Implicit Pressure Explicit Saturation Research Articles

Numerical simulation of flow of multicomponent mixtures in porous media

A study is made of the isothermal flow of multicomponent mixtures in a porous medium, accompanied by phase transitions, interphase mass exchange, and change in the physicochemical properties of the phases [1–3], It is assumed that at each point of the flow region, phase equilibrium is established instantaneously and the flow velocities of the separate phases conform to Darcy's law. Approximate solutions of problems of displacing oil by high-pressure gas were obtained in [1]. By generalizing the theory developed in [4], a study is made in [5] of the structure of the exact solutions of the problems of the flow of three-component systems which describe the displacement of oil by different reactants (gases, solvents, micellar solutions). The numerical solutions of the problems of multicomponent system flow are considered in [2, 3, 6, 7]. This paper presents a numerical method which is distinguished from the well-known ones [2, 3, 6, 7] by the following characteristics. The flow equations are approximated by a completely conservative finite-difference scheme of the implicit pressure-explicit saturation type, the calculation being carried out using Newton's method of iteraction with spect to both the pressure and the composition of the mixture. The minimum derivative principle [8] is used in the approximation of the divergence terms of the equations. The phase equilibrium is calculated using the equation of state.

General Purpose Compositional Model

Abstract A direct sequential method has been developed to simulate isothermal compositional systems. The solution technique is the same as that of the implicit pressure, explicit saturation (IMPES) method: one pressure is treated implicitly and (instead of the phase saturation) the component masses/moles are treated explicitly. A "volume balance" equation is used to obtain the pressure equation. A weighted sum of the conservation equations is used to eliminate the nonlinear saturation/concentration terms from the accumulation term of the pressure equation. The partial mass/mole volumes are used as "constants" to weight the mass/mole conservation equations. The method handles uniformly a range of cases from the simplified compositional (i.e., black-oil) models to the most complicated multiphase compositional models of incompressible and compressible fluid systems. The numerical solution is based on the integrated finite-difference method that allows one- (ID), two- (2D), and three-dimensional (3D) grids of regular or irregular volume elements to be handled with the same ease. The mathematical model makes it possible to develop modular versatile computer realizations; thus the model is highly suitable as a basis for general-purpose models.

Instability in Runge-Kutta schemes for simulation of oil recovery

Runge-Kutta timestep schemes have been used recently in an attempt to increase the maximum stable timestep for IMPES (implicit pressure, explicit saturation) simulation of oil recovery. It has been claimed that anm-stage method will increase the real stability boundary bym2. A rigorous stability analysis for a Buckley-Leverett problem shows that this claim is false, and that previous stabilized IMPES schemes are never more efficient than ordinary IMPES. Test calculations support these results.

Multigrid Solution of the Pressure Equation in Reservoir Simulation

Abstract The multigrid technique is compared with the incomplete Cholesky conjugate gradient/modified incomplete Cholesky gradient (ICCG/M1CCG) methods for solving implicit pressure, explicit saturation (IMPES)-type pressure equations. The numerical results of several test problems with widely varying transmissibilities are presented. The multigrid algorithm is enhanced by pattern relaxation and acceleration. Optimization of the ICCG/MICCG algorithms is investigated by high-order (up to tenth) decompositions and different ordering schemes. For large problems the multigrid method is superior in terms of scalar work. The multigrid scheme can also be highly vectorized, and is an O(N) algorithm even for problems with large jump discontinuities in equation coefficients. Results of the multigrid method applied to nonsymmetric problems are also presented.

