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ECMOR XVII
- Conference date: September 14-17, 2020
- Location: Online Event
- Published: 14 September 2020
81 - 100 of 145 results
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Non-Linear Solver Optimisation for Multiphase Porous Media Flow Based on Machine Learning
Authors V.L.S. Silva, P. Salinas, C.C. Pain and M.D. JacksonSummaryNumerical simulation of multiphase flow in porous media is of paramount importance to understand, predict and manage subsurface reservoirs with applications to hydrocarbon recovery, geothermal energy resources, CO2 geological sequestration, groundwater sources and magma reservoirs. However, the numerical solution of the governing equations is very challenging due to the non-linear nature of the problem and the strong coupling between the different equations. Newton methods have been traditionally used to solve the non-linear system of equations, although, the Picard iterative method has been gaining ground in recent years. The Picard method is attractive because the multiphysics problem can be subdivided and each subproblem solved separately, which gives wide flexibility and extensibility.
Rapid convergence of the non-linear solver is of vital importance as it strongly affects the overall computational time. Therefore, a great deal of effort has been put on obtaining robust and stable convergence rates. At the same time, machine learning (ML) is gaining more and more attention with revolutionary results in areas such as computer vision, self-driving cars and natural language processing. The success of ML in different fields has inspired recent applications in reservoir engineering and geosciences. Here, we present a Picard non-linear solver with convergence parameters dynamically controlled by ML. The ML is trained based on the parameters of the reservoir model scaled to a dimensionless space. In the approach reported here, data for the ML training is generated using simulation results obtained for multiphase flow in a two-layered reservoir model which captures many of the flow features observed in models of natural reservoirs. The presented method significantly reduces the computational effort required by the non-linear solver as it can adjust itself to the complexity/physics of the system. We demonstrate its efficiency under a variety of numerical tests cases, including gravity, capillary pressure and extremely heterogeneous models.
Technical contributions:
- – Significantly reduces the computational cost of the non-linear solver.
- – The ML model is trained very efficiently based on a two-layered reservoir model and dimensionless numbers.
- – Enables us to carry out large-scale and/or physically demanding numerical simulations.
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Distributed Quasi-Newton Derivative-Free Optimization Method for Optimization Problems with Multiple Local Optima
More LessSummaryFor highly nonlinear problems, the objective function f(x) may have multiple local optima and it is desired to locate all of them. Analytical or adjoint-based derivatives may not be available for most real optimization problems, especially, when responses of a system are predicted by numerical simulations. The distributed-Gauss-Newton (DGN) optimization method performs quite efficiently and robustly for history-matching problems with multiple best matches. However, this method is not applicable for generic optimization problems, e.g., life-cycle production optimization or well location optimization.
In this paper, we generalized the distribution techniques of the DGN optimization method and developed a new distributed quasi-Newton (DQN) optimization method that is applicable to generic optimization problems. It can handle generalized objective functions F(x,y(x))=f(x) with both explicit variables x and implicit variables, i.e., simulated responses, y(x). The partial derivatives of F(x,y) with respect to both x and y can be computed analytically, whereas the partial derivatives of y(x) with respect to x (sensitivity matrix) is estimated by applying the same efficient information sharing mechanism implemented in the DGN optimization method. An ensemble of quasi-Newton optimization tasks is distributed among multiple high-performance-computing (HPC) cluster nodes. The simulation results generated from one optimization task are shared with others by updating a common set of training data points, which records simulated responses of all simulation jobs. The sensitivity matrix at the current best solution of each optimization task is approximated by either the linear-interpolation (LI) method or the support-vector-regression (SVR) method, using some or all training data points. The gradient of the objective function is then analytically computed using its partial derivatives with respect to x and y and the estimated sensitivities of y with respect to x. The Hessian is updated using the quasi-Newton formulation. A new search point for each distributed optimization task is generated by solving a quasi-Newton trust-region subproblem for the next iteration.
The proposed DQN method is first validated on a synthetic history matching problem and its performance is found to be comparable with the DGN optimizer. Then, the DQN method is tested on different optimization problems. For all test problems, the DQN method can find multiple optima of the objective function with reasonably small numbers of iterations (30 to 50). Compared to sequential model-based derivative-free optimization methods, the DQN method can reduce the computational cost, in terms of the number of simulations required for convergence, by a factor of 3 to 10.
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Dynamic Saturation Reconstruction for Multiphase Flow by Time-Of-Flight Fill Functions
By O. MoynerSummaryThe hyperbolic nature of transport equations makes multiphase simulations sensitive to numerical diffusion or smearing due to insufficient grid resolution or long time -steps, in particular for cases with linear or weakly nonlinear displacement fronts. The number of grid cells is often limited by the available computational resources, and is tightly coupled to the geological description.
