Monotone Non-Galerkin Algebraic Multigrid Method Applied to Reservoir Simulations

The windowing technique was introduced in 1993 by Heinemann and Deimbacher (Ref 9). This method allows a locally restricted and time-dependent replacement of grids and parameters during simulation runs. Windows can represent any area of special interest in a reservoir. In most cases this will be the near wellbore area, in order to resolve near wellbore effects and to honor the pattern of flow. Introducing a wellblock with exactly the same dimensions as the real wellbore and gridding the near wellbore area allows to model all sorts of well trajectories, by applying the 3D (k-)PEBI grid method. Introducing small block sizes requires a drastic reduction of the timestep length. Therefore the windowing technique can only be applied if the solution of the equations for the full model and the solution of the window area is decoupled. This can be done based on a special kind of Domain Decomposition. In a first step the fully implicit solution for the full-scale model will be calculated but the inner blocks of the windows are solved for the pressure only, without updating the saturations and mole fractions, providing the boundary influx for the windows, which are then solved for the same overall time step with up to 1000 small local steps. The windowing technique can find numerous applications. Keeping overall CPU time consumption in he order of magnitude of a conventional model, the windowing technique is a potential approach to replace the analytical Peaceman (Ref. 23) model in full field reservoir simulation, while increasing the quality of the results. A further application of this technique is the simulation of well tests in full field reservoir models. This paper presents the advances in the gridding and dynamic solution process. Several examples to demonstrate the advantages of this technique for detail - and full field reservoir simulation will be included. Introduction Special treatment and high resolution, both in time and space, of regions of interest, as for example the near wellbore area, has always been an important question in reservoir simulation. But using small timesteps and local grid refinement unavoidably lead to long and uneconomical CPU times. In the attempt to solve this problem several techniques have been developed in the reservoir simulation literature, such as patch refinement. true refinement or multigrid methods (Ref. 2–6), but none of these approaches could fully satisfy the requirements of local grid replacement by flexible and irregular grids and its time dependent incorporation in full field reservoir simulation. In 1986 Aziz and Pedrosa introduced the concept of "hybrid grids" in a Cartesian block system (Ref. 1), trying to overcome the insufficiencies of the Peaceman model. Gridding the near wellbore area by hybrid grids honors the radial nature of flow, small block sizes help handling large saturation changes caused by high production rates. The paper showed that the solution of the equations in the radially gridded well region can be decoupled from the solution of the equations of the conventional grid system by either applying an iterative or a direct solution method.

The key idea with multiscale methods for reservoir simulation is to construct a set of prolongation operators that interpolate solutions from a coarse spatial resolution to the grid resolution. Efficient multiscale methods need prolongation operators that accurately represent flow at the grid resolution. For high-contrast models, it is especially important that this flow interpolation is confined within high-contrast boundaries. In this paper, we present an improved algorithm to construct multiscale prolongation operators that better capture strong contrasts in geological properties. Specifically, to construct effective prolongation operators, the improved algorithm first finds dominant flow directions by comparing the values of connection transmissibility in a neighborhood, then emphasizes the interpolation along these dominant directions and ignores the interpolation in transverse direction if connection transmissibility is weak. The new algorithm is implemented in a commercial reservoir simulator that also provides a commercial implementation of a state-of-the-art multiscale method. The advantage of the new algorithm is demonstrated using synthetic and real reservoir models with high-contrast features. We also analyze the interpolation errors of poorly constructed prolongation operators for such models to identify the root cause of the slow linear solver convergence rate. With the new algorithm, we obtain better linear and nonlinear convergence rates in the pressure solver and shorter simulation time than with a previously published state-of-the-art multiscale method. For completeness, we also benchmark our multiscale pressure solver performance against a standard algebraic multigrid (AMG) fine-scale pressure solver, and we highlight differences in linear solver convergence and computational efficiency. Finally, we demonstrate that the new algorithm is beneficial for a real high-contrast heterogeneous field model.

