Reservoir Domain Research Articles

Learning a CNN with the finite volume method for modelling fluid seepage flow under closed boundary

Physics-informed neural networks (PINNs) are an efficient technique for incorporating physics law in models based on neural networks (NNs). In this paper, we propose a local refinement physics-informed convolutional neural network (LR-PICNN) to solve the seepage equation in closed boundary reservoirs. LR-PICNN employs a local refinement method to enhance focus on the region containing the well. First, the finite volume (FV) method is used to discretize the seepage equation on the global reservoir domain and on the local domain around the well to construct the loss functions respectively. Second, the boundary between the global and local domains are used as boundary conditions to couple the two systems. Finally, two loss functions are added to train the CNN to solve the seepage equation in autoregressive way and coupled without any labeled data. Experimental results show that LR-PICNN can more accurately solve seepage flow equation. Due to the coupling of solving equations and that LR-PICNN focuses on the behaviors of the well, compared to physics-informed Deep Convolutional neural network (PIDCNN) under the same well grid size, LR-PICNN has a faster solving speed that is at least 8 times faster, higher solution accuracy for BHP that is at least twice as high, and more stable solution, which could be used in numerical well testing.

Combined mechanistic and machine learning method for construction of oil reservoir permeability map consistent with well test measurements

We introduce a novel method for estimating the spatial distribution of absolute permeability in oil reservoirs, consistent with well logging and well test measurements. The primary objective is to create a permeability map, incorporating the well test interpretation results and achieving hydrodynamic similarity to the actual permeability distribution around each well. This enhancement aims to improve the accuracy of reservoir modeling outcomes in reproducing real data. We utilize Nadaraya-Watson kernel regression to parameterize the two-dimensional spatial distribution of rock permeability. The kernel regression parameters are optimized by minimizing the discrepancies between actual and predicted values of permeability at well locations, the integral permeability of the reservoir domain around each well, and skin factors. This inverse optimization problem is addressed by repeatedly solving forward problems, where an artificial neural network (ANN) predicts the integral permeability of the formation surrounding a well and skin factor. The ANN is trained on a physics-based dataset generated through a synthetic well test procedure, which includes the numerical modeling of the bottomhole pressure decline curve in a reservoir simulator and its interpretation using a semi-analytical reservoir model. The proposed method is tested on the “Egg Model”, a synthetic reservoir with significant heterogeneity due to highly permeable channels. The permeability map created by our approach demonstrates hydrodynamic similarity to the original map. Numerical reservoir simulations, corresponding to the constructed and original permeability maps, yield comparable pore pressure and water saturation distributions at the end of the simulation period. Additionally, we observe a notable match in flow rates and total volumes of produced oil, water, and injected water between simulations. The developed approach outperforms kriging in terms of numerical reservoir modeling outcomes. This research advances existing geostatistical interpolation techniques by fusing well logging and well test data to build the reservoir permeability map through an optimization framework coupled with machine learning. Unlike traditional variogram-based geostatistical simulation algorithms, our method provides a permeability distribution that is hydrodynamically similar to the actual one, enhancing initial guess in the history matching process. The novel incorporation of well test interpretation results into the permeability map represents a significant improvement over existing methods, offering an innovative approach that can benefit the petroleum industry. We also provide recommendations for further development of the proposed algorithm to account for geological realism.

A novel streamline simulation method for fractured reservoirs with full-tensor permeability

