Hydraulic Conductivity Field Research Articles

Evaluating the effect of aquifer heterogeneity on multiobjective optimization of in-situ groundwater bioremediation

In-situ bioremediation is a cost-effective technique to eliminate petroleum hydrocarbons from groundwater. In previous studies, this has been mostly applied under the approximation of a homogeneous porous medium, even though in reality aquifers are often characterized by significant heterogeneities. Here, different heterogeneous and equivalent homogeneous hydraulic conductivity fields are considered to study the effect of aquifer heterogeneity on optimal in-situ bioremediation. For this, the multiobjective simulation-optimization (S/O) model referred to as BIOEFGM-NSGA II is proposed — which uses the element-free Galerkin method (EFGM) for discretization of the governing equations, and the non-dominated sorting genetic algorithm II (NSGA II) for multiobjective optimization. This model is then applied to different heterogeneous conductivity fields generated through a pseudo-random correlated field generator for different combinations of variance and correlation lengths with constant mean. Results show that the optimal pumping policy for an equivalent homogeneous conductivity field violates the constraint of maximum allowable concentration in all the studied heterogeneous fields. To satisfy this constraint, in-situ bioremediation cost increases by 8.47% to 56.83% to that of a homogeneous field. It shows the significance of aquifer heterogeneity in designing an optimized in-situ bioremediation system and hence, should be incorporated in the S/O model for in-situ groundwater bioremediation.

Joint Estimation of Adsorptive Contaminant Source and Hydraulic Conductivity Using an Iterative Local Updating Ensemble Smoother with Geometric Inflation Selection

The joint estimation of groundwater contaminant source characteristics and hydraulic conductivity is of great significance for reactive contaminant transport models in heterogeneous subsurface media. The accurate determination of the sorption parameters of such contaminants is also a key prerequisite for estimating the parameters of the groundwater system. In this study, to investigate the impact of the sorption parameter field on the accuracy of hydraulic conductivity and source characteristics estimation, numerical experiments were conducted in a synthetic aquifer considering the contaminant sorption process in groundwater models with varying sorption parameter settings. Iterative local updating ensemble smoother with geometric inflation selection (ILUES-GEO) was employed to assimilate hydraulic head and contaminant concentration data to jointly estimate the contaminant source information and hydraulic conductivity in a heterogeneous aquifer. The results indicated that the ILUES-GEO successfully recovers contaminant source information simultaneously with hydraulic conductivity, and its performance improves as more accurate sorption parameters are introduced. Furthermore, the influence of the ILUES algorithm parameters and ensemble size is investigated to improve the estimation accuracy. Additionally, the characterization of contaminant sources and hydraulic conductivity fields is influenced by the number and locations of measurements. This study can help to understand the significance of sorption parameter setting for the joint estimation of reactive contaminant source and hydraulic parameters.

Groundwater contamination source identification and high-dimensional parameter inversion using residual dense convolutional neural network

Data assimilation for high-dimensional parameter joint inversion of multiple time-varying source strength and hydraulic conductivity fields can be computationally intensive as a large number of forward model runs are usually required. In this study, a deep neural network-based surrogate is proposed to replace the forward model in the data assimilation process to efficiently achieve high-dimensional parameter inversion. The deep convolutional encoding-decoding network architecture is used to leverage the advantages of the convolutional network in processing image-like data, where the high-dimensional input and output fields of the forward model are expressed as images. The encoding network extracts the main features of the multiple time-varying source input image and high-dimensional hydraulic conductivity fields, and the decoding network refines the extracted features to generate the output hydraulic head and contaminant concentration fields. The Residual Dense Convolutional Neural Network (RDCNN) is proposed by combining residual learning with a densely-connected convolutional neural network to solve the image regression problem for surrogate construction. Iterative local updating ensemble smoother (ILUES) is utilized to assimilate hydraulic head and contaminant concentration data to inverse high-dimensional parameters. The proposed method is evaluated by jointly inversing the high-dimensional hydraulic conductivity and source parameters for a synthetic multiple-source contaminated aquifer. The results indicate that compared to the surrogate constructed by Dense Convolutional Neural Network (DCNN) without addressing residual learning, the RDCNN surrogate captures the model input-output relationship better with relatively high training efficiency. The RDCNN surrogate-coupled ILUES achieves state-of-the-art performance in terms of inversion accuracy of 9 pollution source parameters and 255 hydraulic conductivity field parameters and computational efficiency in comparison to the ILUES based on the forward model.