Apart from increasing the grid resolution, several approaches have been taken to remedy the problem. The first is to use a more accurate scheme for the transport equations, e.g., in the form of a high-resolution finite-volume scheme, or by adding more degrees of freedom in the form of higher-order finite elements. Such schemes are well developed on rectilinear and curvilinear grids, but more challenging to formulate on general polytopal grids. A second approach is to use some form of upscaling to generate new pseudo-relative permeability/mobility functions, since the simulation grid in many cases is formed by upscaling an underlying finer geocellular grid.
Herein, we present a novel approach to two-phase flow, based on dynamic reconstruction of saturations, that combines the two approaches. The key idea is to solve the transport on a coarser grid, but use a set of numerically computed filling functions to reconstruct fine-scale saturation variations. These fill functions are computed by solving local flow and time-of-flight problems before the simulation. Each fill- function accounts for the local velocity field by a simple superposition of solutions, and ensures that any 1D solution can be mapped onto the underlying fine-scale cells while preserving the average saturation within the containing coarse block. By assuming that the local solution is a self-similar solution of a Riemann problem, we can approximate the fine-scale saturation distribution at any point in the coarse block. We demonstrate that this can give highly accurate results for both linear and Buckley-Leverett type flux functions for a range of heterogeneous test cases. A comparison is made with different levels of implicitness and a WENO scheme at both coarse and fine scales.
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Flow Diagnostics for Model Ensembles
Authors F. Watson, S. Krogstad and K. LieSummaryEnsembles of geomodels provide an opportunity to investigate the range of parameters and possible operational outcomes for a reservoir of interest. Full-featured dynamic modelling of all ensemble members is often computationally unfeasible, however some form of dynamic modelling, allowing us to discriminate between ensemble members based on their flow characteristics, is required.
Flow diagnostics involve simplified analysis of steady flow scenarios, single-phase or multiphase, and can be run in a much shorter time than a full dynamic multiphase simulation.
Fundamental quantities calculated for flow diagnostics include travel times, volumetric partitions, inter-well communications, and measures of dynamic heterogeneity. Heterogeneity measures like the dynamic Lorenz coefficient and sweep efficiency can be used as proxies for oil recovery in order to rank models. More advanced flow diagnostic techniques can also be used to estimate recovery.
We present two different forms of flow diagnostics metrics and investigate how well they perform in an ensemble setting. The first are based on volume-averaged travel times, which are calculated on a cell by cell basis from a given flow field. These measures are inexpensive to calculate and yield good results for relative rankings of models in the ensemble. The second use residence time distributions, which lead to more accurate results allowing for better estimation of recovery volumes. In addition, we have developed new metrics for better correlation between diagnostics and simulations when models have non-uniform initial saturations.
Three different ensembles of models are analysed; Egg, Norne, and Brugge. Very good correlation, in terms of model ranking and recovery estimates, is found between flow diagnostics and full simulations for all three ensembles. In the Egg and Norne examples, we consider uniform initial saturation and evenly spread well locations. Simulation results in terms of model ranking are well characterised by flow diagnostics based on volume-averaged travel times and residence time distributions, which are calculated using average initial saturations.
For the Brugge example, we consider producers placed in an oil cap, and demonstrate how the diagnostics results can be localized to the region of interest. We observe good correlations between simulations and simple flow diagnostic proxies for oil sweep. In addition, we also obtain good approximations for recovery when mapping saturation to the backward time-of-flight variable and solving 1D transport equations with the inter-well residencetime- distributions as source terms.
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Particle Transport Scheme for Embedded Discrete Fracture Models
Authors R. Monga, R. Deb, D.W. Meyer and P. JennySummaryEmbedded Discrete Fracture Models (EDFMs) for fractured porous media are preferable over Discrete Fracture Models if complex fracture geometries are to be fully resolved and the fractures and matrix discretizations are conformal. Lagrangian particle-tracking schemes offer convenient means for solute transport modeling because in EDFM frameworks, an orthogonal grid can be used irrespective of the fracture geometries. However, the absence of resolved fracture-matrix interfaces and different dimensionalities of the matrix and fracture continua motivate the use of a stochastic framework for particle-tracking. In this work, we developed a stochastic, time-adaptive particle-tracking scheme for EDFM models of fractured media with a permeable matrix. We formulated the probabilities of inter-continuum particle transfer, which have dependency on the particle travel time through the matrix/fracture control volumes. We showcase the conservative nature of the proposed particle-tracking scheme and additionally, illustrate the estimation of averaged solute concentration field. Such an illustration hints at the potential extensions of the tracking scheme, e.g., modeling of solute transport with kinetic reactions, and its incorporation into random walk models for dispersion in fractured media.
