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Keywords: matrix

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Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Petroleum Computer Conference, June 2–5, 1996

Paper Number: SPE-36007-MS

... geometries (like Voronoi, median, boundary adapting grids, etc.). The entire problem is divided into subsystems like geometry, gridnodes, gridnode connectivity, grid, reservoir fluid flow, and

**matrix**. Each of these subsystems have objects which are closely related. The dependency of these subsystems...
Abstract

Abstract This paper describes the design of FLEX, an object-oriented, flexible grid, black-oil reservoir simulator helps in dealing with the complexity of this problem. This approach is particularly useful because of the difficulties associated with generation and use of flexible grid geometries (like Voronoi, median, boundary adapting grids, etc.). The entire problem is divided into subsystems like geometry, gridnodes, gridnode connectivity, grid, reservoir fluid flow, and matrix. Each of these subsystems have objects which are closely related. The dependency of these subsystems is established. A detailed analysis of each subsystem leads to identifying the classes, which are a set of objects having similar behavior. Attributes and behavior of the classes are assigned. After establishing relationships between the classes, they are arranged into hierarchies. About one hundred major classes have been identified and designed to achieve the desired behavior from FLEX. The programming language used is C++. Introduction Reservoir simulators are inherently complex. A simulator has to deal with issues such as reservoir and grid geometry, fluids, flow calculation, matrix computations, several well and production constraints, visualization, etc. The most important feature of FLEX, a black oil simulator, is its ability to handle complexities arising from flexible grids. Verma and Aziz (1996) give a description of flexible grids in reservoir simulation. The flexibility in grids increases geometrical complexities as well as complexities in flow calculation. These complexities need sophisticated data structures (and associated procedures) to simplify the problem. It is expected that FLEX will change with time to incorporate new features. One of the important considerations in designing the simulator is the ease with which the simulator can be expected to handle new problems. All these factors combined to make the development process of FLEX quite complex. This paper describes the advantages of using an object-oriented approach for the development of reservoir simulators. The philosophy followed in designing FLEX is that advocated by Booch (1994) and Cheriton (1995). Basic Features of FLEX FLEX solves flow equations based on the control volume formulation (see Verma and Aziz, 1996). It uses the Newton-Raphson method to iteratively solve for the variables. A connection-based approach is employed to form the Jacobian matrix and the residual vector (see Lim, Shiozer and Aziz, 1995 and Verma and Aziz, 1996). Presently the simulator is developed to handle only two immiscible phases. The gridnodes can be located so that they represent reservoir geometry, wells, faults, etc. Figure 1 is an example of the flexible grid generation capabilities of FLEX. Why Object-Oriented? An object-oriented approach was followed in the design of FLEX to handle the complexities associated with a flexible grid simulator, and to provide for future enhancements.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Petroleum Computer Conference, June 2–5, 1996

Paper Number: SPE-35998-MS

..., the parallel speedup is 1.85. ILU factorization Consider the linear system, Ax=b As a preconditioner of ORTHOMIN, ILU factorization provides a

**matrix**M, a "good" approximation to coefficient**matrix**A and easy to factor, convergence may be accelerated by solving the equivalent system, [M1A]x=M1b...
Abstract

Abstract This paper presents a new parallel ILU preconditioner. It is based on the technique of Sequential Staging of Tasks (SST) which overcomes the difficulty of recursiveness arising from ILU factorization. This new parallel ILU preconditioner is easy to reconstruct from the sequential version. The only requirement is to insert the synchronization codes of stages of tasks into the sequential version, and any other modifications to the original serial codes are not needed. The characteristic of matching various ordering schemes is still maintained, and the new merit of handling different numbers of processors is obtained. Numerical results were obtained using a thermal model with different grid system. The parallel speedup is satisfactory. Introduction Simulation of thermal recovery processes using a fully implicit treatment of component concentrations, phase saturations, pressure and temperature requires solution of large systems of linear equations. Currently, the most robust techniques for solving this large system of linear equations are preconditioned conjugate gradient like methods such as ORTHOMIN which is widely used in traditional solvers for sequential computers. The major computation of ORTHOMIN is vector inner product and is easy to be paralleled. However, the most robust preconditioners such as ILU factorization and nested factorization are not suitable for parallel computer because of their intrinsical recursiveness. It is difficult to parallel ILU preconditioner directly. At present, the general methods to parallel preconditioners are the use of new preconditioning methods such as parallel nested factorization preconditioning. These new preconditioning methods have high parallel efficiency in parallel computers, but also have limitations: limited types of ordering schemes; comprehensive modifications of the sequential codes; slow convergence. This paper presents a new parallel ILU preconditioner based on the technique of Sequential Staging of Tasks (SST). Using the SST technique, the new preconditioner exploits the small scale parallelism of ILU factorizations, and achieves a temporal, larger scale parallelism within certain computing domain, consequently obtains an applicable parallel preconditioner. The new parallel preconditioner maintains all the characteristics of the sequential version, and is easy to reconstruct from the sequential version. The new preconditioner is applicable to computers with different number of processors. The numerical experiments show that the parallel speedup is satisfactory. On an NP 1/52 Mini-Supercomputer System (produced by GOULD Co. in 1989, shared main memory, symmetrical operation system UTX/32) with two processors, the parallel speedup is 1.85. ILU factorization Consider the linear system, Ax=b As a preconditioner of ORTHOMIN, ILU factorization provides a matrix M, a "good" approximation to coefficient matrix A and easy to factor, convergence may be accelerated by solving the equivalent system, [M1A]x=M1b Such preconditionings should offset the added cost of factoring M and performing a forward and back solution with each matrix-vector multiplication by reducing the number of iterations substantially. Main stages of ILU factorization are as follow: A symbolic factorization, defining the non-zero structure of the incomplete factorization. For k=l, NB Do Inverting the main elements (1) P. 135