Identification of hydraulic conductivity field of a karst aquifer by using transition probability geostatistics and discrete cosine transform with an ensemble method

AbstractKnowledge of the spatial distribution characteristics of hydraulic parameters is essential for the management and protection of karst groundwater resources. In this study, we propose a workflow integrating the transition probability geostatistics (T‐PROGS) and the discrete cosine transform (DCT) with the ensemble smoother with multiple data assimilation (ES‐MDA) method to map the hydraulic conductivity of karst aquifers that usually follow a multimodal distribution. The priori parameter ensemble is constructed from the T‐PROGS and transformed into an approximately Gaussian distributed coefficient field containing critical spatial structural features about the aquifer using DCT. The ES‐MDA method is employed to update the DCT coefficients by assimilating the measurements. In addition, a postprocessing process based on cumulative distribution function (CDF) mapping is used to address the problem of parameters gradually tending to a Gaussian distribution after updating using the ES‐MDA and the inappropriate selection of initial parameter values. In practice, the limited amount of available data makes it difficult to fully capture the spatial distribution of the parameters in the initial ensemble of a single realization. Therefore, we suggest a new strategy for structuring the initial ensemble by mixing samples from multiple realizations. We then apply the proposed approach to four single realization models and a combined multiple realizations model in a field hydraulic tomography investigation of a real karst aquifer. The computed results show that this method can effectively identify the characteristics of the spatial distribution of hydraulic conductivity in karst aquifers. Compared with an individual realization model, the uncertainty of the hydraulic conductivity estimated by the combined multiple realizations model is significantly reduced. Therefore, including more uncertainty of the aquifer in the initial ensemble is beneficial to improve the accuracy of the parameter estimation.

Evaluating the Effects of Multiscale Heterogeneous Sediments on Solute Mixing and Effective Dispersion

AbstractThe heterogeneity of sedimentary aquifers can be characterized by parameters related to the underlying sedimentary structure such as physical attributes of sediments (e.g., volume proportions and lengths of facies types). These parameters often reflect multiscale variability typically observed in the hydraulic conductivity field and the resulting fluid velocity. Although there have been many studies of solute transport processes (e.g., dispersion, mixing, and dilution) in heterogeneous media, there is still a lack of understanding of the controlling influence of sedimentary architecture on the time‐evolution of the solute plume's spreading and mixing behavior and their uncertainty. In this work, we used high‐resolution and three‐dimensional numerical simulations to investigate the time‐evolution of the effective dispersion and mixing behavior of the solute through a heterogeneous aquifer that displays multiscale sedimentary architecture. Transport simulations are performed by using the random walk particle tracking method. Our numerical experiments allow us to understand which scale of sedimentary architecture is most relevant to explaining effective dispersion and mixing. We further analyze whether or not the uncertainty in solute spreading and mixing are equally affected by the different spatial organization of sedimentary facies types. We also perform an extensive parametric study to better understand the effect of physical sediment attributes and univariate statistics of hydraulic conductivity (e.g., mean and variance) at the facies scale on both effective dispersion and mixing. Our results indicate that meter‐scale heterogeneity plays a major role in controlling solute transport processes. Moreover, the effective dispersion is more sensitive to the spatial organization of sedimentary facies type.

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.

GenES-MDA: A generic open-source software package to solve inverse problems via the Ensemble Smoother with Multiple Data Assimilation

Ensemble Kalman filter methods have been successfully applied for data assimilation and parameter estimation through inverse modeling in various scientific fields. We have developed a new generic software package for the solution of inverse problems implementing the Ensemble Smoother with Multiple Data Assimilation (genES-MDA). It is an open-source, platform-independent Python-based program. Its aim is to facilitate the management and configuration of the ES-MDA through several programming tools that help in the preparation of the different steps of ES-MDA. genES-MDA has a flexible workflow that can be easily adapted for the implementation of different variants of the ensemble Kalman filter and for the solution of generic inverse problems. This paper presents a description of the package and some application examples. genES-MDA has been tested in three synthetic case studies: the solution of the reverse flow routing for the estimation of the inflow hydrograph to a river reach using observed water levels and a calibrated forward model of the river system, the identification of a hydraulic conductivity field using piezometric observations and a known forward flow model, and the estimation of the release history of a contaminant spill in an aquifer from measured concentration data and a known flow and transport model. The results of all these tests have demonstrated the flexibility of genES-MDA and its capabilities to efficiently solve different types of inverse problems.