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Gauss-Newton Trust Region Search Optimization Method for Least Squares Problems with Singular Hessian
Authors G. Gao, F. Saaf, J. Vink, M. Krymskaya and T. WellsSummaryAlthough the Gauss-Newton trust-region sub-problem (GNTRS) solver using inverse-quadratic model (GNTRS-IQ) performs more efficiently than other direct solvers using matrix factorization and more robustly than available iterative solvers, it cannot compute the desired GN search step for many least-squares problems in which the Hessian is (near) singular. A popular approach to handle a singular matrix is to apply singular-value-decomposition (SVD). However, it is quite expensive to compute the SVD of a large matrix.
In this paper, we developed an integrated GNTRS solver by combining different methods together. For problems with a positive-definite Hessian, the GN search step is computed by solving a linear equation with N=min(Nd,Nm) unknowns, where Nd is the number of data and Nm the number of unknown parameters. Otherwise, we apply a linear transformation to reduce the dimension from Nm to r (the rank of Hessian) and then compute the GN step by solving a linear equation with r unknowns. When r=Nd< Nm, we use the sensitivity matrix J as the transformation matrix. When r is smaller than min(ND,Nm), we first compute the compact-SVD of J and then use the compact form of the right-singular matrix as the transformation matrix. For performance comparison, we also developed two GNTRS solvers using the traditional-SVD and compact-SVD formulation.
The three GNTRS solvers are validated and their performances are benchmarked on three sets of synthetic test problems. Each set contains 500 problems with different number of parameters and observed data. For small-scale and intermediate-scale problems, the solutions obtained by the three solvers have comparable accuracy. However, for large-scale problems, the solutions obtained by the solver using the traditional-SVD deviate from the solutions obtained by other solvers with unacceptably large errors. The integrated GNTRS solver performs most efficiently, and it can reduce the CPU time by a factor ranging from 1 to 173 and from 1 to 14585 when compared to the two solvers using the compact- and traditional-SVD. Our numerical tests confirm that the integrated GNTRS solver is efficient, robust, and applicable to least squares problems with singular Hessian. Finally, we applied the newly developed GN trust region search optimization method using the integrated GNTRS solver to a Gaussian-Mixture-Model (GMM) fitting problem. Compared with the one using the old version GNTRS-IQ solver, the new optimizer can reduce the CPU time used to construct an acceptable GMM by a factor of 8.
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Upscaling Low Salinity Water Flooding in Heterogenous Reservoirs
Authors H. Al-Ibadi, K. Stephen and E. MackaySummaryModelling the dynamic fluid behaviour of Low Salinity Water Flooding (LSWF) at the reservoir scale is a challenge which requires a coarse grid enable prediction in a feasible timescale. However, evidence shows that using low resolution models will result in a considerable mismatch compared with an equivalent fine scale model with the potential of strong numerically induced oscillations. This work examines two new upscaling methods in a heterogenous reservoir where viscous crossflow takes place to improve the precision of predictions.
We apply two approaches to upscaling of the flow to improve precision. In the first upscaling method, we shift the effective salinity range for the coarse model based on algorithms that we have developed to correct for numerical dispersion. The second upscaling method uses appropriate pseudo relative permeability curves that we derive. The shape of this new set of relative permeability is designed based on a modified fractional flow analysis of LSWF that we have developed and captures the relationship between dispersion and the waterfront velocities. This approach removes the need for explicit simulation of salinity transport. We applied these approaches in layered models and for permeability distributed as a correlated random field.
Upscaling by shifting the effective salinity range of the coarse model gave a good match to the fine case scenario, while considerable mismatch was observed for traditional upscaling of the absolute permeability only using averaging methods. For highly coarsened models, this method of upscaling reduces the oscillations appear, but they can be apparent. On the other hand, upscaling by using a single (pseudo) relative permeability produced more robust results with a very promising match to the fine scale scenario. These methods of upscaling showed promising results where they were used to upscale fully communicating and non-communicating layers as well as models with randomly correlated permeability.
Unlike documented methods in literate, these newly derived methods take into account the crucial effect of numerical dispersion and effective concentration on fluid dynamic using mathematical tools. These methods could be applied for other models where the phase mobilities change as a result of an injected solute, such as surfactant flooding and alkaline flooding. Usually these models use two sets of relative permeability and switch from one to another as a function of the concentration of the solute.
