History Matching Algorithm Research Articles

A novel meshless method for numerical simulation of fractured-vuggy reservoirs

This paper proposes a novel meshless numerical simulation method for fractured-vuggy reservoirs. Initially, model discretization involves extracting fracture and vug feature nodes along the contours of the reservoir bodies and in alignment with fracture orientations, tailored to the characteristics of the fractured-vuggy oil deposit. Subsequently, a connection element model is established within the influence domain. Based on the seepage equations of the connection system, a computational method for the parameters of the connection elements (connection transmissibility and connection volume) is defined. In addition, a sequential method is employed to solve for the pressure and saturation, achieving rapid prediction of the reservoir's production dynamics. On this basis, an automatic history matching algorithm is utilized for real-time fitting of oil–water dynamic indicators, inverting the characteristic parameters of the connection element model, and quantitatively characterizing the injection–production connection elements. The research findings indicate that the method is capable of rapidly fitting and predicting the dynamics of oil reservoir production. Furthermore, conceptual model examples have substantiated that it achieves comparable computational efficiency and superior computational accuracy when compared to the traditional finite difference (volume) method under the same nodal conditions. Additionally, the node arrangement for fractures and vugs is comparatively flexible, and it can ensure the integrity of the flow paths with a limited number of nodes to enhance predictive accuracy. Therefore, this method can effectively meet the rapid prediction requirements for fractured-vuggy reservoirs and also provides a novel meshless computational approach for the numerical simulation of such reservoirs.

A Local Parameterization-Based Probabilistic Cooperative Coevolutionary Algorithm for History Matching

A Local Parameterization-Based Probabilistic Cooperative Coevolutionary Algorithm for History Matching

The importance of mechanical conditions in the testing of excitation abnormalities in a population of electro-mechanical models of human ventricular cardiomyocytes.

Background: Populations of in silico electrophysiological models of human cardiomyocytes represent natural variability in cell activity and are thoroughly calibrated and validated using experimental data from the human heart. The models have been shown to predict the effects of drugs and their pro-arrhythmic risks. However, excitation and contraction are known to be tightly coupled in the myocardium, with mechanical loads and stretching affecting both mechanics and excitation through mechanisms of mechano-calcium-electrical feedback. However, these couplings are not currently a focus of populations of cell models. Aim: We investigated the role of cardiomyocyte mechanical activity under different mechanical conditions in the generation, calibration, and validation of a population of electro-mechanical models of human cardiomyocytes. Methods: To generate a population, we assumed 11 input parameters of ionic currents and calcium dynamics in our recently developed TP + M model as varying within a wide range. A History matching algorithm was used to generate a non-implausible parameter space by calibrating the action potential and calcium transient biomarkers against experimental data and rejecting models with excitation abnormalities. The population was further calibrated using experimental data on human myocardial force characteristics and mechanical tests involving variations in preload and afterload. Models that passed the mechanical tests were validated with additional experimental data, including the effects of drugs with high or low pro-arrhythmic risk. Results: More than 10% of the models calibrated on electrophysiological data failed mechanical tests and were rejected from the population due to excitation abnormalities at reduced preload or afterload for cell contraction. The final population of accepted models yielded action potential, calcium transient, and force/shortening outputs consistent with experimental data. In agreement with experimental and clinical data, the models demonstrated a high frequency of excitation abnormalities in simulations of Dofetilide action on the ionic currents, in contrast to Verapamil. However, Verapamil showed a high frequency of failed contractions at high concentrations. Conclusion: Our results highlight the importance of considering mechanoelectric coupling in silico cardiomyocyte models. Mechanical tests allow a more thorough assessment of the effects of interventions on cardiac function, including drug testing.

