Heterogeneous Permeability Fields Research Articles

Rapid Inference of Reservoir Permeability from Inversion of Traveltime Data Under a Fast Marching Method-Based Deep Learning Framework

Summary The fast marching method (FMM) is a highly efficient numerical algorithm used to solve the Eikonal equation. It calculates traveltime from the source point to different spatial locations and provides a geometric description of the advancing front in anisotropic and heterogeneous media. As the Eikonal solution, the diffusive time of flight (DTOF) can be used to formulate an asymptotic approximation to the pressure diffusivity equation to describe transient flow behavior in subsurface porous media. For the infinite-acting flow that occurs in porous media with smoothly varying heterogeneity, traveltime of the pressure front from the active production or injection well to the observation well can be directly estimated from the DTOF using the concept of radius (or depth) of investigation (ROI or DOI), which is defined as the moment when a maximum magnitude of the partial derivative of pressure to time occurs. Based on the ROI or DOI definition, we propose a deep neural network called the inversion neural network (INN) to inversely estimate heterogeneous reservoir permeability by inverting the traveltime data. The INN is trained by traveltime data created for a large data set of distinct permeability fields from FMM simulations, which can be two orders of magnitude faster than conventional reservoir simulators. A convolutional neural network (CNN), the U-Net architecture, is incorporated into the INN, which establishes a nonlinear mapping between the heterogeneous permeability fields and the traveltime data collected at sparse observation wells. The loss function used for the INN is defined as the root mean square error (RMSE) between the logarithm of the predicted permeability and the logarithm of the true permeability. The performance of the INN is tested on reservoir models with both smoothly varying heterogeneity and high-contrast media properties. For the 2D smoothly varying heterogeneous models with a grid size of 49×49, the permeability predicted by the INN has an average estimation error of 8.73% when a set of 7×7 uniformly distributed observation wells is used to collect “observational” traveltime data from the FMM simulation. For models with the same grid size and observation well density but with high-contrast media properties, the INN can still capture the general heterogeneity distribution, although with reduced prediction accuracy. Using a graphics processing unit (GPU) for training and prediction allows the entire inverse modeling process for a 2D 49×49 reservoir model to be completed within 7 minutes.

Augmenting Deep Residual Surrogates with Fourier Neural Operators for Rapid Two-Phase Flow and Transport Simulations

Summary Accurate numerical modeling of multiphase flow and transport mechanisms is essential to study varied, complex physical phenomena including flow in subsurface oil and gas reservoirs and subsurface aquifers subject to CO2 sequestration. State-of-the-art complete physics-based solvers suffer from many computational challenges. High-fidelity data-driven surrogate models that solve the governing partial differential equations (PDEs) have the potential to optimize the time to solution and increase confidence in critical business and engineering decisions through better quantification of solution statistics. We leverage the recently proposed Fourier neural operators (FNOs) with quasilinear time complexity to capture the spectral information from feature maps to solve the coupled porous flow and transport PDEs. Embedding Fourier layers within the residual blocks results in a highly effective structure that, while achieving competitive accuracy, also enables efficient training of deeper networks with a dramatically reduced number of trainable parameters. The resulting novel deep-learning (DL) architecture is coined as FResNet++. FResNet++ uses squeeze and excitation blocks, atrous spatial pyramid pooling (ASPP), and attention blocks to increase its sensitivity to the relevant features and capture multiscale information, and it is specifically tuned to operate optimally to learn from and predict numerically simulated flow (pressure and saturation) fields. We demonstrate the ability of FResNet++ to generalize over multiple high-dimensional input parameter spaces that describe subsurface permeability and porosity heterogeneity. The resulting DL architecture accurately captures the complex interplay between viscous forces and highly heterogeneous permeability and porosity fields. We investigate two-phase flow in porous media, which is the archetypal problem for reservoir simulation giving rise to a system of nonlinearly coupled PDEs with highly heterogeneous coefficients. We show in blind tests that FResNet++ predicts saturation fields more accurately compared to ResU-Net and original FNO with fully connected linear layers. We additionally investigate the effects of using alternative loss functions and an alternative way of utilizing FResNet++ to increase its effectiveness. For the first time in the literature, we show that the spatiotemporal evolution of pressure and saturation fields can be jointly predicted with good accuracy using a single FResNet++ network over long time horizons in response to previously unseen permeability and porosity fields. After a moderate training investment on graphics processing units (GPUs), FResNet++ yields a speedup of at least four orders of magnitude compared to a conventional numerical PDE solver and operates with notably fewer trainable parameters compared to the original FNO. Our numerical experiments validate that FNOs can be utilized in various convolutional neural network-based architectures and can effectively substitute for repetitive physics-based forward simulations for scenario testing.