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Novel Stabilizations for A Piecewise Constant Lagrangian Formulation of Frictional Contact Mechanics with Hydraulically Active Fractures
Authors A. Franceschini, N. Castelletto, J. White, R. Settgast and H. TchelepiSummaryMany reservoir engineering applications involve tight coupling between fluid flow processes and poromechanical deformation. In particular, accurate simulation of phenomena like fault reactivation and fracture propagation strongly depends on the two-way coupled fluid-structure interaction. In this work, we focus on modeling frictional contact mechanics coupled with hydraulically active fractures. Specifically, fluid flow occurs inside the fracture, with the fluid pressure acting as an external load for the continuous body, and the conductivity of the fracture is a strong function of the bulk rock deformation.
In our numerical model we adopt a single conforming computational grid for both mechanical and flow processes. A cell-centered finite-volume scheme is used to solve the pressure field inside the fracture while the displacement field in the surrounding rock is approximated through first-order continuous finite elements. Contact conditions on the fracture are imposed through Lagrange multipliers, which represent the contact tractions. For the Lagrange multipliers we employ the same piecewise-constant interpolation (component-wise) used for the pressure approximation.
While this approximation space is convenient from a modeling perspective, the combination of linear displacement and piecewise constant traction/pressure variables is not uniformly inf-sup stable and requires a suitable stabilization. Hence, starting from a macroelement analysis, we develop three novel techniques, one local and two global, which aim at stabilizing the traction jumps across the elements discretizing the fracture surface. Effectiveness and robustness of proposed stabilization strategies are demonstrated and compared against complex analytical two- and three-dimensional benchmarks from the literature.
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Lattice Boltzman Method Assisting WAG Hysteresis and Trapped Non-Wetting Phase Simulations
Authors L.G Rodrigues, F. Munarin, H. Vasquez and S. LucenaSummaryThe numerical formulation of an oil reservoir is a formidable task that requires the contribution of several areas of expertise, often unrelated, at different scales. Since this is a hierarchical problem, errors introduced in one step will interfere with the next step, increasing inaccuracies. Our objective is to use predictive numerical methods in different simulation scales to study the oil-trapping relationship which is influenced by wettability, flow rate, interfacial tension, and saturation histories during WAG (Water Alternate Gas) process. This complex behavior requires rigorous models to considering the simultaneous flow of all three phases and, in addition, the reversibility of drainage and imbibition scanning curves is removed. The three-phase hysteresis model implemented during numerical reservoir simulation is based on the work of Larsen and Skauge and comparative scenarios were done for parameters that came from the Lattice Boltzmann method and typical values of literature. The Lattice Boltzmann method is used to simulate two-phase flow of water-oil and oil-gas through a porous medium in order to determine capillary pressure and relative permeability curves in a pore network. Molecular simulation of fluid properties (PVT, viscosity and interfacial tension) are performed to ensure the accuracy of the state equations used in the model. In this scale, density x pressure curves and viscosity x pressure curves similar to those obtained in experimental tests of differential liberation (1 to 400 bar) of reservoir fluids were obtained through molecular simulation models. Besides, the effect of the density ratio between the fluids and contact angle on the shape of the capillary pressure and relative permeability curves are investigated in the porous scale. Hysteresis is observed in all studied cases, becoming more apparent with large density differences. The density ratio is found to influence the pressure required to remove fluids from porous media and the volume of residual fluids trapped in it. The results are important for the study of these curves of a reservoir and confirm that the multi-component Lattice Boltzmann method can supply mesoscale information to take effect at the macroscale studies using reservoir simulation software.
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Fractured Reservoir Characterization in Brazilian Pre-Salt Using Pressure Transient Analysis with a Probabilistic Approach
Authors C.K. Quispe Cerna, D.J. Schiozer, G. Soares Oliveira, A. De Lima and R. B. Z. L. MorenoSummaryThe integration of dynamic data in the characterization of a fractured carbonate reservoir contributes to uncertainties reduction and construction of more reliable simulation models. This paper proposes the inclusion of pressure transient analysis with a probabilistic approach, in the characterization of a fractured carbonate reservoir to generation and calibration of stochastic discrete fracture network (DFN) models. The process aims to reduce uncertainty through the calibration of a set of realizations considering the well testing interpretation.