In this work, we develop a novel streamline (SL) simulation method that integrates seamlessly within the embedded discrete fracture model (EDFM). The novel SL-based method is developed based on a hybrid of two-point flux approximation (TPFA) and mimetic finite difference (MFD) methods, which is applicable to a two-phase anisotropic flow in fractured reservoirs. We refer to this novel approach as EDFM-TPFA-MFD-SL. The approach is operated in an IMplicit Pressure Explicit Saturation (IMPES) manner. First, this work establishes a novel EDFM utilizing a hybrid TPFA-MFD scheme to solve the pressure equation for phase flux approximation. Subsequently, we introduce a practical streamline tracing workflow designed to derive the distribution of streamlines within the reservoir domain and the time-of-flight distribution along each individual streamline. This feature allows for the parallel computation of water saturation along the streamlines. Two numerical examples are presented to validate the superiority of the proposed EDFM-TPFA-MFD-SL method compared to the existing streamline-based EDFM on cases with full-tensor permeability. The results show that the proposed method could significantly mitigate the numerical dissipation and reduce the computation costs. Another numerical example demonstrates the effectiveness of the proposed method in dealing with complex fracture networks and providing rapid flow diagnostics, indicating its significant potential for real-world field applications.

A novel hydraulic fracturing model for the fluid-driven fracture propagation in poroelastic media containing the natural cave

Hydraulic fracturing is an efficient technology to extract hydrocarbon within natural caves. However, these caves can markedly affect the fracture propagation behavior. This paper proposes a novel hydraulic fracturing model to simulate the fracture propagation in poroelastic media containing the natural cave, utilizing the strengths of the phase-field method. By coupling the Reynolds flow with cubic law in fracture domain, free flow in cave domain, and low-permeability Darcy flow in reservoir domain, the fracture-cave-reservoir flow governing equations are established. The Biot poroelasticity theory and fracture width are the links of hydro-mechanical coupling. The smooth phase-field is introduced to diffuse not only the sharp fracture but also the sharp cave edge. The fully coupling model is solved by a staggered scheme, which independently solves the pressure field and displacement field in inner cycle, and then independently solves the phase field in outer cycle. The proposed model is verified by comparing with the Khristianovic–Geertsma–de Klerk (KGD) model and Cheng's hydraulic fracturing model. Then, the interaction between hydraulic fracture and natural cave is investigated through several two-dimensional and three-dimensional cases. The result shows that the cave effect can make the hydraulic fracture deflect and raise its propagation velocity. Increasing the fracture-cave distance, injection rate, and in situ stress difference can all decline the cave effect. The displayed cases also substantiate the capability and efficiency of the proposed model.

Electrochemical Model for a Three-Electrode Lead Acid Cell: Boundary Conditions for Asymmetric Configurations

A one-dimensional (1D) electrochemical model is developed for a lead-acid demonstration cell comprising two positive electrodes engaging a single negative electrode. Classical 1D models, which analyze a single repeat unit in a full battery, are extended to this non-standard configuration by considering additional porous electrode and electrolyte reservoir domains. The non-standard model necessitates appropriate boundary conditions to couple electrochemical variables in different domains, since symmetry boundary conditions typically employed at the current collector grids are no longer applicable. The modified model, with new boundary conditions, is simulated using standard techniques. Salient modeling and simulation differences compared to a standard lead-acid model are discussed, as are trends in predicted variables. The non-standard model predicts reduced polarization and more uniform utilization of the non-limiting electrodes, in addition to non-zero reaction currents and active material utilization in the terminal electrode faces, which are typically ignored in “unit-cell” analyses. This model is expected to be of substantial utility in guiding evaluations of new lead-acid battery designs through improved prediction of performance of such test configurations. The boundary conditions introduced herein are also useful for 1D modeling of non-standard configurations characterized by multiple, electrochemically interacting porous domains, particularly where spatial current distributions are initially unknown.

A Mechanical Picture of Fractal Darcy’s Law

The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.

An Integrated Approach for History Matching of Complex Fracture Distributions for Shale Oil Reservoirs Based on Improved Adaptive Particle Filter