Using 3D-printed fluidics to study the role of permeability heterogeneity on miscible density-driven convection in porous media

Miscible density-driven convection in porous media has important implications to the long-term security of geological carbon sequestration. Laboratory investigations of miscible density-driven convection in heterogeneous porous media have been greatly limited due to the challenge in constructing well-controlled heterogeneous permeability fields. In this study, three-dimensional (3D) printing was used to solve the challenge. Particularly, elementary sediment blocks were 3D printed to construct heterogeneous permeability fields having the desired mean, standard deviation, and spatial correlation length of permeability. A methanol-ethylene-glycol (MEG) solution was placed at the top of the permeability field to trigger miscible density-driven downward convection. Results showed that permeability heterogeneity caused noticeable uncertainty in the total MEG mass transferred into the permeability field, and the uncertainty increased with increasing correlation length of the permeability field. In a heterogeneous permeability field with a larger spatial correlation length, a larger effective vertical permeability is in general favorable for solute mass transfer into the underlying porous medium. Conversely, in a heterogeneous permeability field with a shorter spatial correlation length, a larger effective vertical permeability does not necessarily lead to a higher mass transfer rate. This is because mass transfer across the top boundary through miscible density-driven convection depends on the flow recirculation near the interface. A large effective vertical permeability does not necessarily lead to fast flow recirculation because the former is measured in the vertical direction whereas the latter depends more on the internally-connected, high-permeability streaks within the domain. The 3D-printed elementary sediment blocks can be re-distributed to construct another permeability field easily, which greatly reduces the experimental time and thus significantly increases the total number of experiments that can be conducted, thereby approaching the ergodicity requirement when a large number of random permeability fields is needed.

Efficient simulation of CO2 migration dynamics in deep saline aquifers using a multi-task deep learning technique with consistency

CO2 sequestration and storage in deep saline aquifers is a promising technology for mitigating the excessive concentration of the greenhouse gas in the atmosphere. However, accurately predicting the migration of CO2 plumes requires complex multi-physics-based numerical simulation approaches, which are prohibitively expensive due to highly nonlinear coupled governing equations and uncertainties in heterogeneous spatial parameter distributions. To address this challenge, we developed an end-to-end deep learning workflow employing encoder–decoder architectures with residual network (ResNet) to efficiently predicts the spatial–temporal evolution of the solution CO2-brine ratio (Rs) and gas saturation (Sg) – the two essential tasks for quantifying the amount of trapped CO2 – given heterogeneous permeability fields as input. Specifically, we introduce a general multi-task learning with consistency (MTLC) framework to simultaneously predict Rs and Sg. The MTLC model leverages related tasks with less computational expensive labeled datasets to improve generalization ability. In our study, predictions for multiple tasks from the same permeability realization are not independent but expected to be consistent, as the proposed framework utilizes data-driven cross-task consistency constraints to augment learning of related tasks. Our deep learning model is trained based on physical trapping mechanisms, which play a dominant role in the CO2 migration process. The results demonstrate that the MTLC model with joint learning yields more accurate predictions and improved generalization for predicting CO2 migration in several test cases. Furthermore, our workflow is 105 times faster than a high-fidelity physics-based numerical simulator, making it a viable alternative for field-scale applications.