Three-Dimensional Geothermal Reservoir Simulation

Abstract This paper presents the development of a three-dimensional, finite-difference model for the simulation of geothermal reservoirs. The model is designed to simulate geothermal reservoirs that contain water in any of its vapor or liquid states and provisions are included for properly treating changes provisions are included for properly treating changes of state during a time step. This logic provides for a stable calculation of state change and eliminates pressure, heat balance, and material-balance errors. pressure, heat balance, and material-balance errors. Mass and energy balances are solved simultaneously using an implicit pressure-explicit saturation (IMPES) formulation. An implicit treatment of production rates, capillary pressure, and transmissibilities is included as an option. Thus, entire field, cross-sectional, or individual well studies can be performed efficiently. Example problems are presented to demonstrate the model's utility and to provide insight into the nature of geothermal production under various conditions. In particular, an example is presented of a reservoir initially containing subcooled liquid where fluid conditions near the production well range from subcooled liquid to saturated steam and then to superheated steam. Introduction Geothermal energy represents an essentially untapped, alternate source of energy world-wide. Exploration for this type of energy has increased, however, and should lead to the discovery and development of new geothermal areas. Notable producing geothermal fields include Wairekei in producing geothermal fields include Wairekei in New Zealand, Cero Prieto in Mexico, Matsukawa in Japan, Larderello in Italy, and The Geysers in California. Recently, industry literature has dealt with zero-(one-cell), one-, and two-dimensional, geothermal reservoir simulation. Most studies treat steam in only one state or mention the difficulty of simulating blocks that change from one state to another. Whiting and Ramey presented a zero-dimensional, geothermal model that included water influx. They discussed reservoir performance during the various states that steam can assume. Their study also included a successful match of the pressure/ production performance of the Wairekei geothermal production performance of the Wairekei geothermal reservoir in New Zealand from 1956 to 1966. Coats developed a three-dimensional, finite-difference, steam-flooding model that simultaneously solved the mass and energy equations. An IMPES formulation was used to solve for pressures implicitly followed by a solution for temperatures, saturations, and so forth. Saturations calculated using explicit rates, capillary pressures, and transmissibilities were recalculated at the end of each time interval, using an implicit formulation for these variables. Brigham and Morrow discussed the P/Z behavior of geothermal steam reservoirs. Their model was similar to Whiting and Ramey's model, but they allowed for the presence of a vapor and liquid zone. Mass and energy balances were written for both zones, and logic for a constant or falling liquid level was included. Martin analyzed internal steam drive in geothermal reservoirs by assuming that temperature, pressure, and fluid saturation gradients as well as capillary pressure and gravity were negligible. Gould described the development of a vertical two-phase, pressure-drop calculation for flow in geothermal wells. He discussed the effect of heat transfer to the surrounding formation and the effect of steam quality on production. Mercer and Faust developed a two-dimensional (areal), two-phase geothermal model using a Galerkin finite-element formulation in space and finite-difference formulation for time. They chose pressure and enthalpy as dependent variables and pressure and enthalpy as dependent variables and solved for these two variables simultaneously. They presented a 10-month simulation of a hypothetical hot-water reservoir with initial conditions similar to those in the Wairekei reservoir. No provisions were included during the course of a run for changing states - for example, from subcooled liquid to saturated steam. SPEJ P. 151

A Three-Phase, Two-Dimensional Compositional Thermal Simulator For Steam Injection Processes

Abstract A numerical model for simulating steam injection into a porous medium is described. The model simulates three-phase, three-dimensional, multi-component flow, with interphase mass and heat transfer, and thus accounts for changes in oil composition during steam injection. The example problems consider a system made up of water (or steam) and three hydrocarbon components. The partial differential equations describing the mass balance for each component were solved, together with a heat balance and auxiliary relations. Examples are presented illustrating the use of the model for cyclic steam stimulation and steamflooding. The sensitivity of the predictions to the input parameters is examined. It was found that steam distillation and gravity segregation are important mechanisms in the steam injection processes. Simultaneous solution of heat and mass balances and compositional constraint equations appear to represent the best procedure for efficient simulation. Other solution schemes tested are also discussed. Introduction EARLY EFFORTS in the mathematical modeling of steam injection processes utilized modifications of the Buckley-Leverett equation to allow for temperature variations(1,2), flow-pattern representation by means of stream tubes and subsequent temperature4 dependent Buckley-Leverett type displacement (3,4), and still simpler treatments where oil recovery was based only on the oil saturation reduction in the steam zone. Three-phase, one- and two-dimensional models for steam injection were first presented by Shutler (5,6). He solved the three-phase (oil, water and gas) mass balances simultaneously using Newtonian iteration. The energy balance equation was solved separately by the non-iterative ADIP. Interphase mass transfer between water and gas phases was allowed, but the oil phase was assumed to be nonvolatile, and the hydrocarbon gas was insoluble in the liquid phases. The implicit-pressure, explicit-saturation (IMPES) method was used by Abdalla and Coats'" to simulate steam drive. They developed a two-dimensional, three-phase (oil, water and steam) model. A new stabilized IMPES method was described by Vinsome(8). He simulated steam drive and steam soak by means of a three-phase (oil, water and steam) model. This model did not account for steam distillation. Coats, George, Chu and Marcum(10) presented a three-phase (oil, water and steam), three-dimensional numerical model for the steam injection process. The mass and energy balance equations were solved simultaneously. Water and steam equations were combined to avoid iteration on the mass transfer condensation term. No distillation of the oil was assumed. Weinstein, Wheeler and Woods(11) developed a model that describes three-phase (oil, water and gas) flow in one dimension and accounts for inter-phase mass transfer. The model does not include gravity and capillary pressure effects. "Sequential solution" (Spillette, Hillestad and Stone 12) was basically used as the method of solution. Recently, Coats(13) described a three-dimensional, three-phase (oil, water and gas) model that accounted for steam distillation. Sequential solution(15) was basically used as the method of solution. The present work describes a compositional simulator designed to simulate the steam injection processes in two-dimensional geometry.

