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Paper presented at the SPE Symposium on Numerical Simulation of Reservoir Performance, February 19–20, 1976
Paper presented at the SPE Symposium on Numerical Simulation of Reservoir Performance, February 19–20, 1976
Paper presented at the SPE Symposium on Numerical Simulation of Reservoir Performance, February 19–20, 1976

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Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation of Reservoir Performance, February 19–20, 1976

Paper Number: SPE-5724-MS

..., Galerkin's process yields results superior, to those obtained by the finite difference technique. production monitoring PVT measurement interpolation production control reservoir simulation difference technique flow in porous media

**Fluid****Dynamics**spe 5724 simulator relative permeability...
Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. THIS PAPER IS SUBJECT TO CORRECTION Abstract This study presents a development of an Alternating Direction Galerkin (ADG) method following some theoretical work of Douglas and Dupont. The technique is applied to the construction of a two-phase coning simulator in cylindrical coordinates. Solutions are approximated in smooth bicubic Hermite subspaces. The non-linear coefficients in the differential equations are interpolated into a Hermite subspace with a mesh three times finer than the solution mesh. This interpolation scheme provides an order of accuracy consistent with and as high as that of the method of solution. Also it makes a substantial reduction in computer time. A novel method has been devised to simplify the notation and programming effort compared to work previous done in this area. In addition, a permutation matrix is employed to bring the matrix problem for each sweep into a simpler consistent form such that a block solution technique can be readily applied. Finally, some computational results for hypothetical coning problems are presented and discussed. These are compared to those problems are presented and discussed. These are compared to those obtained from a non-alternating Galerkin formulation and comparable finite difference model. Introduction For the past two decades reservoir engineers have been confronted with the problem of describing and predicting reservoir behavior. The lack of analytical solutions to the partial differential equations generated by such problems, along with recent advances in high speed electronic computers, has led to numerical methods using finite difference techniques. Despite the relative success of these methods, it was soon discovered that not all problems are adequately amenable to such treatment. Certain reservoir phenomena involving sudden changes in the dependent variables are poorly simulated by finite difference techniques. For example, to describe the zone of rapid change occurring in Buckley-Leverett type displacement, one must resort to excessive grid refinement. Computation of pressure and saturation distributions around wellbores must be approached in the same manner. The main drawback is that only discrete solutions in time and space are obtained by finite difference methods. Furthermore, these techniques are frequently limited by stability and convergence considerations. Because of these problems attention has shifted to numerical methods which do not require spatial discretization of derivatives. Galerkin's method is one of these. Combined with piecewise polynomial approximations, Galerkin's process yields results polynomial approximations, Galerkin's process yields results superior, to those obtained by the finite difference technique.

Proceedings Papers

#### Numerical Solution of Slightly Compressible Single Phase Transient Flow Problems by Spline Functions

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation of Reservoir Performance, February 19–20, 1976

Paper Number: SPE-5733-MS

... heat Panton and Salle on a one dimensional heat conduction equation. They reported that integral methods give higher accuracy in the solution but programming is quite laborous. pressure transient analysis production logging

**Fluid****Dynamics**coefficient production monitoring pressure...
Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract Two dimensional spline functions were utilized in the solution of the linear diffusion equation both with and without a sink within the domain. A collocation method of solution was used and several types of spline representations were investigated. Results were compared with finite difference solutions and with the analytical solutions. B-splines with an extended region of definition were concluded to be superior. For problems with a sink the B-splines alone were not accurate, however, supplementing the splines with a log term produced very good results. Introduction In engineering design, ship builders and others often use a spline, an elastic ruler which can be bent so that it passes through a given set of points. Under certain assumptions the curve can be approximately described as being built up of different third degree polynomials (cubic polynomials) in such a way polynomials (cubic polynomials) in such a way that the function and its first two derivatives are everywhere continuous. The third derivative however, can have discontinuities at the given points. Such a function is called cubic spline. In recent years, spline functions have found wide applications in approximation theory, data fitting, interpolation and so on 4, 9, 15, 16, 17. A smaller amount of work has been devoted to the solution of ordinary and partial differential equations. partial differential equations. Application of spline functions in reservoir problems was initiated by Culham and Varga. problems was initiated by Culham and Varga. These authors used spline functions to approximate the space derivative in a one dimensional nonlinear parabolic partial differential equation which describes the real gas flow in a porous media. The time derivative, however, was approximated by means of various finite difference schemes. They employed Galarkin's and some non-Galarkin type methods to obtain the solutions. Cubic spline basis functions were recommended as the optimum choice for a single-phase flow. When a spline function is used to approximate the solution of a differential equation, two different methods may be used to eliminate the space dependency. These are namely, Galarkin's method (or Integral Methods) and collocation method. Comparisons of these two different solution methods were made by Panton and Salle on a one dimensional heat Panton and Salle on a one dimensional heat conduction equation. They reported that integral methods give higher accuracy in the solution but programming is quite laborous.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation of Reservoir Performance, February 19–20, 1976

