Practical considerations for adaptive implicit methods in reservoir simulation

This paper describes an Adaptive Implicit Method (AIM) for reservoir simulation. The method is based on an extension of the generalized compositional solution procedure. In addition to the creation of a robust method, we have tried to minimize the overhead associated with the method, so that our implementation is almost as fast as an IMPES (implicit-pressure-explicit-saturation) method when few implicit blocks are needed. Several test problems demonstrate important aspects of the switching logic and the tradeoffs between large highly implicit time steps and smaller time steps with less implicitness. We find that in most cases the greatest efficiency occurs when only a small number of blocks (less than 5 percent) are treated implicitly. In many cases the most efficient method is IMPES with time steps selected using the numerical stability calculations of AIM, Smaller time steps lead not only to higher efficiency, but also to solutions with less numerical dispersion. In general. AIM appears to benefit compositional problems more frequently than black oil problems.

We propose a new, efficient, adaptive algebraic multigrid (AMG) solver strategy for the discrete systems of partial differential equations arising from structured or unstructured grid models in reservoir simulation. The proposed strategy has been particularly tailored to linear systems of equations arising in adaptive implicit methods. The coarsening process of the AMG method designed automatically employs information on the physical structure of the models; as a smoother, an adaptive ILUT method is employed, taking care of an efficient solution of the hyperbolic parts whilst providing adequately smooth errors for the elliptic parts. To achieve a good compromise of high efficiency and robustness for a variety of problem classes – ranging from simple, small black-oil to challenging, large compositional models – an automatic, adaptive ILUT parameter and AMG solver switching strategy, α-SAMG, has been developed. Its efficiency is demonstrated for eight industrial benchmark cases by comparison against standard one-level and AMG solvers as well as the pure one- level variant of the proposed new strategy. In addition, very promising results of first parallel runs are shown.

Modern petroleum reservoir simulation serves as a primary tool for quantitatively managing reservoir production and planning new fields. It involves repeatedly solving the Jacobian of a set of strong nonlinear partial differential equations governing the mass and energy conduction and conservation. Most of the existing reservoir simulators adopt iterative solver with multiple stages of preconditioners, in which the incomplete LU (ILU) factorization is an outstanding universal smoother. However, it turns out that when the degree of freedom of each grid grows, ILU usually becomes the bottleneck of the solver. Moreover, ILU is difficult to parallelize due to its inherent data dependency. In this paper, we developed a sparse iterative solver with parallelized ILU and triangular solve using block-wise data structure. Compared with the state of art iterative solver on 14 industrial reservoir simulation matrices, the proposed ILU is 5.2x faster (on average) than the state of art iterative solver because of the block-wise data structure, which leads to 2.2x speedup on the total solver runtime. In addition, parallel ILU and triangular solve are developed to further accelerate the solver. To tackle the strong data dependency in ILU and triangular solve, we first partition the algorithm into separated tasks and construct a data flow graph to represent the data dependency. Then, tasks are scheduled in parallel according to the topological order of the data flow graph. On an 8-thread multicore architecture, we achieved another 3.6x speedup on ILU factorization, and 3.3x on triangular solve with good scalability.

The estimation of the initial oil in place is a crucial topic in the period of exploration, appraisal, and development of the reservoir. In the current work, two conventional methods were used to determine the Initial Oil in Place. These two methods are a volumetric method and a reservoir simulation method. Moreover, each method requires a type of data whereet al the volumetric method depends on geological, core, well log and petrophysical properties data while the reservoir simulation method also needs capillary pressure versus water saturation, fluid production and static pressure data for all active wells at the Mishrif reservoir. The petrophysical properties for the studied reservoir is calculated using neural network technique from 13 cored and logged wells. The results showed that the reservoir simulation method gave a value of Initial Oil in Place that agrees and is close to the value of Initial Oil in Place obtained from the volumetric method with a percentage different around 2%. However, the estimation of Initial Oil in Place by reservoir simulation method offered accurate results during good history matching with observed data as well as making appropriate adjusting for Pc vs. Sw values for the whole reservoir from October, 1976 until December, 2020. MB21 unit own most Initial Oil in Place equal to 525*106 SM3 while MB12 has lowest initial oil in place equal to 2*106 SM3. Finally, the calculation of Intial Oil in Place by both volumetric and simulation methods presented good results while comparing with previous study at 2013 with discovered different around 1.5% and 0.6% respectively.