Estimation of heterogeneous permeability using pressure derivative data through an inversion neural network inspired by the Fast Marching Method

Reservoir heterogeneity significantly impacts the fluid flow behavior in porous media. In subsurface communities including hydrocarbon or geothermal recovery, geological storage of CO2 or H2, and hydrology, transient pressure data are often used to infer the subsurface rock properties (e.g., permeability field) due to its ready availability and quick response. The accurate estimation of such properties is critical to accurately predict fluid flow in porous media.In this work, we propose a deep learning (DL) approach inspired by the Fast Marching Method (FMM), namely the Inversion Neural Network (INN), to inversely infer heterogeneous reservoir model parameters using transient pressure data. The forward model used to generate training data for the INN is established based on a semi-analytic asymptotic solution to the diffusivity equation using the diffusive time of flight (DTOF). The FMM solves the Eikonal equation for the DTOF and provides the partial derivative of pressure drop to the natural logarithm of time (hereafter pressure derivative). The pressure derivative data are then used to predict reservoir model parameters by the INN. In the homogeneous scenario, the INN architecture is a relatively simple fully connected neural network as a proof-of-concept to validate the feasibility, and it directly correlates the permeability value with the pressure derivative. In the heterogeneous scenario, as the heterogeneous permeability field is estimated based on sparse observational data of pressure derivatives, we adopt the convolutional neural network (CNN) to flexibly deal with the image-based properties, and leverage transfer learning to efficiently train a robust INN in the heterogeneous scenario.We first validated that INN can infer the homogeneous permeability fields through numerical experiments by testing root-mean-square-error (RMSE) around 1.086 md. Inspired by that, observational data with different sparsity are used to train convolutional INN and predict heterogeneous permeability fields. As the number of observational locations (nobs) increases from 3 × 3 to 48 × 48, the testing RMSE in heterogeneous scenarios decreases from 187.510 md to 18.080 md. Besides, we found that transfer learning significantly improves the predictive accuracy at low nobs, with relative error decreased by 17.25%. Finally, noisy pressure derivative data under different nobs are used to history match the heterogeneous reservoir model, and INN can infer the permeability field with a low error of 9.6% at nobs=7×7. Without an iterative procedure for parameter estimation, the INN demonstrates to perform permeability inversion with CPU time in the magnitude of 9.2×10−4 s on a single GPU (NVIDIA Quadro P2200). The FMM-inspired INN sets up the basis for the accurate characterization of reservoir model heterogeneity by inverting pressure derivative data with both decent predictive accuracy and computational efficiency.

Transient dispersion regimes in heterogeneous porous media: On the impact of spatial heterogeneity in permeability and exchange kinetics in mobile–immobile transport

Transport processes in the subsurface are coupled with the heterogeneity of the porous structure. Obtaining an accurate description of mass transport in such media at all time scales remains a crucial task, as the subsurface is strongly impacted by human activity. Spreading of pollutants in the transient but also in the asymptotic regime, as well as the time to reach a critical location (e.g. aquifer, well) significantly depend on the underlying heterogeneity of the permeability field. Moreover, in the case of solute retention, transport is also characterized by local exchange kinetics that depend on the local aquifer properties. Consequently, exchange (retention) times are expected to be spatially heterogeneous. In this work we focus on the influence of spatially heterogeneous permeability and exchange times on the transient and asymptotic transport regimes and provide a parametric study. To this goal, we simulate in a first part the transport in a two-dimensional heterogeneous medium under spatially varying permeability and mobile–immobile mass transfer parameters. Equations are solved using a Lattice-Boltzmann two-relaxation-time (TRT) algorithm. We assume the following relation between the local permeability K and the local exchange time: τ∝Kγ. Taking into account this relation, we investigate the impact of the Damköhler number (Da, ratio of the advection and exchange time scales), the disorder of the permeability field and the value of the exponent of the coupling function (γ) on the spatial evolution of the concentration field and breakthrough curve. We show that, depending on the parameters (Da, γ, etc.), we can observe transient, non-Fickian dispersion, which is characterized by non-exponential tails of the solute breakthrough curves and a non-linear evolution of the spatial variance of the solute distribution.In a second part, we present a new continuous time random walk (CTRW) model to upscale these transport behaviors. The model is based on a spatial Markov model for particle velocities that couples advective–dispersive transport and heterogeneous mass transfer through a compound Poisson process. The upscaled model can be fully parameterized by the statistics of permeability and the hydraulic gradient (with no fitting parameter), that is, in terms of medium and flow characteristics. The results of the CTRW model fully capture the non-Fickian transient transport regimes both for the breakthrough curves and spatial concentration variance. In the longtime limit, as expected from the central limit theorem, the CTRW model predicts normal, Fickian behavior. Finally we show, that the time to reach the asymptotic regimes in heterogeneous media (e.g. heterogeneous exchange times) is parameter dependent and in average is two orders of magnitude larger than for the respective homogeneous case. To summarize, the coupling between the heterogeneous permeability field and the local mass transfer properties can strongly influence transient and asymptotic transport regimes and potentially explain experimentally observed non-Gaussian behaviors.

