History matching can play a key role in improving geological characterization and reducing the uncertainty of reservoir model predictions. Application of reservoir history matching is restricted by the huge computational cost by amongst others the many runs of the full model. Surrogate models with a reduced complexity are therefore used to reduce the computational demands. This paper presents an efficient surrogate-assisted deterministic inversion framework to primarily explore the possibility of applying deep neural network (DNN) surrogate to approximate the gradient of large-scale history matching by using auto-differentiation (AD). In combination with the deep neural network model, the AD enables us to evaluate the gradients efficiently in a parallel manner. Furthermore, the benefits of using stochastic gradient optimizers in the deep learning practice, instead of full gradient optimizers in conventional deterministic inversions, is investigated as well. Numerical experiments are conducted on a 3D benchmark reservoir model in the context of a water-flooding production scenario. The quantity of interest, e.g., dynamic saturation for an ensemble of test models, can be accurately predicted. The proposed surrogate-assisted inversion with stochastic gradient optimizer obtains a very quick convergence rate against the model and data noise for the high-dimensional history matching problem with a large number of data and parameters. In addition, we also conduct several comparisons and evaluations with our previously proposed projection-based subdomain POD-TPWL approach in terms of computational efficiency and accuracy. The subdomain POD-TPWL constructs a local surrogate model, which is repeatedly reconstructed a number of times for maintaining a satisfactory accuracy, while DNN constructs a global surrogate model based on the entire training data and generally does not require additional reconstructions. The subdomain POD-TPWL is very sensitive to how the domain is decomposed, increasing the training samples does not infinitely improve the history matching results by a fixed decomposition. Overall, these two kinds of surrogate models have demonstrated great potential in solving large-scale history matching problem. The DNN surrogate is particularly useful to generate multiple posteriors for model uncertainty quantification.

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We present an efficient adjoint model based on the deep-learning surrogate to address high-dimensional inverse modeling with an application to subsurface transport. The proposed method provides a completely code nonintrusive and computationally feasible way to approximate the model derivatives, which subsequently can be used to derive gradients for inverse modeling. This conceptual deep-learning framework, that is, an architecture of deep convolutional neural network through combining autoencoder and autoregressive structure, efficiently produces an analogously analytical adjoint with the help of auto-differentiation module in the popular deep-learning packages. We intentionally retain training data at the specific time instances where the measurements are taken, the storage of the intermediate states and computation of their adjoint, therefore, are completely avoided. This proposed adjoint state method is tested on a synthetic two-dimensional model for parameter estimations. The preliminary results reveal the feasibility of the proposed adjoint state method in term of computational efficiency and programming flexibility.

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In the community of petroleum engineering, the use of surrogate modelling techniques have recently gained more and more popularity to improve the efficiency of history matching. However, it is still not possible to fully utilize their potential in realistic applications. One of the challenges is to retain high accuracy while increasing the computational efficiency using a surrogate model. This dissertation proposed a projection-based reduced-order model and a data-driven deep convolutional neural network. In the first part of the thesis, a non-intrusive subdomain POD-TPWL method for solving gradient-based reservoir history matching problems is presented. It is a projection-based reduced-order modelling approach wherein the adjoint model of the original high-dimensional non-linear model is approximated by a subdomain reduced-order linear model. Furthermore, by introducing domain decomposition for the reduced-order model and by restricting the number of uncertain parameter patterns to the subdomains, the number of full order simulations required for the derivation of this surrogate model is reduced drastically. In the second part of the thesis, we propose two kinds of deep-learning inversion frameworks for efficiently solving large-scale history matching problems. The first deep-learning deterministic inversion framework primarily explores the possibility of applying a DNN surrogate to approximate the gradient of the objective function by making use of auto-differentiation (AD). In combination with the DNN surrogate, the AD enables us to evaluate the gradients efficiently in a parallel manner and without the need of explicitly coding of the adjoint model. The second framework is the deep-learning stochastic inversion which constructs a deep-learning surrogate based on an image-oriented distance parameterization for ensemble-based seismic history matching. Instead of directly assimilating spatially dense seismic data, image-oriented distance parameterization is employed to extract valuable information from the water fronts. Inspired by the methodologies developed for image segmentation in the field of computer vision and image processing, we propose an advanced image segmentation network for accurately predicting water fronts with highly-complex spatial discontinuities. In comparison with the conventional workflows entirely based on high-fidelity simulation models, experimental results show that the proposed surrogate-supported workflow achieves an accuracy equal to or better than the conventional workflow at significantly lower cost.@en