SummaryRecently, graphics processing units (GPUs) have been demonstrated to provide a significant performance benefit for black-oil reservoir simulation, as well as flash calculations that serve an important role in compositional simulation. A comprehensive approach to compositional simulation based on GPUs has yet to emerge, and the question remains as to whether the benefits observed in black-oil simulation persist with a more complex fluid description. We present a positive answer to this question through the extension of a commercial GPU-based black-oil simulator to include a compositional description based on standard cubic equations of state (EOSs). We describe the motivations for the selected nonlinear formulation, including the choice of primary variables and iteration scheme, and support for both fully implicit methods (FIMs) and adaptive implicit methods (AIMs). We then present performance results on an example sector model and simplified synthetic case designed to allow a detailed examination of runtime and memory scaling with respect to the number of hydrocarbon components and model size, as well as the number of processors. We finally show results from two complex asset models (synthetic and real) and examine performance scaling with respect to GPU generation, demonstrating that performance correlates strongly with GPU memory bandwidth.NOTE: This paper is also published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

This project will reactivate ARCO`s idle Pru Fee lease in the Midway-Sunset field, California and conduct a continuous steamflood enhanced oil recovery demonstration aided by an integration of modern reservoir characterization and simulation methods. Cyclic steaming will be used to re-establish baseline production within the reservoir characterization phase of the project. During the demonstration phase, a continuous steamflood enhanced oil recover will be initiated to test the incremental value of this method as an alternative to cyclic steaming. Other economically marginal Class III reservoirs having similar producibility problems will benefit from insight gained in this project. The objectives of the project are: (1) to return the shut-in portion of the reservoir to commercial production; (2) to accurately describe the reservoir and recovery process; and (3) to convey the details of this activity to the domestic petroleum industry, especially to other producers in California, through an aggressive technology transfer program. A summary of technical progress covers: geological and reservoir characterization, and reservoir simulation.

The purpose of the article is to present and evaluate usefulness of a new preconditioning technique (for the Gauss-Seidel algorithm), namely incomplete WZ factorization, for iterative solving of sparse and singular linear equations systems, which arise during modeling with Markov chains. The incomplete WZ factorization proposed here will be compared with the incomplete LU factorization in respect of amount of fill-in (newly created non-zeros) and in respect of the accuracy improvement of preconditioned algorithms in relation to not preconditioned ones. In the paper, the results of some numerical experiments will be presented, which were conducted for various matrices representing Markov chains. The experiments show that the incomplete WZ factorization can be a real alternative - because it is faster than incomplete LU factorization and the fill-in generated in the process is smaller (the output matrices are sparser).

The exploitation of an oil field is a complex and multidisciplinary task, which demands a lot of prior knowledge, time, and money. A good reservoir characterization is deemed essential in the accomplishment of Enhanced Oil Recovery (EOR) processes in order to estimate accurately the properties of the porous medium affecting the flow properties. Several techniques at a field scale are currently being used to determine these properties, which are time and money consuming. But these alone do not guarantee the success of the project. Reservoir simulation and numerical techniques were then included in the pre-development and follow-up studies as an effective tool to determine the productivity and future behavior of the oil field. As the computational power increased, more advanced and detailed models were developed, including different chemical and physical phenomena. But alongside this process, there was an active research in the area of reservoir simulation, improving the accuracy and efficiency of the numerical schemes used for the flow, transport, and energy equations. The aim of this review is to address the topics described. Firstly, the origin of an oil recovery process, the economic factors and field tests involved are introduced. Secondly, the oil and porous medium origin and characterization as well as an introduction to the fundamental concepts and equations are associated to reservoir simulation. Finally, a brief description and analysis of the techniques are used in reservoir simulation employing finite difference methods, their downsides and possible ways to overcome these problems.

