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We propose a novel approach based on Deep Learning methods using a convolutional neural networks (CNNs) to compare observed four-dimensional (4-D) seismic data with reservoir simulation model results and select the simulation models with the best match. Our approach advantages are twofold: 1) it is not a pixel-based approach, as it learns spatial features through training examples and 2) it does not require the simulation model and 4-D seismic maps to be in the same domain, enabling a direct comparison of maps in the seismic domain with maps in the numerical simulation domain. Additionally, we present a new methodology to quantitatively assess the performance of each method relative to the expected results. For this purpose, we build an annotated dataset containing the maps that best match the 4-D seismic in different realistic scenarios while taking into account the specialists' knowledge. We apply our methodology to evaluate if a method is able to perform according to what specialists expect, considering the annotated data. We also propose an improvement to traditional methods using a statistical method to convert maps of different physical properties into a common property, which can be used as an alternative to avoid forward and/or inversion procedures commonly used to bring 4-D seismic and simulation data to the same domain. Finally, we present an extensive comparison, qualitative and quantitative, of our approaches with other methods in the literature, such as the traditional pixelwise strategy for maps in the same domain, and methods based on binarization of maps in different domains. The results show that our proposed methods are capable of more accurately identifying the best-matched models, according to the specialists' answers.

Characterizing natural fractures has a decisive effect on production forecasts in fractured oil and gas reservoirs. Discrete Fracture Networks (DFN) constitutes the main modeling framework for fractured geosystems. However, myriads of uncertainties are enclosed prior modeling representative stochastic or deterministic DFN ensembles. This paper presents a novel methodology for DFN calibration and an efficacious field application, which incorporates Well-testing interpretation, Embedded Discrete Fracture Model (EDFM) framework, and numerical reservoir simulation. The proposed workflow starts with the DFN generation from seismic data, imaging logging data and core data. After multiple DFNs are modeled, well-test analysis is employed to calibrate the intrinsic properties of fractures at different locations. Then, these fracture networks are characterized dynamically by EDFM, which promotes capturing the optimal fracture model quickly screened. Finally, pressure and production history match are reached for the DFN realization that honors the optimal fracture model.

Accurate estimation of permeability is critical for oil and gas reservoir development and management, as it controls production rate. After assessing numerical techniques ranging from pore network modeling (PNM) to the lattice Boltzmann method (LBM), an AI-based workflow is developed for a quick and accurate estimation of the permeability of a complex carbonate rock from its X-ray micro-computed tomography (micro-CT) image. Following features engineering using both image processing and PNM, we trained and tested the workflow on thousands of segmented 3D micro-CT images using both shallow and deep learning algorithms to assess the permeability. A broad variety of supervised learning algorithms are implemented and tested, including linear regression, support vector regression, improved gradient boosting, and convolutional neural networks. Additionally, we explored a hybrid physics-driven neural network that takes into account both the X-ray micro-CT images and petrophysical properties. Finally, we found that the predicted permeability of a complex carbonate by machine learning (ML) agrees very well with that of a more computationally-intensive voxel-based direct simulation. In addition, the ML model developed here provides a substantial reduction in computation time by roughly three orders of magnitude compared to that of the LBM. This paper highlights the crucial role played by features engineering in predicting petrophysical properties by machine and deep learning. The proposed framework, integrating diverse learning algorithms, rock imaging, and modeling, has the potential to quickly and accurately estimate petrophysical properties to aid in reservoir simulation and characterization.

We have developed a machine-learning (ML) accelerated optimization method for challenging real-life well-location optimization (WLO) applications. The optimization protocol encompasses a stage at which the objective function is computed by a proxy ML model trained to accurately approximate the objective function that would be otherwise calculated using a computationally more intensive reservoir simulation run. As ML algorithms for proxy development, we have used two variants of the support-vector regression (SVR) technique, namely, e-SVR and ν-SVR. In the proposed proxy-accelerated optimization workflow, WLO starts with the conventional approach of using the reservoir simulator as the objective-function evaluator for a relatively small number of iterations of a global optimizer that conducts explorative search. This initial simulation investment is utilized not only to deliver optimization improvements but also to train an SVR proxy for the objective function of the investigated WLO problem. Subsequent to initial iterations, the SVR proxy is activated to serve as the objective-function evaluator for the global optimizer. Upon convergence, optimal model parameters estimated by the proxy-based optimization protocol are input to the flow simulator to access the full set of simulation results and also to re-evaluate the proxy accuracy. Optionally, the optimal model configuration can be supplied as initial guess to an efficient local optimizer to evaluate the further improvement potential for a few iterations using the reservoir simulator as the objective-function evaluator. The proposed SVR-accelerated optimization method is tested on a real field WLO problem. We have observed a 40 to 60% reduction in the overall computational cost with the SVR-accelerated protocol. It is important to note that the training time of the SVR proxy is negligibly short for small datasets (with 100s to 1000s data points) such as the ones generated over the course of WLO iterations. However, the main challenge associated with the proposed method is tuning the hyper-parameters of the SVR proxy for a given problem of interest and striking a good balance between accuracy and generalizability of proxy predictions. While some experimentation is still required with SVR hyper-parameters, especially in the first application, we accelerate this step through an auditable optimization process in our implementation. Results of our study demonstrate that the SVR-accelerated WLO method is applicable for real-life problems.

As the main way to reproduce flow response in subsurface reservoirs, the reservoir simulation could drastically assist in reducing the uncertainties in the geological characterization and in optimizing the field development strategies. However, the challenges in providing efficient and accurate solutions for complex field cases constrain further utilization of this technology. In this work, we develop a new reservoir simulation framework based on advanced spatial discretization and linearization schemes, the mimetic finite difference (MFD) and operator-based linearization (OBL), for fully implicit temporal discretization. The MFD has gained some popularity lately since it was developed to solve for unstructured grids and full tensor properties while mimicking the fundamental properties of the system (i.e. conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus). On the other hand, in the OBL the mass-based formulations are written in an operator form where the parametric space of the unknowns is treated in a piece-wise manner for the linearization process. Moreover, the values of these operators are usually precomputed into a nodal tabulation and with the implementation of multi-linear interpolation, the values of these operators and their derivatives during a simulation run can be determined in an efficient way for the Jacobian assembly at any time-step. This saves computational time during complex phase behavior computations. By first coupling these two schemes within a parallel framework, we can solve large and complex reservoir simulation problems in an efficient manner. Finally, we present a benchmark case that compares the numerical solutions to a Buckley-Leverett analytical solution to assure their accuracy and convergence. Moreover, we test three challenging field cases to demonstrate the performance of the advanced parallel framework for complex reservoir simulation.

This study presents a workflow to optimize the location of CO2 injectors in order to maximize stored volume and prevent fault reactivation due to increases of pore pressure. We combine coupled reservoir flow and geomechanics simulations with neural network surrogate models and find the best injector location based on net present environmental value (NPEV). The surrogate models can complement the numerical reservoir simulations and efficiently optimize the well location and injection rate via the steepest ascent algorithm. We apply the workflow to the Brugge field, a closed synthetic reservoir that incorporates multiple faults, as an example. The surrogate model predicts a NPEV of approximately 0.19 billion carbon credits for the Brugge field, which is consistent with the corresponding reservoir simulation. NPEV will change under different safety factors that serve as an additional injection control to enhance the safety of injection. This proposed workflow could be readily applied to other similar reservoir models.