Inversion Framework of Reservoir Parameters Based on Deep Autoregressive Surrogate and Continual Learning Strategy

Summary History matching is a crucial process that enables the calibration of uncertain parameters of the numerical model to obtain an acceptable match between simulated and observed historical data. However, the implementation of the history-matching algorithm is usually based on iteration, which is a computationally expensive process due to the numerous runs of the simulation. To address this challenge, we propose a surrogate model for simulation based on an autoregressive model combined with a convolutional gated recurrent unit (ConvGRU). The proposed ConvGRU-based autoregressive neural network (ConvGRU-AR-Net) can accurately predict state maps (such as saturation maps) based on spatial and vector data (such as permeability and relative permeability, respectively) in an end-to-end fashion. Furthermore, history matching must be performed multiple times throughout the production cycle of the reservoir to fit the most recent production observations, making continual learning crucial. To enable the surrogate model to quickly learn recent data by transferring experience from previous tasks, an ensemble-based continual learning strategy is used. Together with the proposed neural network–based surrogate model, the randomized maximum likelihood (RML) is used to calibrate uncertain parameters. The proposed method is evaluated using 2D and 3D reservoir models. For both cases, the surrogate inversion framework successfully achieves a reasonable posterior distribution of reservoir parameters and provides a reliable assessment of the reservoir’s behaviors.

Effect of Temperature on Two-Phase Gas/Oil Relative Permeability in Viscous Oil Reservoirs: A Combined Experimental and History-Matching-Based Analysis

Summary Thermal enhanced oil recovery (TEOR) is the most widely accepted method for exploiting the heavy oil reservoirs in North America. In addition to improving the mobility of oil due to its viscosity reduction, the high temperature down in the hole due to the injection of the vapor phase may significantly alter the fluid flow performance and behavior, as represented by the relative permeability to fluids in the formations. Therefore, in TEOR, the relative permeabilities can change with a change in temperature. Also, there is no model that accounts for the change in temperature on two-phase gas/oil relative permeability. Further, the gas/oil relative permeability and its dependence on temperature are required data for the numerical simulation of TEOR. Very few studies are available on this topic with no emerging consensus on a general behavior of such effects. The scarcity of such studies is mostly due to experimental problems to make reliable measurements. Therefore, the primary objective of this study was to overcome the experimental issues and investigate the effect of temperature on gas/oil relative permeability. Oil displacement tests were carried out in a 45-cm-long sandpack at temperatures ranging from 64°C to 210°C using a viscous mineral oil (PAO-100), deionized water, and nitrogen gas. It was found that the unsteady-state method was susceptible to several experimental artifacts in viscous oil systems due to a very adverse mobility ratio. However, despite such experimental artifacts, a careful analysis of the displacement data led to obtaining meaningful two-phase gas/oil relative permeability curves. These curves were used to interpret the relative permeability curves for gas/heavy oil systems using the experimentally obtained displacement results. We noted that at the end of gasflooding, the “final” residual oil saturation (Sor) still eluded us even after several pore volumes (PVs) of gas injection. This rendered the experimentally determined endpoint gas relative permeability (krge) and Sor unreliable. In contrast, the irreducible water saturation (Swir) and the endpoint oil relative permeability (kroe) were experimentally achievable. The complete two-phase gas/heavy oil relative permeability curves are inferred with a newly developed systematic history-matching algorithm in this study. This systematic history-matching technique helped us to determine the uncertain parameters of the oil/gas relative permeability curves, such as the two exponents of the Corey equation (No and Ng), Sor and krge. The history match showed that kroe and Swir were experimentally achievable and were reliably interpreted, except these four parameters (i.e., Corey exponents, true residual oil saturation, and gas endpoint relative permeability) were interpreted from simulations rather than from experiments. Based on our findings, a new correlation has been proposed to model the effect of temperature on two-phase gas/heavy oil relative permeability.

Reservoir Modeling, History Matching, and Characterization with a Reservoir Graph Network Model