Paper Number: SPE-5725-MS

... number of basis functions. flow in porous media node basis function formulation reservoir simulation two-phase flow saturation finite element method spe 5725 Upstream Oil & Gas boundary condition matrix equation Artificial Intelligence

**Fluid****Dynamics**buckley-leverett problem...
Abstract

American Institute of Mining, metallurgical, and Petroleum Engineers Inc. Abstract A finite element approximation to the differential equations for flow of two compressible fluids (oil and gas) through a porous medium is presented. Following a Galerkin approach, the algebraic equations are derived using matrix notation. Time integration is done by a simple finite difference scheme, and the resulting non-linear equations are solved by a chord slope technique. Some simple means for stabilizing the solution are pointed out and incorporated in the method. Numerical results are presented for a displacement and a gas percolation problem, and the finite element results are shown to be slightly superior to those of a conventional model. Introduction The finite element method (FEM) has come to be recognized as an effective analysis tool for a wide range of problems. In particular, it has become the dominating numerical particular, it has become the dominating numerical method within static and dynamic structural analysis, for which it was originally developed some twenty years ago. The FEM gained immediate popularity because of its systematic formulation, ability to handle irregularly shaped boundaries and general appeal to engineers. It was soon recognized as an extension of the variational methods. The basic idea of variational calculus is that the solution is found as the function which results in an extremum value of some integral, the functional, often related to the energy of the system. The problem is to identify a functional which problem is to identify a functional which will yield the same result as direct solution of the differential equation. If a functional exists, an approximate solution can be found by finding the extremum value when the solution is restricted to linear combinations of a finite number of basis functions.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper Number: SPE-5743-MS

... content pressure gradient on the energy content of the output gas. Also the subsurface energy conversion efficiency is discussed. SAGD

**Fluid****Dynamics**concentration thermal method steam-assisted gravity drainage coal gasification numerical simulation flow in porous media Upstream Oil...
Abstract

To mathematically describe the long wall generator process of underground coal gasification the coalbed is considered as solid carbon with a system of vertical micro-fissures along which combustion reactions occur. Mass and energy balances for the solid and gas phases and a mass balance for each hydrocarbon component are derived. Two overall reactions are considered resulting in seven coupled, nonlinear partial differential equations. partial differential equations. The primary mechanisms of heat transfer are conduction in the solid phase and convection in the gas phase. Also, convective transfer-between the two phases along the fissure walls, phases along the fissure walls, expansion of the fissure system and effective Darcy permeability as a function of distance and time, and temperature-dependent reaction rates are considered. The equations are written in implicit finite-difference form and solved stepwise in one dimension by direct methods with systematic updating of coefficients. Solutions are obtained for the solid and gas temperature, total gas density, and mole fractions of oxygen, carbon dioxide, and carbon monoxide as functions of distance and time. Boundary conditions include input and output pressures, inlet gas and/or solid temperatures, and inlet oxygen concentration. Typical sets of temperature and gas component concentrations along the generator length are shown. Curves are shown indicating the effect of the dimensions of the seam and the applied pressure gradient on the energy content pressure gradient on the energy content of the output gas. Also the subsurface energy conversion efficiency is discussed.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper Number: SPE-5736-MS

... direct finite difference simulation spe 5736 equation reservoir permeability contrast ratio

**Fluid****Dynamics**gas well expression fracture SCX:IETY OF PETroLEUM ENGINEERS OF AIME PAPER NUMBER s p E 5 7 3 6 Direct Finite Difference Simulation Wei I w1th a Fin1te Capacity Vertical of A Gas Fracture...
Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract Numerical simulation of a well in a vertically fractured reservoir is described. Real gas potential is the dependent variable. A fractional step numerical method which simultaneously integrates the difference expressions of the reservoir and fracture in an economical manner is used. Simulations of well performance with constant well potential and at several fracture extents are shown and discussed. Introduction The increasing demand for an adequate supply of natural gas has caused the development of extremely low permeability gas reservoirs. This is now possible because hydraulic fracturing can produce fractures with radii of 500 feet or more and thus stimulate production to levels that are economic. Estimation of the benefits to result from the hydraulic generation of a fracture in a particular formation is desirable. It is the authors' belief that the numerical model given here is sufficiently novel and advantageous to merit attention. Real gas potential is the dependent variable. The fracture is taken to occupy a negligible volume when compared to the volume of the reservoir. Then there is not the difficult problem of giving numerical representation of the small fracture width and the large reservoir extent in the one model. Yet one-dimensional Darcy flow is modeled in the fracture and two-dimensional Darcy flow is modeled in the reservoir. The numerical replacements of those differential expressions are solved simultaneously with relative ease. The numerical method used is from a class of methods called fractional step methods. Some recent results show how these methods lend themselves to the treatment of third type boundary conditions such as occur for this problem. Also, it is recognized that numerical methods of this type may yield solutions that exhibit peculiar behavior near boundaries. peculiar behavior near boundaries.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper Number: SPE-5728-MS