The bubble point pressure (Pb) is a crucial pressure–volume–temperature (PVT) property and a primary input needed for performing many petroleum engineering calculations, such as reservoir simulation. The industrial practice of determining Pb is by direct measurement from PVT tests or prediction using empirical correlations. The main problems encountered with the published empirical correlations are their lack of accuracy and the noncomprehensive data set used to develop the model. In addition, most of the published correlations have not proven the relationships between the inputs and outputs as part of the validation process (i.e., no trend analysis was conducted). Nowadays, deep learning techniques such as long short-term memory (LSTM) networks have begun to replace the empirical correlations as they generate high accuracy. This study, therefore, presents a robust LSTM-based model for predicting Pb using a global data set of 760 collected data points from different fields worldwide to build the model. The developed model was then validated by applying trend analysis to ensure that the model follows the correct relationships between the inputs and outputs and performing statistical analysis after comparing the most published correlations. The robustness and accuracy of the model have been verified by performing various statistical analyses and using additional data that was not part of the data set used to develop the model. The trend analysis results have proven that the proposed LSTM-based model follows the correct relationships, indicating the model’s reliability. Furthermore, the statistical analysis results have shown that the lowest average absolute percent relative error (AAPRE) is 8.422% and the highest correlation coefficient is 0.99. These values are much better than those given by the most accurate models in the literature.

Time-lapse (4D) seismic analysis plays a vital role in reservoir management and reservoir simulation model updates. However, 4D seismic data are subject to interference and tuning effects. Being able to resolve and monitor thin reservoirs of different quality can aid in optimizing infill drilling or in locating bypassed hydrocarbons. Using 4D seismic data from the Maui field in the offshore Taranaki Basin of New Zealand, we generate typical seismic attributes sensitive to reservoir thickness and rock properties. We find that spectral instantaneous attributes extracted from time-lapse seismic data illuminate more detailed reservoir features compared with those same attributes computed on broadband seismic data. We have developed an unsupervised machine-learning workflow that enables us to combine eight spectral instantaneous seismic attributes into single classification volumes for the baseline and monitor surveys using self-organizing maps (SOMs). Changes in the SOM natural clusters between the baseline and monitor surveys suggest production-related changes that are caused primarily by water replacing gas as the reservoir is being swept under a strong water drive. The classification volumes also facilitate monitoring water saturation changes within thin reservoirs (ranging from very good to poor quality) as well as illuminating thin baffles. Thus, these SOM classification volumes indicate internal reservoir heterogeneity that can be incorporated into reservoir simulation models. Using meaningful SOM clusters, geobodies are generated for the baseline and monitor SOM classifications. The recoverable gas reserves for those geobodies are then computed and compared with production data. The SOM classifications of the Maui 4D seismic data seem to be sensitive to water saturation change and subtle pressure depletions due to gas production under a strong water drive.

We explore and develop a Proper Orthogonal Decomposition (POD)-based deflation method for the solution of ill-conditioned linear systems, appearing in simulations of two-phase flow through highly heterogeneous porous media. We accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method achieving speed-ups of factors up to five. The up-front extra computational cost of the proposed method depends on the number of deflation vectors. The POD-based deflation method is tested for a particular problem and linear solver; nevertheless, it can be applied to various transient problems, and combined with multiple solvers, e.g., Krylov subspace and multigrid methods.

In order to make full use of the information provided in the physical reservoirs, including the production history and environmental conditions, the whole life cycle of reservoir discovery and recovery should be considered when mapping in the virtual space. A new concept of reservoir digital twin and the exploratory multi-scale framework is proposed in this paper, covering a wide range of engineering processes related with the reservoirs, including the drainage, sorption and phase change in the reservoirs, as well as extended processes like injection, transportation and on-field processing. The mathematical tool package for constructing the numerical description in the digital space for various engineering processes in the physical space is equipped with certain advanced models and algorithms developed by ourselves. For a macroscopic flow problem, we can model it either in the Navier-Stokes scheme, suitable for the injection, transportation and oil-water separation processes, or in the Darcy scheme, suitable for the drainage and sorption processes. Lattice Boltzmann method can also be developed as a special discretization of the Navier-Stokes scheme, which is easy to be coupled with multiple distributions, for example, temperature field, and a rigorous Chapman-Enskog expansion is performed to show the equivalence between the lattice Bhatnagar-Gross-Krook formulation and the corresponding Navier-Stokes equations and other macroscopic models. Based on the mathematical toolpackage, for various practical applications in petroleum engineering related with reservoirs, we can always find the suitable numerical tools to construct a digital twin to simulate the operations, design the facilities and optimize the processes. Cited as: Zhang, T., Sun, S. An exploratory multi-scale framework to reservoir digital twin. Advances in Geo-Energy Research, 2021, 5(3): 239-251, doi: 10.46690/ager.2021.03.02

Fault modeling has become an integral element of reservoir simulation for structurally complex reservoirs. Modeling of faults in general has major implications for the simulation grid. In turn, the grid quality control is very important in order to attain accurate simulation results. We investigate the dynamic effects of using stair-step grid (SSG) and corner-point grid (CPG) approaches for fault modeling from the perspective of dynamic reservoir performance forecasting. We have performed a number of grid convergence and grid-type sensitivity studies for a variety of simple, yet intuitive faulted flow simulation problems with gradually increasing complexity. We have also explored the added value of the multipoint flux approximation (MPFA) method over the conventional two-point flux approximation (TPFA) to increase the accuracy of reservoir simulation results obtained on CPGs. Effects of fault seal modeling on grid-resolution convergence and grid-type sensitivity have also been briefly examined. For simple geometries, both SSG and CPG can be used for fault modeling with similar accuracy in conjunction with the pillar-grid approach. This is evidenced by the fact that simulation results from SSG and CPG converge to identical solutions. SSG and CPG yield different results for more complex geometries. Simulation results approach to a converged solution for relatively fine SSGs. However, a SSG only provides an approximation to the fault geometry and reservoir volumes when the grid is coarse. On the other hand, non-orthogonality errors are increasingly evident in relatively more complex faulted models on CPGs and such errors cannot be addressed by grid refinement. It has been observed that MPFA partially addresses the discretization errors on non-orthogonal grids but only from the total flux accuracy perspective. However, transport related errors are still evident. Grid convergence behaviors and grid effects are quite similar with or without fault seal modeling (i.e., dedicated fault-zone modeling by use of scaled-up seal factors) for simple geometries. However, in more complex test cases, we have observed that it is more difficult to achieve converged results in conjunction with fault seal modeling due to increased heterogeneity of the underlying problem.

A forecasting method of oil well production based on multivariate time series (MTS) and vector autoregressive (VAR) machine learning model for waterflooding reservoir is proposed, and an example application is carried out. This method first uses MTS analysis to optimize injection and production data on the basis of well pattern analysis. The oil production of different production wells and water injection of injection wells in the well group are regarded as mutually related time series. Then a VAR model is established to mine the linear relationship from MTS data and forecast the oil well production by model fitting. The analysis of history production data of waterflooding reservoirs shows that, compared with history matching results of numerical reservoir simulation, the production forecasting results from the machine learning model are more accurate, and uncertainty analysis can improve the safety of forecasting results. Furthermore, impulse response analysis can evaluate the oil production contribution of the injection well, which can provide theoretical guidance for adjustment of waterflooding development plan.

Reservoir simulation is critically important for optimally managing petroleum reservoirs. Often, many of the parameters of the model are unknown and cannot be measured directly. These parameters must then be inferred from production data at the wells. This is an inverse problem which can be formulated within a Bayesian framework to integrate prior knowledge with observational data. Markov Chain Monte Carlo (MCMC) methods are commonly used to solve Bayesian inverse problems by generating a set of samples which can be used to characterize the posterior distribution. In this work, we present a novel MCMC algorithm which uses a sequential transition kernel designed to exploit the redundancy which is often present in time series data from reservoirs. This method can be used to efficiently generate samples from the Bayesian posterior for time-dependent models. While this method is general and could be useful for many different models. We consider a Bayesian inverse problem in which we wish to infer fault transmissibilities from measurements of pressure at wells using a two-phase flow model. We demonstrate how the sequential MCMC algorithm presented here can be more efficient than a standard Metropolis-Hastings MCMC approach for this inverse problem. We use integrated autocorrelation times along with mean-squared jump distances to determine the performance of each method for the inverse problem.