SummaryThe application of horizontal well drilling technology and volume fracturing technique makes the economic development of shale oil reservoirs feasible. The unknown fracture networks lead to severe nonlinearity and high uncertainty during fracture characterization. Moreover, the reservoir parameters usually exhibit a highly non-Gaussianity. Therefore, the key challenges for history matching in fractured shale oil reservoirs are effectively representing the fracture network and coping with the non-Gaussian distribution of reservoir-model parameters. In this work, a new characterization method for complex fracture networks is established, in which the distribution of connected fractures of the reservoir domain is represented by some statistical parameters such as fracture dip angle, fracture azimuth, and fracture half-length and some deterministic parameters such as the coordinates of fracture center points. In the uncertainty quantification and history-matching process, an integrated approach that combines the particle filter and an improved kernel density estimation (KDE) based on its Shannon entropy (SE) for estimating fracture distributions and physical parameters is presented. An adaptive mechanism based on Kullback-Leibler divergence (KLD) is introduced in the proposed history matching workflow, which automatically adjusts the number of particles to reduce the computational burden. Two examples of 3D shale oil production were constructed to validate the efficiency and accuracy of the proposed method. Results showed that the method was capable of capturing the main features of the fracture distributions in the reference cases. The proposed method has the potential to be applied in more complex cases such as multiple wells and multiphase flow.

A Meshless Numerical Modeling Method for Fractured Reservoirs Based on Extended Finite Volume Method

Summary In this paper, a meshless numerical modeling method named mesh-free discrete fracture model (MFDFM) of fractured reservoirs based on the newly developed extended finite volume method (EFVM) is proposed. First, matching and nonmatching point cloud generation algorithms are developed to discretize the reservoir domain with fracture networks, which avoid the gridding challenges of the reservoir domain in traditional mesh-based methods. Then, taking oil/water two-phase flow in fractured reservoirs as an example, MFDFM derives the EFVM discrete scheme of the governing equations, constructs various types of connections between matrix nodes and fracture nodes, and calculates the corresponding transmissibilities. Finally, the EFVM discrete scheme of the governing equations and the generalized finite difference discrete scheme of various boundary conditions form the global nonlinear equations, which do not increase the degree of nonlinearity compared with those in the traditional finite volume method (FVM)-based numerical simulator. The global equations can be solved by the existing nonlinear solver in the FVM-based reservoir numerical simulator by only adding the linear discrete equations of boundary conditions, which reduce the difficulty of forming a general purpose MFDFM-based fractured reservoir numerical simulator. Several numerical test cases are implemented to illustrate that the proposed MFDFM can achieve good computational performance under matching and nonmatching point clouds, and for heterogeneous reservoirs, complex fracture networks, complex boundary geometry, and complex boundary conditions, by comparing the computational results of MFDFM with embedded discrete fracture model (EDFM). Thus, MFDFM retains the computational performances of the traditional mesh-based methods and can avoid the difficulties of handling complex geometry and complex boundary conditions of the computational domain, which is the first meshless numerical framework to model fractured reservoirs in parallel with the mesh-based discrete fracture model (DFM) and EDFM.

A fully coupled multidomain and multiphysics model considering stimulation patterns and thermal effects for evaluation of coalbed methane (CBM) extraction

A deep understanding of the process of coalbed methane (CBM) extraction is of significance for both the unconventional energy supply and mine safety. The effects of stimulation patterns and thermal effects (including thermal shrinkage, non-isothermal sorption, and gas density change) significantly affect the CBM extraction process. Although thermal effects on CBM extraction have been investigated in previous studies, these effects have not been coupled with stimulation patterns, leading to an insufficient evaluation accuracy of previous models and a limited understanding of gas extraction in stimulated CBM reservoirs. This study developed a multidomain and multiphysics (thermal-hydraulic-mechanical, THM) model to fully couple different physical processes (including gas flow, coal deformation, gas desorption, and heat transfer) within a CBM reservoir framework considering various stimulation patterns to improve the simulation accuracy and obtain insights into CBM extraction. Three domains with distinct properties, i.e., the stimulated reservoir domain (SRD), non-stimulated reservoir domain (NSRD), and radial primary fracture (PF), were integrated into the CBM reservoir model to characterize the stimulation pattern. A group of partial differential equations (PDEs) was derived to characterize CBM transport within each domain and across different domain boundaries. A stimulated coalbed was defined as an assembly of three interacting porous media: coal matrix, continuous fractures (CF), and PF. The matrix and CF constituted a dual-porosity dual-permeability system, while the PF was simplified as a 1-D fractured medium. The finite element method (FEM) was employed to numerically solve the above PDEs. The proposed model was verified against two sets of field-measured CBM production data and compared to three previously published numerical models to reveal its advantage. The verified model was applied to investigate multidomain effects of stimulation treatment and evaluate the thermal effects of gas depletion on gas extraction in stimulated CBM reservoirs. The simulation results suggested (1) that the distinct properties of the different domains result in different permeability evolutions, which in turn influences CBM production; (2) that increasing the fracture complexity, enlarging the SRD size, and improving the SRD permeability with suitable stimulation techniques constitute effective approaches to enhance the CBM recovery; and (3) that ignoring thermal effects in CBM extraction can either overestimate or underestimate production, which mainly depends on the net effects of the thermal shrinkage strain and non-isothermal sorption during different production periods.