Fourier Neural Operator for Fluid Flow in Small-Shape 2D Simulated Porous Media Dataset

Machine Learning (ML) and/or Deep Learning (DL) methods can be used to predict fluid flow in porous media, as a suitable replacement for classical numerical approaches. Such data-driven approaches attempt to learn mappings between finite-dimensional Euclidean spaces. A novel neural framework, named Fourier Neural Operator (FNO), has been recently developed to act on infinite-dimensional spaces. A high proportion of the research available on the FNO has focused on problems with large-shape data. Furthermore, most published studies apply the FNO method to existing datasets. This paper applies and evaluates FNO to predict pressure distribution over a small, specified shape-data problem using 1700 Finite Element Method (FEM) generated samples, from heterogeneous permeability fields as the input. Considering FEM-calculated outputs as the true values, the configured FNO model provides superior prediction performance to that of a Convolutional Neural Network (CNN) in terms of statistical error assessment based on the coefficient of determination (R2) and Mean Squared Error (MSE). Sensitivity analysis considering a range of FNO configurations reveals that the most accurate model is obtained using modes=15 and width=100. Graphically, the FNO model precisely follows the observed trend in each porous medium evaluated. There is potential to further improve the FNO’s performance by including physics constraints in its network configuration.

A fully-implicit hybridized discontinuous Galerkin spectral element method for two phase flow in petroleum reservoirs

A hybridized discontinuous Galerkin method with high order spectral element discretization is presented for modeling multiphase immiscible 2D fluid flow in petroleum reservoir simulations. The time dependent nonlinear governing equations for reservoir simulation are presented, and a fully coupled implicit time stepping procedure is implemented using a high order W-Rosenbrock algorithm. For incompressible fluids considered in the present study, the governing PDE's include an elliptic PDE for the pressure field and a parabolic convective diffusion equation for the saturation (water volume fraction) in a two phase oil-water system. The PDE systems are coupled through the highly nonlinear relative permeability and capillary pressure functions relating the fluid velocities of individual species to the individual species pressures and saturations. With HDG/spectral element (HDGSEM) spatial discretization of the coupled governing equations, the system constitutes an index 1 differential algebraic equation. It is shown that the W-Rosenbrock algorithm in combination with the HDGSEM method yields an efficient and effective solution procedure which demonstrates robust, high order temporal accuracy.Numerous physical tests cases are presented demonstrating that the versatility of this approach is sufficient to address the challenges in modeling heterogeneous permeability fields and geologic features of arbitrary shape with discontinuous permeability functions. It is also capable of resolving the sharp time dependent displacement fronts in oil-water systems encountered in secondary oil recovery. It is demonstrated that the high order polynomial basis functions (up to 12th order) used in the HDGSEM algorithm have the ability to resolve complex geometry and achieve accurate results utilizing a simple Cartesian mesh for non-conforming geological features. Eliminating the need to re-grid and conform the mesh to the boundaries of different geological features greatly simplifies the workflow for petroleum engineers.