... dominates the computer cost. flow in porous media criterion iteration grid point matrix Upstream Oil & Gas three-dimensional reservoir problem LSOR three-dimensional problem

**Fluid****Dynamics**convergence Watt correction computer time reservoir simulation overrelaxation submatrix...
Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract A relaxation method called Slice Successive Overrelaxation has been developed for the solution of matrix problems occurring in three-dimensional reservoir simulation. The SSOR technique represents a significant improvement over previously used techniques and is considered to be a general method. The mathematics of SSOR is a straight-forward extension of block SOR techniques in which the submatrix, or block, corresponds to a vertical plane (slice) through the reservoir. This technique has become practical by the application of a reordering practical by the application of a reordering scheme to the direct solution of each submatrix. On iterations subsequent to the first, the forward elimination on submatrices is not required, making succesive iteratons comparable to LSOR in speed. The Watts correction method is applied in its generalized manner similar to LSOR and dramatically enhances the speed of convergence. Test results are shown for various reservoir problems. The method is shown to be particularly advantageous in a typical three-dimensional problem where vertical permeability barriers exist in portions of the permeability barriers exist in portions of the reservoir and high vertical transmissibilities occur elsewhere. It is demonstrated that the convergence rate of applying SSOR to a three-dimensional problem is essentially the same as the convergence rate of applying LSOR to the two-dimensional projection of the same problem. A method is presented which can be used to estimate the computer time required for an SSOR solution. Introduction One of the persistent problems in reservoir simulation is that of solving the matrix problem which arises from the "pressure problem which arises from the "pressure equation." Over the years, a number of methods have been developed and utilized with success for various types of problems. For most reservoir simulation problems, we find that the solution of the matrix problems, we find that the solution of the matrix problem is very efficient and represents problem is very efficient and represents only a minor part of the cost of a reservoir simulation computer run. On other problems, however, the solution method can problems, however, the solution method can become sufficiently slow so that the matrix problem dominates the computer cost. problem dominates the computer cost.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation of Reservoir Performance, January 11–12, 1973

Paper Number: SPE-4287-MS

.... production control reservoir simulation Thickness flow in porous media water zone Reservoir Surveillance

**Fluid****Dynamics**time step fraction permeability ratio oil production production rate water production well penetration production monitoring Upstream Oil & Gas oil zone aquifer...
Abstract

1973 Third Symposium on Numerical Simulation of Reservoir Performance Houston, Texas January 11–12, 1973 Abstract Most research on water coning has been directed toward minimizing water production by reduced well penetration or production rate control. However, complete depletion of an oil column underlain by water will necessarily be accompanies by considerable water production, at least in the latter stages of depletion. By means of numerical simulation a systematic study was made of the effects of reservoir and well parameters on water coning performance. Included were the effects of aquifer thickness, well penetration, pressure drawdown, and horizontal to vertical permeability ratio. permeability ratio. These studies revealed that higher water-oil ratios, at every stage of depletion, resulted from increasing well penetration or increasing pressure drawdown at the producing well. Accompanying the increase in pressure drawdown at the producing well. Accompanying the increase in water-oil ratio is an increase in oil production rate of the same magnitude thus decreasing the operating life of the well. There is no indication that the ultimate oil recovery is adversely affected by increased production rates. As a matter of fact it appears that high production production rates. As a matter of fact it appears that high production rates could lead to improved economic oil recovery. Produced water-oil ratio at every stage of depletion is strongly influenced by the ratio of horizontal to vertical permeability and to a lesser degree by aquifer thickness. Introduction Water coning, during which bottom water gradually or abruptly breaks into an oil- or gas-producing well, has long been a problem in oil fields. It is a very complex problem and has been investigated by many researchers.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation of Reservoir Performance, January 11–12, 1973

Paper Number: SPE-4290-MS

... of these investigations more than one fracture system may be found. The area of an individual fracture plane may vary from a few square inches to several hundred square feet. complex reservoir hydraulic fracturing element method flow in porous media