To increase the efficiency of appraisal and development of hydrocarbon fields, the most effective technology is the integration of geological and field information into modern software systems for reservoir simulation, starting from the input of seismic data to field development planning based on a reservoir models. Building a three-dimensional geocellular models is an integral part of hydrocarbon reservoir management. After all, the reservoir model built to solve the main tasks of reservoir management such as: achieve the highest recovery and maximum economic effect. The modern level of software and the advances in computer processing power and graphics allows you to operate with big data, provides an opportunity for its systematization, statistical processing and detecting the basic laws between them. The availability of reliable data plays important role in the creation of a database for reservoir models. The input data for reservoir modelling of the fields, which were discovered in the 1960s, are the results of complex studies, characterized by insufficient accuracy and low quality. Lack of reliable and high-quality data brings significant uncertainty in all stages of modelling from volumetric estimation to the history match and forecasting. The necessity of using available geological and field information has led to the development of new methodological principles and approaches to reservoir modelling in conditions of limited initial information. According to the results of numerous studies, a significant number of methods have been developed, which are successfully used in the design of simulation models of hydrocarbon reservoirs of JSC “Ukrgazvydobuvannya”. Thanks to the use of new approaches to the building of 3D-models, high accuracy and reliability of the simulation results under such conditions is achieved.

Volume rendering techniques are essential to inspect volume data. In this work, we consider volume rendering for unstructured meshes. We focus on the visualization of black oil reservoir simulation models, represented by irregular hexahedral meshes with geometry distortions and discontinuities. We conduct an experimental study comparing three different strategies to compute the volume integration. The first strategy resamples the model in a regular grid; the second linearizes the model geometry and the variation of the scalar field within a hexahedron cell; the third uses an accurate evaluation of the scalar field at each sample. We compare performance and image quality delivered by each strategy by running a set of experiments with different models. We also investigate the gain in perception applying directional ambient occlusion shading effects. All strategies were entirely implemented on the graphics card.

Reservoir simulator is widely known in the petroleum industry for analysing and predicting the fluid flow behaviour through porous media. Conventional mathematical approach, which is the mostly used approach in reservoir simulation, assumes several unrealistic assumptions. Introducing the memory formalism with classical mathematical approach can eliminate this shortcoming. Because, the assumptions of fundamental laws (e.g., mass conservation, equation of state, and constitutive equation) used in reservoir simulation are reduced through incorporation of memory formalism. Memory-based flow model accounts the time varying nature of rock-fluid properties. To date, flow models do not yet exist where all the possible reservoir properties are considered as a function of time and space. The present study shows a comprehensive workflow in developing a memory-based reservoir simulator. The fundamentals of memory formalism and fractional calculus along with the use of fractional order derivative to formulate memory formalism are summarized. The proposed memory approach is validated using an application. A comparison of memory approach and multi-continuum approach is shown to depict the significance of the memory formulation for anomalous diffusion. This research is expected to open a door for introducing non-linear solvers in reservoir simulation study where cloud points solutions will be obtained instead of a single point solution. The outcome of this study will help engineers and researchers to develop more transparent simulator instead of creating a black box.

SummaryUsing fast and reliable proxies instead of sophisticated and time-consuming reservoir simulators is of great importance in reservoir management. The capacitance-resistance model (CRM) as a fast proxy has been widely used in this area. However, the inadequacy of this proxy for simplifying complex reservoirs with a limited number of parameters has not been addressed appropriately in related works in the literature. In this study, potential uncertainties in the modeling of the waterflooding process in the reservoir by the producer-based version of CRM (CRMP) are formulated, leading to embedding a new error-related term into the original formulation of the proxy. Considering a general form of the model error to represent both white and colored noises, a system of a CRMP-error equation is introduced analytically to deal with any type of intrinsic model imperfection. Two approaches are developed for the problem solution including the following: tuning the additional error-related parameters as a complementary stage of a classical history-matching procedure, and updating these parameters simultaneously with the original model parameters in a data-assimilation approach over model training time. To validate the model and show the effectiveness of both solution schemes, the injection and production data of a water-injection procedure in a three-layered reservoir model are used. Results show that the error-related parameters can be matched successfully along with the model original variables either in a routine model calibration procedure or in a data-assimilation approach by using the ensemble-based Kalman filter (EnKF) method. Comparing the average of the obtained range for the liquid rate as the problem output with true data demonstrates the effectiveness of considering model error. This leads to substantial improvement of the results compared with the case of applying the original model without considering the error term.

Summary When performing classic uncertainty reduction according to dynamic data, a large number of reservoir simulations need to be evaluated at high computational cost. As an alternative, we construct Bayesian emulators that mimic the dominant behavior of the reservoir simulator, and which are several orders of magnitude faster to evaluate. We combine these emulators within an iterative procedure that involves substantial but appropriate dimensional reduction of the output space (which represents the reservoir physical behavior, such as production data), enabling a more effective and efficient uncertainty reduction on the input space (representing uncertain reservoir parameters) than traditional methods, and with a more comprehensive understanding of the associated uncertainties. This study uses the emulation-based Bayesian history-matching (BHM) uncertainty analysis for the uncertainty reduction of complex models, which is designed to address problems with a high number of both input and output parameters. We detail how to efficiently choose sets of outputs that are suitable for emulation and that are highly informative to reduce the input-parameter space and investigate different classes of outputs and objective functions. We use output emulators and implausibility analysis iteratively to perform uncertainty reduction in the input-parameter space, and we discuss the strengths and weaknesses of certain popular classes of objective functions in this context. We demonstrate our approach through an application to a benchmark synthetic model (built using public data from a Brazilian offshore field) in an early stage of development using 4 years of historical data and four producers. This study investigates traditional simulation outputs (e.g., production data) and also novel classes of outputs, such as misfit indices and summaries of outputs. We show that despite there being a large number (2,136) of possible outputs, only very few (16) were sufficient to represent the available information; these informative outputs were used using fast and efficient emulators at each iteration (or wave) of the history match to perform the uncertainty-reduction procedure successfully. Using this small set of outputs, we were able to substantially reduce the input space by removing 99.8% of the original volume. We found that a small set of physically meaningful individual production outputs were the most informative at early waves, which once emulated, resulted in the highest uncertainty reduction in the input-parameter space, while more complex but popular objective functions that combine several outputs were only modestly useful at later waves. The latter point is because objective functions such as misfit indices have complex surfaces that can lead to low-quality emulators and hence result in noninformative outputs. We present an iterative emulator-based Bayesian uncertainty-reduction process in which all possible input-parameter configurations that lead to statistically acceptable matches between the simulated and observed data are identified. This methodology presents four central characteristics: incorporation of a powerful dimension reduction on the output space, resulting in significantly increased efficiency; effective reduction of the input space; computational efficiency, and provision of a better understanding of the complex geometry of the input and output spaces.