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Petroleum Computer Conference, July 11–14, 1993

Paper Number: SPE-26231-MS

... that the sediments were deposited, (this involves determination of the structure at present day, removal and decompaction of the sediments to obtain the time-depth history of the sediments, and restoration of any blocks that may have moved along faults). P. 107^

**matrix**finite element analysis spe 26231...
Abstract

Abstract Object-oriented software design has provided many benefits to the development of scientific software by encapsulating the attributes and behaviour of models into separate entities or objects that can interact with other objects in the software project. Objects communicate via message passing not by external reading and writing of an object's attributes. This results in more maintainable code since variables can only be modified by the object's own methods. Furthermore, inheritance creates opportunities for extensibility. Finite Element Analysis is a powerful tool for modelling heat and fluid transfer in, and the deformation of, complex shapes. The method achieves generality by allowing element definition to proceed to some extent without regard to the geometry of the problem. It is only after the elements are 'embedded' into the problem space that their geometry is set. This separation of element definition and general methods for solution of the global matrices suggests that the Finite Element Method can benefit from an object oriented structure. A more complex element can be created by inheriting properties from a simpler element and adding any specific behaviour. Thus elements can be designed to match specific problem requirements more easily. Much of the element's complexity is hidden from the rest of the program making overall development simpler. We present a prototype object oriented library for applying the Finite Element Method and analyse its general structure and its benefits. Finally, we present an example of its application to 2-dimensional heat transfer within a basin. Introduction Basin modelling is a process whereby exploration geologists investigate the hydrocarbon potential of a geological basin area. Oil and gas are produced when organic material is subjected to an elevated temperature over a sustained period of time. Once produced, hydrocarbons may migrate due to pressure gradients in the source rock to a new location in the basin, the reservoir. Basin modelling is a four fold process: Reconstruction of the basin geometry back through to the age that the sediments were deposited, (this involves determination of the structure at present day, removal and decompaction of the sediments to obtain the time-depth history of the sediments, and restoration of any blocks that may have moved along faults). P. 107^

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Petroleum Computer Conference, June 25–28, 1990

Paper Number: SPE-20365-MS

... implicit model spe 20365

**matrix**reservoir simulation implicit black-oil model flow equation test problem equation implicit black-oil simulation society of petroleum engineers linear equation different number simulator computer simulation grid block derivative artificial intelligence...
Abstract