Numerical simulations of high viscosity DNAPL recovery in highly permeable porous media under isothermal and non-isothermal conditions

We developed a decimetric size model based on coupling generalized Darcy's law and heat-transfer equations to model viscous dense non-aqueous phase liquid (DNAPL) pumping through highly permeable porous media under non-isothermal conditions. The presence of fingering and non-wetting phase ganglia was modeled through an unsteady capillary diffusion coefficient and an arbitrary heterogeneous permeability field. The model was validated using existing experimental data of a simple case, an oil injection in a 2D tank packed with glass beads. Next, we compared the results of this model against a DNAPL extracting situation in the 2D tank to better understand the two-phase flow behavior in highly permeable porous media. We found that natural convection during heating plays an essential role in heat transfer, especially in the wetting phase zone. By adding the dynamic effect (unsteady conditions) we were better able to describe the presence of the ganglia in porous media. We observed good agreement between modeled and experimental oil saturation curves until the breakthrough point, with a mean relative error of about 10% for low and high flow rates, and 8% and 16% after breakthrough for low and high flow rates, respectively. Extracting viscous oil at low flow rates and high temperature generates less fingering and is well described by the generalized Darcy's law. The remobilization of residual non-wetting ganglia after the breakthrough point at the outlet is, however, difficult to simulate using the generalized Darcy's law. In the end, we treated this issue by using a perturbed permeability field to simulate the observed fingering in the 2D tank.

Continuous conditional generative adversarial networks for data-driven solutions of poroelasticity with heterogeneous material properties

Machine learning-based data-driven modeling can allow computationally efficient time-dependent solutions of PDEs, such as those that describe subsurface multiphysical problems. In this work, our previous approach (Kadeethum et al., 2021d) of conditional generative adversarial networks (cGAN) developed for the solution of steady-state problems involving highly heterogeneous material properties is extended to time-dependent problems by adopting the concept of continuous cGAN (CcGAN). The CcGAN that can condition continuous variables is developed to incorporate the time domain through either element-wise addition or conditional batch normalization. Moreover, this framework can handle training data that contain different timestamps and then predict timestamps that do not exist in the training data. As a numerical example, the transient response of the coupled poroelastic process is studied in two different permeability fields: Zinn & Harvey transformation and a bimodal transformation. The proposed CcGAN uses heterogeneous permeability fields as input parameters while pressure and displacement fields over time are model output. Our results show that the model provides sufficient accuracy with computational speed-up. This robust framework will enable us to perform real-time reservoir management and robust uncertainty quantification in poroelastic problems.

Minimum principle for the flow of inelastic non-Newtonian fluids in macroscopic heterogeneous porous media

Non-Newtonian fluids are found in many applications related to porous or fractured media. At a macroscopic scale, inelastic non-Newtonian fluid obeys a nonlinear Darcy's equation. This paper show thats the solution of such a nonlinear equation in a heterogeneous permeability field obeys a minimum principle similar to the minimum dissipation principle of Stokes flow.

Computational multiscale method for parabolic wave approximations in heterogeneous media

In this paper, we develop a computational multiscale method to solve the parabolic wave approximation with heterogeneous and variable media. Parabolic wave approximation is a technique to approximate the full wave equation. One benefit of the method is that one wave propagation direction can be taken as an evolution direction, and one then can discretize it using a classical scheme like backward Euler method. Consequently, one obtains a set of quasi-gas-dynamic (QGD) models with possibly different heterogeneous permeability fields. For coarse discretization, we employ constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to perform spatial model reduction. The resulting system can be solved by combining the central difference in time evolution. Due to the variable media, we apply the technique of proper orthogonal decomposition (POD) to further the dimension of the problem and solve the corresponding model problem in the POD space instead of in the whole multiscale space spanned by all possible multiscale basis functions. We prove the stability of the full discretization scheme and give the convergence analysis of the proposed approximation scheme. Numerical results verify the effectiveness of the proposed method.