Recent experience in applying recurrent neural networks (RNNs) to interpreting Permannt Downhole Gauge (PDG) records has highlighted the potential utility of machine learning algorithms to learn reservoir behavior from data. The power of the RNN resides in its ability to retain information in a form of memory of previous patterns and information contained in the previous behavior of phenomena being modeled. This memory plays a role of informing the decision at the present time by using what happened in the past. This property suggests RNNs as a suitable choice to model sequences of reservoir information, even when the reservoir modeler is faced with incomplete knowledge of the underlying physical system. Convolutional Neural Networks (CNNs) are another variant of machine learning algorithms that have shown promise in sequence modeling domains, such as audio synthesis and machine translation. In this study, RNN and CNN were applied to tasks that traditionally would be modeled by a reservoir simulator. This was achieved by formulating the relationship between physical quantities of interest from subsurface reservoirs as a sequence mapping problem. In addition, the performance of a CNN layer as compared to RNN was evaluated systematically to investigate their capabilities in a variety of tasks of interest to the reservoir engineer. Preliminary results suggest that CNNs, with specific design modifications, are as capable as RNNs in modeling sequences of information and as reliable when making inferences to cases that has not been seen by the algorithm during training. Design details and reasons pertaining to the way these two seemingly different architectures process information, and handle memory are also discussed.

The optimization of oil field development scheme considering the uncertainty of reservoir model is a challenging and difficult problem in reservoir engineering design. The most common method used in this regard is to generate multiple models based on statistical analysis of uncertain reservoir parameters and requires a large number of simulations to efficiently handle all uncertainties, thus requiring a huge amount of computational power. In order to reduce the computational burden, a method which combines reservoir simulation, an economic model, polynomial chaos expansion with response surface methodology, and Levy flight particle swarm optimization (LFPSO) algorithm is proposed to determine the optimal injection-production parameters with reservoir uncertainties at a reasonable computational cost. This approach is applied to a five-spot well pattern optimization design for obtaining the optimal parameters, including oil-water well distance, injection rate, and bottom hole pressure, while considering the uncertainties of porosity, permeability, and relative permeability. The results of the case study indicated that the integrated approach is practical and efficient for performing reservoir optimization with uncertain reservoir parameters.

Bubble point pressure is of great significance in reservoir engineering calculations affecting the success of reservoir simulation. For determining this valuable parameter, experimental tests are the most reliable techniques; however, these measurements are costly and time-consuming. So, it is crucial to propose an empirical model for estimating bubble point pressure. The existing correlations mainly have large errors and develop based on restricted database from a specific geographical location. As a result, development of an all-inclusive correlation is essential. In current article, gene expression programming (GEP) was used to create a generalized model for bubble point pressure estimation. To do this, an all-inclusive source of data was utilized for training and testing the model from the petroleum industry. Several statistical approaches including both illustration tools and diverse error functions were utilized to show the supremacy of the developed GEP model. Consequently, the recommended model is the most accurate as compared to the similar correlations in literature with the average absolute relative error (AARE = 11.41%) and determination coefficient (R 2 = 0.96). Furthermore, the solution gas-oil ratio shows to be the most influencing variable on determining bubble point pressure according to sensitivity analysis. The results of contour map analysis demonstrate that most portions of the experimental region are predicted via the GEP equation with fewer errors as compared to two well-known literature correlations. Finally, the proposed GEP model can be of high prominence for accurate bubble point pressure estimation. Cited as : Rostami, A., Daneshi, A., Miri, R. Proposing a rigorous empirical model for estimating the bubble point pressure in heterogeneous carbonate reservoirs. Advances in Geo-Energy Research, 2020, 4(2): 126-134, doi: 10.26804/ager.2020.02.02

A method to manage geological uncertainty as part of an expensive simulation-based optimization process is presented. When the number of realizations representing the uncertainty is high, the computational cost to optimize the system can be considerable, and often prohibitively, as each forward evaluation is expensive to evaluate. To overcome this limitation, an iterative procedure is developed that selects a subset of realizations, based on a binary nonlinear optimization subproblem, to match the statistical properties of the target function at known sample points. This results in a reduced-order model that is optimized in place of the full system at a much lower computational cost. The result is validated over the ensemble of all realizations giving rise to one new sample point per iteration. The process repeats until the stipulated stopping conditions are met. Demonstration of the proposed method on a publicly available realistic reservoir model with 50 realizations shows that comparable results to full optimization can be obtained but far more efficiently.

Simulations and seismic inversions exhibit good performance in reservoir modeling task for the steady performance of conventional techniques. However, they still hardly meet the high demand of petroleum exploration since the impossibility of reaching high resolution both in vertical and lateral directions. Furthermore, simulations can only provide high-resolution results near loggings, while seismic inversions usually contain band-limited problems. Therefore, we present the modified generative adversarial nets with a decoder (DeGAN) as a novel approach, which is integrated with sequential simulation to realize super-resolution reservoir simulation. Specifically, the proposed method provides a geological model with high vertical resolution and optimized by the Zeoppritz function, introducing logging and seismic data simultaneously. After resampling and warping, DeGAN can be trained by these data sets and supplies a structure to generate high-frequency parts for reconstructing a super-resolution simulation of a subsurface profile. The proposed method presents the three-flow architecture of DeGAN to balance the contributions of three neural network models and utilizes this strategy in an offshore area successfully. By introducing multiple data sets, density experiments demonstrate that this approach can provide density profile with super-resolution for revealing possible thin layers, and the frequency distribution is in accord with loggings. The positive result verifies the effectiveness of this approach for providing a super-resolution simulation to supply a solution to the problem of the band-limited profile in seismic inversion.

Natural gas reservoir simulation, as a physics-based numerical method, needs to be carried out with a high level of precision. If not, it may be highly misleading and cause substantial losses, poor estimation of ultimate recovery factor, and wasted effort. Although simple simulations often provide acceptable approximations, there is a continued desire to develop more sophisticated simulation strategies and techniques. Given the capabilities of Machine Learning (ML) and their general acceptance in recent decades, this chapter considers the application of these techniques to gas reservoir simulations. The aspiration ML technics should be capable of providing some improvements in terms of both accuracy and speed. The simulation of gas reservoirs (dry gas, wet gas, and retrograde gas-condensate) is introduced along with its fundamental concepts and governing equations. More specific and advanced concepts of applying ML in modern reservoir simulation models are described and justified, particularly with respect to history matching and proxy models. Reservoir simulation assisted by machine learning is becoming increasingly applied to assess suitably of reservoirs for carbon capture and sequestration associated with enhanced gas recovery. Such applications, and the ability to improve reservoir performance via production efficiency, make ML-assisted reservoir simulation a valuable approach for improving the sustainability of natural gas reservoirs. The concepts are reinforced using a case study applying two ML models providing dew point pressure predictions for gas condensate reservoirs. Banner headline Reservoir simulation methods applied to gas reservoirs are reviewed and the key influencing variables identified. Machine Learning (ML) methods can be applied in various ways to improve the performance of gas reservoir simulations, especially in respect to history matching and proxy modeling. Additionally, ML can assist the CO2 sequestration and enhanced gas recovery, well placement optimization, production optimization, estimation of gas production, dew point prediction in gas condensate reservoirs, and pressure and rate transient analysis.

When coupling reservoir simulators to surface network solvers, an often used strategy is to perform a rule or priority-driven allocation based on individual well and group constraints, augmented by back-pressure constraints computed periodically by the network solver. The allocation algorithm uses an iteration that applies well-established heuristics in a sequential manner until all constraints are met. The rationale for this approach is simply to maximize performance and simulation throughput; one of its drawbacks is that the computed allocation may not be feasible with respect to the overall network balance, especially in cases where not all wells can be choked individually. In the work presented here, the authors integrate the well allocation process into the network flow solver, in the form of an optimization engine, to ensure that the solution conforms to the network rate and pressure balance equations. Results for three stand-alone test cases are discussed.