Imaging-type monitoring techniques are used in monitoring dynamic processes in many domains, including medicine, engineering, and geophysics. This paper aims to propose an efficient workflow for application of such data for the conditioning of simulation models. Such applications are very common in e.g. the geosciences, where large-scale simulation models and measured data are used to monitor the state of e.g. energy and water systems, predict their future behavior and optimize actions to achieve desired behavior of the system. In order to reduce the high computational cost and complexity of data assimilation workflows for high-dimensional parameter estimation, a residual-in-residual dense block extension of the U-Net convolutional network architecture is proposed, to predict time-evolving features in high-dimensional grids. The network is trained using high-fidelity model simulations. We present two examples of application of the trained network as a surrogate within an iterative ensemble-based workflow to estimate the static parameters of geological reservoirs based on binary-type image data, which represent fluid facies as obtained from time-lapse seismic surveys. The differences between binary images are parameterized in terms of distances between the fluid-facies boundaries, or fronts. We discuss the impact of the choice of network architecture, loss function, and number of training samples on the accuracy of results and on overall computational cost. From comparisons with conventional workflows based entirely on high-fidelity simulation models, we conclude that the proposed surrogate-supported hybrid workflow is able to deliver results with an accuracy equal to or better than the conventional workflow, and at significantly lower cost. Cost reductions are shown to increase with the number of samples of the uncertain parameter fields. The hybrid workflow is generic and should be applicable in addressing inverse problems in many geophysical applications as well as other engineering domains.

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In the process of the exploitation of deep oil and gas resources, shale wellbore stability control faces great challenges under complex temperature and pressure conditions. It is difficult to reflect the micro mechanism and process of the action of inorganic salt on shale hydration with the traditional experimental evaluation technology on the macro effect of restraining shale hydration. Aiming at the characteristics of clay minerals of deep shale, the molecular dynamics models of four typical cations (K⁺, NH₄⁺, Cs⁺ and Ca²⁺) inhibiting the hydration of clay minerals have been established by the use of the molecular dynamics simulation method. Moreover, the micro dynamics mechanism of typical inorganic cations inhibiting the hydration of clay minerals has been systematically evaluated, as has the law of cation hydration inhibition performance in response to temperature, pressure and ion type. The research indicates that the cations can promote the contraction of interlayer spacing, compress fluid intrusion channels, reduce the intrusion ability of water molecules, increase the negative charge balance ability and reduce the interlayer electrostatic repulsion force. With the increase in temperature, the inhibition of the cations on montmorillonite hydration is weakened, while the effect of pressure is opposite. Through the molecular dynamics simulation under different temperatures and pressures, we can systematically understand the microcosmic dynamics mechanism of restraining the hydration of clay in deep shale and provide theoretical guidance for the microcosmic control of clay hydration.

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A reduced order modeling algorithm for the estimation of space varying parameter patterns in numerical models is proposed. In this approach domain decomposition is applied to construct separate approximations to the numerical model in every subdomain. We introduce a new local parameterization that decouples the computational cost of the algorithm from the number of global principal components and therefore provides attractive scaling for models with a very large number of uncertain parameter patterns. By defining uncertain parameter patterns only in the various subdomains the number of full order simulation required for the derivation of the reduced order models can be reduced drastically. To avoid non-smoothness at the boundaries of the subdomains, the optimal local parameters patterns are projected onto global parameter patterns. The computational effort of the new methodology hardly increases when the number of parameter patterns increases. The number of training models depends primarily on the maximum number of local parameters in a subdomain, which can be decreased by refining the domain decomposition. We apply the new algorithm to a large-scale reservoir model parameter estimation problem. In this application 282 parameters could be estimated using only 90 full order model runs.

