Diffusive Time Of Flight Research Articles

Well Trajectory Optimization under Geological Uncertainties Assisted by a New Deep Learning Technique

Summary The large number of geological realizations and well trajectory parameters make field development optimization under geological uncertainty a time-consuming task. A novel deep learning-based surrogate model with a novel well trajectory parametrization technique is proposed in this study to optimize the trajectory of wells under geological uncertainty. The proposed model is a deep neural network with ConvLSTM layers to extract the most salient features from highly channelized and layered reservoirs efficiently. ConvLSTM layers are used because they can extract spatiotemporal features simultaneously since layered reservoirs can be regarded as a time series of spatially distributed reservoir properties. The proposed surrogate model could predict the individual objective function with a coefficient of determination of 0.96. After verifying the validity of the surrogate model, four approaches were used to optimize well trajectories. Two of the approaches consumed all available realizations (surrogate model-based and simulation-based approaches), while the remaining two used a subset of realizations. The selection of the subset was based on the cumulative oil production (COP) and the diffusive time of flight (DTOF). Results showed that although the surrogate model used all realizations, it could provide similar results to the simulation-based optimization with only a 5% computational cost of the simulation-based approach. The novelty of this work lies in its proposal of an innovative surrogate model to improve the analysis of channelized and layered reservoirs and its introduction of a novel well trajectory optimization framework that effectively addresses the challenge of optimizing well trajectories in complex three-dimensional spaces, a problem not adequately tackled in previous works.

Hybrid Approach Proves Effective in Multiwell Forecasting in Unconventionals

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper URTeC 3855422, “RGNet for Multiwell Forecasting in Unconventional Reservoirs,” by Zhenyu Guo, SPE, and Sathish Sankaran, SPE, Xecta Digital Labs, and Ying Li, The University of Tulsa. The paper has not been peer reviewed. _ The aim of the complete paper is to propose a hybrid approach that combines physics with data-driven approaches for efficient and accurate forecasting of the performance of unconventional wells under codevelopment. The method the authors propose is the reservoir graph network (RGNet) model. By reducing system complexity while maintaining fundamental physics, the model provides an efficient and accurate way to model, history-match, and predict unconventional wells. Compared with a full-physics model that takes from hours to days to run, the described model only takes from seconds to minutes. Performance Analysis of Unconventional Reservoirs Developing unconventional reservoirs is a complex process involving well targeting, timing, spacing, and completion design for horizontal wells with hydraulic fractures. Among commonly used methods, decline curve analysis (DCA) can execute quick well-performance analysis without considering the complex physics in unconventional reservoirs piece by piece. The convenient aspect of DCA is that practitioners can obtain forecast results quickly by providing historical production data. However, because DCA has some strong assumptions on operational conditions and does not incorporate the key physics of the flow, it may fail to predict accurately in many situations. As a simple, fast tool for analyzing the capacity of a reservoir, rate transient analysis (RTA) has been used to model and forecast unconventional wells. Both pressure and rate data are considered to generate solutions that are more credible than those generated by DCA, where only rate information is used. The drawback of RTA is related to significant assumptions used to derive analytical solutions (e.g., homogeneous reservoir and simple planar fractures with uniform properties). Moreover, it is not an easy task to acquire some of the information required by RTA. Full-physics reservoir simulation models are used as the most-rigorous approach to understand dynamics of unconventional reservoirs because they allow for different complexities for modeling unconventionals. However, because it takes tremendous effort for information gathering, geological/fracture modeling, and history matching, applying this approach to large-scale reservoirs is not tractable. Based on the idea of diffusive time of flight (DTOF), the RGNet model was developed for reservoir modeling, history matching, forecasting, and optimization. It can model multiple wells with communications. In this work, the authors incorporate pressure dependency of pore volumes and transmissibility into RGNet while modeling unconventional reservoirs. Methodology The basic idea of the described model comes from the DTOF transformation that converts the 3D diffusivity problem into a 1D problem. The complete paper provides several equations that enable this conversion.