Objectives/Scope Methods for efficient probabilistic history matching and forecasting have been available for complex reservoir studies for nearly 20 years. These require a surprisingly small number of reservoir simulation runs (typically less than 200). Nowadays, the bottleneck for reservoir decision support is building and maintaining a reservoir simulation model. This paper describes an approach which does not require a reservoir simulation model, is data driven, and includes a physics model based on material balance. It can be useful where a full simulation model is not economically justified, or where rapid decisions need to be made. Methods, Procedures, Process Previous work has described the use of proxy models and Hamiltonian Markov Chain Monte Carlo to produce valid probabilistic forecasts. To generate a data driven model, we take historical measurements of rates and pressures at each well, and apply multi-variate time series to generate a set of differential-algebraic equations (DAE) which can be integrated over time using a fully implicit solver. We combine the time series models with material balance equations, including a simple PVT and Z factor model. The parameters are adjusted in a fully Bayesian manner to generate an ensemble of models and a probabilistic forecast. The use of a DAE distinguishes the approach from normal time-series analysis, where an ARIMA model or state space model is used, and is normally only reliable for short term forecasting. Results, Observations, Conclusions We apply these techniques to the Volve reservoir model, and obtain a good history match. Moreover, the effort to build a reservoir model has been removed. We demonstrate the feasibility of simple physics models, and open up the possibility of combinations of physics models and machine learning models, so that the most appropriate approach can be used depending on resources and reservoir complexity. We have bridged the gap between pure machine learning models and full reservoir simulation. Novel/Additive Information The approach to use multi-variate time series analysis to generate a set of ordinary differential equations is novel. The extension of previously described probabilistic forecasting to a generalised model has many possible applications within and outside the oil and gas industry, and is not restricted to reservoir simulation.

The mismatching between the multi-scale feature of complex fracture networks (CFNs) in unconventional reservoirs and their current numerical approaches is a conspicuous problem to be solved. In this paper, the CFNs are divided into hydraulic macro-fractures, induced fractures, and natural micro-fractures according to their mode of origin. A hybrid model coupling various numerical approaches is proposed to match the three-dimensional multi-scale fracture networks. The macro-fractures with high-conductivity and wide-aperture are explicitly characterized by a mimetic Green element method-based hierarchical fracture model. The induced fractures and natural micro-fractures that have features of low-conductivity and small-openings are upscaled to the dual-medium grid and enhanced matrix grid through the equivalent continuum-medium method, respectively. Subsequently, some benchmark cases are implemented to confirm the high-precision and high-robustness of the proposed hybrid model that indeed accomplishes accurate modeling of fluid flow in multi-scale CFNs by comparing with commercial software tNavigator®. Furthermore, an integrated workflow of simulation modeling for multiscale CFNs combined with a field example in Sichuan from China is used to analyzing the production information of fractured horizontal wells in shale gas reservoirs. Compared with the field production data from this typical well, it can be proved that the hybrid model has strong reliability and practicability.

A compositional reservoir simulation model with ten-million grids is successfully computed using parallel processing techniques. The load balance optimization principle for parallel calculation is developed, which improves the calculation speed and accuracy, and provides a reliable basis for the design of reservoir development plan. Taking M reservoir as an example, the parallel numerical simulation study of compositional model with ten million grids is carried out. When the number of computational nodes increases, message passing processes and data exchange take much time, the proportion time of solving equation is reduced. When the CPU number increases, the creation of Jacobian matrix process has the higher acceleration ratio, and the acceleration ratio of I/O process become lower. Therefore, the I/O process is the key to improve the acceleration ratio. Finally, we study the use of GPU and CPU parallel acceleration technology to increase the calculation speed. The results show that the technology is 2.4 ∼ 5.4 times faster than CPU parallel technology. The more grids there are, the better GPU acceleration effect it has. The technology of parallel numerical simulation for compositional model with ten-million grids presented in this paper has provided the foundation for fine simulation of complex reservoirs. Cited as : Lian, P., Ji, B., Duan, T., Zhao, H., Shang, X. Parallel numerical simulation for a super large-scale compositional reservoir. Advances in Geo-Energy Research, 2019, 3(4): 381-386, doi: 10.26804/ager.2019.04.05

The numerical simulation of multicomponent compressible flow in porous media is an important research topic in reservoir modeling. Traditional semi-implicit methods for such problems are usually conditionally stable, suffer from large splitting errors, and may accompany with violations of the boundedness requirement of the numerical solution. In this study we reformulate the original multicomponent equations into a nonlinear complementarity problem and discretize it using a fully implicit finite element method. We solve the resultant nonsmooth nonlinear system of equations arising at each time step by a parallel, scalable, and nonlinearly preconditioned semismooth Newton algorithm, which is able to preserve the boundedness of the solution and meanwhile treats the possibly imbalanced nonlinearity of the system. Some numerical results are presented to demonstrate the robustness and efficiency of the proposed algorithm on the Tianhe-2 supercomputer for both standard benchmarks as well as realistic problems in highly heterogeneous media.

It is common in ensemble-based methods of history matching to evaluate the adequacy of the initial ensemble of models through visual comparison between actual observations and data predictions prior to data assimilation. If the model is appropriate, then the observed data should look plausible when compared to the distribution of realizations of simulated data. The principle of data coverage alone is, however, not an effective method for model criticism, as coverage can often be obtained by increasing the variability in a single model parameter. In this paper, we propose a methodology for determining the suitability of a model before data assimilation, particularly aimed for real cases with large numbers of model parameters, large amounts of data, and correlated observation errors. This model diagnostic is based on an approximation of the Mahalanobis distance between the observations and the ensemble of predictions in high-dimensional spaces. We applied our methodology to two different examples: a Gaussian example which shows that our shrinkage estimate of the covariance matrix is a better discriminator of outliers than the pseudo-inverse and a diagonal approximation of this matrix; and an example using data from the Norne field. In this second test, we used actual production, repeat formation tester, and inverted seismic data to evaluate the suitability of the initial reservoir simulation model and seismic model. Despite the good data coverage, our model diagnostic suggested that model improvement was necessary. After modifying the model, it was validated against the observations and is now ready for history matching to production and seismic data. This shows that the proposed methodology for the evaluation of the adequacy of the model is suitable for large realistic problems.

During the solution of fully implicit reservoir simulation time steps, it is often observed that the computed Newton updates may be very sparse, considering computer precision. This sparsity can be as high as 95% and can vary largely from one iteration to the next. In recent works, a mathematically sound framework was developed to predict the sparsity pattern before the full linear system is solved. The theory is restricted to general, scalar nonlinear advection-diffusion-reaction problems in multidimensional and heterogeneous settings. This theory had been applied to reduce the size of the linear systems that were computed during sequential implicit time steps for two-phase flow. The results confirmed that the linearization computations and the linear solution processes may be localized by as much as 95% while retaining the exact Newton convergence behavior and final solution. Inspired by the great success of that methodology, this work develops algorithmic extensions in order to devise localization algorithms for fully implicit coupled multicomponent problems. We propose, apply, and test a novel algorithm to resolve a system of hyperbolic equations obtained from an Equation of State–based compositional simulator. When applied to various fully implicit flow and multicomponent transport problems, involving six thermodynamic species, on the SPE 10 geological model, the observed reduction in computational effort ranges between six and 49-fold depending on the level of locality present in the system. We apply this algorithm to several injection and depletion scenarios with and without gravity and capillarity in order to investigate the adaptivity and robustness of the proposed method to the underlying heterogeneity and complexity. We demonstrate that the algorithm enables efficient and robust full-resolution fully implicit simulation without the errors introduced by adaptive discretization methods or the stability concerns of semi-implicit approaches.