Generalized finite difference method (GFDM) based analysis for subsurface flow problems in anisotropic formation

This paper applies the meshless generalized finite difference method (GFDM) to analyze the subsurface flow problem in anisotropic formations for the first time, and develops the treatment methods of anisotropic permeability tensor in continuous function and discrete distribution expressions respectively. In particular, in case of the common discrete distribution expression of anisotropic permeability tensor in reservoir simulation, this paper suggests combining general difference operators and harmonic average scheme to simply and efficiently obtain the meshless discrete scheme of the anisotropic flow equation. Three numerical examples of steady and transient flow in the rectangular computational domain, transient flow with complex boundary, and steady flow with strongly discontinuous anisotropic permeability are used to verify the high accuracy and good convergence of GFDM to handle anisotropic flow problems. Compared with the traditional mesh-based reservoir numerical simulation method that replies on laborious gridding to discretize the reservoir domain and complicated multi-point flux approximation (MPFA) to handle discrete distribution anisotropic permeability tensor, the meshless GFDM can be more practically applied to the anisotropic flow problems in the reservoir domain with complex geometry and complex boundary conditions. Besides, it is found that the radius of the influence domain has little impact on the accuracy of steady flow. For the transient flow, it generally holds that the larger the radius of the influence domain, the lower the calculation accuracy in case of Cartesian collocation. In sum, this work provides an efficient and simple meshless solver to handle anisotropic flow problems in porous media under GFDM framework, which reveals the great application potential of GFDM in reservoir numerical simulation.

A multi-scale quadruple-continuum model for production evaluation of shale gas reservoirs considering complex gas transfer mechanisms and geomechanics

Shale gas reservoirs are uniquely characterized by extremely low values of permeability, porosity, and complex fracture network which adequately hinders gas transfer and production evaluation. At the reservoir scale, the analysis of complex gas flow and transfer mechanisms from one domain to another is still a challenge. In this regard, a fully coupled multi-scale quadruple-continuum model is proposed, where the gas transfer mechanisms across the four domains (Kerogen, inorganic matrix, natural fractures, and hydraulic fractures) are adopted and the flow terms are corrected to suit real gas and account for shale deformation considering dynamic effective stress. The complexity of modeling gas transfers from the kerogen to the inorganic matrix to fracture system (natural and hydraulic fractures) and the production well is overcome by using the Warren-Root and Vermeulen transfer terms, respectively. The current model matches field data from typical shale reservoirs and accurately predicts gas production. Numerical simulation results show that increasing the kerogen pore volume and stress sensitivity coefficient decreases cumulative gas production. Increasing domain permeability, porosity, number of natural fractures, and the width and length of natural fractures improve the flow of gas, leading to higher cumulative gas production. Moreover, crisscrossing natural fractures and larger stimulated reservoir domain (SRD) could extend the area of contact with the unstimulated reservoir domain (USRD), thus allowing more gas to flow to the production well. Overall, more gas is produced when the four domains are considered than when either of them is ignored. This study introduces a comprehensive understanding of the four domains' primary flow and transfer mechanisms. It provides a practical baseline for evaluating gas production in fractured shale reservoirs.