**Fluid****Dynamics**fracture conductivity conductivity finite...
Abstract

Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract The finite element method has been used to investigate mathematical models of idealized systems of vertical fractures in an effort to extend our understanding of the behavior of naturally fractured reservoirs. Regular systems of fractures have been chosen in such a way that the length of the fracture can be characterized in terms of the dimensions of a representative elemental area associated with that fracture. By changing the permeability of the fracture relate to the matrix, steady state solutions demonstrate how the effective permeability of the fractured system depends on both fracture conductivity and dimensionless parameters representing fracture density. Horizontal shale lenses, scattered through a reservoir can be modelled in the same way using finite elements of very low conductivity. Comparison of our results with those from a recent analytic solution for fractured systems indicate excellent agreement. Introduction There are a significant number of petroleum reservoirs where discontinuities such as fractures or joints in the porous rock matrix are the main path transmitting fluids to the producing well. These naturally fractured systems producing well. These naturally fractured systems characteristically have a low matrix permeability and one or more well developed fracture systems. The published reports on naturally fractured reservoirs show that vertical joint planes are a dominant feature. Daniel studied three fractured limestone reservoirs of the Middle East. He reports that in all three cases, vertical and near vertical fractures were widespread, and for the most part, there was a complete absence of low dipping fracture planes. Wilkinson's work on the Spraberry reservoir in Texas revealed that vertical joints are the largest, both in extent and width, and by far the most abundant type of fractures. This dominance of vertical fractures has also been reported by other authors. Investigations of non-reservoir rocks have also revealed the same conclusions. Kelly and Clinton, Parker, and Hodgson have made extensive regional investigations on fractures and joints. They report that vertical joints constitute the most dominant mode of fracture systems. Furthermore, they report that in the areas studied, the fractures are not randomly oriented but are systematically arranged in sets. The same finding has also recently been reported by Louis and Perrot and Mahtab et al. In all of these investigations more than one fracture system may be found. The area of an individual fracture plane may vary from a few square inches to several hundred square feet.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation, February 5–6, 1970

Paper Number: SPE-2803-MS

... accuracy time step fluid modeling

**Fluid****Dynamics**time step size diffusivity equation multiphase flow ev technique problem 2 equation problem mesh point problem 1 Saul error norm half time step Upstream Oil & Gas non iterative technique ADI technique parabolic problem 2 error map...
Abstract

Abstract This paper presents a new noniterative difference technique which may be used to solve parabolic partial differential equation parabolic partial differential equation problems, including those arising in the prediction problems, including those arising in the prediction of fluid flow in a porous media. It is shown that this difference technique is unconditionally stable and is suitable to solve engineering problems. A comparison is made between the new problems. A comparison is made between the new method and two other stable noniterative difference techniques, the Saul'ev technique and the noniterative ADI technique. Of the two problems worked as examples, the new method is problems worked as examples, the new method is most efficient for the elongated grid problem, and noniterative ADI is most efficient for the square grid problem. Introduction Since the introduction of high speed computers, a reservoir engineering research goal has been to develop increasingly efficient methods of predicting reservoir fluid movement. Various methods have been developed or used by petroleum engineers to predict the performance petroleum engineers to predict the performance of an oil field. The major emphasis in the solution of partial differential equations simulating multiphase fluid flow has been the finite difference approach, which can be summarized by the following series of steps. The reservoir or section of a reservoir is characterized by a series of mesh points wit varying rock and fluid properties simulated at each point. The partial differential equations which describe the fluid flow are written. At each mesh point the partial differential equations are replaced by difference equations. The resulting equations are solved [sometimes iteratively to obtain an approximate solution to the original partial differential equation problem. This paper presents a new technique for replacing the differential equation at each mesh point by a difference equation. Difference equations are evaluated in terms of the stability, consistency and order of accuracy. Usually these items are discussed in regard to solution of the diffusivity equation, a linear partial differential equation for sore uniform rectangular mesh covering the region of interest. The same approach is used here. However, it should be noted that the conclusions reached considering a linear partial differential equation and a uniform rectangular mesh are not always applicable to petroleum problems where the differential equations are nonlinear and the mesh is often not uniform. DESCRIPTION OF THE DIFFERENCE TECHNIQUE In this paper a new two-step noniterative difference technique is presented.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation, February 5–6, 1970

Paper Number: SPE-2814-MS

... uniformity as indicated by the SP portion of the log and the core analysis. There is one noticeable exception to the uniformity and that is the portion marked "barrier" on Fig. 1. portion marked "barrier" on Fig. 1. conformance improvement Reservoir Characterization enhanced recovery