Summary Efficient reservoir models are more desirable for fast-paced reservoir management. Moreover, due to the complexity of flow underground, it is also essential to capture the fundamental physics for model reliability. Although they are fast, pure data-driven models frequently have issues associated with interpretability, physical consistency, and ability to forecast. On the other hand, we have used full-physics simulation models to mimic and investigate hydrocarbon systems for over several decades. However, considering its infrequent model updates related to high model complexity, it is a big challenge to manage reservoirs using full-physics models in short cycles. The objective here is to propose an approach that blends reservoir physics with data-driven models to fit in the framework of dynamic reservoir management. We propose to use a reservoir graph network (RGNet) modeling approach based on a diffusive time-of-flight (DTOF) concept to simulate reservoir behaviors. By assimilating field observation data (such as pressure and rates), an RGNet model can be used for future predictions, scenario studies, and well-control optimizations. By discretizing DTOF of a 3D system with multiple wells, RGNet simplifies the system into a graph network represented by a set of 1D grid blocks that significantly reduces the system complexity and run time. RGNet can also handle multiple flow problems with various types of physics. In this work, we propose to use two methods to develop reliable and parsimonious models scalable to large-scale systems. In addition, we propose a more robust method to assimilate pressure data. We applied the proposed approach to a synthetic and a field example. Two different history-matching algorithms, the ensemble smoother with multiple data assimilation (ES-MDA) and an adjoint-based method, are compared. While ES-MDA provides the capability for uncertainty analysis, an adjoint-based method generally requires fewer simulation runs to generate a posterior model. With the proposed methods for generating interwell connections, RGNet model calibration can be achieved without system redundancy and spurious long-distance well connectivity. Also, by using a more stable pressure-matching technique, we show that pressure data are better matched and reservoir volume is accurately characterized. RGNet provides a novel hybrid physics and data-driven reservoir modeling method to fit in closed-loop reservoir management (CLRM). As RGNet models are combined with fundamental flowing physics, the calibrated model parameters are easy to interpret and understand. An RGNet model runs with far less computational cost than required by a full-physics model, which allows it to be a more practical solution to history match, predict, and optimize real assets.

History-Matching and Forecasting Production Rate and Bottomhole Pressure Data Using an Enhanced Physics-Based Data-Driven Simulator

Summary In this study, we present a novel application of our newly developed physics-based data-driven interwell numerical simulator (INSIM) referred to as INSIM-BHP to history match highly variable real-life (oscillatory) oil rate and bottomhole pressure (BHP) data acquired daily in multiperforated wells produced from an oil reservoir with bottomwater drive mechanism. INSIM-BHP provides rapid and accurate computation of well rates and BHPs for history matching, forecasting, and production optimization purposes. It delivers precise BHP calculations under the influence of a limited aquifer drive mechanism. Our new version represents the physics of two-phase oil-water flow more authentically by incorporating a harmonic-mean transmissibility computation protocol and including an arithmetic-mean gravity term in the pressure equation. As the specific data set considered in this study contains a sequence of highly variable oil rate and BHP data, the data density requires INSIM-BHP to take smaller than usual timesteps and places a strain on the ensemble-smoother multiple data assimilation (ES-MDA) history-matching algorithm, which utilizes INSIM-BHP as the forward model. Another new feature of our simulator is the use of time-variant well indices and skin factors within the simulator’s well model to account for the effects of well events on reservoir responses such as scaling, sand production, and matrix acidizing. Another novel modification has been made to the wellhead term calculation to better mimic the physics of flow in the wellbore when the production rate is low, or the well(s) is(are) shut in. We compare the accuracy of the history-matched oil rate and BHP data and forecasted results as well as computational efficiency for history matching and future prediction by INSIM-BHP with those from a high-fidelity commercial reservoir simulator. Results show that INSIM-BHP yields accurate forecasting of wells' oil rates and BHPs on a daily level even under the influence of oscillatory rate schedules and changing operational conditions reflected as skin effects at the wells. Besides, it can help diagnose abnormal BHP measurements within simulation runs. Computational costs incurred by INSIM-BHP and a high-fidelity commercial simulator are evaluated for the real data set investigated in this paper. It has been observed that our physics-based, data-driven simulator is about two orders of magnitude faster than a conventional high-fidelity reservoir simulator for a single forward simulation. The specific field application results demonstrate that INSIM-BHP has great potential to be a rapid approximate capability for history matching and forecasting workflow in the investigated limited-volume aquifer-driven development.