Coupled Geomechanics and Flow Modeling of Fractured Reservoirs considering Matrix Permeability Anisotropy

The effect of geomechanics is crucial in the modeling of fractured reservoirs since fractures can be more stress-sensitive than the rock matrix. In fractured reservoirs, the microscale fractures are often homogenized into the matrix continuum leading to matrix permeability anisotropy. The embedded discrete fracture model (EDFM) is becoming increasingly popular for numerical simulation of fractured reservoirs and has been successfully employed in the simulation of coupled geomechanics and flow systems. However, the anisotropic permeability of matrix cannot be considered in traditional EDFM. An approach of coupled geomechanics and flow modeling is proposed for fractured reservoirs considering anisotropic matrix permeability. In order to calculate the fluid exchange between fractures and rock matrix in an anisotropic formation, an integrally embedded discrete fracture model (IEDFM) is used. The geomechanics model of the proposed approach uses an equivalent continuum approach, which introduces an equivalent material to represent the overall deformation of the fractured rock under normal and shear stresses. The constitutive relations of the equivalent continuum are derived rigorously from stress-strain analysis, where the stress-dependent moduli of natural fractures are included. The coupled geomechanics and flow system is solved using the fixed-stress split iterative coupling strategy with the dynamic hydraulic parameters of matrix and fractures updated separately. Several examples are performed to demonstrate the applicability of the proposed approach for modeling the coupled geomechanics and flow system in fractured reservoirs considering anisotropic permeability. The effect of anisotropy is investigated, which indicates that the dynamic behavior of a fracture is highly orientation-related due to initial stress anisotropy and matrix permeability anisotropy. Simulations also show that the anisotropic matrix permeability affects the compaction in the reservoir domain, which reflects on the performance of production.

An efficient finite element model for dynamic analysis of gravity dam-reservoir-foundation interaction problems

In this paper, a finite element (FE)-based model that efficiently evaluates the dynamic behavior of dam-reservoir-foundation interaction (DRFI) problem was proposed including the radiation of waves to the unbounded rock and reservoir domains. Lagrangian fluid elements were used to discretize the near-field reservoir domain, while the presented infinite fluid elements were used to discretize the far-field reservoir domain. The fully coupled equation of motion for DRFI problem was solved by direct method. A two-dimensional (2D) plane-strain FE formulation of the problem is written in FORTRAN 90 programming language. Investigations were conducted on the effect of near-field domain size (length and depth) on the dynamic behavior of DRFI, dam-foundation interaction (DFI), and dam-reservoir interaction (DRI) problems. The results of this study demonstrate that the proposed model outperforms many other models that have been evolved in the literature in terms of accuracy and speed. The reflected hydrodynamic pressures at the far-field reservoir domain were efficiently absorbed by the suggested infinite fluid elements. The near-field domains size has a noticeable impact on the dynamic behavior of the dam. Making an exact choice about the size is more challenging. However, it was observed that the size of 1.5H is the physically appropriate response.

A reply to “Relevant factors in the eutrophication of the Uruguay River and the Río Negro”

A recent paper by Beretta-Blanco and Carrasco-Letelier (2021) claims that agricultural eutrophication is not one of the main causes for cyanobacterial blooms in rivers and artificial reservoirs. By combining rivers of markedly different hydrological characteristics e.g., presence/absence and number of dams, river discharge and geological setting, the study speculates about the role of nutrients for modulating phytoplankton chlorophyll-a. Here, we identified serious flaws, from erratic and inaccurate data manipulation. The study did not define how erroneous original dataset values were treated, how the variables below the detection/quantification limit were numerically introduced, lack of mandatory variables for river studies such as flow and rainfall, arbitrary removal of pH > 7.5 values (which were not outliers), and finally how extreme values of other environmental variables were included. In addition, we identified conceptual and procedural mistakes such as biased construction/evaluation of model prediction capability. The study trained the model using pooled data from a short restricted lotic section of the (large) Uruguay River and from both lotic and reservoir domains of the Negro River, but then tested predictability within the (small) Cuareim River. Besides these methodological considerations, the article shows misinterpretations of the statistical correlation of cause and effect neglecting basic limnological knowledge of the ecology of harmful algal blooms (HABs) and international research on land use effects on freshwater quality. The argument that pH is a predictor variable for HABs neglects overwhelming basic paradigms of carbon fluxes and change in pH because of primary productivity. As a result, the article introduces the notion that HABs formation are not related to agricultural land use and water residence time and generate a great risk for the management of surface waterbodies. This reply also emphasizes the need for good practices of open data management, especially for public databases in view of external reproducibility.