**Fluid**...
Abstract

Abstract In reservoirs with bottom-water drives, water coning has historically been a producing problem. Prior to the use of high-speed problem. Prior to the use of high-speed computers, the coning problem practically defied solution. This paper discusses a study of water coning in the Oil Creek reservoir of the North Antioch field, which was made utilizing an r-z, two-phase compressible coning model. The model matches water-cut history and predicts future performance. The study points up the importance performance. The study points up the importance of variation in vertical permeability on the coning phenomenon. More particularly, it shows the effects of a continuous 4-ft zone of low permeability which occurs in the reservoir. permeability which occurs in the reservoir. Coning characteristics of wells in which the original oil-water contact was located above the zone are compared with those in which the original contact was below the zone. The study indicated that the reservoir could safely be produced at rates that would greatly increase produced at rates that would greatly increase the present worth value of the reserves without materially affecting the ultimate recovery. Introduction The North Antioch field is located in the northwest part of Garvin County, Okla. The field was discovered in May, 1965, with the drilling of Coastal States' J. R. Winchester No. 1. As the field developed, it became obvious that the oil reserves in the Oil Creek reservoir were of substantial proportions. Development was essentially complete by Oct., 1966. Because of a rather unusual combination of reservoir and fluid characteristics and the obvious possibilities for increasing recoverable reserves, possibilities for increasing recoverable reserves, reducing costs and increasing present worth value through proper operation, the decision was made to utilize some advanced forms of reservoir modeling to predict the results of various possible methods of operation of the field. This paper presents the results obtained from modeling and recommendations for future operations in the field. RESERVOIR CHARACTERISTICS The Oil Creek reservoir of the North Antioch field is made up of approximately 108 ft of fairly uniform and clean Ordovician sandstone found at a depth of approximately 6,500 ft subsea or 7,500 ft subsurface. Through coring and analysis of logs, it was determined that the average porosity and permeability of the sand are 17.32 percent and 350 md, respectively. Fig. 1 is a portion of a typical electric log of the Oil Creek sand in the field. Table 1 shows the results of core analysis on two cores taken from the Oil Creek reservoir. The most striking of the characteristics of the sand is its uniformity as indicated by the SP portion of the log and the core analysis. There is one noticeable exception to the uniformity and that is the portion marked "barrier" on Fig. 1. portion marked "barrier" on Fig. 1.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation, February 5–6, 1970

Paper Number: SPE-2810-MS

... to evaluate such reservoirs in a routine manner. reservoir simulation well rate Modeling & Simulation

**Fluid****Dynamics**gas reservoir Henderson low permeability reservoir equation flow in porous media Simulation Technique time step block pressure well 2 Dempsey History 24-hour flow...
Abstract

Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract This paper represents an effort to apply simulation techniques to low permeability gas reservoirs. Problems encountered are illustrated by an actual field example. Techniques normally employed for gas reservoirs are not easily extended to the low permeability case. The results show that it is permeability case. The results show that it is possible to rigorously calculate the possible to rigorously calculate the performance of such reservoirs, but to do performance of such reservoirs, but to do so requires data that make application to large fields impractical. However, certain adjustments can be made that approximate performance of such reservoirs, making large performance of such reservoirs, making large applications possible under some restricted conditions. Introduction Natural gas transmission companies must invest millions of dollars in surface pipeline systems and compression facilities in pipeline systems and compression facilities in order to produce natural gas and transport it thousands of miles to the market area. These surface systems are designed to fit the specific reservoirs with which they are connected. In the mid-continent gas supply area, and specifically the Anadarko Basin, many gas reservoirs are characterized by low permeability. Reasonable gas rates that permeability. Reasonable gas rates that satisfy both the transmission company's market requirements and the producer's economics result in large pressure transients existing over extended periods of time. While the pipeline company is basically interested in stabilized well performance, the individual well rates are often determined by short term transient well tests. This makes it necessary to calculate both highly transient and stabilized flow. Numerical simulation techniques would appear to offer the best solution to such a problem. This paper deals with experience gained from simulating a low permeability reservoir and is illustrated with an actual field example. THE PROBLEM AND DATA AVAILABLE The simulation of gas reservoirs in the Anadarko Basin requires consideration of low permeability. The extension of techniques permeability. The extension of techniques used for high permeability fields has proved to be not entirely adequate. This study represents an effort to calculate the complete performance of a reservoir that is characterized performance of a reservoir that is characterized by low permeability, then to develop a technique to evaluate such reservoirs in a routine manner.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation, February 5–6, 1970

Paper Number: SPE-2804-MS

... production monitoring multiphase flow psia production control production logging Upstream Oil & Gas injection pressure viscosity pressure distribution regime prediction stability algorithm two-phase flow Tech lifting potential curve operation flow string