Miscible fluids mixing via alternating injection in a radial Hele-Shaw cell: experimental and numerical studies

Abstract An alternating injection scheme is experimentally employed to study the mixing performance of miscible viscous fluids in a radial Hele-Shaw cell. The concept of the covered area is introduced to quantify the mixing efficiency and various numerical simulations are performed to support and verify the experimental results. It has been observed that the alternating injection can improve the mixing efficiency, depending on the prominence of the channeling interactions. The alternating injection can effectively improve the mixing efficiency under certain conditions, where weaker fingering interactions are dominated by diffusive mixing. However, the results under the high Péclet number conditions are inconsistent, i.e. the mixing efficiency is reduced due to the emergence of orderly viscous fingers prevailing random chaotic interactions. This phenomenon is similar to the channeling interaction in the heterogeneous permeability fields in which less viscous fluid flows through favorable paths due to the local pressure difference between two successively emerging fingers.

Numerical modeling as a tool for evaluating the renewability of geothermal resources: the case study of the Euganean Geothermal System (NE Italy).

Renewable natural resources are strategic for reducing greenhouse gas emissions and the human footprint. The renewability of these resources is a crucial aspect that should be evaluated in utilization of scenario planning. The renewability of geothermal resources is strictly related to the physical and geological processes that favor water circulation and heating. In the Veneto region (NE Italy), thermal waters of the Euganean Geothermal System are the most profitable regional geothermal resource, and its renewability assessment entails the evaluation of fluid and heat recharge, regional and local geological settings, and physical processes controlling system development. This renewability assessment is aimed at defining both the importance of such components and the resource amount that can be exploited without compromising its future preservation. In the second part of the twentieth century, the Euganean thermal resource was threatened by severe overexploitation that caused a sharp decrease in the potentiometric level of the thermal aquifers. Consequently, regulation for their exploitation is required. In this work, the renewability of the Euganean Geothermal System was assessed using the results from numerical simulations of fluid flow and heat transport. The simulations were based on a detailed hydrogeological reconstruction that reproduced major regional geological heterogeneities through a 3D unstructured mesh, while a heterogeneous permeability field was used to reproduce the local fracturing of the thermal aquifers. The model results highlight the role played by the resolved structural elements, in particular the subsurface high-angle faults of the exploitation field, and by the anomalous regional crustal heat flow affecting the central Veneto region.

Utilizing Environmental Tracers to Reduce Groundwater Flow and Transport Model Parameter Uncertainties

AbstractNon‐uniqueness in groundwater model calibration is a primary source of uncertainty in groundwater flow and transport predictions. In this study, we investigate the ability of environmental tracer information to constrain groundwater model parameters. We utilize a pilot point calibration procedure conditioned to subsets of observed data including: liquid pressures, tritium (3H), chlorofluorocarbon‐12 (CFC‐12), and sulfur hexafluoride (SF6) concentrations; and groundwater apparent ages inferred from these environmental tracers, to quantify uncertainties in the heterogeneous permeability fields and infiltration rates of a steady‐state 2‐D synthetic aquifer and a transient 3‐D model of a field site located near Riverton, Wyoming (USA). To identify the relative data worth of each observation data type, the post‐calibration uncertainties of the optimal parameters for a given observation subset are compared to that from the full observation data set. Our results suggest that the calibration‐constrained permeability field uncertainties are largest when liquid pressures are used as the sole calibration data set. We find significant reduction in permeability uncertainty and increased predictive accuracy when the environmental tracer concentrations, rather than apparent groundwater ages, are used as calibration targets in the synthetic model. Calibration of the Riverton field site model using environmental tracer concentrations directly produces infiltration rate estimates with the lowest uncertainties, however; permeability field uncertainties remain similar between the environmental tracer concentration and apparent groundwater age calibration scenarios. This work provides insight on the data worth of environmental tracer information to calibrate groundwater models and highlights potential benefits of directly assimilating environmental tracer concentrations into model parameter estimation procedures.