Parallel multi-stage preconditioners with adaptive setup for the black oil model

ESA: An efficient sequence alignment algorithm for biological database search on Sunway TaihuLight

Compositional reservoir simulation of gas injection processes is commonly performed with a reduced number of components to minimize computation time. However, in simulations of downstream processes, more detailed fluid descriptions are required to design and optimize separation processes. To connect up and down-stream processes, delumping approaches have been proposed to deconvolute the well/production streams from a reservoir simulator into a more detailed fluid description. The accuracy of delumping is closely tied to the lumping approach that is used to generate equation of state (EOS) input for the reservoir simulator. To this end, we have previously introduced a flow-based lumping scheme that includes information from the dynamics of a given displacement process into the grouping of detailed components. In this paper, we investigate the accuracy of the delumped production streams from various gas injection processes and compare results from different lumping schemes. We investigate the benefit of applying flow-based lumping schemes in the selection of pseudo-components for compositional reservoir simulation and find an improvement in terms of uniqueness and overall accuracy of the composition of the delumped well streams. Examples are provided from 3D displacement calculations of enhanced recovery by CO2 and separator gas injection for multicomponent reservoir fluids where a substantial amount of PVT data is available (CME, DDE, swelling test and slimtube MMP). Using a commercial reservoir simulator, the delumped compositions of the produced fluids are compared to detailed displacement calculations for three different lumping schemes and two different gas injection processes. We demonstrate that the proposed lumping/delumping strategy provides additional control of the predictive quality over currently available lumping/delumping approaches. In addition, the proposed workflow provides a unique answer to the question of how to lump a detailed fluid description for displacement calculations. A majority of oil fields currently under production as well as new developments are candidates for EOR processes such as gas floods. The approach presented in this paper are directly applicable to the study and design of such EOR processes an offers an improved workflow for integration of up and down-stream processes. This is, to our best knowledge, the first paper that presents a comparison of delumping results for flow-based lumping schemes in the context of gas injection processes in with emphasis on the accuracy of delumped production stream relative to other available lumping/delumping methods.

The parallel multigrid method is expected to play an important role in large-scale scientific computing on post-peta/exa-scale supercomputer systems, and it also includes serial and parallel communication processes which are generally expensive. In the present work, new format for sparse matrix storage based on sliced Ellpack-Itpack (ELL) format is proposed for optimization of serial communication in data transfer through memories, and hierarchical coarse grid aggregation (hCGA) is introduced for optimization of parallel communication by message passing. The proposed methods are implemented for pGW3D-FVM, a parallel code for 3D groundwater flow simulations using the multigrid method, and the robustness and performance of the code was evaluated on up to 4,096 nodes (65,536 cores) of the Fujistu FX10 supercomputer system at the University of Tokyo. The parallel multigrid solver using the sliced ELL format provided performance improvement in both weak scaling (25%–31%) and strong scaling (9%–22%) compared to the code using the original ELL format. Moreover, hCGA provided excellent performance improvement in both weak scaling (1.61 times) and strong scaling (6.27 times) for flat MPI parallel programming model.