**Fluid****Dynamics**bottom hole...
Abstract

Abstract A numerical simulation for vertical multiphase systems has been developed, based on the original concepts presented by J. Orkiszewski. The rather marked variability for different flow regimes in a given well and the simultaneous coexistence of these various regimes at different depths are properly accounted for in the interative calculation of the pressure distributions. The method features the most reliable and proven correlations available for each of the flow regimes encountered. A programmed data package in support of the above correlations is described. The concept of "Lifting Potential" introduced in an earlier paper has been programmed, permitting applications in programmed, permitting applications in a variety of multiphase flow problems. These include both steady and unsteady state applications. In general, the applications involve the study of "heading," "loading," "unloading," and "dying" of gas wells subject to liquid production, as well as different well production, as well as different well engineering and design optimization problems encountered in gas-lift and problems encountered in gas-lift and condensate production. Introduction The occurrence of two-phase flow in vertical flow systems is common to many engineering applications. Problems ranging from oil field technology to boiler design, from nuclear reactors fluidized beds, from special heat exchangers to thermosiphon reboilers frequently encounter conditions of multiphase flow. The main problem of engineering interest in vertical two-phase flow may be defined as follows: Knowing the physical properties of each phase, physical properties of each phase, geometry of the flow system and conditions prevailing at one terminus to predict the pressure distribution along predict the pressure distribution along the pipe. The literature available on vertical two-phase flow includes a large variety of correlations, analytical and empirical methods developed for prediction of energy losses as well as the flow regimes. It is significant to note that the references quoted above belong to two notably different schools of thought. The first group of papers, references 1 through a correlate friction losses by a unique energy loss factor without any particular regard to "slippage." particular regard to "slippage."

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Symposium on Numerical Simulation, February 5–6, 1970

Paper Number: SPE-2805-MS

... convection Aziz

**Fluid****Dynamics**numerical simulation difference approximation reservoir simulation temperature distribution vector potential three-dimensional natural convection flow in porous media partial differential equation three-dimensional solution boundary condition derivative equation...
Abstract

Abstract The equations of motion and the energy equation are developed for the prediction of temperature and velocity fields in the problem of natural convection in porous media. The equations are made suitable for numerical computation by introducing a vector potential which is the counterpart of stream function in two dimensions. This vector potential has previously been used by one of the authors to solve previously been used by one of the authors to solve the Navier-Stokes equation in three dimensions. The partial differential equations are solved by finite difference methods employing Alternating Direction Implicit Procedure [ADIP] and Successive Over Relaxation [SOR] methods. The model may be used when the heated side is vertical, horizontal or inclined. The detailed structure of three-dimensional cells predicted by this model is compared with previous two-dimensional studies conducted by the authors. Analytical solutions are possible and available for the linearized partial differential equations for this problem. Such solutions only predict the onset of convection and fail to reveal what happens after convection sets in. This is the first attempt in solving these equations for the case where nonlinear terms are not neglected and the predictions are carried out as a function of time and all three space directions. The results are presented in the form of contour maps and show the interesting structure of convection cells in porous media. Introduction Natural convection is a phenomena conceivable in a petroleum reservoir subjected to fire or steam flooding. It is the result of adverse temperature gradients generating buoyancy forces that in turn cause fluid motion. A mathematical analog of the physical phenomena may be obtained by writing mass, force and energy balances on a differential volume of porous media. If Darcy's law is employed to porous media. If Darcy's law is employed to relate fluid flow to potential gradients and the Boussinesq approximations [fluid properties independent of temperature except where they contribute to buoyancy] are assumed to be applicable, the three balances reduce to two nonlinear partial differential equations. The two-dimensional form of these equations has been solved by a number of investigators. The reader is referred to Karra's thesis for a comprehensive literature review on the subject. In order to apply the mathematical model with some degree of confidence, an experimental verification is desirable if not necessary. As any experimental verification would presumably involve a three-dimensional apparatus, a study as to the effect of the third dimension on the solution was clearly indicated.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Numerical Simulation Symposium, April 22–23, 1968

Paper Number: SPE-2021-MS

... Gauss elimination grid reservoir simulation fluid flow simulation equation convergence coefficient matrix

**Fluid****Dynamics**calculation iteration parameter spe 2021 equation ABSTRACT This paper presents the methods used to solve the finite difference equations which were developed...
Abstract

ABSTRACT This paper presents the methods used to solve the finite difference equations which were developed in a companion paper (1). Various possible methods of solution are discussed. Experience has narrowed the number of suitable numerical methods that are practical to three: Gauss elimination, successive overrelaxation, and the iterative alternating direction implicit process. The final sections of the paper are devoted to a presentation of computational techniques which are vital to actual use of each of the above-mentioned methods. All the terms are defined in the paper. Here, however, we have dropped the subscript denoting the pressure, p, as an oil pressure. Further breakdown requires definition of the numerical solution to be used. This paper describes the breakdown and solution processes most often used in the MUFFS program. Sufficient detail is given so that computer programming can be done. Contrary to popular opinion, economic simulation has been found to require the development of several solution methods, rather than relying on a single one. This requires that the computer subprogram for generating coefficients (A's and O's) be written as a distinct, separate entity to supply the coefficients in Equation (1). Furthermore, it is necessary to be able to obtain these coefficients automatically in column-by-column, row-by-row, or point-by-point form, in any order required by a numerical solution. Columns, rows, and points refer to the columns, rows, and points of the finite difference grid. A program that can generate coefficients in several forms is a simple but important concept, for it allows the easy insertion and modification of experimental methods. The computing inefficiencies that may be incurred within a general coefficient generator are small in comparison to the computing time saved by using the fastest of several solution techniques.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Numerical Simulation Symposium, April 22–23, 1968