Field Application of a Novel Multiresolution Multiwell Unconventional Reservoir Simulation: History Matching and Parameter Identification

Summary This study focuses on developing an efficient workflow by integrating a multiresolution simulation model and a multiobjective evolutionary algorithm (MOEA) for application to multiwell unconventional reservoirs. In this approach, hydraulic fractures are represented using a dual porosity, dual permeability system facilitated by an embedded discrete fracture model (EDFM). A novel fast-marching simulation method is used to cut down on computational expenses by an order of magnitude, greatly accelerating the history-matching process. A variety of integrated monitoring technologies were implemented to map out the hydraulic fracture network. Insights into hydraulic fracture locations were gleaned from warm-back analysis of distributed temperature sensing data, and these locations were then assimilated into the simulation model as embedded discrete fractures. For the simulation, a fast-marching-based multiresolution model was used to partition the reservoir into local and shared domains guided by the diffusive-time-of-flight (DTOF) principle. The local domain maintained the original 3D grids near the wells while transforming the remaining area into 1D grids to accelerate the simulation process. Before history matching, a thorough sensitivity analysis was conducted to pinpoint the most impactful parameters. Subsequently, the model was fine-tuned using production data through an MOEA. The most sensitive parameters in history matching were identified as fracture geometry and conductivity, fluid saturations, and rock compressibility in the stimulated reservoir volume (SRV) areas. After history matching, there was a noteworthy reduction in the uncertainty of these tuning parameters. The calibrated parameters are valuable to evaluate the effectiveness of the well completion design. Overall, this work emphasizes the innovative combination of techniques applied, the efficiency gains in the history-matching process, and the scalability of the approach to other oilfield applications.

Artificial Lift Timing and Selection Optimized With Reduced-Physics Models

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 213089, “Optimizing Artificial Lift Timing and Selection Using Reduced-Physics Models,” by Hardikkumar Zalavadia, SPE, Metin Gokdemir, and Utkarsh Sinha, SPE, Xecta Digital Labs, et al. The paper has not been peer reviewed. _ The complete paper presents an artificial lift timing and selection (ALTS) methodology based on a hybrid data-driven and physics-based work flow. The proposed method predicts future unconventional reservoir inflow performance relationship (IPR) consistently and allows for continuous evaluation of ALTS scenarios in unconventional reservoirs with multiple lift types and designs. Continuous use of this process has been shown to improve production, reduce deferred production, and extend the life of lift equipment. Introduction The intent of the work flow is to maximize the positive economic impact of a well. The authors write that, to their knowledge, incumbent methods do not include the effect of subsurface performance. In the proposed approach, feedback is injected to the reservoir and a closed-loop response is obtained implicitly because of the selection of artificial lift type and operational parameters of that type. The hybrid reservoir modeling methodology is based on identifying transient well performance (TWP). The method is based on a novel formulation that combines diffusive time of flight (DTOF), succession of pseudosteady-state material balance, and transient productivity-index (PI) concepts for estimating dynamic reservoir deliverability. (Equations associated with these processes are provided in the complete paper.) This is combined with well-deliverability estimation for different artificial lift methods and their operating parameters to perform continuous nodal analysis and forecast phase rates using novel PI-based forecasting techniques. TWP Analysis This work establishes a practical method to estimate transient-well PI for every well at field scale to understand unconventional well performance. This transient PI then can be used in a variety of applications, including well forecasting, artificial lift planning, production optimization, and field development planning. The complete paper is focused primarily on using the TWP work flow for optimizing ALTS in unconventional reservoirs that is practical in its application at a field scale. Fig. 1 depicts the suggested work flow, which combines reduced-physics and data-driven methodologies to describe well performance over a series of steps. For TWP calculations, material balance is used through a sequence of pseudosteady states on the drainage volume evolution of a closed system.

Rapid Inference of Reservoir Permeability from Inversion of Traveltime Data Under a Fast Marching Method-Based Deep Learning Framework