Improving history matching performance in a 3D field-scale case study by combining tracer and production data

Reservoir models are often subject to uncertainties, which, if not properly taken into account, may introduce biases to the subsequent reservoir management process. To improve reliability and reduce uncertainties, it is crucial to condition reservoir models on available field datasets through history matching. There are different types of field data. Among others, production data are the most common choice, but they are subject to a major limitation of carrying relatively low value of information. On the other hand, inter-well tracer data have been shown to provide additional information about well-to-well connectivity and reservoir dynamics. However, jointly history matching production and inter-well tracer data still remains challenging due to the lack of a coherent quantitative workflow to fully integrate them. This work can be considered a step towards tackling this noticed problem. To this end, we propose a non-intrusive and derivative-free ensemble history matching workflow, in which reservoir models are more coherently conditioned on both production and inter-well tracer data with the help of a recently developed technique (correlation-based adaptive localization). The workflow is successfully implemented in the Brugge benchmark case. Our study indicates that the history matching algorithm matches the production data well, regardless of the presence or absence of the tracer data. Nevertheless, by including tracer data as an additional source of information, we are able to improve the quality of the estimated reservoir models, in terms of both improved data match and reduced model discrepancies. As such, the finding of this study can help to achieve a better understanding of the impacts of tracer data on history matching performance, and the proposed workflow could serve as a useful too for more proper uncertainty quantification, and more coherent utilization of different types of field data in real case studies in general.

Опыт адаптации многопластовой гидродинамической модели Ныдинского участка Медвежьего месторождения

Introduction. History matching of a multi-layer model, where several layers are combined into one object and developed using a unified well grid, is complicated by the mutual influence of parameters (porosity, aquifer volume and intensity, permeability, productivity) on reservoir pressure, balance of production from the producing layers, and advancement of the water front. A change in any of the adjustable parameters of one layer entails a change in the controlled parameters of all interconnected layers. Materials and research methods. The model of the Lower Cretaceous gas condensate deposits of the Nydinsky section of the Medvezhye deposit is a multi-layer one. The development objects consist of 3-4 layers. Each object is developed by an independent grid of wells. Each deposit is layer-uplifted, flooding occurs in the lateral direction. The analysis by the P/z method was carried out, according to the results of which the effect of the water pressure regime on the 2nd and 4th development site is observed, which must be reproduced in the model. Also, this method can indirectly indicate the amount of drained reserves. Research results and their discussion. The configurable parameters for history matching are: reservoir pressure, the front of the reservoir water advance, gas production profiles for each well based on the PGI using a flow meter. The setting of the initial inflow profile is reproduced based on the results of the primary GIS. The correspondence of the inflow profiles of further GIS should be ensured by the correct distribution of the initial gas reserves across the layers inside the operational facility. Further, while maintaining the balance of the inflow profile, the formation pressure and the water advance front are adjusted by iteratively changing the volume and intensity of the aquifer and the pore volume. Conclusions. The algorithm for history matching of a multi-layer model, in which several layers are developed by a single grid of wells, should begin with adapting the inflow profile to the initial date. Further redistribution of the production profile should occur iteratively due to the balance of reserves of deposits, volume and intensity of the aquifer, while taking into account the front of the advance of reservoir waters based on the results of observations on the well stock. Adjusting the inflow profile by changing kh is not recommended, since changing the inflow profile over time is most often not the function of kh. At the same time, it is necessary to take into account the mutual influence of the above parameters of one formation on all other layers inside the operational facility.

A simulation study of in-situ NAPL remediation treatment by using surfactant and foam processes in a military base South Korea