The Multiscale Robin Coupled Method (MRCM) is a recent multiscale numerical method based on a non-overlapping domain decomposition procedure. One of its hallmarks is that the MRCM allows for the independent definition of interface spaces for pressure and flux over the skeleton of the decomposition. The accuracy of the MRCM depends on the choice of these interface spaces, as well as on an algorithmic parameter β in the Robin interface conditions imposed at the subdomain boundaries. This work presents an extensive numerical assessment of the MRCM in both of these aspects. Two types of interface spaces are implemented: usual piecewise polynomial spaces and informed spaces, the latter obtained from sets of snapshots by dimensionality reduction. Different distributions of the unknowns between pressure and flux are explored. Two non-dimensionalizations of β are tested. The assessment is conducted on realistic, high contrast, channelized permeability fields from a SPE benchmark database. The results show that β , suitably non-dimensionalized, can be fixed to unity to avoid any indeterminacy in the method. Further, with both types of spaces it is observed that a balanced distribution of the interface unknowns between pressure and flux renders the MRCM quite attractive both in accuracy and in computational cost, competitive with other multiscale methods from the literature

Multiphase flow simulation in fractured reservoirs at field scale is a significant challenge. Despite recent advances and a wide range of applications in both hydrology and hydrocarbon reservoir engineering, discussing efficient methods that cover computational complexity, accuracy, and flexibility aspects is still of paramount importance for a better understanding of these complex media. In this work, we present a new method that handles both the topological and computational complexities of fractures, taking into consideration advantages of various existing approaches, which include hybrid-grid and crossflow equilibrium models. The hybrid-grid (HG) model consists of representing fractures as lower-dimensional objects that still are represented as control volumes in a computational grid. The HG model is equivalent to a single-porosity model with a practical solution for the small control volumes at the intersection between fractures; however, the overall simulation run time is still dominated by the remaining fracture small control volumes. To overcome single-porosity computational challenges, a crossflow equilibrium (CFE) concept between discrete fractures and a small neighborhood in the matrix blocks can be employed. The CFE model consists of combining fractures with a small fraction of the neighborhood matrix blocks on either side in larger elements to achieve a better computational efficiency than conventional single-porosity models. The implementation of a CFE model at field scale is not practical because of the fracture topological challenges associated with the construction of an accurate computational grid for the CFE elements. In this work, we propose a method based on a combination of HG and CFE models to overcome the challenges associated with the HG fracture small control volumes and field-scale CFE computational grid construction. First, we assess the performance of the existing CFE model, and we propose an improved model. In addition, we suggest an input data handling method that is sufficient to account for fractional flow inside the CFE elements for flow in homogeneous fractured reservoirs without the need of any change in the simulator. Second, we describe the uniqueness of the proposed method, and we discuss different numerical examples to assess both the accuracy and computational efficiency. The results obtained are very accurate, and, computationally, one to two orders of magnitude speedup can be achieved. The improved CFE results are superior over the traditional CFE model. Combined with the HG model, the results are significantly improved while retaining a very good performance.

Traditionally, production optimization is performed based on repeatedly running a full-scale reservoir simulation model, which is quite time-consuming. Recently, we developed a computationally efficient data-driven model, which is referred to as interwell-numerical-simulation model (INSIM), for history matching, characterizing interwell connectivity, monitoring, and optimization for a waterflood. In this research, we develop a new workflow for fast optimization of waterflooding performance which only requires one single INSIM forward run. Specifically, the optimized well injection rate and production rate are determined with the help of the new INSIM-derived injection efficiency and the existing injector-centered allocation factor. The basic idea of the proposed optimization workflow is to recover more oil by injecting more water to those injector-producer pairs that have more remaining oil left; whereas decrease water injection for those connections without much oil left. The major criterion to tune the injection rate is based on the injection efficiency calculated with INSIM, a value to measure the effectiveness of injected water from an injector. The larger the injection efficiency is, the more oil can be produced by injecting a unit amount of water. The advantage of this method is that the well controls are optimized based on the reservoir-engineering point of view which is easy to understand and visualize. Also, the new workflow is very easy to implement and is much more computationally efficient than the traditional optimization workflow. We test the proposed workflow with two examples including a large-scale field example. The computational results indicate that, by using the proposed workflow, the field oil production is increased with less producing water.

In reservoir engineering, history matching is the technique of conditioning a reservoir simulation model to the available production data. The reservoir properties have uncertainties which lead to discrepancies between the observed data and the reservoir simulator response, making history matching an indispensable tool in the petroleum industry. The standard Ensemble Smoother with Multiple Data Assimilation (ES-MDA) application has become popular in history matching problems. However, in the standard methodology, the number of iterations must be previously defined by the user, what makes it a determinant parameter in the ES-MDA results. One way to solve this problem is to perform adaptive algorithms: these algorithms keep iterating until they reach desirable matchings with the real data. Furthermore, to apply this method in large-scale reservoirs, it is necessary to use some localization technique to prevent spurious updates and high uncertainty reduction after the ES-MDA is applied. This work evaluates the influence of the distance-based Kalman Gain localization in an adaptive Ensemble Smoother, by applying the mentioned methodology in a large-scale synthetic reservoir model. The experiments showed a connection between the localization parameters and the number of iterations required by the adaptive algorithm. Moreover, the results presented significant reduction in the production data mismatch, regardless the localization, being the mismatch of the prior and posterior models an important parameter to determine the history matching quality.

New sequential fully implicit (SFI) methods for compositional flow simulation have been recently investigated. These SFI schemes decouple the fully coupled problem into separate pressure and transport problems and have convergence properties comparable with those of the fully implicit (FI) method. The pressure system is a parabolic problem with fixed overall compositions and the transport system is a hyperbolic problem with fixed pressure and total velocity. We discuss some aspects of how to design optimal SFI schemes for compositional flow with general equation-of-states by localizing the computations. The different systems are solved sequentially and the fully implicit solution is recovered by controlling the a posteriori splitting errors due to the choice of decoupling. When the parabolic and the hyperbolic operators are separated, it is possible to design nonlinear domain decomposition schemes taking the advantage of the specific properties of each operator. Usually, for reservoir simulation models, most of the reservoir is converged with SFI methods in one outer iteration. However, in some localized regions with strong coupling between the pressure and the compositions, the SFI algorithms may need several outer iterations. Here, we propose a domain decomposition method based on a predictor-corrector strategy. As a first step, the nonlinear parabolic pressure equation is solved on the whole domain with the Multiscale Restriction-Smooth Basis (MsRSB) method used as a linear domain decomposition solver. In a second step, the compositions system is solved. At the end of this first outer iteration, most of the reservoir is converged. Based on a posteriori splitting errors of the SFI scheme in volume and velocity, we define local regions where additional global outer iterations would be required in the conventional SFI scheme. We then fix Dirichlet boundary conditions for the pressure and the compositions and solve local problems in these non converged regions. After convergence of these smaller nonlinear problems, if the boundary conditions are changed by the updated regions, the global pressure problem is revisited. An additional post-processing of local transport iterations makes sure mass is conserved everywhere. The resulting algorithm converges to the same solution as the FI solver, with all simultaneous updates to composition and pressure in localized regions. We demonstrate the robustness of this nonlinear domain decomposition algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.