This work is supported by the pressure transient analysis performed in a reservoir located in Santos basin in Brazil´s pre-salt. The proposed methodology integrates the well testing interpretation considering their uncertainties, in the calibration and generation of stochastic sub-seismic fault models based on fractal hypothesis. We choose some realizations considering the faults density that crosses the wellbore be consistent with the borehole image logs and calibrated these realizations with the well testing. Later, we upscaled these models, imported the properties into numerical simulation models, and compared their results with those obtained by the simulation models generated before the proposed calibration.
Well test interpretation results showed characteristics of a fractured reservoir, presence of heterogeneities and boundaries. The analytical model used in the well test interpretation is supported by the borehole image logs, petrophysical data and seismic information. The inclusion of these results in the generation and calibration of DFN models allows us to obtain simulation results consistent with the well tests history, improving simulation models’ reliability. Likewise, this procedure reduced the high variability of the generated simulation models compared to simulation models corresponding to DFN models not calibrated. Additionally, the interpretation results enable us to estimate parameters of the reservoir and the well used in the numerical simulation model and also improve the characterization of the reservoir.
The main contribution of this work relies on the integration of pressure transient analysis considering uncertainties in its interpretation, into the calibration of stochastic DFN models. This methodology provides an alternative to the DFN models calibration that tries to reduce the variability and generate simulation models consistent with production data. Besides, we compare the DFN models calibrated by the proposed methodology with the DFN models not calibrated, revealing positive and negative aspects of this methodology.
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Estimation of the Chance of Success of A Four-Dimensional Seismic Project for A Developed Oil Field
Authors A.T.F.S. Gaspar, S.M.G. Santos, C.J. Ferreira, A. Davolio and D.J. SchiozerSummaryDeveloped oil fields often present challenges for further exploitation owing to existing production facilities. Frequently with a long production history, new wells cannot be drilled as freely when compared to earlier phases. As more knowledge is acquired along the course of field development, there is less room for changes, with a potential end point for data acquisition. This paper is based on a workflow originally conceived for quantifying the chance of success (CoS) of a four-dimensional (4D) seismic project for an oil field at the beginning of the development phase, when a complete infrastructure must be defined. Here, we apply this workflow to a developed oil field combined with an assisted production strategy optimization process proposed to optimize large-scale problems using a multilevel approach, allowing to estimate CoS, within a global and integrated decision analysis framework The optimization process is composed of steps to define and optimize decision variables of an oil production strategy, involving a given set of uncertain reservoir models, within a viable number of simulation runs through the use of automatic methods and reservoir engineering analyses. The information provided by 4D seismic data can be used to identify the most-likely reservoir model and, combined with numerical reservoir simulation, to optimize the control and field revitalization variables of the production strategy. This work compares the chance of success and the expected value of information (EVoI) methodologies. We use representative models selected from an ensemble of reservoir models based on cross plots of technical and economic objective functions, the associated risk curves and the probability distribution function of the uncertain attributes. The use of representative models makes the production strategy optimization and CoS and EVOI quantification processes practical, considering all the uncertainties and decision variables involved in the same run, where limiting computational costs is essential. We also analyze the influence of the number of representative models on these estimates. The results of this study showed that, although this oil field presented limited room for changes because of the late stage of development, 4D seismic information effectively impacted decisions regarding production strategy. Besides, our methodology showed that the expected economic gains from improved decisions are higher than the acquisition and processing costs related to information acquisition.
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Physics Based Deep Learning for Nonlinear Two-Phase Flow in Porous Media
Authors O. Fuks and H. TchelepiSummaryThere is growing interest in employing Machine Learning (ML) strategies to solve forward and inverse computational physics problems. The physics-informed machine learning (PIML) frameworks developed by Raissi et al.[ 1 ] and Zhu et al.[ 2 ] are prominent examples. The basic idea is to encode the partial differential equations (PDE) that govern the flow physics into the neural network. This encoding is achieved by enriching the loss function with the governing conservation equation. Using the initial and boundary conditions, the network is then able to learn the solution of the forward problem without any labeled data. The scarcity of site-specific “labeled” data presents serious challenges to modeling of Enhanced Oil Recovery (EOR) processes. Thus, if PIML approaches can be used to model the nonlinear flow and transport that govern EOR processes, then they could change the practice of reservoir simulation.