Directly Generating Spatially Periodic, Heterogeneous Groundwater Flow Fields: A Finite Volume Approach

AbstractWe present a finite volume approach for generating doubly‐periodic, 2D heterogeneous groundwater velocity fields, given an arbitrary hydraulic conductivity field discretized on a rectangular lattice. The method conserves mass, allows direct specification of any desired mean flow direction and computes the corresponding field with a single solve operation, and solves for inter‐cell fluxes that may be used directly for particle tracking with the Pollock method without further interpolation. We demonstrate an open‐source Python implementation of the approach that we created.

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.

High‐Dimensional Groundwater Flow Inverse Modeling by Upscaled Effective Model on Principal Components

AbstractThe main computational costs of gradient‐based inverse methods for high‐resolution groundwater flow inverse problems include costly forward model simulations and the large number of such simulations required to determine the Jacobian matrix. We develop an upscaling‐based inverse approach, named upscaled principal component inverse approach (UPCIA), which achieves dimensionality reduction and reduces computational cost of forward model simulations by evaluating the Jacobian through upscaled effective models on a coarse‐resolution grid constructed from upscaled principal components. UPCIA integrates downscaling into the inverse problem by estimating principal component coefficients based on the coarse‐resolution forward model, which are then used to generate high‐resolution parameter fields. Various numerical experiments demonstrate the effectiveness and efficiency of UPCIA, including 2‐D/3‐D high‐dimensional steady‐state and transient hydraulic tomography with known storativity to estimate multi‐Gaussian transmissivity or hydraulic conductivity fields. Results show that the hydraulic head is insensitive to small‐scale variability of conductivity, and UPCIA provides high‐quality inversion results similar to inverse methods with high‐resolution forward model simulations and significantly reduces computation time by orders of magnitude. In addition to supporting the characterization of heterogeneity in sufficient detail, UPCIA can also be used to examine whether finer resolution is necessary and possibly to determine an optimal inverse resolution.

Hydrogeological multiple-point statistics inversion by adaptive sequential Monte Carlo

For strongly non-linear and high-dimensional inverse problems, Markov chain Monte Carlo (MCMC) methods may fail to properly explore the posterior probability density function (PDF) given a realistic computational budget and are generally poorly amenable to parallelization. Particle methods approximate the posterior PDF using the states and weights of a population of evolving particles and they are very well suited to parallelization. We focus on adaptive sequential Monte Carlo (ASMC), an extension of annealed importance sampling (AIS). In AIS and ASMC, importance sampling is performed over a sequence of intermediate distributions, known as power posteriors, linking the prior to the posterior PDF. The AIS and ASMC algorithms also provide estimates of the evidence (marginal likelihood) as needed for Bayesian model selection, at basically no additional cost. ASMC performs better than AIS as it adaptively tunes the tempering schedule and performs resampling of particles when the variance of the particle weights becomes too large. We consider a challenging synthetic groundwater transport inverse problem with a categorical channelized 2D hydraulic conductivity field defined such that the posterior facies distribution includes two distinct modes. The model proposals are obtained by iteratively re-simulating a fraction of the current model using conditional multiple-point statistics (MPS) simulations. We examine how ASMC explores the posterior PDF and compare with results obtained with parallel tempering (PT), a state-of-the-art MCMC inversion approach that runs multiple interacting chains targeting different power posteriors. For a similar computational budget, ASMC outperforms PT as the ASMC-derived models fit the data better and recover the reference likelihood. Moreover, we show that ASMC partly retrieves both posterior modes, while none of them is recovered by PT. Lastly, we demonstrate how the power posteriors obtained by ASMC can be used to assess the influence of the assumed data errors on the posterior means and variances, as well as on the evidence. We suggest that ASMC can advantageously replace MCMC for solving many challenging inverse problems arising in the field of water resources.