Summary The fast marching method (FMM) is a highly efficient numerical algorithm used to solve the Eikonal equation. It calculates traveltime from the source point to different spatial locations and provides a geometric description of the advancing front in anisotropic and heterogeneous media. As the Eikonal solution, the diffusive time of flight (DTOF) can be used to formulate an asymptotic approximation to the pressure diffusivity equation to describe transient flow behavior in subsurface porous media. For the infinite-acting flow that occurs in porous media with smoothly varying heterogeneity, traveltime of the pressure front from the active production or injection well to the observation well can be directly estimated from the DTOF using the concept of radius (or depth) of investigation (ROI or DOI), which is defined as the moment when a maximum magnitude of the partial derivative of pressure to time occurs. Based on the ROI or DOI definition, we propose a deep neural network called the inversion neural network (INN) to inversely estimate heterogeneous reservoir permeability by inverting the traveltime data. The INN is trained by traveltime data created for a large data set of distinct permeability fields from FMM simulations, which can be two orders of magnitude faster than conventional reservoir simulators. A convolutional neural network (CNN), the U-Net architecture, is incorporated into the INN, which establishes a nonlinear mapping between the heterogeneous permeability fields and the traveltime data collected at sparse observation wells. The loss function used for the INN is defined as the root mean square error (RMSE) between the logarithm of the predicted permeability and the logarithm of the true permeability. The performance of the INN is tested on reservoir models with both smoothly varying heterogeneity and high-contrast media properties. For the 2D smoothly varying heterogeneous models with a grid size of 49×49, the permeability predicted by the INN has an average estimation error of 8.73% when a set of 7×7 uniformly distributed observation wells is used to collect “observational” traveltime data from the FMM simulation. For models with the same grid size and observation well density but with high-contrast media properties, the INN can still capture the general heterogeneity distribution, although with reduced prediction accuracy. Using a graphics processing unit (GPU) for training and prediction allows the entire inverse modeling process for a 2D 49×49 reservoir model to be completed within 7 minutes.

Estimation of heterogeneous permeability using pressure derivative data through an inversion neural network inspired by the Fast Marching Method

Reservoir heterogeneity significantly impacts the fluid flow behavior in porous media. In subsurface communities including hydrocarbon or geothermal recovery, geological storage of CO2 or H2, and hydrology, transient pressure data are often used to infer the subsurface rock properties (e.g., permeability field) due to its ready availability and quick response. The accurate estimation of such properties is critical to accurately predict fluid flow in porous media.In this work, we propose a deep learning (DL) approach inspired by the Fast Marching Method (FMM), namely the Inversion Neural Network (INN), to inversely infer heterogeneous reservoir model parameters using transient pressure data. The forward model used to generate training data for the INN is established based on a semi-analytic asymptotic solution to the diffusivity equation using the diffusive time of flight (DTOF). The FMM solves the Eikonal equation for the DTOF and provides the partial derivative of pressure drop to the natural logarithm of time (hereafter pressure derivative). The pressure derivative data are then used to predict reservoir model parameters by the INN. In the homogeneous scenario, the INN architecture is a relatively simple fully connected neural network as a proof-of-concept to validate the feasibility, and it directly correlates the permeability value with the pressure derivative. In the heterogeneous scenario, as the heterogeneous permeability field is estimated based on sparse observational data of pressure derivatives, we adopt the convolutional neural network (CNN) to flexibly deal with the image-based properties, and leverage transfer learning to efficiently train a robust INN in the heterogeneous scenario.We first validated that INN can infer the homogeneous permeability fields through numerical experiments by testing root-mean-square-error (RMSE) around 1.086 md. Inspired by that, observational data with different sparsity are used to train convolutional INN and predict heterogeneous permeability fields. As the number of observational locations (nobs) increases from 3 × 3 to 48 × 48, the testing RMSE in heterogeneous scenarios decreases from 187.510 md to 18.080 md. Besides, we found that transfer learning significantly improves the predictive accuracy at low nobs, with relative error decreased by 17.25%. Finally, noisy pressure derivative data under different nobs are used to history match the heterogeneous reservoir model, and INN can infer the permeability field with a low error of 9.6% at nobs=7×7. Without an iterative procedure for parameter estimation, the INN demonstrates to perform permeability inversion with CPU time in the magnitude of 9.2×10−4 s on a single GPU (NVIDIA Quadro P2200). The FMM-inspired INN sets up the basis for the accurate characterization of reservoir model heterogeneity by inverting pressure derivative data with both decent predictive accuracy and computational efficiency.

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.

Rapid Simulation of Unconventional Reservoirs by Multidomain Multiresolution Modeling Based on the Diffusive Time of Flight