A new methodology based on finite element method (FEM) for generation of the probability field of rock types from subsurface

In this paper, we present a novel methodology by employing the finite element method (FEM) for creating probability fields of rock types (facies) present in a reservoir. The method is developed for a hydrocarbon reservoir, which, situated thousands of meters underground, has a high uncertainty regarding its geology. However, from the exploration phase, are collected facies observations (hard data) about the rock types present at some sparse locations of the reservoir domain. These observations are projected into a continuous probabilistic space, which is the environment of the FEM. The novelty consists of the fact that the probability field of a facies type is viewed as a deformation field of a plate. For each facies type, the FEM constructs a deformation profile which will be a probability field. The fields generated with the FEM verify the conditions of the probability theory without post-regularization techniques. We test the methodology for a two-dimensional reservoir model with three facies types. We also show the applicability of the new method by introducing the probability fields in the truncated pluri-Gaussian simulation method for simulating facies fields which are the inputs in the assisted history matching of the reservoir.

Modification of direct-FE method for nonlinear seismic analysis of arch dam-reservoir-foundation system considering spatially varying ground motion

The present paper aims to modify an existing direct finite element method for nonlinear seismic analysis of high arch dam-reservoir-massed foundation systems to consider the effects of spatially varying ground motion at the bottom of the foundation rock with the non-stationary frequency characteristics. Viscous-damper boundaries are used to model the semi-unbounded foundation rock and the reservoir domains. In the existing direct finite element method, it is assumed that incident waves propagate vertically upwards from beneath the foundation rock towards the ground surface, and the effective seismic forces starting from the control motion on the foundation surface are applied to the bottom and the side vertical boundaries of the foundation rock. Three components of the motion recorded at Parkfield fault zone 16 during the 1983 Coalinga earthquake are selected as the free-field control motion at the surface of the foundation rock. On the other hand, in the modified direct FE method, fully non-stationary spectrum-compatible ground motion time histories, including both wave passage and incoherency effects, are simulated at a number of locations at the foundation bottom. Then the generated ground motions are used as multiple input incident motions to compute effective seismic forces applied to the bottom and the side boundaries of the foundation rock in the dam-reservoir-massed foundation system. Step-by-step procedures for computing the effective earthquake forces in the existing and the modified direct FE methods are presented. The nonlinearity originates from opening/sliding of the vertical contraction joints within the dam body. The reservoir–structure interaction is accounted for via the finite element method assuming compressible reservoir. The Karoun-I, a double curvature high arch dam, is selected as a case study. Several numerical validation models are conducted and compared with the benchmark solutions to indicate the accuracy of the existing direct FE method. It was seen that applying effective earthquake forces determined from the modified direct finite element method increases the nonlinear seismic response of the dam-reservoir-massed foundation system in terms of the crest displacements and stress levels.

