparison to PFOS results. The results confirm that the attenuation and subsequent mass removal of the nonreactive tracers and PFOS were influenced by mass transfer between the hydraulically less accessible zone and the coarser matrix (i.e., back diffusion). A mathematical model was used to simulate flow and transport, with the values for all input parameters determined independently. The model predictions provided good matches to the measured breakthrough curves, as well as to plots of reductions in mass flux as a function of mass removed. These results reveal the importance of molecular diffusion and pore water velocity variability even for systems with relatively minor hydraulic conductivity heterogeneity. The impacts of the diffusive mass transfer limitation were quantified using an empirical function relating reductions in contaminant mass flux (MFR) to mass removal (MR). Multi-step regression was used to quantify the nonlinear, multi-stage MFR/MR behavior observed for the heterogeneous experiments. The MFR/MR function adequately reproduced the measured data, which suggests that the MFR/MR approach can be used to evaluate PFOS removal from heterogeneous media.

A multiscale finite element method for coupled heat and water transfer in heterogeneous soils

• A MsFEM scheme is designed for heat and water transfer in heterogeneous soils. • Two MsFEM basis functions along with a Petrov-Galerkin formulation is applied. • The MsFEM scheme is more accurate than the standard FEM schemes. • MsFEM is used to model soil heat and water transfer in a partially coupled version. Large-scale modeling of coupled heat and water transfer (CHWT) is challenging due to the spatial variabilities of soil hydraulic and thermal properties. A multi-scale finite element method (MsFEM) has been designed for simulating liquid water fluxes in unsaturated soils. In this study, the MsFEM approach is expanded as a new scheme that can handle CHWT in soils. Two groups of MsFEM basis functions are constructed to incorporate the heterogeneities in soil hydraulic conductivity and thermal conductivity, and a Petrov-Galerkin formulation is adopted to implement the proposed MsFEM scheme. The MsFEM scheme is also exploited as a sequential solver when heat transfer and water transfer are expressed in a partially coupled formulation (Wang et al., 2022a). Numerical examples illustrate that, without essentially increasing the computing load, the MsFEM scheme can improve the accuracy by up-to 30% compared to the standard finite element method (FEM), especially for thermally driven water transfer. Some fine scale spatial variations in soil temperature can only be revealed with the MsFEM scheme. If the MsFEM sequential solver is applied, the coupled heat and water transfer model can be rewritten into a sequence of modules, where liquid water transfer, heat transfer and vapor transfer are solved step-by-step, but the thermally driven liquid water is omitted due to its relatively small value (Wang et al., 2022a). With the MsFEM sequential solver, a flexible modeling architecture can be achieved at the cost of a relatively small increase of error (<5%). Therefore, the MsFEM scheme presented in this study is an effective numerical approach to simulating CHWT in heterogeneous soils.

Fast inverse estimation of hydraulic conductivity field based on a deep convolutional-cycle generative adversarial neural network

This study proposes a novel deep convolutional-cycle generative adversarial neural network (DC-CGAN) for estimating the hydraulic conductivity field K from the observed head field H. The estimation of the heterogeneous hydraulic conductivity is apparently a high-dimensional inverse problem. Although the indirect approach is popularly utilized to solve the inverse problem, when compared with the indirect approach, the direct approach can conduct the inversion task faster without any iteration process. The key of the direct approach is to establish the inverse operator which maps the observed head to hydraulic conductivity. However, the complex inverse mapping relationship and high-dimensional estimation have always challenged the direct approach. Owing to the potential ability to handle high-dimensional images, six residual blocks with convolutional kernels have been designed as the generators of DC-CGAN to capture the potential distribution for enhancing the generation of estimation images similar to the real hydraulic conductivity. However, the traditional adversarial pattern of one-way generator and discriminator might suffer from the equifinality from different parameters. Therefore, to alleviate the equifinality, a cycle adversarial training pattern involving two generators and discriminators has been employed to supervise the map. Particularly, this pattern performs a forward map from H to K and then the reconstructed K back to H. The reconstruction process has been a supervision to the forward map process. A hypothetical case with 40×30 unknown K values has been designed to evaluate the performance of DC-CGAN, and the results indicate that DC-CGAN has achieved a desired generalization accuracy of MRE of 9.25 % and SSIM of 0.85 with the swift running speed of 0.2 s, which outperforms indirect approach. The application of the proposed DC-CGAN shows that it can provide a reliable and fast high-dimensional estimation task for the heterogenous hydraulic conductivity.

Decoupled Finite Particle Method With Normalized Kernel (DFPM‐NK): A Computationally Efficient Method for Simulating Solute Transport in Heterogeneous Porous Media

AbstractPrevious studies found that Finite Particle Method (FPM), an improved Smoothed Particle Hydrodynamics (SPH) method, yields more accurate solutions of the advection‐dispersion equation (ADE) than SPH method does when simulating solute transport in heterogeneous porous media. FPM however is computationally more expensive than SPH because FPM needs to solve a correction matrix equation for each particle. While decoupled FPM (DFPM) can reduce computational cost of FPM by using diagonal terms of the matrix equation, DFPM is less accurate than FPM due to discarding off‐diagonal terms of the matrix equation. This study develops the Decoupled Finite Particle Method with Normalized Kernel (DFPM‐NK) to improve computational accuracy of DFPM by using normalized kernels in DFPM. We evaluate computational performance of SPH, FPM, DFPM, and DFPM‐NK using two numerical experiments of ADE with non‐reactive and reactive solute transport in a heterogeneous aquifer. Results of the two experiments indicate that, among the four methods, SPH has the least computational time, but has the worst computational accuracy. DFPM‐NK is substantially more efficient than FPM, and has similar accuracy of FPM. On the other hand, DFPM‐NK and DFPM have similar computational cost, but DFMP‐NK is significantly more accurate than DFPM, especially for heterogeneous hydraulic conductivity fields. We thus recommend to use DFPM‐NK for computationally expensive ADE problems without sacrificing computational accuracy.

Analytical Study on the Tidal Wave Propagation Coupled with Vertical Leakage from a Vertically Heterogeneous Aquitard in a Coastal Leaky Aquifer System

Abstract Analytical studies on the tidal wave propagation in a coastal leaky aquifer commonly assume that the low-permeability aquitard is homogeneous. The aquitard is, however, vertically heterogeneous in nature due to varying soil types along the burial depth as can be frequently seen from borehole logs. In this study, an innovative analytical model is developed to explore the vertical heterogeneity in aquitard hydraulic conductivity (K) and specific storage (Ss) on the tidal wave propagation coupling with vertical leakage from the aquitard. The novelty behind the newly derived analytical solution is attributed to the fact that the aquitard along the burial depth can be divided into any number of homogeneous zones, each of which is associated with distinct K and Ss values, so that arbitrary vertical heterogeneity pattern of aquitard K and Ss can be captured. Theoretical analysis results reveal that an interlayer within the aquitard with a smaller K can significantly enhance the amplitude and phase shift of periodic groundwater head fluctuations in the leaky aquifer, while a larger Ss of the interlayer will weaken the amplitude and phase shift provided a relatively small aquitard K. Buried locations of the interlayer also implement nonnegligible effects on the tidal wave propagation. For the scenario of exponentially decaying aquitard K and Ss, which is commonly encountered for a thick aquitard, a larger decay exponent results in smaller amplitude and phase shift. This analytical study highlights the importance of vertical aquitard heterogeneity on tidal wave propagation in a coastal leaky aquifer system.