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Seismic history matching can play a key role in geological characterization and uncertainty quantification. However, challenges related to intensive computational demands and complexity restricts its application in many practical cases. This paper presents a conceptual deep-learning-based framework fully deployed in the popular Pytorch architecture to accelerate the seismic history matching. We introduce a surrogate model based on a deep convolutional neural network with a stack of dense blocks, specifically a conditional deep convolutional autoencoder-decoder architecture (cDCAE). The surrogate model allows us to fully deploy data assimilation algorithms in Pytorch architecture and hence to easily make full use of the efficient computing units, in particular GPU’s for the matrix-matrix and matrix-vector multiplications. The feature of built-in automatic differentiation (AD) provided by Pytorch also makes is possible to evaluate gradient information efficiently in a parallel manner. Furthermore, it has been acknowledged to benefit from the deep learning practice of using stochastic gradient (SG) optimizers, e.g., Adam, instead of full gradient optimizers, e.g., Quasi-Newton, as is most common in conventional big-data assimilation. The proposed framework is tested on a benchmark 3D model in the context of petroleum engineering. This surrogate model is demonstrated to be capable of accurately predicting the quantity of interest, e.g., dynamic saturation maps for new geological realizations. Assessments demonstrating high surrogate-model accuracy are presented for an ensemble of test models. The robustness and dramatic speedup provided by the surrogate model suggests that it can help facilitate the application of large-scale seismic history matching.@en

This paper introduces an efficient surrogate model with the aim of accelerating joint estimation of subsurface geological properties and relative permeability parameters for high-dimensional inversion problems. We fully replace the high-fidelity model with a set of subdomain linear models through integrating model linearization with smooth local parameterization where the Gaussian geological parameters and non-Gaussian facies indicators are locally parameterized. These subdomain linear models with smooth local parameterization, referred to as SLM-SLP, are constructed in each subdomain individually using only a few high-fidelity model simulations. An adaptive scheme, that is, weighting smooth local parameterization (WSLP), is introduced as well to mitigate the negative effects of inappropriate domain decomposition schemes by adaptively optimizing the domain decomposition strategy. The computational advantages of the proposed algorithm are demonstrated on a synthetic non-Gaussian facies model and a real-world high-dimensional Gaussian model. The amount of computational cost has been drastically reduced while reasonable accuracy remains. Specifically, SLM-SLP only needs 220 fidelity simulations to optimize 302 parameters. Compared to ensemble smoother with multiple data assimilation (ES-MDA), SLM-SLP effectively and efficiently mitigates the ensemble collapse problem in the course of uncertain quantification.@en

Shale gas resources (SGR), as a representative of natural gas hydrate reservoirs, have been the main energy supply for the energy consumption currently. The multi-scale pore structure of shale, complicated seepage mechanisms, including Knudsen diffusion, matrix deformation, stress sensitivity, non-Darcy flow and spatial fracture network stimulated by hydraulic fracturing technology have posed huge challenges to an accurate prediction and assessment of shale gas recovery. A full understanding of gas seepage mechanism of shale gas is the critical and scientific issue to develop carbon hydrogen energy resources effectively. It is very urgent to establish a comprehensive mathematical model to analyze the productivity capacity through simultaneously considering various flow mechanisms and fractures network system. To fill this gap, this paper presents a comprehensive numerical model of hydraulic fracturing horizontal well with discrete fracture network where embedded discrete fracture model (EDFM) is employed to characterize the coupled phenomenon between discrete fracture network and fractured SGR. And then two numerical discretization methods, e.g., finite difference and finite-volume, are used to numerically discretize the equations, subsequently, the Newton-Raphson iterative method is adopted to obtain the final solutions. Finally, the sensitivity analysis experiments are employed to investigate the effects of the key parameters. The results can provide some certain guidance for the optimization of stimulated treatment in natural gas hydrate reservoirs.