A coupled permeability thresholding with effective medium theory upscaling technique for highly heterogeneous reservoirs

Reservoir simulations are performed with fine grid geological models as primary input into the reservoir simulators. These fine grid geological models contain ten million to one hundred million cells while current serial reservoir simulators can only handle one hundred thousand to one million cells. This disparity in scales prohibits direct input of fine grid geological models into serial simulators. However, traditional upscaling techniques are unable to reproduce the fine-scale dynamic flow behavior at the coarse-scale and up-scaled permeability values are dependent on the applied boundary conditions. The problem is exacerbated if isolated flow units are present. These problems motivated the need to couple the permeability thresholding and effective medium theory (EMT) approaches to develop a practical and accurate quasi-global single-phase permeability upscaling technique that uses the reservoir simulator in the upscaling process. The workflow involves identifying and making the no-flow cells slightly permeable. This is followed by defining two models for each region to be scaled up, one of which includes the heterogeneous permeability field in the region and the other the upscaled value. Also, a fine-scale gridded ring is constructed around the target region. Then the upscaled value of the permeability is resolved by minimizing the difference between the well water production rates (WWPR) of the two runs. The technique is shown to mimic the channel fine-scale permeability distribution at the coarse-scale. It is also shown to reproduce the physics of flow for two-phase flow problems as evident in the well oil production rate, injection well pressure and pressure distribution profile. Sensitivity analyses on different scale-up factors, well locations and sequentially active wells show the superiority of the novel technique over the upscaling and downscaling with effective medium theory (UDEMT) and local upscaling techniques - albeit at the cost of more CPU time. The new method takes more CPU time than local upscaling method but the accurate results justifies the extra CPU time. Our workflow should be extended to three dimensions and fractured reservoirs as well as coupled with existing upscaling techniques for near wellbore and relative permeability upscaling.

Modeling, simulation, and optimization of geothermal energy production from hot sedimentary aquifers

Geothermal district heating development has been gaining momentum in Europe with numerous deep geothermal installations and projects currently under development. With the increasing density of geothermal wells, questions related to the optimal and sustainable reservoir exploitation become more and more important. A quantitative understanding of the complex thermo-hydraulic interaction between tightly deployed geothermal wells in heterogeneous temperature and permeability fields is key for a maximum sustainable use of geothermal resources. Motivated by the geological settings of the Upper Jurassic aquifer in the Greater Munich region, we develop a computational model based on finite element analysis and gradient-free optimization to simulate groundwater flow and heat transport in hot sedimentary aquifers, and numerically investigate the optimal positioning and spacing of multi-well systems. Based on our numerical simulations, net energy production from deep geothermal reservoirs in sedimentary basins by smart geothermal multi-well arrangements provides significant amounts of energy to meet heat demand in highly urbanized regions. Our results show that taking into account heterogeneous permeability structures and a variable reservoir temperature may drastically affect the results in the optimal configuration. We demonstrate that the proposed numerical framework is able to efficiently handle generic geometrical and geological configurations, and can be thus flexibly used in the context of multi-variable optimization problems. Hence, this numerical framework can be used to assess the extractable geothermal energy from heterogeneous deep geothermal reservoirs by the optimized deployment of smart multi-well systems.