Surfactant and foam processes have been widely used to enhance oil recovery from petroleum-bearing geological formations, and also been recently extended to remediate non-aqueous-phase-liquid (NAPL) contaminants from the shallow subsurface. This study investigates the potential of using surfactant and foam processes for the in-situ remediation treatments within a military base in South Korea: first, optimizing the subsurface permeability and net-to-gross (NTG) values based on history matching and machine-learning algorithm; second, performing simulations that successfully predict the surfactant/foam processes applied in the field; and third, expanding simulations that evaluate different scenarios that might have been used for field tests.The site for the pilot-scale testing, located in the existing fuel-distribution facilities within a military base, has 5 m × 5 m treatment area with 3 m depth, prepared with 3 injection wells and 3 extraction wells. The NAPL of interest is a mixture of various oil compounds showing the average oil saturation of 5%. The overall remediation process applied consists of two major steps: the first is a 20-day injection of surfactant solution (Tween 80) to mobilize the oleic phase trapped by capillary force, and the second is a 3-day injection of foam (i.e., gas and surfactant co-injection) to control the mobility of injected gas and overcome the heterogeneity of the underground system.The major findings of this study are (i) surfactant/foam processes can be an effective means of NAPL recovery from shallow subsurface recovering more than 90% of contaminants, and (ii) computer simulations can be a useful tool for evaluating the in-situ treatment and improving the design of similar operations.

Automatic History Matching of Parent-Child Wells in Shale Gas Reservoirs with Complex Fracture Hits

Abstract It is excessively crucial for performing production match of parent-child wells automatically by using assisted history matching approach, especially considering the interwell interference scenario. We explored a new EDFM-AI (Embedded Discrete Facture Model-Artificial Intelligence) workflow by taking into account the multiple uncertainty parameters that were composed of fracture height, fracture half-length, fracture conductivity, fracture water saturation, fracture cluster efficiency, and natural fracture conductivity. The history matching algorithm implemented in this study is advanced machine learning model called XGBoost. It could overcome the issue that the proxy model is always imprecise due to the limited training sample data. Specifically, we established a field-scale reservoir model that includes parent-child shale gas wells with fracture hits in the Sichuan basin. By using this sophisticated workflow, two wells’ history matching results are excellent in terms of the inversed solution of the hydraulic fracture properties and natural fracture properties. And then, we perform the production forecasting for parent and child wells, respectively. Consequently, the results show that the best match value for estimated ultimate recovery (EUR) of parent well is 287.4 million cubic meters whereas the best match value for EUR of child well is 187.6 million cubic meters. The reliable reason is that effective fractured area of parent well is larger than that of child well. Besides, the natural fractures have a significant impact on the performance of shale gas wells with fracture hits observed, especially in the short-term production period.

Information content in 4D seismic data: Effect of correlated noise

Several types of data used in history matching for subsurface reservoir characterization have errors that are spatially or temporally correlated. Although it is often assumed that correlated observation errors decrease information content, using a simplified flow problem, we show that for data that are spatially dense (such as 4D seismic data), correlated observation errors result in higher information content than data with similar levels of errors, but without correlation. Unfortunately, correlations in the observation error are often unrecognized and difficult to estimate, especially if the correlation scale of the errors is similar to a characteristic length of the signal. In addition, many history matching algorithms are incapable of accounting properly for correlated observation error, so data are either thinned or the observation error is inflated to partially account for the lack of proper treatment. We show that neglecting correlations in the observation errors or inflating the variance, result in loss of information content. Finally, we show that it is possible to iteratively estimate the correlated observation errors through analysis of residuals after history matching. • Measurement errors in seismic data are generally correlated. • Correlated errors can increase information content when measurements are dense. • Neglecting correlations during model calibration results in information loss. • Correlations in seismic data can be estimated from data mismatch after calibration.

Integrated characterization of wormhole network by use of a modified pressure-gradient-based (PGB) sand failure criterion and ensemble-based history matching during CHOPS processes