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Shale gas reservoirs (SGR) have been a central supply of carbon hydrogen energy consumption and hence widely produced with the assistance of advanced hydraulic fracturing technologies. On the one hand, due to the inherent ultralow permeability and porosity, there is stress sensitivity in the reservoirs generally. On the other hand, hydraulic fractures and the stimulated reservoir volume (SRV) generated by the massive hydraulic fracturing operation have contrast properties with the original reservoirs. These two phenomena pose huge challenges in SGR transient pressure analysis. Limited works have been done to take the stress sensitivity and spatially varying permeability of the SRV zone into consideration simultaneously. This paper first idealizes the SGR to be four linear composite regions. What is more, the SRV zone is further divided into subsections on the basis of nonuniform distribution of proppant within the SRV zone which easily yields spatially varying permeability away from the main hydraulic fracture. By means of perturbation transformation and Laplace transformation, an analytical multilinear flow model (MLFM) is obtained and validated as a comparison with the previous models. The flow regimes are identified, and the sensitivity analysis of critical parameters is conducted to further understand the transient pressure behaviors. The research results provided by this work are of significance for an effective recovery of SGR resources.

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Stress interference of multiplied fractures has significant influences on the propagation behavior of hydraulic fractures in roads, bridges, clay formations, and other forms of engineering. This paper establishes a crossing criterion and initiation angle model with comprehensive consideration of remote stress, stress intensity near the tip of fracture, and stress interference of multiplied fractures. Compared with the existing crossing criterion and initiation angle model, the ability to cross natural fractures decreases. Furthermore, the secondary initiation angle decreases with consideration of multiplied fracture propagation. The length of hydraulic fractures and natural fractures has little influence on the secondary initiation angle. With the increase in fracture space, the stress interference between fractures decreases, and as a result, the initiation angle begins to increase and then decrease. Differing from the propagation behavior of single fracture, the initiation angle basically does not vary with the increasing of net pressure under the high intersection angle between hydraulic fractures and natural fractures. Under a low intersection angle condition, the bigger the net pressure is, the smaller the initiation angle is. These results have great significance when analyzing the propagation behavior of multiplied fractures in real-world applications.

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Based on fractal theory (FT) and fractional calculus (FC), a new fractally fractional diffusion model (FFDM) of composite dual-porosity has been developed to evaluate performance of multiple fractured horizontal wells (MFHWs) with stimulated reservoir volume (SRV) in tight gas reservoirs (TGRs). More specifically, FT is used to characterize the complex and heterogeneous fracture network (FN) both inside and outside of SRV, while anomalous behavior of diffusion processes both inside and outside of SRV is quantified by applying the temporal fractional derivatives. The FFDM is then solved by the Laplace transformation, line source function, the numerical discrete method, and superposition principle. The transient pressure responses are then inversely converted from Laplace domain into real time domain with the Stehfest algorithm, and the FFDM is also validated, and type curves are generated as well. Flow stages are subsequently identified together with analysis on characteristics of the type curves, especially the anomalous features different with those generated from the conventional Euclidean model. Sensitivity analyses of some related parameters have also been discussed as well. And the FFDM is then also matched with the real field well-testing data of a MFHW with SRV in a TGR. The proposed FFDM provides a new understanding of the performance of MFHWs with SRV in TGRs, which can be used to interpret the field pressure data more accurately and appropriately.