The objective of this study is to evaluate CO2 storage potential of a small gas field located in Northern Oman using advanced reservoir simulation techniques. The field is to be used for CO2 storage in the future when this reservoir is depleted. The study includes the estimation of structural, residual, solubility and mineral trapping mechanisms, as well as determination of safe operating pressure limits for CO2 injection. The study utilizes various reservoir simulation methods to investigate each trapping mechanism, taking into account the hydrodynamic, physical, and chemical conditions within the reservoir. Specifically, solubility trapping is analyzed using Henry's law, while residual trapping is modeled using the critical gas saturation via Corey correlations. To analyze risks of faults reactivation and leakage into overlying formations, the geomechanical analysis was applied in the study. At the end of the study, a sensitivity analysis is performed to assess the influence of uncertain parameters for each technical approach. The results of the case study provided both quantitative and qualitative evaluations of each trapping mechanism for the gas field. Potential deformation and fracturing of the caprock trap were predicted to ensure it remains intact to prevent CO2 leakage. Additionally, the study demonstrated the practical application of various reservoir simulation techniques for modelling CO2 storage, adding further value to the study. In spite of the fact, that the gas field has been well studied during exploration and production periods, including previous reservoir simulations, a sufficient level of uncertainty still exists in reservoir description, fluid and rock properties and even was increased with adding CO2 storage approaches. This study took into account these uncertain parameters, and probabilities were calculated for the estimations of trapping mechanisms. This case study presents a complete modelling cycle for CO2 storage within a depleted gas reservoir, employing a range of diverse reservoir simulation techniques. It serves as a valuable guide for those interested in modelling CO2 storage through the application of various reservoir simulation methods.

When solving large linear systems of equations arising from the discretization of elliptic boundary value problems, a combination of iterative methods and preconditioners based on incomplete LU factorizations is frequently used. Given a model problem with variable coefficients, we investigate a class of incomplete LU factorizations depending on a relaxation parameter. We show that the associated preconditioner and the factorization itself both are numerically stable. The theoretical results are complemented by numerical experiments.

A highly efficient and robust scheme is proposed for analyzing electromagnetic scattering from electrically large arbitrary shaped conductors in a half space. This scheme is based on the electric field integral equation (EFIE) with a half-space Green’s function. The precorrected fast Fourier transform (p-FFT) is first extended to a half space for general three-dimensional scattering problems. A novel enhanced dual threshold incomplete LU factorization (ILUT) is then constructed as an effective preconditioner to improve the convergence of the half-space EFIE. Inspired by the idea of the improved electric field integral operator (IEFIO), the geometrical-optics current/principle value term of the magnetic field integral equation is used as a physical perturbation to stabilize the traditional ILUT perconditioning matrix. The high accuracy of EFIE is maintained, yet good calculating efficiency comparable to the combined field integral equation (CFIE) can be achieved. Furthermore, this approach can be applied to arbitrary geometrical structures including open surfaces and requires no extra types of Sommerfeld integrals needed in the half-space CFIE. Numerical examples are presented to demonstrate the high performance of the proposed solver among several other approaches in typical half-space problems.

Summary A new switching criterion for adaptive-implicit reservoir simulation, based on the numerical stability of the local amplification matrix, is discussed. Adaptive-implicit methods can be used to achieve substantial savings in computing time and storage. The success of their application, however, depends critically on an appropriate switching criterion. The standard criterion discussed in the literature is the application of threshold changes in the primary variables, which are usually determined by some preliminary testing and chosen conservatively because of their empirical nature. Further, backward switching (implicit to explicit) is not normally feasible. With the switching criterion discussed in this paper, these disadvantages are eliminated. The method has been applied to a variety of black-oil reservoir simulation problems, including the first and second SPE Comparative Solution Projects and a water-injection problem. In all cases tested, the new criterion is shown to perform satisfactorily. For the waterflood problem in particular, the new criterion with backward switching was found to decrease computing time and storage significantly compared with the standard switching criterion. This criterion can easily be extended to other types of reservoir simulators.