In this work, we explore the application of a particular PIML approach to solve the nonlinear hyperbolic equation that describes nonlinear immiscible two-phase flow in porous media. Specifically, we are concerned with the forward solution of a Riemann problem - a nonlinear conservation law together with piecewise constant data having a single discontinuity. It is well known that it is hard to solve this nonlinear transport problem, especially with a non-convex flux function, due to emergence of saturation shocks in the domain. The focus is on the pure forward problem, i.e., the absence of previously simulated (so-called labeled) data in the interior of the domain. The PIML framework breaks down for this nonlinear hyperbolic problem with non-convex flux function. We have found that it is essential to add a diffusion term to the underlying nonlinear PDE. That is, we used the parabolic form of the equation with a finite Peclet number. When the loss function includes a finite amount of diffusion, the neural network can actually produce reasonable approximations of the forward solution when shocks and mixed waves (shocks and rarefactions) are present.
For the obtained neural networks we also analyze the training process and provide 2-D visualizations of the loss landscape, then we discuss possible reasons for the observed behavior.
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Investigation of the Accuracy and Efficiency of the Operator-based Linearization through an Advanced Reservoir Simulation Framework
Authors A. Al-Jundi, L. Li and A. AbushaikhaaSummaryAny complex phase behavior computation is a main challenge in reservoir simulation since it introduces high nonlinearities. To overcome this, an operator-based linearization (OBL) has been introduced recently. In OBL, an operator format is applied to represent the mass-based formulations. By computing the values of the operators related to rock and fluid properties on pre-defined status, the values of the operators and their derivatives on any status, which emerges during a simulation run, can be determined by interpolation. Obviously, the accuracy of the results is mainly controlled by the pre-defined status. In this work, we present a detailed investigation of the accuracy and efficiency of the OBL. To guarantee an objective evaluation, a novel advanced parallel framework is applied for reservoir simulation. In this framework, we implement a multipoint linearization method that is capable to provide accurate, robust, and convergent solutions for reservoir simulation. The number of points in the parametric space of each nonlinear known is defined as resolution. By running simulations at different resolutions, we compare the numerical solutions with analytical solutions. It shows that the resolution has a large effect on the accuracy of numerical solutions. We also investigate the robustness of the OBL by running simulations on several models with different complexity of the phase behavior. Besides, by looking into the convergent process, we also study the efficiency of the OBL method. Finally, we test several filed cases to show the performance of the OBL method for general-purpose reservoir simulations.
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An Advanced Parallel Framework for Reservoir Simulation with Mimetic Finite Difference Discretization and Operator-based Linearization
Authors L. Li and A. AbushaikhaaSummaryReservoir simulation is the only way to reproduce flow response in subsurface reservoirs that drastically assists in reducing the uncertainties in the geological characterization and in optimizing the field development strategies. However, it is always challenging to provide efficient and accurate solutions for field cases which in turn further constrains the utilization of reservoir simulation. In this work, we develop a novel reservoir simulation framework based on advanced spatial discretization and linearization scheme, 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 nonlinear unknowns is treated piece-wisely 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 coupling these two novel schemes within a parallel framework, we can solve large and complex reservoir simulation problems in an efficient manner. Finally, we benchmark these methods with analytical solutions to assure their robustness, accuracy, and convergence. We also test several field cases to demonstrate the performance and scalability of the advanced parallel framework for reservoir simulation.
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Discontinuous Control Volume Finite Element Method for Multiphase Flow in Porous Media on Challenging Meshes
Authors J. Al Kubaisy, H. Osman, P. Salinas, C. Pain and M. JacksonSummaryControl volume finite element methods (CVFEM) are gaining increasing popularity for modeling multi-phase flow in porous media due to their inherited geometric flexibility for modeling complex shapes. Nonetheless, classical CVFEM suffer from two key problems; first, mass conservation is enforced by the use of control volumes that span element boundaries. Consequently, when modeling flow in regions with discontinuous material properties, control volumes that span geologic domain boundaries result in non-physical leakage that degrades the numerical solution accuracy. Another challenge is to provide an accurate solution for distorted elements; elements with high aspect ratio that are part of the discretized heterogeneous domain. In fact, most numerical methods struggle to provide a converged pressure solution for high aspect ratio elements of the domain.
Here, we introduce a numerical scheme that removes non-physical leakage across geologic domains and addresses the accuracy of classical control volume finite element method (CVFEM) in high aspect ratio elements. The scheme utilizes the frameworks of double-CVFEM (DCVFEM) where pressure is discretized CV-wise rather than element-wise. In addition, it introduces discontinuous control volumes by allowing pressure to be discontinuous between elements. The resultant finite element pair has an equal-order of velocity and pressure, with discontinuous linear elements for both the pressure and velocity fields P1DG–P1DG. This type of element pair is LBB unstable. The instability issue is circumvented by global enrichment of the finite element velocity interpolation space with an interior bubble function, given by the new element pair P1(BL)DG–P1DG. This element pair resolves both issues addressed earlier.