In this paper, a damped iterative ensemble Kalman filter (IEnKF) technique has been developed and integrated with a recently modified pressure-gradient-based (PGB) sand failure criterion to implicitly and inversely characterize the dynamic wormhole network during cold heavy oil production with sand (CHOPS) processes. With an improved upscaling process, not only has a microscopic PGB sand failure criterion been properly upscaled to the grid-scale simulation, but also it can be dynamically, implicitly, and inversely determined. Then, such a sand failure model together with three-phase relative permeability is validated and applied to reproduce wormhole network in a laboratory experiment conditioned to its sand production profile. Good agreements between the observed and simulated cumulative production profiles including oil, gas, and sand have been achieved with satisfactory convergence by integrating the ensemble-based history matching method. The simulated wormhole network has shown its great similarity to the experimental observations, while the convergence of the ensemble-based history matching algorithm is greatly improved with a widespread of initial ensembles in each grid. Furthermore, deviations between the observed and updated production profiles can be significantly minimized by dividing the entire production period into three sequential stages, each of which has its distinct relative permeability curves due to the major effect (i.e., foamy oil flow, sand failure, and slurry flow) on its multiphase flow behaviours, respectively. By integrating the ensemble-based sand failure model with the damped-IEnKF method, the dynamic wormhole network during a CHOPS process can be accurately characterized with satisfactory convergence, demonstrating its robustness and practicality of dynamically and inversely simulating the preferential wormhole network. Such an ensemble-based sand failure model can be seamlessly incorporated with any commercial reservoir simulators under various conditions. • A hybrid IEnKF-PGB technique is developed to characterize wormhole network for CHOPS wells. • Good agreements between the observed and simulated cumulative production profiles are achieved. • Simulated wormhole network shows its great similarity to the experimental observations. • Dividing the production period into three stages minimizes deviations between observed and updated production profiles. • This hybrid technique is robust and can be seamlessly coupled with any reservoir simulators.

Stimulated Reservoir Volume Characterization and Optimum Lateral Well Spacing Study of Two-Well Pad: Midland Basin Case Study

This paper introduces a flow simulation-based reservoir modeling study of a two-well pad with long production history and identical completion parameters in the Midland Basin. The study includes building geologic model, history matching, well performance prediction, and finding optimum lateral well spacing in terms of oil volume and economic metrics. The reservoir model was constructed based on a geologic model, integrating well logs, and core data near the target area. Next, a sensitivity analysis was performed on the reservoir simulation model to better understand influential parameters on simulation results. The following history matching was conducted with the satisfactory quality, less than 10% of global error, and after the model calibration ranges of history matching parameters have substantially reduced. The population-based history matching algorithm provides the ensemble of the history-matched model, and the top 50 history-matched models were selected to predict the range of Estimate Ultimate Recovery (EUR), showing that P50 of oil EUR is within the acceptable range of the deterministic EUR estimates. With the best history-matched model, we investigated lateral well spacing sensitivity of the pad in terms of the maximum recovery volume and economic benefit. The results show that, given the current completion design, the well spacing tighter than the current practice in the area is less effective regarding the oil volume recovery. However, economic metrics suggest that the additional monetary value can be realized with 150% of current development assumption. The presented workflow provides a systematic approach to find the optimum lateral well spacing in terms of volume and economic metrics per one section given economic assumptions, and the workflow can be readily repeated to evaluate spacing optimization in other acreage.

Handling Big Models and Big Data Sets in History-Matching Problems through an Adaptive Local Analysis Scheme

SummaryIn applications of ensemble-based history matching, it is common to conduct Kalman gain or covariance localization to mitigate spurious correlations and excessive variability reduction resulting from the use of relatively small ensembles. Another alternative strategy not very well explored in reservoir applications is to apply a local analysis scheme, which consists of defining a smaller group of local model variables and observed data (observations), and perform history matching within each group individually. This work aims to demonstrate the practical advantages of a new local analysis scheme over the Kalman gain localization in a 4D seismic history-matching problem that involves big seismic data sets. In the proposed local analysis scheme, we use a correlation-based adaptive data-selection strategy to choose observations for the update of each group of local model variables. Compared to the Kalman gain localization scheme, the proposed local analysis scheme has an improved capacity in handling big models and big data sets, especially in terms of computer memory required to store relevant matrices involved in ensemble-based history-matching algorithms. In addition, we show that despite the need for a higher computational cost to perform model update per iteration step, the proposed local analysis scheme makes the ensemble-based history-matching algorithm converge faster, rendering the same level of data mismatch values at a faster pace. Meanwhile, with the same numbers of iteration steps, the ensemble-based history-matching algorithm equipped with the proposed local analysis scheme tends to yield better qualities for the estimated reservoir models than that with a Kalman gain localization scheme. As such, the proposed adaptive local analysis scheme has the potential of facilitating wider applications of ensemble-based algorithms to practical large-scale history-matching problems.