Summary The fast marching method (FMM)-based rapid flow simulation has been shown to accelerate simulation efficiency by orders of magnitude by transforming 3D simulation to equivalent 1D simulation using the concept of the “diffusive time of flight” (DTOF). However, the 1D transformation does not directly apply to multiwell problems. In this paper, we propose a novel DTOF-based multidomain multiresolution discretization scheme to accelerate multiwell simulation of unconventional reservoirs. Our method formulates multiwell simulation problems based on the DTOF which displays the pressure front propagation in unconventional reservoirs. The DTOF contours are used to partition the reservoir into local and shared domains. A local domain is where the flow is dominated by a single well, and the shared domain is where the fluid flow is influenced by multiple wells. The DTOF contours expand independently in local domains and interfere in the shared domain. After the partitioning, each domain is discretized using a multiresolution scheme whereby the original 3D fine mesh is preserved near the wells to account for detailed physics including gravity, and the rest of the domain is discretized into 1D mesh based on the DTOF contours to alleviate the simulation workload. The power and efficacy of our approach are demonstrated using synthetic and field-scale simulation models with different degrees of geologic and well-completion complexity. The simulation results, number of active cells, and computation time for the proposed discretization scheme are compared with the original high-fidelity 3D model for each case. The results show that the proposed method is suitable for multiwell simulation problems in unconventional reservoirs and can accelerate flow simulations by orders of magnitude with minimal loss of accuracy. The novelty of this work is the creation of DTOF-derived multiresolution discretization with local and shared domains to simplify and accelerate the calculation of subsurface flow problems, especially in unconventional reservoirs. Our workflow can be easily interfaced with commercial simulators, making it suitable for large-scale field applications.

An efficient method for injection well location optimization using Fast Marching Method

Well placement is one of the challenging steps in field development optimization. Finding optimal locations with fewer function evaluations and less time is crucial when dealing with giant models. In this study, we propose a method based on the combination of Fast Marching Method and Particle Swarm Optimization to decrease the number of function evaluations in optimizing the location of vertical injection wells. The idea is to restrict the optimization algorithm's search space to a region around the drainage boundary of producers. Thereby, the optimization algorithm can find good locations with fewer function evaluations compared to direct optimization. Drainage boundaries are predicted based on the diffusive time of flight (arrival time) calculated by the Fast Marching Method. First, the performance of the proposed method in predicting the drainage boundaries is evaluated by verifying the drainage boundary of four producers on two synthetic models using the fast marching method and streamline simulator which showed a good match. Next, the approach is applied to two- and three-dimensional example cases, including four synthetic models and a real field case to test the robustness and efficiency of the proposed method. Results are compared with the direct optimization approach and showed that the proposed approach can provide higher Net Present Values with fewer function evaluations most of the time. Finally, a comparison between the computational efficiency of the fast marching method and streamline simulator showed that the time needed by the fast marching method to predict the drainage boundary of the wells was orders of magnitude lower than the streamline simulator. • PSO-FMM is used to facilitate the process of finding good positions for injection wells. • PSO-FMM could find appropriate locations for the injectors with fewer function evaluations compared to direct optimization. • FMM could predict the drainage boundary of producers with satisfactory accuracy compared to streamline simulation. • Computational time required by FMM to estimate the drainage boundary was much lower than streamline simulation.

Scenario Reduction of Realizations Using Fast Marching Method in Robust Well Placement Optimization of Injectors

Choosing a representative subset of realizations can reduce significantly the number of simulations and the computational cost associated with optimization under geological uncertainty. Methods that use dynamic criteria, such as full flow simulators, can choose effectively representative realizations and reduce the number of simulations during optimization. However, these methods are expensive computationally. This study aims at investigating the effect of using diffusive time of flight (DTOF) as a feature to select a representative subset of realizations for well placement optimization under uncertainty. The proposed approach is based on the calculation of DTOF for pressure propagation in a reservoir; it was calculated using the fast marching method (FMM). To determine reservoir connectivity, a threshold was used for a number of grid blocks at specific time intervals; it can be used as a measure to evaluate and select the realizations. The proposed methodology was utilized to optimize the location of vertical injection wells on three numerical models, which are three dimensional with equally probable realizations. The optimization results of the proposed method were compared with the results of optimization using the full set and the subset realizations obtained using the K-means clustering method. In this study, 10 realizations were selected from the full set as the representative subset. The selection was based on K-means clustering and FMM. Results show that the FMM-based approach outperforms the clustering method and it can capture the uncertainty range with only a small subset of realizations with a much lower computational burden compared to the full set.