A Fully Implicit Three-Dimensional Model in Curvilinear Coordinates

Abstract Because of saturation instabilities, conventional multiphase three-dimensional simulators cannot use large time-step sizes if grid blocks are small. Therefore, to be economical, such models do not use a fine grid near wells, and they are inadequate for accurately describing reservoir phenomena around producing wells. Only radial circular models can be used inexpensively to describe coning behavior. However, they are not always reliable because there usually is no axial symmetry around producing wells. producing wells. We propose a three-dimensional reservoir model capable of simultaneously describing flows at a distance from and around wells in a part of an oil reservoir subject to lateral drive from an aquifer or a gas cap. The model uses an asymmetrical curvilinear grid whose mesh sizes increase with distance from the wells. It is a fully implicit model that can use large time-step sizes. A convergence acceleration device is used for equation solving. The results of this model have been checked against those obtained with a large laboratory model. Wells were in a line pattern and in a staggered one. pattern and in a staggered one Introduction In an oil reservoir subject to lateral drive from an aquifer or a gas cap, no axial symmetry exists around the producing wells. As a result of such production and of lateral drive, the transition zone becomes deformed and, in particular, the gas moves along the top of the layer toward the wells. Pressures (especially if the problem is with imposed Pressures (especially if the problem is with imposed bottom-hole pressures) and coning phenomena near the wells in such transition zones is very difficult to describe with existing numerical reservoir models. The grids and numerical methods usually used for this are not properly adapted to solve the problem. FROM THE STANDPOINT OF PATTERN A circular radial coning model is not representative. Also, a conventional three-dimensional model with parallelepipedic grids has the disadvantage of using large blocks corresponding to producing wells (this will be called well blocks in the remainder of the paper). The average pressure and saturation values are quite different from the actual well values since it is in the vicinity of these wells that pressures and saturation vary the most in space. One way of solving this would be to adopt a highly irregular grid with small cells in the vicinity of the wells. The cost of simulation with such a system would be very high because of the excessive number of grids, including some perfectly useless ones in distant zones with low gradients. FROM THE NUMERICAL STANDPOINT From the numerical standpoint a conventional model of the type described above using the finite-difference technique and "implicit-pressure explicit-saturation" concept requires time steps that become smaller as mesh size decreases. This is an additional reason for trying to avoid any tightening up of the grid. As a result, while maintaining a relatively loose grid, some authors have tried to find special processing methods for the individual points made up by the wells:arbitrarily reducing permeability in the well blocks to adjust bottom-hole pressures,approximating pressures by polynomial functions of a high order pressures by polynomial functions of a high order in the vicinity of wells and of a lower order elsewhere,estimating bottom-hole pressure by extrapolation (based on Darcy's law written in radial form and integrated for steady-state conditions) from grid blocks adjacent to the well block, andinserting a radial-circular coning model in well blocks. (This last solution is not a good one because of the unsatisfied symmetry conditions and assumptions on the linking of both models.) We propose here a three-dimensional reservoir model capable of simultaneously describing flows at a distance and in the vicinity of wells. This model is discretized according to a curvilinear grid in the plane of the layer, enabling small meshes to be formed around the well and large meshes in zones farther away without creating any difficulties of linking between meshes. SPEJ P. 361