SPE Member Abstract For many difficult reservoir simulation problems, the fully implicit black-oil models are the only efficient and viable means to the solution. Because of their stringent demands on computer's power. The fully implicit simulators are generally run on mini, main or super computers. However, the current trend of 32-bit microprocessors being used in the main-stream Personal Computers (PC) opens a new horizon for reservoir simulation applications. This paper examines the feasibility of a fully implicit black-oil simulator run on these PC's. The simulator uses a general black-oil formulation and can solve 3-dimensional, 3-phase problems with multi-layered wells. problems with multi-layered wells. Two aspects which affect the applicability of person a computer environment in reservoir simulation are addressed. First, various considerations for models's efficiency in program design and coding are discussed PC systems are program design and coding are discussed PC systems are compared. The calculation examples show that the fully implicit black-oil model presented can be run efficiently on the current 1386 type of PC's for problems up to several thousands grid blocks. For putting the advances of personal computer environment in perspective, some runs personal computer environment in perspective, some runs made on the new IBM RISC/6000 320 desktop are also presented. presented. BACKGROUND Even though many black-oil models use IMPES or sequential formulation for the sake of computational efficiency, the fully implicit model is still the only efficient and viable method to solve many difficult reservoir simulation problems. Because of their stringent demands for the problems. Because of their stringent demands for the computer's speed and memory, the fully implicit simulators are usually run on the mini, mainframe or super computers. To be able to run a program on personal computers is always tempting. Because they are easy to access and use, and economic. There also exists a large numbers of utility programs for PC's which are of great help in preparing programs for PC's which are of great help in preparing and reporting the work being done. One can use pc either to run an entire simulation or to prototype the input dataset for a complex field simulation to be run on a large computer. For a long time (under computer industrial's standard), few reservoir simulation applications have been run in the personal computer environment. The reason is simple personal computer environment. The reason is simple they are not powerful enough to meet requirements of intensive computations and large amount of data storage associated with the reservoir simulation calculations. The 16-bit (98 bit with earlier PC's) word length of 80286 processor is inherently inefficient to the floating point (single or processor is inherently inefficient to the floating point (single or double precision) representation. The processor's speed is low and the storage is limited. P. P.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Petroleum Computer Conference, June 27–29, 1988

Paper Number: SPE-17792-MS

... combines pre-stimulation production performance data, geological and reservoir data to match a well to one of four stimulation operations, namely, hydraulic fracturing,

**matrix**acid fracturing, acid fracturing and recompletion. In the second module, the design of the stimulation operation selected from...
Abstract

SPE Members ABSTRACT: Optimization methods of linear and non-linear programming have been used in the oil industry to study the scheduling of oil well production and drilling, and to determine the optimal investment policies. There are few studies published in the area of optimization of well stimulation policies. This paper describes the application of non-linear mathematical programming methods to optimize the design and evaluation of oil and gas wells for stimulation. The model presented is modular and it has three sub modules: SCREEN, DESIGN, and OPTIMIZE. The screening module combines pre-stimulation production performance data, geological and reservoir data to match a well to one of four stimulation operations, namely, hydraulic fracturing, matrix acid fracturing, acid fracturing and recompletion. In the second module, the design of the stimulation operation selected from the screening module is performed. The program uses widely accepted equations and correlations to design hydraulic fracturing, matrix acidizing, acid-fracturing and recompletion. In the optimization module, analytical decline curve and economic discounting techniques are used to predict the production profile after the stimulation operation, and to determine the time value of money and production. The optimization model combines the reservoir and economic data to determine the optimal well stimulation strategy. The constraints of the optimization model, include those imposed by the reservoir, the production facilities, operating costs, and operating budget of the field. Program development and validation procedures, and the application to field data are presented in the paper. Introduction Optimizing well stimulation deals with the problem of allocating resources among competing stimulation techniques to maximize production and ultimate recovery. It requires the prediction of cost and the corresponding well performance resulting from the different well stimulation strategies. Optimization of well stimulation requires a systems approach to be meaningful. Not only are the reservoir aspects of the formation evaluated but we must also consider the interactions with other sub-systems like production facilities and economics. Optimization depends greatly on the type of stimulation treatment and format-ion parameters, and consequently, the number of significant factors could range from a limited set to one including a large number of variables. Langenkamp defines well stimulation as a technique of getting more production from a downhole formation. Three points are suggested by this definition. First, well stimulation involves a technique. This could be any of the treatment models discussed in this study. Second, a reservoir that reacts to the effect of the technique is needed. The third is the interaction between the reservoir and the technique. The relationship between the technique, reservoir, cost and constraints provide the basis for optimization. To develop an optimization module to evaluate stimulation, three basic model requirements must be satisfied. The first requirement is a reservoir model that determines the production rates and recoveries given a stimulation treatment. There are two ways of characterizing a reservoir for this purpose. The first and more sophisticated method is the use of a finite difference reservoir simulator. The simulator is run and the production rates and recoveries resulting from various stimulation schemes are evaluated. The second method is to use production decline curves to describe the reservoir. The latter has advantages in some ways. It is simpler, requires less computer time, and lends itself to the development of simple closed form analytical solutions. P. 149^

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Petroleum Computer Conference, June 27–29, 1988

Paper Number: SPE-17777-MS

...) and were solved on main frame computers. Another recent application of LP is equation-of-state matching of laboratory PVT data. This problem leads to a smaller, denser LP