Improving well productivity – Ways to reduce the lag between the diffusive and convective time of flight in shale wells

This short communication explains how the key to improve shale well productivity lies in reducing the gap between the diffusive and convective time of flight. Diffusive time of flight in a hydrocarbon reservoir controls which reservoir regions will be affected by the pressure transient at a certain time after the onset of production. The convective time of flight is a measure of tracer front advance as well as for fluid withdrawal rates due to a well system draining the reservoir. The difference between the advance rate of a diffusive pressure front and a convective tracer front has been long recognized. However, only recently has it become clear that a thorough grasp of the two concepts is crucial for better decisions about fracture and well spacing in low permeability reservoirs. Although the diffusive time of flight is highlighted in the traditional pressure depletion models, the convective time of flight has a more direct impact on well productivity because it controls the advance rate, to the well system, of hydrocarbon fluids to be produced. The two concepts, diffusive and convective time of flight, are briefly highlighted below, with emphasis on their relevance for decision-making in shale field development solutions.

Integration of Pressure Transient Data into Reservoir Models Using the Fast Marching Method

SummaryCalibration of reservoir model properties by integration of well-test data remains an important research topic. Well-test data have been recognized as an effective tool to describe transient flow behavior in petroleum reservoirs. It is also closely related to the drainage volume of the well and the pressure-front propagation in the subsurface. Traditional analytic means of estimating reservoir permeability relies on an interpretation of the diagnostic plot of the well pressure and production data, which usually leads to a bulk average estimation of the reservoir permeability. When more detailed characterization is needed, a forward model that is sensitive to the reservoir heterogeneity needs to be established, and a numerical inversion technique is required.We use the concept of the diffusive time of flight (DTOF) to formulate an asymptotic solution of the diffusivity equation that describes transient flow behavior in heterogeneous petroleum reservoirs. The DTOF is obtained from the solution of the Eikonal equation using the fast marching method (FMM). It can be used as a spatial coordinate that reduces the 3D diffusivity equation to an equivalent 1D formulation. We investigate the drainage-volume evolution as a function of time in terms of the DTOF. The drainage volume might be directly related to the well-test derivative, which can be used in an inversion calculation to calibrate reservoir model parameters.The analytic sensitivity coefficients of the well-test derivative with respect to reservoir permeability are derived and incorporated into an objective function to perform model calibration. The key to formulating the sensitivity coefficients is to use the functional derivative of the Eikonal equation to derive the analytic sensitivity of the DTOF to reservoir permeability. Its solution is implemented by tracking the characteristic trajectory of the local Eikonal solver within the FMM. The major advantage of calculating the sensitivity coefficients using the FMM is its significant computational efficiency during the iterative inversion process.This inverse-modeling approach is tested on a 2D synthetic heterogeneous reservoir model and then applied to the 3D Brugge Field, where a single well with constant flow rate is simulated. The well-test derivative is shown to be inversely proportional to the drainage volume and is treated as the objective function for inversion. With an additional constraint to honor the prior model, our inverse-modeling approach will adjust the reservoir model to obtain permeability as a function of distance from the well within the drainage volume. It provides a modification of reservoir permeability both within and beyond the depth of investigation (DOI).

Fast Models of Hydrocarbon Migration Paths and Pressure Depletion Based on Complex Analysis Methods (CAM): Mini-Review and Verification

A recently developed code to model hydrocarbon migration and convective time of flight makes use of complex analysis methods (CAM) paired with Eulerian particle tracking. Because the method uses new algorithms that are uniquely developed by our research group, validation of the fast CAM solutions with independent methods is merited. Particle path solutions were compared with independent solutions methods (Eclipse). These prior and new benchmarks are briefly summarized here to further verify the results obtained with CAM codes. Pressure field solutions based on CAM are compared with independent embedded discrete fracture method (EDFM) solutions. The CAM method is particularly attractive because its grid-less nature offers fast computation times and unlimited resolution. The method is particularly well suited for solving a variety of practical field development problems. Examples are given for fast optimization of waterflood patterns. Another successful application area is the modeling of fluid withdrawal patterns in hydraulically fractured wells. Because no gridding is required, the CAM model can compute the evolution of the drained rock volume (DRV) for an unlimited (but finite) number of both hydraulic and natural fractures. Such computations of the DRV are based on the convective time of flight and show the fluid withdrawal zone in the reservoir. In contrast, pressure depletion models are based on the diffusive time of flight. In ultra-low permeability reservoirs, the pressure depletion zones do not correspond to the DRV, because the convective and diffusive displacement rates differ over an order of magnitude (diffusive time of flight being the fastest). Therefore, pressure depletion models vastly overestimate the drained volume in shale reservoirs, which is why fracture and well spacing decisions should be based on both pressure depletion and DRV models, not pressure only.