We demonstrate that the developed numerical method is mass conservative, and it accurately preserves sharp saturation changes across different material properties or discontinuous permeability fields as well as improves convergence to the pressure solution for distorted mesh, i.e. elements with high aspect ratio. We show the effectiveness of the presented formulation on realistic highly heterogeneous models.
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A Modeling Workflow for Geological Carbon Storage Integrated with Coupled Flow and Geomechanics Simulations
Authors J. Torres, I. Bogdanov and M. BoissonSummaryGeological Carbon Storage (GCS) is called to play a critical role for accelerating the transition toward a lowcarbon economy during the remainder of the century. However, for making a significant contribution, GCS targets should be scaled-up from megatons (per year) to gigatons levels. Among other challenges, the ability to perform reliable Reservoir Modeling and Simulation is an important aspect needed for an optimal management of potential risks. The injection of large volumes of fluid has raised public awareness about potentially induced seismicity. Subsurface mechanisms that may trigger seismicity are numerous and complex, and affect the prediction capabilities of the standard simulation tools. Understanding and simulating the physics of these complex phenomena is an important part of TOTAL’s CCUS R&D program, as it is the company’s ambition to become a major actor in CO2 storage activity, while ensuring risk management and mitigation. To improve their understanding, we studied fundamental aspects related with Reservoir Modeling and Simulation in the GCS context.
This study is the first step towards enhancing our understanding of these complex phenomena. We revisited the role that the presence of fractures and faults has on reservoir containment. A coupled flow and geomechanics simulation was developed to investigate the influence of different parameters on the fault reactivation of critically stressed faults. We evaluated different experimental scenarios, which resulted in values of axial displacements in the range of from 15 m to 170 m, within a lab model that has a characteristic length of 20m. The worst-case scenario relates to a situation with pre-existing vertical fractures. Results indicate that the dynamics of the fault permeability is a critical factor, among those that have to be taken into account for CO2 injection scenarios in complex geological formations. In our view, these results confirm that uncertainty in the fault characterization needs to be taken into account for improved risk assessments associated with CO2 injection.
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Optimization of Reservoir Surveillance Strategies Under Uncertainty: An Application to the Design of Sparse Monitoring Surveys
Authors E. Barros and O. LeeuwenburghSummaryReservoir monitoring or surveillance is crucial for a responsible and efficient use of subsurface reservoirs. In both production and storage systems, operators need to demonstrate that their assets can be managed in a safe way. Effective monitoring practices also help operators unlock additional value from their assets, by revising expectations, gaining confidence on their projected potentials and allowing development strategies to be adjusted.
The design of monitoring systems can be a challenge for both subsurface operators and regulators. The main difficulties range from the technical uncertainties on subsurface characterization and the unavailability of direct measurements to the lack of technologies to support monitoring design decisions.
As a step to bridge this gap, we have developed a quantitative value-of-information (VOI) methodology within the context of conformance management in CO2 storage operations. It is a practical model-based approach that uses a Bayesian framework to derive a measure of the expected contribution of (future) measurements from candidate monitoring strategies for discrimination of conformance and non-conformance situations.
In this work we integrate our practical VOI approach into an optimization workflow. This novel workflow is applied to determine the optimal design of time-lapse seismic surveys in a realistic CO2 storage case study. Obviously, the larger the spatial coverage of the survey, the more informative it will be. Therefore, the search for cheaper sparse survey designs is by nature a bi-objective problem which needs to consider both the accuracy requirements and the costs of the survey in the same optimization. Our optimization workflow also accounts for the uncertainties associated with the reservoir system by using ensembles of plausible measurement outcomes and model realizations.
Our results show that sparse survey designs can be optimized to reduce costs while keeping accuracy levels comparable to denser designs. Our results also suggest that optimal survey configurations lie on a Pareto front of the two objectives considered, corroborating the idea that the design of cost-efficient monitoring strategies has a multi-objective character.
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Application of Diffuse Source Basis Functions for Improved Near Well Upscaling
More LessSummaryNear well flow can have significant impact on the accuracy of the upscaling of geologic models. A recent benchmark study has shown that these errors may dominate over other aspects of upscaling in commercial reservoir simulators. This same study showed the advantage of "Diffuse Source" (DS) upscaling over previous approaches. We now demonstrate the application of the DS basis functions to the calculation of the upscaled well index and the well cell intercell transmissibility.