Using commercial numerical reservoir simulators to build a full field reservoir model and simultaneously history match multiple dynamic variables for a highly complex, offshore mature field in Malaysia, had proven to be challenging, manpower intensive, highly expensive, and not very successful. This field includes almost two hundred wells that have been completed in more than 60 different, non-continuous reservoir layers. The field has been producing oil, gas and water for decades. The objective of this article is to demonstrate how Artificial Intelligence (AI) and Machine Learning is used to build a purely data-driven reservoir simulation model that successfully history match all the dynamic variables for all the wells in this field and subsequently used for production forecast. The model has been validated in space and time. The AI and Machine Learning technology that was used to build the dynamic reservoir simulation and modeling is called spatio-temporal learning. Spatio-temporal learning is a machine-learning algorithm specifically developed for data-driven modeling of the physics of fluid flow through porous media. Spatio-temporal learning is used in the context of Deconvolutional Neural Networks. In this article Spatio-temporal Learning and Deconvolutional Neural Networks will be explained. This new technology is the result of more than 20 years of research and development in the application of AI and Machine Learning in reservoir modeling. This technology develops a coupled reservoir and wellbore model that for this particular oil & gas field in Malaysia uses choke setting, well-head pressure and well-head temperature as input and simultaneously history matches Oil production, GOR, WC, reservoir pressure, and water saturation for more than a hundred wells through a completely automated process. Once the data-driven reservoir model is developed and history matched, it is blind validated in space and time in order to establish a reliable and robust reservoir model to be used for decision making purposes and opportunity generation to maximise the field value. The concepts and the methodology of history match of multiple wells, individual offshore platforms, and the entire field will be presented in this article along with the results of blind validation and production forecasting. Results of using this model to perform uncertainty quantification will also be presented. A case study of a highly complex mature field with large number of wells and years of production has been used to be studied and simulated by this data-driven approach. Time, efforts, and resources required for the development of the dynamic reservoir simulation models using AI and Machine Learning is considerably less than time and resources required using the commercial numerical simulators. It is validated that the TDM developed model can make very reasonable prediction of field behavior and production from the existing wells based on modification of operational constraints and can be a prudent complementary tool to conventional numerical simulators for such complex assets.

This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set of system unknowns comprises both element and face pressures, in addition to the cell saturations, resulting in a 3×3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3\ imes 3$$\\end{document} block-structured Jacobian matrix with a 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} inner pressure problem. The CPR method is one of the most established techniques for reservoir simulations, but most research focused on solutions for Two-Point Flux Approximation (TPFA)-based discretizations that do not readily extend to our problem formulation. Therefore, we designed a dedicated two-stage strategy, inspired by the CPR algorithm, where a block preconditioner is used for the pressure part with the aim at exploiting the inner 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} structure. The proposed preconditioning framework is tested by an extensive experimentation, comprising both synthetic and realistic applications in Cartesian and non-Cartesian domains.

The application of parallel computers to reservoir simulation is studied with the aid of two recently introduced Multiple Instruction, Multiple Data (MIMD) computers. The problems posed by reservoir simulators are shown to be particularly well suited for such parallel computers* Fundamental parallel programming techniques using Fortran are presented for the Hetrogeneous Element Processor (HEP) produced by Denelcor and the iPSC Hypercube produced by Intel. These techniques are then applied to two of the most time consuming tasks in reservoir simulation: forming the matrix coefficients and producing a sparse matrix solution. The problem of parallel formation of sparse matrix coefficients is addressed for the single and multiphase cases using both black oil and compositional fluid models. Parallel sparse matrix solution is addressed by considering D4 ordered Gaussian elimination, and the multigrid, conjugate gradient, and Successive Over-Relaxation (SOR) methods. Three parallel algorithms based on SOR are developed to illustrate the possible diversity of parallel algorithms. The red-black and multicolored SOR methods which are often used on vector machines are shown to be easily adaptable to parallel MIMD machine. Matrix partitioning and a new method of iteration pipelining are also presented as parallel SOR methods. The various SOR algorithms, in parallel form, are mapped onto the HEP to illustrate the speedups currently possible on an MIMD machine. Projections are made on the future importance of these types of machines to reservoir simulation.

This paper presents a new approach for the simulation of fractured or partially fractured reservoirs with a general-purpose reservoir simulator. In many fractured reservoirs, fractures could be found in limited zones or "sweetspots". Hence, not all parts of the reservoir will typically require dual porosity or dual permeability treatment. The method presented in this paper allows creation of dual porosity or dual permeability cells wherever the attributes of the fracture geological model suggest to do so. The remaining - unfractured - part of the model can be treated like conventional single porosity areas. This approach will be called adaptive dual continuum. As a result, the generated simulation grid block models will resemble the real geological situation more closely than any other modeling approach. This will result in maximum reliability of the simulation model and allow better estimation of recovery for the various recovery processes modeled. All fracture and matrix-fracture transfer properties are derived from the geological information and fracture intensity. The simulator - that applies an unstructured gridding method - will store them on a cell by cell basis. The paper will discuss the new workflow from generating the geological model by combining the strengths of continuous and discrete models, thereby achieving a high level of data integration until the consistent data export into the reservoir simulator. The adaptive dual continuum approach was successfully implemented into a commercial reservoir simulator and several examples will prove the capability of this reservoir modeling approach for accurate and economic simulation of fractured or partially fractured reservoirs.