An Interactively Corrected Smoothed Particle Hydrodynamics (IC‐SPH) for Simulating Solute Transport in a Nonuniform Velocity Field

AbstractThe Smoothed Particle Hydrodynamics (SPH) method is a Lagrangian approach that has been widely used to eliminate numerical dispersion for solving advection‐dispersion equation (ADE) of groundwater solute transport under advection‐dominated situations. It has been found that accuracy of SPH results is severely deteriorated, when particles are irregularly distributed in a model domain with heterogeneous hydraulic conductivity. To resolve this problem, we developed a new approach called Interactively Corrected SPH (IC‐SPH), which is an improved version of the Corrected SPH (C‐SPH) method. IC‐SPH uses an interactively corrected kernel gradient to construct concentration gradients used to solve ADE. This correction is made for each particle by using not only the particle's neighbor particles within the particle's support domain but also the particles within each neighbor particle's support domain. We evaluated IC‐SPH performance in two numerical studies. One considers diffusive transport with an analytical solution, and the other considers advection‐dispersion transport in a heterogeneous field of hydraulic conductivity. For each numerical study, several numerical experiments were conducted using multiple sets of irregularly distributed particles with different levels of particle irregularity. The numerical experiments indicate that, while IC‐SPH is more computationally expensive than SPH and C‐SPH, IC‐SPH produces more accurate ADE solutions, and converges faster to the analytical solution. IC‐SPH is mathematically general, and can be applied to a wide range of problems that require solving ADE.

Optimized Pilot Point Emplacement Based Groundwater Flow Calibration Method for Heterogeneous Small-Scale Area

Groundwater flow modeling in a small-scale area requires practical techniques to obtain high accuracy results. The effectiveness of the model calibration is the most challenging for simulating the hydraulic head. In pursuit of this, we proposed an optimized groundwater flow calibration method based on the pilot point emplacement technique for a 3D small-scale area in this work. Subsequently, two emplacement structures were tested during the experimentation, the regular pilot point placement, and the middle head measurement down gradient (MHMDG) placement with two different densities. The parameter estimation (PEST) numerical code applying the kriging interpolation was used to estimate the hydraulic conductivity field by MODFLOW. Moreover, geological SGrid models were chosen for the conceptual model. Thirty-seven observation wells were used for experimental simulations to test the proposed method in a heterogeneous confined aquifer. The result shows that the small-scale modeling was complicated, and the studying area presented a significant heterogeneity in horizontal hydraulic conductivity. The middle head measurement down gradient (MHMDG) pilot point case with the larger density gave the best R-squared 0.901 and minimum residual error of 0.0053 m compared to 0.880 and 0.078 m, respectively, for the regular placement. The calibration accuracy depended on the frequency and the emplacement of the pilot point. Therefore, the initial value should be technically selected to minimize the computation burden. The proposed techniques help to improve the groundwater flow model calibration based on the pilot point methodology for groundwater resources management.

The coastal aquifer recovery subject to storm surge: Effects of connected heterogeneity, physical barrier and surge frequency

Storm surge, a worldwide phenomenon triggering the vertical saltwater infiltration, is likely to exacerbate coastal groundwater salinization due to geologic heterogeneity, anthropogenic engineering and climate change. This study analyzed the combined effects of connected heterogeneity, physical barrier and surge frequency on the coastal aquifer recovery. A series of modeling cases were investigated using HydroGeoSphere in the heterogeneous and equivalent homogeneous aquifer. The heterogeneity setting is composed of different connectivity level of hydraulic conductivity field. The simulation results of single storm surge event demonstrate that the connected heterogeneity elevates the salinized extent and reduces the aquifer recovery time due to a number of preferential flow paths. In comparison to the equivalent homogeneous aquifer, heterogeneity alleviates the maximum salinized extent and vertical intrusion distance due to the accelerated mixing of salinized groundwater with fresh groundwater. Physical barrier, classified as subsurface dam and cutoff wall, leading to different groundwater discharge pattern is tailored to investigate the influences of the permanent subsurface engineering on the aquifer recovery. Our results show that the connectivity level controls the salinization pattern subject to physical barrier. Then, the repetitive storm surge events were simulated to investigate the effects of surge frequency. For the low-frequency surge event, the variation of salinized metric is the repetition of the unimodal curve in the single surge event. Nevertheless, for the high-frequency surge event, the residual salt mass cannot be flushed out over the simulation period, especially for the low-connectivity aquifer. Meanwhile, the high-frequency surge event broadens the differences of aquifer recovery process due to physical barrier. These findings have critical implications for coastal groundwater management which is facing the substantial environmental risks of surge-induced vertical saltwater intrusion derived from geologic heterogeneity and climate changes.