Generative well pattern design—principles, implementation, and test on OLYMPUS challenge field development problem

A novel generative (well pattern) design approach is proposed for a reservoir well pattern design, building upon the observation that automated methods are slow to beat human-designed patterns. The approach relies upon the construction of 3D well patterns as a function of geology in a mindset in which functional requirements derive from reservoir engineering heuristics. The concept of well patterns as graphs is leveraged and expanded. Nodes are well parts and geological features. Edges represent functional requirements driven by economical and physical considerations; they are expressed as 3D functions of geology. Diffusive time of flight and a novel measure are proposed to quantify the relative suitability of model cells to the positioning of wells. A small, but deemed complete, set of requirements is proposed relative to the reservoir engineering domain. The search space for inserting wells is limited to cells non-dominated from the joint perspective of requirements applicable to the considered well type. Tentatively, optimal patterns are built by balancing weights given to each requirement. The process is applied to a single and to multiple realizations enabling consideration of uncertainties. Weights are few and display quasi-linear and independent relations to common objective functions. The approach was tested on the OLYMPUS field development benchmark problem. Results illustrate the potential for initializing optimization with performing candidates while maintaining geographic coverage. The search space dimension was reduced by a factor of ~ 10100. A solution was found within the first 92 investigated settings that reaches a net present value (8% discount) of 643M$. Such performances are of a nature to ensure systemic superiority over purely human-driven optimization processes and, upon integration in iterative search processes, over competing global parameterization schemes whenever few calls are made to the objective function.

Optimizing CO2- and Field-Gas-Injection EOR in Unconventional Reservoirs Using the Fast-Marching Method

SummaryRecently, there has been an increasing interest in enhanced oil recovery (EOR) from shale-oil reservoirs, including injection of carbon dioxide (CO2) and field gas. For the performance assessment and optimization of CO2- and field-gas-injection processes, compositional simulation is a powerful and versatile tool because of the capability to incorporate reservoir heterogeneity, complex fracture geometry, and multiphase and multicomponent effects in nanoporous rocks. However, flow simulation accounting for such complex physics can be computationally expensive. In particular, field-scale optimization studies requiring a large number of high-resolution compositional simulations can be challenging and sometimes computationally prohibitive. In this paper, we present a rapid and efficient approach for the optimization of CO2- and field-gas-injection EOR in unconventional reservoirs using a fast-marching-method (FMM) -based flow simulation.The FMM-based simulation uses the concept of diffusive time of flight (DTOF). The DTOF is a representation of the travel time of pressure-front propagation and accounts for geological heterogeneity, well architecture, and complex fracture geometry. The DTOF can be efficiently obtained by solving the Eikonal equation using the FMM. The 3D flow equation is then transformed into an equivalent 1D equation using the DTOF as a spatial coordinate, leading to orders of magnitude faster computation for high-resolution and compositional models as compared to full 3D simulations. The speed of computation enables using robust population-based optimization techniques such as genetic- or evolutionary-based algorithms that typically require a large number of simulation runs to optimize the operational and process parameters.We demonstrate the efficiency and robustness of our proposed approach using synthetic and field-scale examples. We first validate the FMM-based simulation approach using an example of CO2 huff ‘n’ puff for a synthetic heterogeneous dual-porosity model with a multistage hydraulically fractured well. Next, we present a field-scale optimization of operating strategies for gas-injection EOR in the Eagle Ford Formation. The rapid computation of the FMM-based approach enabled a comprehensive evaluation of the EOR project, including sensitivity studies, parameter-importance analysis, and optimal operating strategies using high-resolution geologic models.

Modeling Hydraulically Fractured Shale Wells Using the Fast-Marching Method With Local Grid Refinements and an Embedded Discrete Fracture Model