Paper Number: SPE-2027-MS

... Of PRODUC110~ ~NO \NJEC110N Z , 5 E; -\ 1 -\ -I 9 \0 \I IZ 2 -\ -2 9 \0 \I IZ 2 , 4 1 \0 \1 s thermal method steam-assisted gravity drainage integration net time step procedure enhanced recovery reservoir simulation temperature distribution simulator saturation SAGD

**Fluid****Dynamics**viscosity...
Abstract

American Institute of Mining, Metallurgical and Petroleum Engineers Inc. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers Office. Such discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract Generalized heat and fluid flow models are useful in evaluating the effects of heat upon crude oils in a reservoir environment and more particularly that area immediately surrounding a wellbore. The model described allows fluid and heat flow in the two dimensions of a vertical plane. As in other numerical models, each cell is assumed to be homogeneous in pressure, viscosity, temperature, etc. This paper describes how the flow of heat from one cell to another has been superimposed upon the unsteady-state flow of oil, gas, water, and steam between cells. Solutions to problems using this reservoir simulator are shown. Introduction During the last several years the use of numerical models to simulate reservoir conditions have become very commonplace and simulators of several types and configurations have been developed for solution on high speed digital computers. During these same years, thermal recovery techniques have received a great deal of attention, and numerical simulator solutions applied also to this area. The broad term of thermal recovery covers a spectrum from in situ combustion to hot water alteration of fluid saturations around wellbores. Thermal recovery, like many other popular techniques, has a certain amount of "romance" surrounding it. The benefits can be very substantial; however, like any new technique it is subject to misapplication. The purpose of this paper is to describe a numerical simulator that accounts for both fluid and heat flow, and further to relate some of the observations that have evolved from its use. DISCUSSION OF THE THERMAL MODEL The concept used in designing this thermal model was to combine fluid and heat flow into one model. The continuity equations written for each mobile phase (oil, gas, and water) were summed employing the method proposed by Fogin et al. The equations were arranged for an implicit solution of the potential distributions using the alternating direction procedure of Peaceman and Rachford. Following the calculation Peaceman and Rachford. Following the calculation of the potential distributions, the saturation distribution and temperature distribution are calculated explicitly. An iterative procedure of the entire calculation sequence permits the use of conductance and expansion coefficients calculated forward in time. This forward approximation is particularly important in the model discussed here because expansion and phase changes are not only functions of pressure but also of temperature. This approach permitted the use of relatively long time increments which are essential for the use of the simulator in the solution of practical engineering problems. problems.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Numerical Simulation Symposium, April 22–23, 1968

Paper Number: SPE-2037-MS

... flow in porous media viscosity two-phase linear fluid flow modeling increment combustion enhanced recovery

**Fluid****Dynamics**spe 2037 boundary grid equation Artificial Intelligence formula oil bank situ combustion capillary pressure Upstream Oil & Gas difference equation oil...
Abstract

American Institute of Mining, Metallurgical and Petroleum Engineers Inc. Background Considerable experimental and theoretical effort has been devoted to understanding the heat flow and stoichoimetric problems associated with the underground combustion process of oil recovery. Results of these efforts have been vital to success in the engineering of field applications of underground combustion. Continued success in applying this process to diverse petroleum reservoirs would be aided by a quantitative understanding of the associated fluid flow phenomena. Semiquantitative notions about the oil displacement mechanisms and the nature of the "oil bank" are presently employed in the evaluation of underground presently employed in the evaluation of underground combustion prospects. A quantitative delineation of the fluid flow aspects would facilitate better engineering predictions of: Injectivity history (compressor requirements), Sweep pattern, Fuel content dependence on air flux. A reasonably complete theoretical treatment of the in situ combustion process would require simultaneous solution of the heat flow, distillation, and three-phase fluid flow problems. Further, this should be done for a multidimensional medium which is not necessarily homogeneous or isotropic. While it is perfectly feasible to develop the necessary equations for such a model, their general solution was much too ambitious an undertaking for the computing capacity available when this work was begun. We therefore confined this effort to exploring the numerical methods required and to becoming familiar with the displacement mechanisms that would be evidenced in the solution of a greatly simplified mathematical model. Specifically, the following restrictions were adopted: Two-phase fluid flow (oil and air only). No distillation effects. Temperature profiles obtained from an independent study of the associated heat-flow problem. To provide a basis for evaluating the results of the model study, a special laboratory tube run experiment was conducted. Here the sandpack was initially saturated with oil only, omitting the usual "connate" water. During burning, the only water present was the water of combustion. The experiments were conducted under psuedo-adiabatic conditions for which temperature distributions are known.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Numerical Simulation Symposium, April 22–23, 1968