Characterization of the non-Gaussian hydraulic conductivity field via deep learning-based inversion of hydraulic-head and self-potential data

Accurate characterization of the spatial heterogeneity of hydraulic properties such as hydraulic conductivity (K) is essential for understanding groundwater flow and contaminant transport processes. Deep learning-based ensemble smoother methods have proven to be effective in estimating non-Gaussian K fields by assimilating hydrodynamic measurements. However, traditional hydrogeological investigations (e.g., hydraulic tomography) often suffer from the limited number of invasive borehole data (e.g., hydraulic head), making the non-Gaussian K inversion problem strongly underdetermined. As a non-invasive and high-sampling-density geophysical approach, the self-potential (SP) method can measure fluctuations of the electrical field caused by subsurface fluid movement. Thus, the recorded SP signals are electro-kinetic responses of the groundwater flow and can provide complementary information for delineating non-Gaussian heterogeneity. In this paper, we performed a joint hydrogeophysical inversion of hydraulic-head and SP data to characterize the non-Gaussian K field within a deep learning-based inversion framework. We conducted three synthetic cases in a 3-D non-Gaussian aquifer with 16,000 unknown K, to assess the ability of the proposed joint inversion framework for K imaging by assimilating different types of data. The results show that using limited hydraulic-head data alone may capture the large-scale features of the non-Gaussian reference field but might lose fine features. By integrating the hydraulic-head and SP data, the non-Gaussian K field can be reconstructed with an improved accuracy and reduced estimation uncertainty. The low-cost SP method opens up new opportunities for non-Gaussian K characterization and fluid-dynamic monitoring.

Simultaneous Estimation of a Contaminant Source and Hydraulic Conductivity Field by Combining an Iterative Ensemble Smoother and Sequential Gaussian Simulation

Joint estimation of groundwater contaminant source characteristics and hydraulic conductivity is of great significance for contaminant transport models in heterogeneous subsurface media. As for accurate characterization of hydraulic conductivities, both geostatistical modeling and groundwater inverse modeling are alternative approaches. In this study, an iterative ensemble smoother and sequential gaussian simulation (SGSIM) in geostatistics modeling were combined to realize the simultaneous inversion of contaminant sources and hydraulic conductivities, by using directly measured hydraulic conductivities and indirect hydraulic head and concentration data. To alleviate the high computational cost caused by repetitive evaluations of complex, high-dimensional groundwater models, SGSIM with the pilot points method was used. Considering the characteristics of the proposed method, four scenarios with ten cases were set up in terms of ensemble number and iteration number that affect the performance of the iterative ensemble smoother, the number of pilot points, and the observation data, respectively. The results for the synthetic example indicate that the ensemble size of 2000 and the pilot point number of 80 is an ideal combination of parameters, and the proposed method can successfully recover contaminant source information simultaneously with hydraulic conductivity.

Random Characteristics of Hydraulic Gradient through Three-Dimensional Multilayer Embankment

The distribution characteristics of hydraulic gradient in embankment are closely related to seepage failure. Seepage failures such as flowing soil and piping will lead to serious damage and even the overall failure of embankment. The hydraulic conductivity has strong spatial variability, which changes the distribution of hydraulic gradient in embankment and increases the difficulty for predicting the embankment seepage instability. In this study, the distribution of soil hydraulic conductivity in a section of Shijiu Lake embankment was obtained by the permeability test. Based on Local Average Subdivision technique, a three-dimensional multilayer random field of embankment hydraulic conductivity was generated. Then, the mean and standard deviation of overflow point height and hydraulic gradient were calculated by the Monte Carlo method, which combined the generated three-dimensional random model and the deterministic analysis method of seepage field. Finally, the coefficient of variation (COV) of hydraulic conductivity (0.1, 0.3, 0.5, 0.7, 1.0, 2.0, and 3.0), the fluctuation scale in vertical direction (3 m) and the fluctuation scale in horizontal plane (3 m, 6 m, 12 m, 24 m, 36 m, and 48 m) were selected respectively for analyzing the random characteristics of embankment overflow point height and hydraulic gradient under the influence of different COV and fluctuation scale of embankment soil hydraulic conductivity.