Summary Recently, fast–marching–method (FMM) –based flow simulation has shown great promise for rapid modeling of unconventional oil and gas reservoirs. Currently, the application of FMM–based simulation has been limited to using tartan grids to model the hydraulic fractures (HFs). The use of tartan grids adversely impacts the computational efficiency, particularly for field–scale applications with hundreds of HFs. Our purpose in this paper is to extend FMM–based simulation to incorporate local grid refinements (LGRs) and an embedded discrete fracture model (EDFM) to simulate HFs with natural fractures, and to validate the accuracy and efficiency of the methodologies. The FMM–based simulation is extended to LGRs and EDFM. This requires novel gridding through the introduction of triangles (2D) and tetrahedrons (2.5D) to link the local and global domain and solution of the Eikonal equation in unstructured grids to compute the diffusive time of flight (DTOF). The FMM–based flow simulation reduces a 3D simulation to an equivalent 1D simulation using the DTOF as a spatial coordinate. The 1D simulation can be carried out using a standard finite–difference method, thus leading to orders of magnitude of savings in computation time compared with full 3D simulation for high–resolution models. First, we validate the accuracy and computational efficiency of the FMM–based simulation with LGRs by comparing them with tartan grids. The results show good agreement and the FMM–based simulation with LGRs shows significant improvement in computational efficiency. Then, we apply the FMM–based simulation with LGRs to the case of a multistage–hydraulic–fractured horizontal well with multiphase flow, to demonstrate the practical feasibility of our proposed approach. After that, we investigate various discretization schemes for the transition between the local and global domain in the FMM–based flow simulation. The results are used to identify optimal gridding schemes to maintain accuracy while improving computational efficiency. Finally, we demonstrate the workflow of the FMM–based simulation with EDFM, including grid generation; comparison with FMM with unstructured grid; and validation of the results. The FMM with EDFM can simulate arbitrary fracture patterns without simplification and shows good accuracy and efficiency. This is the first study to apply the FMM–based flow simulation with LGRs and EDFM. The three main contributions of the proposed methodology are (i) unique mesh–generation schemes to link fracture and matrix flow domains, (ii) DTOF calculations in locally refined grids, and (iii) sensitivity studies to identify optimal discretization schemes for the FMM–based simulation.

Novel approach for production transient analysis of shale reservoirs using the drainage volume derivative

Recent research into transient reservoir response has developed a novel formulation of the diffusivity equation to model pressure, rate, and production, especially for unconventional reservoirs. This formulation characterizes the geometry of pressure propagation through a reservoir taking into account the heterogeneity of the reservoir and the boundary conditions which arise from multiple fractures and complex well completions. The formulation is derived from the asymptotic (high frequency) limit of the diffusivity equation for the impulse pressure solution. The formulation requires the solution of the Eikonal equation for the Diffusive Time of Flight (DToF) and the corresponding pore volume geometry contained within a Diffusive Time of Flight contour. Once the pore volume geometry is constructed, it may be used as the basis for either numerical or analytic solution. While numerical solutions are more general, the analytic approach is more readily applicable to production data analysis. The analytic approach may be used to determine two important field characteristics: the drainage volume of a well and the ratio of the produced volume to the drainage volume. In the pseudo steady state (PSS) limit, the latter would be the recovery factor for a reservoir. These quantities may be predicted from a model (for calibration purposes) or calculated directly from production data without resorting to any modeling.In this paper, we work directly with field production and pressure data to infer the drainage volume, and the instantaneous recovery ratio, defined as the ratio of the produced volume to the drainage volume. We also introduce a new w(τ) function that is related to a combined fracture and reservoir surface area. Our work draws upon the commonly used pressure transient and Rate Normalized Pressure (RNP) concepts. However, it specifically generalizes the concept of the drainage volume from PSS flow to transient flow, as is required for unconventional reservoirs, and then infers the underlying flow geometry to help differentiate and analyze different reservoir and fracture properties. The proposed formulation is applied to the analysis of the production data from shale oil reservoirs. The field data has been interpreted to describe the variations in performance characteristics seen in a number of wells.

Fracture Diagnosis in Multiple-Stage-Stimulated Horizontal Well by Temperature Measurements With Fast Marching Method

Summary Downhole-temperature measurement is one of the solutions to understanding downhole-flow conditions, especially in complex well/reservoir domains such as multistage-fractured horizontal wells. In the past, models and methodologies have been developed for fracture diagnosis for multiple-stage-fractured horizontal wells. They are based either on a semianalytical approach for simplicity or on reservoir simulation for generality. The challenges are that semianalytical models are not robust enough to describe complex fracture systems, whereas numerical simulation is computationally expensive and impractical for inversion. To develop a comprehensive approach to translate temperature to flow profile, we adopted the fast marching method (FMM) in simulating both heat transfer and the velocity/pressure field in the interested domain (heterogeneous reservoir with multiple-fractured horizontal wells). FMM is a new approach that is efficient in front tracking. Previous studies show a significant success in the investigation of pressure-depletion behavior and shale-gas production-history match. By the nature of heat transfer in porous media, the thermal-front propagation would lag behind pressure, and the noticeable temperature change in the reservoir only happens near hydraulic/natural fractures. FMM can be used to efficiently track the heat front that is associated with the flow field. In this study, we solve the thermal model in porous media by transforming the general energy-balance equation into a 1D equation, with the diffusive time of flight (DTOF) as the spatial coordinate system. Besides the diffusive heat conduction, the convection, Joule-Thomson effect, and viscous dissipation are considered in the model. The inner boundary of the model is carefully handled, and the drainage volume of each fracture is calculated to identify different inflow temperature related to flow rate at perforation locations. The model was validated by the finite-difference approach. Examples are presented in the paper to illustrate the application of the new method. The approach can be used to quantitatively interpret temperature measurements to fracture profiles in horizontal wells.