Paper Number: SPE-2030-MS

...) and ..........................................(4) Here the vectors will approximate a solution to the equation ..........................................(5) flow in porous media stability analysis convergence matrix Adi Procedure Upstream Oil & Gas wi ii application instabi lity

**Fluid****Dynamics**rachford spe 2030 approach...
Abstract

American Institute of Mining, Metallurgical and Petroleum Engineers Inc. Abstract Solving =, by numerical techniques requires application of ADI procedures to singular matrices which leads to numerical instability for small values of the ADI parameter w. A study of this problem demonstrates that restriction of IRml/w (where Rm is the largest residual) will control these instabilities. Applications to compressible and incompressible Darcy flow are discussed. Introduction The equations of reservoir flow commonly have the following form V (KVP) Q where in two dimensions, ..........................................(1) The ADI (or Peaceman-Rachford) procedure is one of the methods frequently used to solve the associated difference equations. In our mathematical discussion we will write these difference equations approximating the above differential equation In matrix notation as Au = q where u is a vector representing pressure or potential at node points, q is a vector of production rates and points, q is a vector of production rates and A is the matrix of coefficients which depend on the permeabilities or mobilities. In using this procedure numerical instabilities often occur for small values of the parameter w,, particularly for incompressible parameter w,, particularly for incompressible flow with no-flow boundaries. These instabilities have been observed often and techniques to avoid them have been suggested. In this paper we will consider the mathematical and computational origin of these instabilities. This study will allow us to suggest improved methods of handling such problems and will give us a better understanding of the roll of the parameter w in the iteration process. process. MATRIX DESCRIPTION OF THE ADI PROCEDURE Consider three symmetric semi-definite n × n matrices A, H, and V such that ..........................................(2) Let D be a symmetric, positive definite normalizing matrix, q a given vector, Uo an arbitrary initial vector, and w-a number for any positive integer k. Then the ADI procedure can be defined by procedure can be defined by ..........................................(3) and ..........................................(4) Here the vectors will approximate a solution to the equation ..........................................(5)

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the Numerical Simulation Symposium, April 22–23, 1968

Paper Number: SPE-2020-MS

... production control PVT measurement production logging fluid flow simulation equation production monitoring Upstream Oil & Gas Reservoir Surveillance

**Fluid****Dynamics**American Institute of Mining, Metallurgical and Petroleum Engineers Inc. Abstract A fundamental flow equation...
Abstract

American Institute of Mining, Metallurgical and Petroleum Engineers Inc. Abstract A fundamental flow equation can be derived for each of the oil, water, and gas phases by combining the law of conservation of mass, a law of force, and thermodynamic relationships that describe the pressure-volume-temperature behavior of the fluids. The law of conservation of mass states that the sum of mass flow into a cell equals the change of mass within a cell. The law of force describing flow through porousmedia is Darcy's law, which assumes that flow is within the laminar flow regime described by the law. The thermodynamic relationships used to describe the fluid behavior must (because of the lack of accurate analytical expressions) be those found experimentally in a "PVT" study. The first part of this report shows how these laws can be combined to give equations describing fluid flow through porous media. The next step in the development is the cabining of these equations with auxiliary equations so that the number of dependent variables is reduced. The result is an equation that can be assumed to be in terms of oil pressure only, so that it may be solved implicitly pressure only, so that it may be solved implicitly for pressures at the new time level. Separate from this pressure equation are the equations that are solved explicitly for saturations, using the newly computed pressures. Hence, there are three equations developed in final, finite difference form in this paper. 1) a pressure equation, 2) an oil saturation equation, pressure equation, 2) an oil saturation equation, and 3) a water saturation equation. The logic used in the computation of all pertinent terms is given in the final section pertinent terms is given in the final section of this report. Introduction The general aspects of numerical fluid flow simulation has been given elsewhere (1). The present paper gives the derivation of the fluid present paper gives the derivation of the fluid flow simulation equations. The numerical solution of these equations is given in (2). FUNDAMENTAL FLOW EQUATIONS Fig. 1 shows a differential element of porous media. Pounds of oil (w o) are flowing porous media. Pounds of oil (w o) are flowing into or out of two faces, with a well (w op)in the center of the elements. Weight rates of flow, rather than mass rates, are used here to simplify the final algebra; of course, this necessitates the assumption of constant acceleration of gravity. The rate of depletion (ROD) of oil from this element can then be defined as: ..........................................(1) where ROD is the rate of oil removal from the element in lb/day at reservoir conditions.