Well placement optimization is a decisive task for the reliable design of field development plans. The use of optimization routines coupled to reservoir simulation models as an automatic tool is a popular practice, which could improve the decision-making process on well placement problems. However, despite the various automatic techniques developed, there is still a lack of robust computer-added optimization tool, which can solve the well placement problem with high accuracy in reasonable time while handling the technical constraints properly. In this paper, a hybrid intelligent system is proposed to deal with a real well placement problem with arbitrary well trajectories, complex model grids, and linear and nonlinear constraints. In this intelligent approach, a Genetic Algorithm (GA) combined with a hybrid constraint-handling strategy is applied in conjunction with a constrained space-filling sampling design, Gaussian Process (GP) surrogate model, and one proposed adaptive sampling routine. This self-adaptive framework allows to consecutively augment the quality of surrogate, enhance the accuracy of the process, and thus guide the optimization rapidly into the optimal solution. To demonstrate the efficiency of the developed method, a full-field reservoir case is considered. This case covers a real well placement project in a fractured unconventional reservoir of El Gassi, which is a mature field located in Hassi-Massoud, Algeria. The obtained results highlighted the effectiveness of the proposed approach for solving the real well placement problem with high accuracy in reasonable CPU-time. These auspicious features make it a reliable tool to be used on other real optimization projects.

The objective of the conventional well testing technique is to evaluate well- reservoir interaction through determining the flow capacity and well potential on a short-term basis by relying on the transient pressure response methodology. The well testing analysis is a major input to the reservoir simulation model to validate the near wellbore characteristics and update the variables that are normally function of time such as skin, permeability and productivity multipliers. Well test analysis models are normally built on analytical approaches with fundamental physical of homogenous media with line source solution. Many developments in the last decade were made to increase the resolution of transient response derivation to meet the complexity of well and flow media. Semi-analytical modeling for the pressure transient response in complex well architecture and complex reservoirs were adopted in this research. The semi analytical solution was based on coupling the boundary condition of source function to the well segment. Coupling well-reservoir on sliced based technique was used to re-produce homogenous isotropic media from several source functions of different properties. The approach can model different well geometries penetrated complex reservoirs. A computer package was prepared to model the pressure transient response of horizontal, dual-lateral, multi-lateral wells in complex anisotropic reservoirs, multilayered, compartmentalized, system of various boundary conditions such as: bottom support aquifers, edge supported, gas caps, interference of injection. The validity of the proposed model was successfully checked by using the commercial simulator.

Phase equilibrium calculation, also known as flash calculation, has been extensively applied in petroleum engineering, not only as a standalone application for a separation process but also as an integral component of compositional reservoir simulation. Previous research devoted numerous efforts to improve the accuracy of phase equilibrium calculations, which place more importance on safety than speed. However, the equation-of-state-based flash calculation consumes an enormous amount of computational time in compositional simulation and thus becomes a bottleneck to the broad application of compositional simulators. Therefore, it is of vital importance to accelerate flash calculation without much compromise in accuracy and reliability, turning it into an active research topic in the past two decades. With the rapid development of computational techniques, machine learning brings another wave of technology innovation. As a subfield of machine learning, the deep neural network becomes a promising computation...

Summary The interplay of multiphase-flow effects and pressure/volume/temperature behavior encountered in reservoir simulations often provides strongly coupled nonlinear systems that are challenging to solve numerically. In a sequentially implicit method, many of the essential nonlinearities are associated with the transport equation, and convergence failure for the Newton solver is often caused by steps that pass inflection points and discontinuities in the fractional-flow functions. The industry-standard approach is to heuristically chop timesteps and/or dampen updates suggested by the Newton solver if these exceed a predefined limit. Alternatively, one can use trust regions (TRs) to determine safe updates that stay within regions that have the same curvature for numerical flux. This approach has previously been shown to give unconditional convergence for polymer- and waterflooding problems, also when property curves have kinks or near-discontinuous behavior. Although unconditionally convergent, this method tends to be overly restrictive. Herein, we show how the detection of oscillations in the Newton updates can be used to adaptively switch on and off TRs, resulting in a less-restrictive method better suited for realistic reservoir simulations. We demonstrate the performance of the method for a series of challenging test cases ranging from conceptual 2D setups to realistic (and publicly available) geomodels such as the Norne Field and the recent Olympus model from the Integrated Systems Approach for Petroleum Production (ISAPP) optimization challenge.

Corner-point is an industry-standard type grid for application in reservoir simulation. The flexible gridding in corner-point grids provides several advantages over Cartesian grids for the representation of complex geological features. In this work, an embedded discrete fracture model (EDFM) is extended to corner-point grids with a full-permeability-tensor formulation to simulate complex fractures in this type of grids. The developed model is implemented in an IMPEC, compositional reservoir simulator. We first describe the formulations of the full-permeability-tensor implementation together with the modified governing equations for EDFM simulations. Then, we present methodologies for computing matrix-fracture intersections and fracture-fracture connections in the EDFM considering the various block geometries in corner-point grids. Subsequently, case studies are presented to verify the developed model, where the impacts of grid distortion and cross derivatives on simulation results are discussed. Three-dimensional case studies are also shown to illustrate the influence of natural fractures on secondary recovery. The results of this study demonstrate the reliability of the developed model, and they also show the compatibility of the EDFM with different types of numerical solution schemes in existing simulators.

Field development optimisation is a critical task in the modern reservoir management processes. The optimum setting provides the best exploitation strategy and financial returns. However, finding such a setting is difficult due to the non-linearity between the reservoir response and the development strategy parameters. Therefore, growing attention is being paid to computer-assisted optimisation algorithms, due to their capabilities in handling optimisation problems with such complexities. In this paper, the performance of ( μ + λ ) Evolution Strategy (ES) Algorithm is compared to Genetic Algorithm (GA), Particle Swarm Optimisation (PSO) and ( μ , λ ) Covariance Matrix Adaptation Evolution Strategy (CMA-ES) using five different optimisation cases. The 1st and 2nd cases are well placement and trajectory optimisation, respectively, which have rough objective function surfaces and a small number of dimensions. The 3rd Case is well control optimisation with a small number of dimensions, while the 4th case is a high-dimensional control optimisation. Lastly, the 5th case is joint optimisation that includes the number of wells, type, trajectory, and control, which has a high dimensional rugged surface. The results show that the use of ES as the optimisation algorithm delivers promising results in all cases, except case 3. It converged to a higher NPV compared to the other algorithms with the same computational budget. The obtained solutions also outperformed the ones delivered by reservoir engineering judgments.

SummaryAssisted history matching (AHM) using design of experiments (DOE) is one of the most commonly applied history-matching techniques in the oil and gas industry. When applied properly, this stochastic method finds a representative ensemble of history-matched reservoir models for probabilistic uncertainty analysis of production forecasts. Although DOE-based AHM is straightforward in concept, it can be misused in practice because the work flow involves many statistical and modeling principles that should be followed rigorously.In this paper, the entire DOE-based AHM work flow is demonstrated in a coherent and comprehensive case study that is divided into seven key stages: problem framing, sensitivity analysis, proxy building, Monte Carlo simulation, history-match filtering, production forecasting, and representative model selection. The best practices of each stage are summarized to help reservoir-management engineers understand and apply this powerful work flow for reliable history matching and probabilistic production forecasting.One major difficulty in any history-matching method is to define the history-match tolerance, which reflects the engineer's comfort level of calling a reservoir model “history matched” even though the difference between simulated and observed production data is not zero. It is a compromise to the intrinsic and unavoidable imperfectness of reservoir-model construction, data measurement, and proxy creation. A practical procedure is provided to help engineers define the history-match tolerance considering the model, data-measurement, and proxy errors.