Gas migration and residual trapping in bimodal heterogeneous media during geological storage of CO2

This study investigates the effect of heterogeneity on CO2–brine two-phase flow behavior and capillary trapping at the field scale. A model based on macroscopic invasion percolation is developed to simulate CO2 migration and trapping in strongly heterogeneous systems with bimodal permeability distributions. Stochastic simulations are performed on heterogeneous permeability fields generated by considering both transition probability based lithofacies distribution and multi-Gaussian random fields, representative of characteristics often encountered in relevant sedimentary basins. The heterogeneity cases of different lithofacies proportions are designed such that they share the same ensemble mean of upscaled permeability over the simulation domain. We find that the CO2 spreading and trapping is strongly influenced by the heterogeneity. The ensemble means of sweep efficiency and domain-average CO2 saturation after primary drainage show a slight decrease with decreasing sandstone proportion p1 from 1.0 to 0.6 but a much steeper decline when p1 further decreases to 0.4. The upscaled two-phase flow properties such as the relative permeabilities are also influenced by the lithofacies proportion. Residual or capillary trapping and its uncertainty is shown to be dependent on the heterogeneity and the choice of trapping models. Our findings have broad implications within the context of CO2 sequestration, as the gas spreading and entrapment are the fundamental processes leading to long-term storage security.

Ensemble-based stochastic permeability and flow simulation of a sparsely sampled hard-rock aquifer supported by high performance computing

Calibrating the heterogeneous permeability distribution of hard-rock aquifers based on sparse data is challenging but crucial for obtaining meaningful groundwater flow models. This study demonstrates the applicability of stochastic sampling of the prior permeability distribution and Metropolis sampling of the posterior permeability distribution using typical production data and measurements available in the context of groundwater abstraction. The case study is the Hastenrather Graben groundwater abstraction site near Aachen, Germany. A three-dimensional numerical flow model for the Carboniferous hard-rock aquifer is presented. Monte Carlo simulations are performed, for generating 1,000 realizations of the heterogeneous hard-rock permeability field, applying Sequential Gaussian Simulation based on nine log-permeability values for the geostatistical simulation. Forward simulation of flow during a production test for each realization results in the prior ensemble of model states verified by observation data in four wells. The computationally expensive ensemble simulations were performed in parallel with the simulation code SHEMAT-Suite on the high-performance computer JURECA. Applying a Metropolis sampler based on the misfit between drawdown simulations and observations results in a posterior ensemble comprising 251 realizations. The posterior mean log-permeability is −11.67 with an uncertainty of 0.83. The corresponding average posterior uncertainty of the drawdown simulation is 1.1 m. Even though some sources of uncertainty (e.g. scenario uncertainty) remain unquantified, this study is an important step towards an entire uncertainty quantification for a sparsely sampled hard-rock aquifer. Further, it provides a real-case application of stochastic hydrogeological approaches demonstrating how to accomplish uncertainty quantification of subsurface flow models in practice.

Numerical resolution of a pseudo-parabolic Buckley-Leverett model with gravity and dynamic capillary pressure in heterogeneous porous media

We discuss a numerical approach for the simulation of a hydrodynamic model for immiscible incompressible two-phase flow in heterogeneous porous media to deal with effects of dynamic capillary pressure and gravity. The governing system of equations consists of a nonlinear pseudo-parabolic Buckley-Leverett saturation transport model coupled with an elliptic pressure-velocity problem. Upon certain manipulation of the equations, we rewrite the transport model as a nonlinear elliptic reaction-diffusion problem along with a simpler time-dependent relation to the dynamic capillary pressure. Hybridized mixed finite elements and domain decomposition procedures are used for the spatial discretization of the equations. Proper locally conservative finite volume techniques are used for a balancing discretization of the first-order hyperbolic flux and the dispersive behavior inherent of the pseudo-parabolic model. After discretizing in time with an implicit method, the resulting nonlinear algebraic equations is solved with a sequential time-marching approach and the arising systems of linear equations are solved efficiently with an algebraic multigrid method. Thus, we formulate and test numerically a fully coupled and implicit formulation of the transport problem of pseudo-parabolic nature. We find that the new procedure is accurate, robust, efficient and easy for implementation. We simulated the two-dimensional transport of the full nonlinear modified Buckley-Leverett model with a dynamic capillary pressure model in a variety of flow regimes along with heterogeneous permeability fields and domination of gravity. Our numerical experiments demonstrated the viability of the proposed formulation based on computational simulations of typical flow situations and saturation overshoot.