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Five-spot well pattern (FSWP) scheme has shown appealing potentialities to enhance the recovery of coal-bed methane (CBM) from methane hydrocarbon reservoirs (MHR). In this paper, a new framework aimed at systematically investigating productivity performance of FSWP scheme with inter-well pressure interference (IWPI) is presented. First, mathematical models which are used to characterize the fluid flow within MHR and hydraulic fractures (HFs) are separately derived. Second, Laplace transformation and Stehfest numerical algorithm are utilized to couple those two flow systems and obtain the pressure-transient solutions of FSWP scheme. Finally, pressure characteristics are discerned and sensitivity analysis of key parameters is implemented as well. This semi-analytical approach outperforms numerical simulation from the point of computational efficiency. Several common flow regimes, e.g., linear and bi-linear flow regimes, are essentially deformed induced by IWPI. Several significant parameters, including gas rate, fracture half-length, and well spacing on the occurrence of IWPI are systematically analyzed. This work gains some new knowledge about the productivity performance of FSWP scheme with the existence of IWPI when extracting CBM from methane hydrate reservoirs (MHR), which provides energy engineers considerable instructions on optimizing the development of methane hydrate reservoirs.@en

Recently, a multiwell-pad-production (MWPP) scheme has been the center of attention as a promising technology to improve shale-gas (SG) recovery. However, the increasing possibility of multiwell pressure interference (MWPI) in the MWPP scheme severely distorts flow regimes, which strongly challenges the traditional pressure-transient analysis methods that focus on single multifractured horizontal wells (SMFHWs) without MWPI. Therefore, a methodology to identify pressure-transient response of the MWPP scheme with and without MWPI is urgent. To fill this gap, a new semianalytical pressure-transient model of the MWPP scheme is established by use of superposition theory, Gauss elimination, and the Stehfest numerical algorithm. Type curves are generated, and flow regimes are identified by considering MWPI. Finally, a sensitivity analysis is conducted. Our results show that there are good agreements between our proposed model and numerical simulation; moreover, our semianalytical approach also demonstrates a promising calculation speed compared with numerical simulation. Some expected flow regimes are significantly distorted by MWPI. In addition, well rate determines the distortion of pressure curves, whereas fracture length, well spacing, and fracture spacing determine when the MWPI occurs. The smaller the gas rate, the more severely flow regimes are distorted. As the well spacing increases, fracture length decreases, fracture spacing decreases, and the occurrence of MWPI occurs later. The stress-sensitivity coefficient has little to no influence on the occurrence of MWPI. Similar to the concept of the dual-porosity model, three new flow regimes—the single-well flow regime, MWPI flow regime, and MWPP flow regime—are artificially defined to systematically characterize the flow regimes of the MWPP scheme. This work offers some additional insights on pressure-transient response for the MWPP scheme in the SG reservoir, which can provide considerable guidance for fracture-properties estimation and well-pattern optimization for the MWPP scheme.@en

This paper presents a non-intrusive subdomain POD-TPWL (SD POD-TPWL) for reservoir history matching through integrating domain decomposition (DD), proper orthogonal decomposition (POD), radial basis function (RBF) interpolation, and the trajectory piecewise linearization (TPWL). It is an efficient approach for model reduction and linearization of general non-linear time-dependent dynamical systems without accessing to the legacy source code. In the subdomain POD-TPWL algorithm, firstly, a sequence of snapshots over the entire computational domain is saved and then partitioned into subdomains. From the local sequence of snapshots over each subdomain, a number of local basis vectors is formed using POD, and then the RBF interpolation is used to estimate the derivative matrices for each subdomain. Finally, those derivative matrices are substituted into a POD-TPWL algorithm to form a reduced-order linear model in each subdomain. This reduced-order linear model makes the implementation of the adjoint easy and results in an efficient adjoint-based parameter estimation procedure. Comparisons with the classic finite-difference-based history matching show that our proposed subdomain POD-TPWL approach is obtaining comparable results. The number of full-order model simulations required is roughly 2–3 times the number of uncertain parameters. Using different background parameter realizations, our approach efficiently generates an ensemble of calibrated models without additional full-order model simulations.@en