DS upscaling is an extension of pseudo-steady-state (PSS) flow-based upscaling that utilizes the diffusive time of flight to distinguish well-connected and weakly-connected sub-volumes. DS upscaling retains the localization advantage of a PSS calculation: unlike steady state flow, the local upscaling problem does not couple to adjacent regions and local-global iterations are not required. DS upscaling has been developed and utilized for the calculation of the intercell transmissibility, but we now apply it to calculation of the upscaled well index. Consistent with other researchers, we adjust the intercell transmissibility in the near well region.
We also consider the upscaling of the well index for a reservoir model in which the well trajectory and the high resolution geologic model are not simultaneously available. For many practitioners, this remains the most common reservoir modelling workflow. The result is an algebraic well index upscaling calculation, which also improves upon commercial applications.
The industry standard for the well index follows Peaceman. We show that PSS/DS upscaling reduces to Peaceman’s well index on a coarse grid, and is consistent with Peaceman’s numerical convergence analysis. (In contrast, steady state upscaling for the well index reduces to the Dietz well index.) The current approach is a generalization of Peaceman’s well index, but now extended to represent near well reservoir heterogeneity and with arbitrary placement of a well perforation within a simulation well cell.
Consistent with steady state upscaling, we find an advantage in adjusting the intercell transmissibility in the near well region. However, we have found that it is only necessary to do so for the well cell itself, which may be a consequence of the improved localization of the current calculation.
The new methodology performs very well. It is tested for several models, including the SPE10 reference model, the Amellago carbonate outcrop model, and the Equinor Volve full-field model. We compare the results to steady state flow-based upscaling, the algebraic well index upscaling described above, and to algorithms found in commercial applications.
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Deep-Learning Inversion to Efficiently Handle Big-Data Assimilation: Application to Seismic History Matching
Authors C. Xiao, A. Heemink, H. Lin and O. LeeuwenburghSummarySeismic history matching can play a key role in geological characterization and uncertainty quantification. However, challenges related to intensive computational demands and complexity restricts its application in many practical cases. This paper presents a conceptual deep-learning-based framework fully deployed in the popular Pytorch architecture to accelerate the seismic history matching. We introduce a surrogate model based on a deep convolutional neural network with a stack of dense blocks, specifically a conditional deep convolutional autoencoder-decoder architecture (cDCAE). The surrogate model allows us to fully deploy data assimilation algorithms in Pytorch architecture and hence to easily make full use of the efficient computing units, in particular GPU’s for the matrix-matrix and matrix-vector multiplications. The feature of built-in automatic differentiation (AD) provided by Pytorch also makes is possible to evaluate gradient information efficiently in a parallel manner. Furthermore, it has been acknowledged to benefit from the deep learning practice of using stochastic gradient (SG) optimizers, e.g., Adam, instead of full gradient optimizers, e.g., Quasi-Newton, as is most common in conventional big-data assimilation. The proposed framework is tested on a benchmark 3D model in the context of petroleum engineering. This surrogate model is demonstrated to be capable of accurately predicting the quantity of interest, e.g., dynamic saturation maps for new geological realizations. Assessments demonstrating high surrogate-model accuracy are presented for an ensemble of test models. The robustness and dramatic speedup provided by the surrogate model suggests that it can help facilitate the application of large-scale seismic history matching.
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Inclusion of Variable Characteristic Length in Microemulsion Flash Calculations
Authors D. Magzymov and R.T. JohnsSummaryRecent developments in predicting microemulsion phase behavior for use in chemical flooding are based on the hydrophilic-lipophilic deviation (HLD) and net-average curvature (NAC) equation-of-state (EoS). The most advanced version of the HLD-NAC EoS assumes that the three-phase micelle characteristic length is constant as parameters like salinity and temperature vary. In this paper, we relax this assumption to improve the accuracy and thermodynamic consistency of these flash calculations.
We introduce a variable characteristic length in the three-phase region based on experimental data that is monotonic with salinity or other formulation variables, such as temperature and pressure. The characteristic length at the boundary of the three-phase region is then used for flash calculations in the two-phase lobes for Winsor type II-/II+. The functional form of the characteristic length is made consistent with the Gibbs phase rule.
The improved EoS can capture asymmetric phase behavior data around the optimum, whereas current HLD-NAC based models cannot. The variable characteristic length formulation also resolves the thermodynamic inconsistency of existing phase behavior models that give multiple solutions for the optimum. We show from experimental data and theory that the inverse of the characteristic length varies linearly with formulation variables. This important result means that it is easy to predict the characteristic length in the three-phase region, which also improves the estimation of surrounding two-phase lobes. This improved physical understanding of microemulsion phase behavior should greatly aid in the design of surfactant blends and improve recovery predictions in a chemical flooding simulator.
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