Many physical processes such as multiphase flow in heterogeneous porous media are described by partial differential algebraic equations (PDAEs) as a set of partial differential, ordinary differential and algebraic equations. However, due to lack of exact analytical solution for most PDAEs, numerical methods should be used to approximate their solution. The main contribution of this article is a comparison among the features of such numerical methods for oil and gas reservoir simulation. Different gridding techniques have been included in this comparison to find the most suitable numerical scheme to simulate multiphase flow in heterogeneous porous media.

Incomplete LU (ILU) methods are widely used in petroleum reservoir simulation and many other applications. However high complexity often makes them the hotspot in the whole simulation due to high complexity when problem size is large. ILU’s inherent serial nature also makes them difficult to take full advantage of computing power of multi-core and many-core devices. In this paper, a greedy graph coloring method is applied to the ILU(k) factorization and triangular solution phases. This method increases degree of parallelism and improves load balance. A block-wise storage format is employed in our ILU implementation in order to take advantage of hierarchical memory structures. Moreover, a dual intensive parallel model is proposed to further improve the performance of ILU(k) on GPUs. We test the performance of the proposed parallel ILU(k) with a set of Jacobian systems arising from petroleum reservoir simulation. Numerical results suggest that the proposed parallel ILU(k) method is effective and robust on multi-core and many-core architectures. On an Intel Xeon E5 multi-core CPU, the speedup compared with the serial execution time is \(5.6\times \) and \(5.4\times \) for factorization and triangular solution, respectively; on an Nvidia K40c GPU card, the speedup can reach \(8.6\times \) and \(12.7\times \) for factorization and triangular solution, respectively.

A 3-D field scale steam injection simulator is developed and tested. The three-dimensional multi-phase mass and energy flow equations are based on compositional balances. A fully implicit numerical solution method is implemented in the solution of the nonlinear system of equations. The model solves for (nhcd+5) primary variables simultaneously where n, denotes the number of hydrocarbon components permissible. The accuracy and efficiency of the model is verified and benchmarked against the compositional problem of the Fourth SPE Comparative Solution Project: Comparison of Steam Injection Simulators. Large computational time and extensive hardware facility requirements are the two major sources of difficulty on compositional simulation of hydrocarbon reservoirs. Nonetheless, it is important to include the correct physics of the problem to be modeled in fine details. The problem is aggravated more by the highly nonlinear nature of most of the improved oil recovery operations requiring computationally demanding, numerically stable fully implicit solutions. Although explicit formulations or semi-implicit formulations are computationally less demanding, they cannot adequately handle highly nonlinear nature of the chemistry and physics encountered in most of the oil recovery operations making their application almost impossible for this class of problems. In this paper by investigating a trade-off between the degree of implicitness and stability as well as a required level of accuracy of the results and computational time, we present a sensitivity analysis of the research results obtained by solving the mathematical model via various implicit formulations with different sets and numbers of primary variables. Understanding the relationship between the degree of implicitness and the accuracy of the simulation results will help in deciding on the degree of refinement of the grid blocks and level of detail of the actual problem representation in the simulation model. In order to speed up the simulator, for each of the implicit formulations implemented, various primary variable/equation alignment schemes are considered in the construction of the Jacobian. The simulator is tested using a wide range of compiler/CPU combinations. While performing well on a certain computational environment the very same alignment may result in logarithmically singular Jacobian on another compiler/CPU combination. An analysis on the round off error dependence of the mathematical stability of the Jacobian as a function of primary variable/equation alignment schemes is presented. Accordingly, a set of criteria for the optimum primary variable/equation alignment is proposed to increase the efficiency of the model. To the authors' knowledge, the model presented in this paper is one of the more implicit models reported in the literature. The developed simulator is found to be strongly stable and internally consistent even for the most dynamic and stiff systems as the ones studied in this research. This is benchmarked by the model's ability in accommodating large average time step sizes of over 50 days and its high average convergence rates of less than three Newtonian iterations per time step.