A Comprehensive Reservoir Simulator for Unconventional Reservoirs That Is Based on the Fast Marching Method and Diffusive Time of Flight

SummaryModeling of fluid flow in unconventional reservoirs requires accurate characterization of complex flow mechanisms because of the interactions between reservoir rock, microfractures, and hydraulic fractures. The pore-size distribution in shale and tight sand reservoirs typically ranges from nanometers to micrometers, resulting in ultralow permeabilities. In such extremely low-permeability reservoirs, desorption and diffusive processes play important roles in addition to heterogeneity-driven convective flows.For modeling shale and tight oil and gas reservoirs, we can compute the well-drainage volume efficiently with a fast marching method (FMM) and by introducing the concept of “diffusive time of flight” (DTOF). Our proposed simulation approach consists of two decoupled steps—drainage-volume calculation and numerical simulation with DTOF as a spatial coordinate. We first calculate the reservoir drainage volume and the DTOF with the FMM, and then the numerical simulation is conducted along the 1D DTOF coordinate. The approach is analogous to streamline modeling whereby a multidimensional simulation is decoupled to a series of 1D simulations resulting in substantial savings in computation time for high-resolution simulation. However, instead of a “convective time of flight” (CTOF), a DTOF is introduced to model the pressure-front propagation.For modeling physical processes, we propose triple continua whereby the reservoir is divided into three different domains: microscale pores (hydraulic fractures and microfractures), nanoscale pores (nanoporous networks), and organic matter. The hydraulic fractures/microfractures primarily contribute to the well production, and are affected by rock compaction. The nanoporous networks contain adsorbed gas molecules, and gas flows into fractures by convection and Knudsen diffusion processes. The organic matter acts as the source of gas. Our simulation approach enables high-resolution flow characterization of unconventional reservoirs because of its efficiency and versatility. We demonstrate the power and utility of our approach with synthetic and field examples.

Qualitative well placement and drainage volume calculations based on diffusive time of flight

Streamlines have been very effectively used for computing swept volumes and drainage volumes in oil reservoirs. However, there is an incorrect but commonly held perception that streamlines cannot be used for gas reservoirs. In fact, streamlines exist whenever there is an underlying velocity field—both in compressible and incompressible flow. In this paper we present one of the first applications of streamlines to tight gas reservoirs for drainage volume calculations of producing wells, optimal placement of infill wells.Current practice of well placement in tight gas reservoirs generally involves the use of empirical correlations based on reservoir properties and past production histories and/or pressure maps from flow simulation. No rigorous procedure is available to compute well drainage volumes in the presence of heterogeneity and hydraulic fractures, particularly for complex wells. We propose a fast approach for drainage volume calculations based on the streamlines and diffusive time of flight and demonstrate its application to optimize well placement. Our approach relies on a high frequency asymptotic solution of the diffusivity equation and emulates the propagation of a ‘pressure front’ in the reservoir. This allows us not only to rigorously compute the well drainage volumes as a function of time, but also to examine the potential impact of infill wells on the drainage volumes of existing producers. Using these results, we present a systematic approach to optimize well placement to maximize the estimated ultimate recoveries (EUR).We demonstrate the power and utility of our method using both synthetic and field applications. The synthetic example is used to validate our approach by establishing the consistency between the drainage volume calculations from streamlines and the EUR. The field example is from the Wamsutter gas field, one of the largest gas fields in the Rocky Mountain region. We utilize the streamline-based drainage volumes to identify depleted sands and generate a reservoir quality map to optimize future well placement based on the undepleted regions. Field application clearly demonstrates a systematic and efficient approach to optimal well placement in tight gas reservoirs.