•A physics-informed machine learning approach is used to manage reservoir pressures.•A physics model provides theoretical constraints to the machine learning approach.•Tens of millions of model ...evaluations may be required for training.
We evaluate the feasibility of using physics-informed machine learning (PIML) for underground energy-related pressure management. To this end, we develop a PIML framework to manage underground reservoir pressures by training neural networks to determine fluid extraction rates for dedicated extraction wells during fluid injection operations given a range of reservoir conditions (e.g., transmissivity and storativity). We implement an automatically-differentiable analytical physics model of fluid flow in porous media within the PIML framework as a proxy for more complicated models. This allows us to execute a sufficient number of training scenarios to fully evaluate the feasibility of using PIML to support pressure management activities. We quantify the number of physics-model parameters required for automatic differentiation to become more efficient than finite-difference gradient calculations. We use a simple scenario with a single injector, extractor, and critical location for our feasibility analysis. We evaluate the effect of the size of the training dataset (i.e., the number of reservoir condition samples) on the accuracy and efficiency of the PIML framework. For an equivalent number of model evaluations, the larger training dataset took less time to train and produced a neural network that was able to more accurately manage reservoir pressures. We also evaluate the effect of the training dataset batch size (i.e., number of reservoir condition samples used to update the neural network coefficients during training; i.e., how the training dataset is partitioned). While training ran faster with larger batch sizes, they produced neural networks that managed pressures less accurately. We demonstrate the approach on a more complex scenario involving 10 injectors, 10 extractors, and 4 critical locations (a relatively high well density of 20 wells/km2). We provide the number of forward and adjoint model evaluations required in each case as an indication of the feasibility of using PIML for pressure management when more complicated physics models with longer execution times are used.
Geological CO2 sequestration (GCS) has been proposed as an effective approach to mitigate carbon emissions in the atmosphere. Uncertainty and sensitivity analysis of the fate of CO2 dynamics and ...storage are essential aspects of large-scale reservoir simulations. This work presents a rigorous machine learning-assisted (ML) workflow for the uncertainty and global sensitivity analysis of CO2 storage prediction in deep saline aquifers. The proposed workflow comprises three main steps: The first step concerns dataset generation, in which we identify the uncertainty parameters impacting CO2 flow and transport and then determine their corresponding ranges and distributions. The training data samples are generated by combining the Latin Hypercube Sampling (LHS) technique with high-resolution simulations. The second step involves ML model development based on a data-driven ML model, which is generated to map the nonlinear relationship between the input parameters and corresponding output interests from the previous step. We show that using Bayesian optimization significantly accelerates the tuning process of hyper-parameters, which is vastly superior to a traditional trial–error analysis. In the third step, uncertainty and global sensitivity analysis are performed using Monte Carlo simulations applied to the optimized surrogate. This step is performed to explore the time-dependent uncertainty propagation of model outputs. The key uncertainty parameters are then identified by calculating the Sobol indices based on the global sensitivity analysis. The proposed workflow is accurate and efficient and could be readily implemented in field-scale CO2 sequestration in deep saline aquifers.
A limit cycle is a cyclic path of oscillation on which the states of a nonlinear system settle. A considerable number of practical systems such as robots, converters, and heartbeat require generating ...the sustainable oscillatory behaviours. The objective is to design a controller to generate limit cycle with specific behaviours. In this paper, a novel fuzzy control (FC) strategy is introduced to create the limit-cycle for nonlinear complex dynamics with unknown uncertainties. The suggested controller benefits from new interval type-3 fuzzy logic, allowing the control synthesis to improve the quality of closed-loop response and robust performance. The adaptively learned backstepping controller based on FC is employed to analyze the convergence and robustness. Various simulations are proposed to ensure the efficiency of the fuzzy-based control law and adaption rules.
The effect of natural fractures, their orientation, and their interaction with hydraulic fractures on the extraction of heat and the extension of injection fluid are fully examined. A fully coupled ...and dynamic thermo-hydro-mechanical (THM) model is utilized to examine the behavior of a fractured geothermal reservoir with supercritical CO2 as a geofluid. The interaction between natural fracture and hydraulic fracture, as well as the type and location of geofluids, influences the production temperature, thermal strain, mechanical strains, and effective stress in rock/fractures in the reservoir. A mathematical model is developed by using the fully connected neural network (FCN) model to establish a mathematical relationship between the reservoir parameters and the temperature. The response surface methodology is applied for qualitative numerical experimentation. It is found that the developed FCN model can be utilized to forecast the temporal variation of temperature in the production well to a desired level using FCN. Therefore, the numerical simulations developed with the FCN method can be useful tools to investigate the temperature evolution with higher accuracy.