A physically-supported data-driven proxy modeling based on machine learning classification methods: Application to water front movement prediction

Steady-state two-phase relative permeability upscaling in synthetic and X-ray computerized tomography (CT) coal cores is performed with a three-dimensional (3D) reservoir simulator using an automated control procedure to drive a series of steady-state fractional flows. A clear understanding of relative permeability in coal is important for coalbed methane reservoir management from pore scale to sales point, as it is valuable for helping forecast production. Automation control enables greater continuity between physical corefloods and the numerical upscaling of the same coreflood procedure. Absolute permeability is computed for primitive synthetic core types using the reservoir simulator and is compared to an analytical formulation to validate the use of the simulator solution for core scale property determination. Relative permeability was computed for synthetic cores considering several scenarios: fracture geometry/abundance (parallel vs. intersecting), rock-type matrix distribution (homogeneous vs. heterogeneous), ratios of matrix-to-fracture permeability (high vs. low), and injection rate conditions [capillary limit (CL) vs. viscous limit (VL)]. Additionally, injection rate conditions were evaluated in the upscaled relative permeability of an X-ray CT segmented composite coal core. Analysis of the upscaled relative permeability curves in the composite and synthetic cores illustrated the impact of each scenario on upscaling relative permeability and suggests that selected characteristics of unconventional cores can potentially be used to delineate parameter dependence in a manner similar to rock type volume fraction and ordering in conventional cores. The consistency of the developed upscaled results with previous studies confirms the applicability of automated process control in core scale multiphase upscaling using a commercial reservoir simulator at varied injection rates and upscaling conditions.

In reservoir simulation, algebraic multigrid (AMG) solvers are the most effective solvers for linear systems arising from the discretization of pressure equations. The AMG solvers are also important components for many multi-stage preconditioners. In this paper, we introduce our work on developing these solvers on NVIDIA Tesla GPUs. A classical algebraic multigrid solver is developed. Numerical experiments are performed to test the efficiency of this GPU-based parallel AMG solver. The numerical experiments show that the GPU-based algebraic solver is sped up to 12.5 times faster than the corresponding CPU-based algebraic multigrid solver.

A fully-coupled flow-reactive-transport formulation based on element conservation, with application to CO2 storage simulations

We describe our multi-threading parallelization strategy of a general-purpose reservoir simulator (GPRS) based on a flexible Automatic Differentiation (AD) framework. Parallel Jacobian construction is achieved with a thread-safe extension of our AD library. For linear solution, we use a two-stage CPR (Constrained Pressure Residual) preconditioning strategy, combining the parallel multigrid solver XSAMG and the Block Jacobi technique with Block ILU(0) applied locally. The speedup of the full SPE 10 problem (1.1M cells) is about 5.0X on a dual quad-core Nehalem node. We then discuss the GPU parallelization of Nested Factorization (NF). The Massively Parallel NF (MPNF) algorithm was first introduced by Appleyard et al. (2011), where the 3D structured grid is divided into kernels, and each kernel is assigned a color such that no neighbouring kernels share the same color. Then, parallelism is exploited in the concurrent solution of all kernels with the same color. The most important aspects of our algorithm are: 1) coalesced memory access via special ordering of the matrix elements, and 2) application of the twisted factorization technique that further improves parallelism. With a 512-core Tesla M2090 GPU, the speedup of the full SPE10 problem is about 26X for single precision, and 19X for double precision.