Accounting for model errors of rock physics models in 4D seismic history matching problems: A perspective of machine learning

Model errors are ubiquitous in practical history matching problems. A common approach in the literature to accounting for model errors is to treat them as random variables following certain presumed distributions. While such a treatment renders algorithmic convenience, its underpinning assumptions are often invalid. In this work, we adopt an alternative approach, and treat model-error characterization as a functional approximation problem, which can be solved using a generic machine learning method. We then integrate the proposed model-error characterization approach into an ensemble-based history matching framework, and show that, with very minor modifications, existing ensemble-based history matching algorithms can be readily deployed to solve the history matching problem in the presence of model errors.To demonstrate the efficacy of the integrated history matching framework, we apply it to account for potential model errors of a rock physics model in 4D seismic history matching applied to the full Norne benchmark case. The numerical results indicate that the proposed model-error characterization approach helps improve the qualities of estimated reservoir models, and leads to more accurate forecasts of production data. This suggests that accounting for model errors from a perspective of machine learning serves as a viable way to deal with model imperfection in practical history matching problems.

Uncertainty Space Expansion: A Consistent Integration of Measurement Errors in Linear Inversion

Summary In general, a probabilistic framework for a modeling process involves two uncertainty spaces: model parameters and state variables (or predictions). The two uncertainty spaces in reservoir simulation are connected by the governing equations of flow and transport in porous media in the form of a reservoir simulator. In a forward problem (or a predictive run), the reservoir simulator directly maps the uncertainty space of the model parameters to the uncertainty space of the state variables. Conversely, an inverse problem (or history matching) aims to improve the descriptions of the model parameters by using the measurements of state variables. However, we cannot solve the inverse problem directly in practice. Numerous algorithms, including Kriging-based inversion and the ensemble Kalman filter (EnKF) and its many variants, simplify the system by using a linear assumption. The purpose of this paper is to improve the integration of measurement errors in the history-matching algorithms that rely on the linear assumption. The statistical moment equation (SME) approach with the Kriging-based inversion algorithm is used to illustrate several practical examples. In the Motivation section, an example of pressure conditioning has a measurement that contains no additional information because of its significant measurement error. This example highlights the inadequacy of the current method that underestimates the conditional uncertainty for both model parameters and predictions. Accordingly, we derive a new formula that recognizes the absence of additional information and preserves the unconditional uncertainty. We believe this to be the consistent behavior to integrate measurement errors. Other examples are used to validate the new formula with both linear and nonlinear (i.e., the saturation equation) problems, with single and multiple measurements, and with different configurations of measurement errors. For broader applications, we also develop an equivalent formula for algorithms in the Monte Carlo simulation (MCS) approach, such as EnKF and ensemble smoother (ES).