Rheological models are usually used to predict foamed fluid viscosity; however, obtaining the model constants under various conditions is challenging. Hence, this paper investigated the effect of ...different variables on foam rheology, such as shear rate, temperature, pressure, surfactant types, gas phase, and salinity, using a high-pressure high-temperature foam rheometer. Power-law, Bingham plastic, and Casson fluid models fit the experimental data well. Therefore, the data were fed to different machine learning techniques to evaluate the rheological model constants with different features. In this study, seven different machine learning techniques have been applied to predict the rheological models’ constants, including decision tree, random forest, XGBoost (XGB), adaptive gradient boosting, gradient boosting, support vector regression, and voting regression. We evaluated the performance of our machine learning models using the coefficient of determination (R 2), cross-plots, root-mean-square error, and average absolute percentage error. Based on the prediction outcomes, the XGB model outperformed the other ML models. The XGB model exhibited remarkably low error rates, achieving a prediction accuracy of 95% under ideal conditions. Furthermore, our prediction results demonstrated that the Casson model accurately captured the rheological behavior of the foam. Additionally, we used Pearson's correlation coefficients to assess the significance of various properties in relation to the constants within the rheological models. It is evident that the XGB model makes predictions with nearly all features contributing significantly, while other machine learning techniques rely more heavily on specific features over others. The proposed methodology can minimize the experimental cost of measuring rheological parameters and serves as a quick assessment tool.
•We develop an efficient physics-constrained deep learning model to solve the problem of multiphase flow in 3-Dimenionsal heterogeneous porous media;•We implement numerical techniques including ...continuity-based smoother and spatial domain decomposition to improve the prediction accuracy of the state variables, especially pressure;•We penalize the training loss in the pressure and saturation plume regions, to steer the training process such that the model can accurately capture flow in these regions;•We validate the whole workflow through the example of CO2 injection in heterogeneous geological formations.
Physics-based simulators for multiphase flow in porous media emulate nonlinear processes with coupled physics, and usually require extensive computational resources for software development, maintenance and simulation execution. As a result, a huge demand exists for fast modeling of coupled processes in a wide range of subsurface applications including geological CO2 sequestration, hydrocarbon recovery and geothermal energy extraction. In this work, an efficient physics-constrained deep learning model is developed for solving multiphase flow in 3-Dimensional (3D) heterogeneous porous media. The model fully leverages the spatial topology predictive capability of convolutional neural networks, specifically U-Net with successive contracting and expansive steps, and is coupled with an efficient continuity-based smoother to predict flow responses that need spatial continuity. Furthermore, the transient regions are penalized to steer the training process such that the model can accurately capture flow in these regions. The model takes inputs including properties of porous media, fluid properties and well controls, and predicts the temporal-spatial evolution of the state variables (pressure and saturation). While maintaining the continuity of fluid flow, the 3D spatial domain is decomposed into 2D images for reducing training cost, and the decomposition results in an increased number of training data samples and better training efficiency. Additionally, a surrogate model is separately constructed as a postprocessor to calculate well flow rate based on the predictions of state variables from the deep learning model. We use the example of CO2 injection into saline aquifers, and apply the physics-constrained deep learning model that is trained from physics-based simulation data and emulates the physics process. The model performs prediction with a speedup of ∼ 1400 times compared to physics-based simulations, and the average temporal errors of predicted pressure and saturation plumes are 0.27% and 0.099% respectively. Furthermore, water production rate is efficiently predicted by a surrogate model for well flow rate, with a mean error less than 5%. Therefore, with its unique scheme to cope with the fidelity in fluid flow in porous media, the physics-constrained deep learning model can become an efficient predictive model for computationally demanding inverse problems or other coupled processes.
•A robust deep learning (DL) workflow has been developed to predict the long term process of geologic CO2 sequestration.•Different sets of features fed to DL have been evaluated to obtain the most ...accurate prediction for pressure and saturation.•The DL workflow shows great efficiency with a speedup of 250 times compared to full reservoir simulation.
Simulation of multiphase flow in porous media is essential to manage the geologic CO2 sequestration (GCS) process, and physics-based simulation approaches usually take prohibitively high computational cost due to the nonlinearity of the coupled physics. This paper contributes to the development and evaluation of a deep learning workflow that accurately and efficiently predicts the temporal-spatial evolution of pressure and CO2 plumes during injection and post-injection periods of GCS operations. Based on a Fourier Neural Operator, the deep learning workflow takes input variables or features including rock properties, well operational controls and time steps, and predicts the state variables of pressure and CO2 saturation. To further improve the predictive fidelity, separate deep learning models are trained for CO2 injection and post-injection periods due to the difference in primary driving force of fluid flow and transport during these two phases. We also explore different combinations of features to predict the state variables. We use a realistic example of CO2 injection and storage in a 3D heterogeneous saline aquifer, and apply the deep learning workflow that is trained from physics-based simulation data and emulate the physics process. Through this numerical experiment, we demonstrate that using two separate deep learning models to distinguish post-injection from injection period generates the most accurate prediction of pressure, and a single deep learning model of the whole GCS process including the cumulative injection volume of CO2 as a deep learning feature, leads to the most accurate prediction of CO2 saturation. For the post-injection period, it is key to use cumulative CO2 injection volume to inform the deep learning models about the total carbon storage when predicting either pressure or saturation. The deep learning workflow not only provides high predictive fidelity across temporal and spatial scales, but also offers a speedup of 250 times compared to full physics reservoir simulation, and thus will be a significant predictive tool for engineers to manage the long-term process of GCS.