**matrix**. Three methods of LP solution were investigated on microcomputers: the simplex method, the revised simplex method...
Abstract

SPE Members Abstract A common type of mathematical optimization is Linear Programming (LP). An LP solution of aquifer influence functions has recently been reported by Gadjica, et al. (1987) and Targac, et al. Their LP matrices were large and sparse (only 3% of the elements were non-zero) and were solved on main frame computers. Another recent application of LP is equation-of-state matching of laboratory PVT data. This problem leads to a smaller, denser LP matrix. Three methods of LP solution were investigated on microcomputers: the simplex method, the revised simplex method, and the symmetric method. These methods were run on several LP problems ranging from a small dense matrix to large sparse matrices. The different methods have different characteristics which affect the speed, storage requirements and simplicity of coding. The simplex method is straightforward, but usually is slower and requires more storage than the other methods. The results of this study are tabulated with running times and storage requirements for the various LP methods and microcomputers. The computers range from the IBM XT to the Compaq 386. This information serves as a documentation of the LP codes and should be useful for an engineer interested in using LP codes on a microcomputer. Introduction Linear programming (LP) is a mathematical technique which optimizes certain problems which fit in the LP format. The requirement for such a formulation is that (1) an objective function is maximized (or minimized), and (2) constraints are placed on the problem which limit linear combinations of the variables. It is necessary that all of these expressions be written as linear equations involving the variables of the problem. The following is a typical expression of an LP problem: Minimize ...............................................(1) Subject to the constraints ...............................................(2) where xj are the "decision variables" which are being determined by the problem. Eq. 1 is the objective function expressing the minimization of cost". Each value of c is a cost coefficient for a particular decision variable constraints, Eq. 2, are inequalities which express the limiting conditions of the problem. There are a number Of petroleum applications that fit this LP format. Dougherty and Durrer and Slater have reviewed optimization techniques in the industry, many of which are LP problems. P. 47^

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Petroleum Industry Application of Microcomputers, June 23–26, 1987

Paper Number: SPE-16482-MS

... surveillance aid. A streamline program can be used to generate an allocation

**matrix**not only for each pattern but also for each fractional pattern. The allocation**matrix**can then be used with other programs to distribute the production and injection to each well and fractional pattern. Monitoring the project...
Abstract

Abstract Oil production over the last several decades has become more dependent upon secondary and tertiary recovery techniques. There have been significant advances in the use of reservoir modeling to help design and evaluate these processes. Unfortunately, there have not been similar advances made in the tools for reservoir surveillance. Injection costs can be the major factor contributing to the profit or loss of a project, therefore creating the need to maximize injection efficiency. Combining streamlining techniques with the micro-computer can result in an invaluable reservoir surveillance aid. A streamline program can be used to generate an allocation matrix not only for each pattern but also for each fractional pattern. The allocation matrix can then be used with other programs to distribute the production and injection to each well and fractional pattern. Monitoring the project on a fractional pattern basis enables the reservoir surveillance engineer to promptly identify potential problems and to make modifications as necessary to maximize profitability. The injection-withdrawal ratio can be evaluated for each fractional pattern, for individual wells, or for the entire project. Injection requirements can be determined more precisely for each well to maximize utilization of the injectant and minimize costly recycling. This paper reviews how a streamline model has been integrated with an allocation program on a micro-computer to maximize reservoir surveillance. STREAMLINE MODEL Streamlines have been used for many years in the oil and gas industry to determine optimum injection patterns, estimate areas of sweep and to make production forecasts. The development of streamline modeling has been discussed in detail by: Caudle, LeBlanc, Lin, and Wessels. The interested reader should consult these references for a more detailed explanation of streamline modeling. Most streamline models can be modified to generate production and injection allocation parameters used for surveillance of injection projects. The purpose of this paper is to show how a streamline model can be modified to be used by the project engineer as a surveillance tool. Most streamline models utilize a reverse tracking method to follow the streamlines backwards from the producing wells to the individual injection wells. With the addition of a single square matrix, dimensioned for the total number of wells in the streamline model, the program can be utilized to develop a production and injection allocation matrix. Referring to Figure 1, the diagonal of the allocation matrix contains the total number of streamlines for an individual producing well. Injection wells are identified by a zero value for their diagonal entry. For example, in Figure 1 Wells #1 through 6 are producing wells and Wells #7 through 9 are injection wells. Well #1 at location 1-1 indicates that there are a total of five (5) streamlines radiating from this producing well. P. 5^