A fully coupled multidomain and multiphysics model for evaluation of shale gas extraction

Creating a stimulated reservoir domain (SRD) is a necessity to effectively extract natural gas from shale reservoirs. After hydraulic fracturing, a shale reservoir has three distinct domains: SRD, non-stimulated reservoir domain (NSRD) and hydraulic fractures (HFs). In previous studies, the property contrasts and interactions between different domains are often not fully considered. In this study, a fully coupled multi-domain and multi-physics model is developed to incorporate these complexities. The shale reservoir is characterized as an assembly of three distinct components: organic kerogen, inorganic matrix and HF. Furthermore, the kerogen and inorganic matrix are defined as dual-porosity-dual-permeability media, while the HF is simplified as a 1-D cracked medium. Particularly, the inorganic matrix has different properties in each of the SRD and NSRD to reflect the stimulation effect of hydraulic fracturing on the near-HF matrix. Under this framework, a series of partial differential equations (PDEs) fully coupled by mass transfer and mechanical deformation relations were derived to define various processes in the shale reservoir. These PDEs were numerically solved by the finite element method. The proposed model is validated against analytical solutions and verified against gas production data from the field. Sensitivity analyses reveal: (1) that both size and internal structure of the SRD significantly affect gas extraction by improving SRD properties and in creating low-pressure zones around the NSRD; (2) that the NSRD determines the sustainability of gas production; and (3) that the change of mechanical properties in one domain affects the evolution of transport properties in the entire reservoir.

An experiment on reservoir representation schemes to improve hydrologic prediction: coupling the national water model with the HEC-ResSim

ABSTRACT This study experiments with reservoir representation schemes to improve the ability to model active water management in the National Water Model (NWM). For this purpose, we developed an integrated water management model, NWM-ResSim, by coupling the NWM with HEC-ResSim, and two reservoir representation schemes are tested: simulation of reservoir operations and retrieval of scheduled operations. The experiments focus on a pilot reservoir domain in the Russian River basin – Lake Mendocino, California – and its contributing watershed. The evaluation results suggest that the NWM-ResSim improves the simulation performance of reservoir outflow from this managed reservoir over the NWM default level pool routing scheme. The degree of this improvement depends on the suitability of the operation guidance; the reservoir operations simulation scheme could have acceptable errors for the purposes of water resources management, but not for flood operations. Results of the retrieval scheme of scheduled operations demonstrated better performance for sub-daily flood operations.

Space–time FEM with block-iterative algorithm for nonlinear dynamic fracture analysis of concrete gravity dam

In this paper, space–time finite element method is developed for the dynamic fracture analysis of dam–reservoir (DR) system which is supported on a perfectly rigid foundation. In this method, an auxiliary variable q representing the first-order time derivative of hydrodynamic pressure is treated as the primary unknown for the reservoir domain. Similarly, velocity v is the primary unknown for the solid domain. A three point Gauss–Lobatto quadrature rule is employed for computing time integral of those terms which contain stress term. Further, a partitioned method based on a block-iterative scheme is employed to incorporate the material nonlinearity of the concrete and to enforce the coupling between the two domains in a single iteration loop. A co-axially rotating crack model with exponential strain softening rule is employed to model the fracture of the concrete. Afterwards, numerical performance of the proposed scheme is demonstrated by analyzing two situations of dynamic fracture of Koyna concrete gravity dam.

A singularity removal method for coupled 1D\u20133D flow models

In reservoir simulations, the radius of a well is inevitably going to be small compared to the horizontal length scale of the reservoir. For this reason, wells are typically modelled as lower-dimensional sources. In this work, we consider a coupled 1D–3D flow model, in which the well is modelled as a line source in the reservoir domain and endowed with its own 1D flow equation. The flow between well and reservoir can then be modelled in a fully coupled manner by applying a linear filtration law. The line source induces a logarithmic-type singularity in the reservoir pressure that is difficult to resolve numerically. We present here a singularity removal method for the model equations, resulting in a reformulated coupled 1D–3D flow model in which all variables are smooth. The singularity removal is based on a solution splitting of the reservoir pressure, where it is decomposed into two terms: an explicitly given, lower-regularity term capturing the solution singularity and some smooth background pressure. The singularities can then be removed from the system by subtracting them from the governing equations. Finally, the coupled 1D–3D flow equations can be reformulated so they are given in terms of the well pressure and the background reservoir pressure. As these variables are both smooth (i.e. non-singular), the reformulated model has the advantage that it can be approximated using any standard numerical method. The reformulation itself resembles a Peaceman well correction performed at the continuous level.