Simulation of multiphase flow in porous media is crucial for the effective management of subsurface energy and environment-related activities. The numerical simulators used for modeling such ...processes rely on spatial and temporal discretization of the governing mass and energy balance partial-differential equations (PDEs) into algebraic systems via finite-difference/volume/element methods. These simulators usually require dedicated software development and maintenance, and suffer low efficiency from a runtime and memory standpoint for problems with multi-scale heterogeneity, coupled-physics processes or fluids with complex phase behavior. Therefore, developing cost-effective, data-driven models can become a practical choice, and in this work, we choose deep learning approaches as they can handle high dimensional data and accurately predict state variables with strong nonlinearity. In this paper, we describe a gradient-based deep neural network (GDNN) constrained by the physics related to multiphase flow in porous media. We tackle the nonlinearity of flow in porous media induced by rock heterogeneity, fluid properties, and fluid-rock interactions by decomposing the nonlinear PDEs into a dictionary of elementary differential operators. We use a combination of operators to handle rock spatial heterogeneity and fluid flow by advection. Since the augmented differential operators are inherently related to the physics of fluid flow, we treat them as first principles prior knowledge to regularize the GDNN training. We use the example of pressure management at geologic CO2 storage sites, where CO2 is injected in saline aquifers and brine is produced, and apply GDNN to construct a predictive model that is trained with physics-based simulation data and emulates the physics process. We demonstrate that GDNN can effectively predict the nonlinear patterns of subsurface responses, including the temporal and spatial evolution of the pressure and CO2 saturation plumes. We also successfully extend the GDNN to convolutional neural network (CNN), namely gradient-based CNN (GCNN), and validate its capability to improve the prediction accuracy. GDNN has great potential to tackle challenging problems that are governed by highly nonlinear physics and enable the development of data-driven models with higher fidelity.
•A workflow based on Gradient-based deep neural network (GDNN) emulates multiphase flow in porous media with high fidelity.•Differential operators derived from the governing PDEs are used as first principle prior knowledge to regularize the training of GDNN.•GDNN is further successfully extended to image-based approaches such as convolutional neural networks (CNN).
In this paper, we propose a pore-scale lattice Boltzmann model to simulate fluid flow coupled with heterogeneous surface reactions and mineral dissolution. The primary innovation lies in the ...transformation of surface reactions, originally treated as a boundary condition, into a volume-source term through dimensional augmentation within the framework of sharp liquid–solid interfaces. This significantly simplifies the implementation, particularly for reactions occurring in porous media with intricate geometric structures. Several benchmark tests were performed to validate the accuracy of this model, including a reaction–diffusion problem in a rectangular domain, a two-dimensional reaction and dissolution of a circular grain, as well as a three-dimensional calcite crystal dissolution in a micro-channel. All the obtained simulation results agree well with the reference solutions. In addition, a dissolution problem in a three-dimensional porous medium built with the sand pack is then investigated. Cases with different Péclet numbers (Pe) and Damköhler numbers (Da) were simulated, and five dissolution modes were obtained, which were finally summarized in a diagram of Pe and Da.
•A new LB model is proposed to simulate heterogeneous reactions with mineral dissolution.•This model can be used to simulate both linear and nonlinear heterogeneous reactions.•Boundary reactions are treated as source terms, making the implementation very simple.•A dissolution diagram of Pe and Da is summarized based on the 3D simulation results.
The Puga geothermal reservoir is located in the south-eastern part of Ladakh (Himalayan region, India), and it is providing encouraging results towards heat production. We proposed an improved ...mathematical model for the fully coupled thermo-hydro-geomechanical model to examine the variations in the Puga geothermal reservoir at between 4500 m from the surface with three, four, and seven hydraulic fractures in the reservoir along with four-spot, five-spot, seven-spot, and nine-spot well patterns. The distribution of low-temperature region is found in each fracture, and it is low in the reservoir with seven hydraulic fractures. The changes in the rock and fluid properties are examined effectively. Thermal strain is dominated in the fractures, and mechanical strain is impressive in the rock matrix; it is dependent on the number of hydraulic fractures and well patterns. The thermal performance of the Puga reservoir is examined with the geothermal life, reservoir impedance, and heat power and found that the number of hydraulic fractures and well patterns are influenced significantly in the multistage modeling of the Puga geothermal reservoir. Thus, the proposed mathematical model can effectively evaluate and predict the variations that occur in the Puga geothermal reservoir with dynamic rock, fracture, and fluid properties.
•Mathematical model improved by including the dynamic variations in the rock, fracture, and fluid properties.•Examined the performance of Puga geothermal reservoir with different well patterns.•Multistage model was proposed to the development of Puga geothermal reservoir.
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