Impacts of the Scale of Representation of Heterogeneity on Simulated Salinity and Saltwater Circulation in Coastal Aquifers

AbstractNumerical models of variable‐density groundwater flow and salt transport are a primary tool for predicting salinity distributions in coastal aquifers and estimating submarine groundwater discharge (SGD). Models are particularly useful to estimate the saline component of SGD, which can occur far offshore and is difficult to measure directly. Depending on the system and application, the level of geologic detail represented can range from homogeneous or layered to fully heterogeneous hydraulic conductivity fields. These features strongly affect model results, limiting understanding of subsurface salinity distributions and associated density‐driven saltwater circulation along coasts worldwide. In this study, the impact of the scale of representation of heterogeneity on salinity distributions and SGD was investigated using numerical simulations. Upscaling hydraulic conductivity can significantly modify salinity distributions and flow paths, resulting in unpredictable variations in simulated SGD, though the values for homogeneous fields with equivalent hydraulic conductivity show consistent trends. Simulated density distributions control both the rate and direction of subsurface saltwater circulation. The length of the mixing zone perimeter, a measure of salinity distribution complexity, is shown to correlate with both the rate of subsurface saltwater circulation and the amount of groundwater circulating in the reverse direction from homogeneous cases. Overall, the results demonstrate a strong dependence of salinity distributions and saltwater circulation on the scale and distribution of geologic heterogeneity represented in numerical models. This suggests that numerical models with simplified geologic structure may substantially underestimate saltwater circulation, and attempts to calibrate them using salinity distributions or SGD measurements may be problematic.

INVERSE ANALYSIS WITH VARIATIONAL AUTOENCODERS: A COMPARISON OF SHALLOW AND DEEP NETWORKS

Inverse problems are applied to determine unknown properties by matching observational data with a physical model that often takes many parameters as input. To overcome the underconstrained nature of inverse problems and achieve good performance, an approach is presented involving regularization with a technique known as a variational autoencoder (VAE), which is trained to map a high-dimensional parameter space with a complex structure to a low-dimensional latent space with a simple structure. We apply this approach to unconditioned realizations of the parameters (heterogeneous hydraulic fields) for a hydrogeological inverse problem. Two types of hydraulic conductivity fields are used to evaluate the characterization for the different levels of heterogeneity complexity of the physical inputs. This approach keeps the computational cost of generating the training data low. The reason is unconditioned realizations neither rely on the observational data used to perform the inverse analysis nor require any groundwater flow model forward runs. In addition, this approach applies regularization on a low-dimensional latent space from the VAE and increases optimization efficiency through automatic differentiation. Furthermore, two different neural network (NN) structures are tested for their utility in using the VAE for inverse analysis. The performance of a deep, convolutional neural network strongly depends on the dimensionality of the latent space, which requires tuning. In contrast, a shallow, dense neural network provides consistently accurate characterization without tuning. Our approach evaluates the advantages of the shallow, dense neural network over the deep, convolutional one and enables future application to a wide range of inverse problems.

Using elements of machine learning to solve the inverse problem of reconstructing the hydraulic conductivity field for a filtration problem

In the modern world, machine learning methods are widely used. In the oil industry, there is also a noticeable trend to use these methods in the context of digitalization and intellectualization of the entire production process. The present work is devoted to the development of a technique for solving the inverse problem of restoring the permeability field of an oil reservoir with the combined use of machine learning elements and a filtration model. A computational algorithm has been implemented, which implies close mutual integration of the filtration part and the machine learning block, the results of which are used to parameterize the physically meaningful model. A network of radial basis functions is used as a machine learning model. The proposed solution search procedure includes the numerical solution of the direct and adjoint problems for the filtration model. Solving the adjoint problem allows one to apply gradient optimization methods widely used in machine learning methods. The paper presents the results of a numerical experiment. On the example of a symmetrical two-dimensional development element, a solution was obtained for the problem of restoring the permeability field for a set of zonal-heterogeneous oil reservoirs. For the reconstructed fields, the characteristic sizes of inhomogeneities coincide with the initial ones with sufficient accuracy. The fundamental possibility of a qualitative restoration of the porosity-permeability characteristics of the interwell space is shown, which is impossible when using classical interpolation methods without involving additional data. The paper studies the influence of the choice of the type of control parameter on the behavior of the objective function and its derivative, which affects the process of solving the inverse problem. As a result of the study, the use of hydrodynamic resistance as an adaptable parameter in solving the inverse problem is proposed.