Methods for Numerical Simulation of Water and Gas Coning

Abstract This paper describes and evaluates three numerical methods for the simulation of well coning behavior. The first method employs the implicit pressure-explicit saturation (IMPES) analysis with pressure-explicit saturation (IMPES) analysis with the production terms treated implicitly. The second technique is similar to the first model except that the interblock transmissibilities are also treated implicitly in the saturation equation. The third model is fully implicitly with respect to all variables in a manner qualified in the introduction and utilizes simultaneous solution of the difference equations describing the multiphase flow. The use of implicit transmissibilities in the IMPES model results in a several-fold increase in the allowable time-increment size over that attainable with the implicit production IMPES scheme, while the computing time per step is increased by less than 10 percent. The fully implicit model accepts larger time-increment sizes than possible with the first two methods but requires 3.3 times the computing time per time step needed by the second model. The fully implicit model is substantially more efficient for problems involving high capillary forces (treated explicitly in the IMPES methods) and small computing grid blocks at the wellbore. In problems involving moderate capillary forces and larger grid spacings, the fully implicit method and the implicit transmissibility IMPES technique are comparable in computing efficiency. The results of three coning studies are presented: a water-oil problem, a three-phase coning presented: a water-oil problem, a three-phase coning example, and a comparison of simulation results with a laboratory coning experiment. Also presented is an analysis of truncation error and a comparison of computational work requirements. Introduction This study was performed to evaluate three finite-difference schemes for simulating well coning behavior. The basis for this evaluation was the IMPES (implicit pressure-explicit saturation) model with explicit transmissibilities and implicit production terms. This model is referred to hereafter production terms. This model is referred to hereafter as Model 1. The next model evaluated in this work is an IMPES model similar to Model 1, except that the saturation-dependent interblock transmissibilities are treated implicitly rather than explicitly in the saturation equation. The third model is fully implicitly with respect to all variables and terms transmissibilities, pressure, saturation an capillary pressure - and utilizes simultaneous solution of the difference equations describing the multiphase flow. These two models are referred to hereafter simply as Model 2 and Model 3. For the purpose of clarity all models are described in purpose of clarity all models are described in reference to the problem of incompressible, two-phase flow. The techniques are equally applicable, however, to compressible, three-phase flow models. The examples chosen for illustration employ both incompressible and compressible simulation models. SPEJ p. 425

Use and Misuse of Reservoir Simulation Models

In reservoir simulation, the question is not whether, but how and how much. The complexity of the questions being asked, and the amount and reliability of the data available, must determine the sophistication of the system to be used. Introduction Webster defines "simulate" as "to assume the appearance of without the reality". Simulation of petroleum reservoir performance refers to the construction and operation of a model whose behavior assumes the appearance of actual reservoir behavior. The model itself is either physical (for example, a laboratory sandpack) or mathematical. A mathematical model is simply a set of equations that, subject to certain assumptions, describes the physical processes active in the reservoir. Although the model itself obviously lacks the reality of the oil or gas field, the behavior of a valid model simulates (assumes the appearance of) that of the field. The purpose of simulation is to estimate field performance (e.g., oil recovery) under a variety of performance (e.g., oil recovery) under a variety of producing schemes. Whereas the field can be produced producing schemes. Whereas the field can be produced only once and at considerable expense - a model can be produced or "run" many times at low expense over a short period of time. Observation of model performance under different producing conditions, performance under different producing conditions, then, aids in selecting an optimum set of producing conditions for the reservoir. More specifically, with reservoir simulation the following are possible.We can determine the performance of an oil field under water injection or gas injection, or under natural depletion.We can judge the advisability of flank waterflooding as opposed to pattern waterflooding.We can determine the effects of well location and spacing.We can estimate the effect of producing rate on recovery.We can calculate the total gas field deliverability for a given number of wells at certain specified locations.We can estimate the lease-line drainage in heterogeneous oil or gas fields. The tools of reservoir simulation range from the intuition and judgment of the engineer to complex mathematical models requiring use of digital computers. The question is not whether to simulate but rather which tool or method to use. There is no general answer to the question as to when the computerized mathematical model should be employed. After some preliminary discussion of the nature of mathematical models and sources of error, some valid and invalid model applications will be illustrated here with specific examples. It should be noted that this discussion is restricted to models for multidimensional, single or multiphase flow in reservoirs. These models apply to dry gas reservoirs and to oil reservoirs undergoing natural depletion or pressure maintenance (such as natural water drive, waterflood or gas injection). The Mathematical Model In 1959 Douglas, Peaceman and Rachford proposed the "Leap-Frog" and "Simultaneous Solution" methods for solving two-dimensional, two-phase flow problems. In 1960, Stone and Sheldon described an problems. In 1960, Stone and Sheldon described an "Implicit Pressure-Explicit Saturation" method. JPT P. 1391