Multilevel Strategies Improve History Matching of Complex Reservoir Models

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 193895, “Multilevel Strategies and Geological Parameterizations for History Matching Complex Reservoir Models,” by Yimin Liu, SPE, and Louis J. Durlofsky, SPE, Stanford University, prepared for the 2019 SPE Reservoir Simulation Conference, Galveston, Texas, 10–11 April. The paper has not been peer reviewed. The complete paper explores the use of multilevel derivative-free optimization for history matching, with model properties described using principal component analysis (PCA) -based parameterization techniques. The parameterizations applied are optimization-based PCA (O-PCA) and convolutional-neural-network-based PCA (CNN-PCA). Mesh adaptive direct search (MADS), a pattern search method that parallelizes naturally, is used for the optimizations required to generate posterior models. The use of PCA-based parameterization reduces considerably the number of variables that must be determined during history matching, but the optimization problem can still be computationally demanding. The multilevel strategy addresses this issue by reducing the number of simulations that must be performed at each MADS iteration. History-matching results demonstrate that substantial uncertainty reduction is achieved in all cases considered and that the multilevel strategy is effective in reducing the number of simulations required. Overview Parameterization has two key advantages in history matching: Far fewer parameters must be determined, which simplifies greatly the minimization in some cases. Posterior (history-matched) geomodels will be consistent with the geological scenario or training image for which the parameterization was constructed. PCA is the foundation for many geological-dimension-reduction parameterization methods. PCA is based on the eigen-decomposition of the prior geomodel covariance matrix and, thus, honors only two-point spatial statistics. History-matching algorithms largely can be classified as either optimization-based or ensemble-based procedures. Optimization-based algorithms generate one history-matched model at a time, while ensemble-based algorithms update a large set of models simultaneously. The resulting ensemble of posterior models is used to quantify uncertainty. In this study, only optimization-based history matching is considered, though the parameterizations applied also can be used with ensemble-based methods. Optimization-based history-matching algorithms can be classified further into gradient-based and derivative-free optimization (DFO) methods. Gradient-based algorithms often are combined with adjoint procedures that provide the required gradient efficiently. Unlike adjoint-based procedures, DFO algorithms (which are considered in this work) are nonintrusive. They treat the reservoir simulator as a so-called “black box,” and, as a result, they are much easier to implement and can be applied with any simulator. In this study, MADS, a pattern-search DFO, is applied for history matching. In pattern-search (or stencil-based) algorithms, the number of simulations required at each iteration scales with the number of parameters to be determined. Because the latter typically is much smaller than the number of grid-blocks in the model, geological parameterization leads to large computational savings when a pattern search, or other DFO, is used for history matching. The performance of DFO for history matching with non-Gaussian models, a primary topic in the complete paper, has not been studied extensively. The authors devise a multilevel strategy, applicable for PCA-based parameterization methods, to further reduce the computational cost of history matching using DFO. A multilevel strategy lends itself naturally to PCA-based parameterizations because the PCA representation can be ordered from the most-important to the least-important principal component. Thus, PCA parameters can be determined sequentially, level by level, in groups, rather than all at once.

Investigation of different production performances in shale gas wells using assisted history matching: Hydraulic fractures and reservoir characterization from production data

We investigated the difference in production performances of three shale gas wells with different completions and fracturing designs. We used assisted history matching technique to perform production match automatically and obtain multiple history matching solutions. The history matching algorithm used in this study is neural networks Markov chain Monte Carlo. In addition, we utilized embedded discrete fracture model technique to model hydraulic fractures instead of local grid refinement method to speed up the simulation runs. The history matching parameters consist of fracture half-length, fracture height, fracture conductivity, fracture width and fracture water saturation for hydraulic fractures properties and include matrix porosity, matrix permeability and relative permeability curves for matrix properties. We found that fracture width, fracture half-length, fracture conductivity and matrix permeability were significantly different in three shale gas wells while fracture height, fracture water saturation, porosity and relative permeability curves were less different. The information from this study will be used further to improve completion and fracturing design.

Assisted history matching in shale gas well using multiple-proxy-based Markov chain Monte Carlo algorithm: The comparison of K-nearest neighbors and neural networks as proxy model

We performed assisted history matching (AHM) in a real shale gas well using uncertain parameters such as fracture geometry, fracture conductivity, matrix permeability, matrix and fracture water saturation, and relative permeability curves. We also investigated the performance of two proxy models including K-nearest neighbors (KNN) and neural networks (NN) to be used in multiple-proxy-based Markov chain Monte Carlo (MCMC) algorithm. We emphasized the performance of both proxy models by comparing the number of history matching solution found and elapsed time. While, KNN required less elapsed time by half than NN, we found that NN performed better in terms of accuracy and predictability than KNN. In other words, NN required a smaller number of simulations by half than KNN in order to obtain the same number of history matching solutions. Therefore, it depends on what is more important to each problem either number of simulations or elapsed time. For history matching result, both proxy models in the multiple-proxy-based MCMC algorithm have similar results of posterior distribution of uncertain parameters. This confirms the robustness of the proposed history matching algorithm. The benefits of this study are that we can characterize fracture geometry and reservoir properties in a probabilistic manner. These multiple realizations can be further used for a probabilistic production forecast, future fracturing design, and well spacing optimization and planning. This AHM workflow can be applied to any hydraulic-fractured wells with historical production data.