When localization methods used in reservoir history matching were introduced from outside the oil and gas industry, the differences between data assimilation for other industries and reservoir history matching were not considered. However, these differences can be extremely important when considering the validity of these methods. This problem is discussed within the context of the roles of covariance localization in detail in this paper. This paper starts with discussion of relations between two significant problems, ill-posedness and spurious correlations, and their solution methods. This analysis leads to a conclusion about two appropriate roles of the covariance localization method. This conclusion is used to validate the cross-covariance localization methods used in reservoir simulation. In addition, for the cross-covariance localization in reservoir history matching, there are many different processes, especially different correlation functions, proposed by different authors. This makes it difficult to choose a proper method when solving history matching or data assimilation problems. The performance of different cross-covariance localization correlation functions is also analyzed theoretically and experimentally in this paper.

For situations where reservoir management based on developing and running a full-scale reservoir simulation model is not plausible, data-driven models may provide an attractive alternative. Recently, we developed a simple data-driven model, INSIM-FT for reservoir flow and transport as a replacement for a reservoir simulator. However, INSIM-FT only considers two-dimensional flow in a reservoir that contains only vertical wells. In this work, we develop a new data-driven model that extends the interwell numerical simulation model with front-tracking (INSIM-FT) from single-layer reservoirs to full three-dimensional (3D) multi-layer reservoirs. The new model, which is referred to as INSIM-FT-3D, can be used for history matching and reservoir performance predictions for a three-dimensional reservoir under waterflooding. The novelty of the new approach includes: (1) INSIM-FT-3D replaces the original Riemann solver in INSIM-FT by a new Riemann solver based on a convex-hull method that enables the solution of the Buckley–Leverett problem with gravity; (2) unlike the original INSIM-FT model, which assumes all wells are vertical, the INSIM-FT-3D model allows for the inclusion of wells with arbitrary trajectories with multiple perforations; (3) INSIM-FT-3D applies Mitchell's best-candidate algorithm to automatically generate the imaginary wells that are evenly distributed in the reservoir given a set of prefixed actual well nodes and (4) INSIM-FT-3D utilizes our own modification of Delaunay triangulation to build the 3D connection map necessary to use the general INSIM-FT-3D formulation. History matching of the INSIM-FT-3D model parameters is performed using the ensemble-smoother with multiple data assimilations. The model parameters for history matching include the connection-based parameters, the parameters defining the power-law relative permeabilities and the parameters defining the well indices for the perforated zones of a well with multiple perforated well segments. The reliability of the analytical water-saturation solution obtained with INSIM-FT-3D is validated by comparison with corresponding results from an Eclipse [47] model with very fine grids and small time steps. After illustrating the utility and reliability of the method for a three-dimensional synthetic problem, the method is shown to give reasonable history-matching and prediction results for a field example. At last, the proposed method is tested for Brugge field, a field-scale synthetic reservoir simulation model.

The selection of an optimal model from a set of multiple realizations for dynamic reservoir modelling and production forecasts has been a persistent issue for reservoir modelers and decision makers. Current evidence has shown that many presumably good reservoir models which originally matched the true historic data did not always perform well in predicting the future of the reservoir as a result of uncertainties. In this paper, a new method for optimal model selection using Signal Detection Principle (SDP) approach is presented. In principle, SDP approach models the dissimilarity between various realizations as a cross-function of spatial distance, statistical correlation and the inherent noise level in the model; while the existing methods estimate the dissimilarity between parameter values of different realizations as a function of distance (statistical or spatial distance) or quality factor. SDP approach fills the model divergence gap by way of classifying the information-bearing patterns in a model as signals, and identifying random patterns which bears no information as noise. Thus, it quantifies the mismatch uncertainty and the inherent fudge in every realization as a measure of the model's reliability. The SDP approach has been validated with historical data, and the results show that the reliability factor of the best model has a value of one. Thus, if the strongest reservoir model has a reliability factor of one, then that model will make approximately the closest and best predictions as the true system in every forecast; whereas if the reliability factor of the model is less than or greater than one, its predictions will always disperse from historical data and is unsuitable for reservoir forecasts. Two examples based on real field data are used to demonstrate the application of SDP approach. In these case examples SDP was used to analyse the difference between the reservoir models in a set of multiple realizations. The target variable in this study was STOIIP (Stock Tank Oil Initially in Place). The results show that the models with a lower reliability factor made similar predictions as the real reservoir data in almost all the forecasts. Similarly, the models with higher reliability factors made dissimilar and more unreliable predictions in most of the forecasts.

The large computational requirements associated with subsurface flow simulation limit the use of realistic models for demanding applications such as optimization and uncertainty quantification. This has motivated the development of reduced-order models, in particular those based on proper orthogonal decomposition (POD). In this study, we implement a POD-based Gauss–Newton with approximated tensors (GNAT) method for oil–water reservoir simulation. Our formulation, which is described in detail, incorporates promising features from previous implementations involving GNAT and discrete empirical interpolation method (DEIM) procedures. A theoretical analysis of the computational cost of GNAT for our problem, and the speedup it may provide relative to the full-order simulation, suggests a complex dependency of speedup on GNAT parameters and solver implementation. Systematic assessments, involving several comprehensive numerical experiments, are then presented. These include a detailed evaluation of GNAT performance on a 2D model, in which 576 different GNAT parameter combinations are considered, which allows us to elucidate the impact of parameter values on error. Detailed comparisons with an existing reduced-order modeling procedure based on trajectory piecewise linearization, POD-TPWL, are also presented. Around 1500 test cases, with varying levels of perturbation relative to training cases, are considered. We show that GNAT is only slightly more accurate than POD-TPWL for cases with small perturbation, but its accuracy advantage increases for larger perturbations. We also demonstrate the successful application of GNAT to a more realistic 3D model.

One of the most crucial tasks in hydrocarbon field development, it is an optimal wells placement. Simulation-based well placement optimization can be intricate to implement or computationally expensive. In this paper, a novel approach for the reduction of computational challenges by combining population-base optimization algorithm with self-adapt reservoir clusterization method to determine initial well placement position for the similar producing zones is presented. The developed methodology supported initial well placement location and allowed to shrink search space area. As a result, the computational time was decreased by 18% for particle swarm optimization and by 21% for genetic algorithm. Additionally, the project profitability was increased by 0.24% in case of genetic algorithm utilization and 0.59% for particle swarm optimization, what confirms the necessity for precise well placement localization. The average convergence factor for the employed algorithms increased from 18.43% to 60.78% for genetic algorithm and from 41.71% to 46.96% for particle swarm algorithm. An additional advantage of the developed clustering method is a notable reduction in time to reach high solution diversity in particle swarm algorithm. Increasing SOM training time for the large data set is negligible in comparison with full reservoir simulation run. The proposed methodology is based on the typical numerical model without requirement of additional dataset or streamline and can be easily transferred from field to field. The power and the workflow utility are demonstrated by the results of significant improvement for the algorithm's convergence rate and time.

Parallel computations have become standard practice for simulating the complicated multi-phase flow in a petroleum reservoir. Increasingly sophisticated numerical techniques have been developed in this context. During the chase of algorithmic superiority, however, there is a risk of forgetting the ultimate goal, namely, to efficiently simulate real-world reservoirs on realistic parallel hardware platforms. In this paper, we quantitatively analyse the negative performance impact caused by non-contributing computations that are associated with the “ghost computational cells” per subdomain, which is an insufficiently studied subject in parallel reservoir simulation. We also show how these non-contributing computations can be avoided by reordering the computational cells of each subdomain, such that the ghost cells are grouped together. Moreover, we propose a new graph-edge weighting scheme that can improve the mesh partitioning quality, aiming at a balance between handling the heterogeneity of geological properties and restricting the communication overhead. To put the study in a realistic setting, we enhance the open-source Flow simulator from the OPM framework, and provide comparisons with industrial-standard simulators for real-world reservoir models.