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ECMOR II - 2nd European Conference on the Mathematics of Oil Recovery
- Conference date: 11 Sep 1990 - 14 Sep 1990
- Location: Arles, France
- ISBN: 978-27-1080-589-2
- Published: 11 September 1990
21 - 40 of 48 results
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An Analytical Investigation by the Method of Characteristics of Gravity Stabilised Gas Injection
More LessAn Analytical Investigation by the Method of Characteristics of Gravity Stabilised Gas Injection
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Dispersive Mixing in Unstable Displacement
More LessThe stable displacement of miscible fluids through a porous medium that has many small-scale permeability variations exhibits dispersive mixing between the fluids. The unstable displacement exhibits two flow regimes: one in which the displacement is dominated by viscous fingers and one in which the displacement is dominated by dispersive mixing due to the permeability variations. In the viscous finger-dominated regime the mixing zone expands linearly in time; in the dispersive-mixing-dominated regime the mixing zone expands as the square root of time. We have estimated the condition of transition between the two flow regimes. This condition has been tested by monitoring the development of the mixing zone in detailed numerical simulations of our own and by evaluating simulations reported by Araktingi and Orr, by Crump and by Moissis, et al. In addition, we found that when the unstable displacement is dominated by dispersive mixing, the expansion of the dispersive-mixing zone can be calculated according to the analytical model proposed by Kempers (1989). This model should be considered as a better alternative to the conventional Koval or Todd and Longstaff models for displacements with moderately mobility ratios.
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Fluid Flow in Porous Media and Related Rock Mechanics Problems
Authors M. Boutéca and J. P. SardaThe equations used to describe fluid flow in reservoirs are usually worked out without taking rock deformations directly into account. Indeed these de formations are included by correcting the compressibility of the fluid. On the basis of coupled equa tions for rock deformations and fluid flow, this paper analyzes the rock mechanics aspects of the diffusivity equatinn used by reservoir engineering specialists. The paper emphasizes the corresponding restrictive assumptions. It then goes on to explain the effective computing of rock deformations and fluid flow by iteration between a mechanical model of the de formable medium and a flow model.
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Composition Paths in Binary C02-C10 Displacements: Effects of Reservoir Heterogeneity and Crossflow on Displacements with Limited Solubility
Authors K. K. Pande and F. M. OrrMaterial balance equations are formulated for the flow of two-phase, two-component mixtures in a porous medium consisting of two layers with differing permeabilities. Effects of viscous crossflow between the layers are modeled under the assumption that enough crossflow has taken place that fluids in the two layers are in vertical pressure equilibrium. The resulting set of coupled hyperbolic partial differential equations is solved using the method of characteristics. Example solutions are reported for displacements of decane by 002. Three layer permeability ratios are considered, 1.5, 3.0, and 10.0, and the solutions are compared with the corresponding solutions without fluid crossflow.
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Mathematical and Numerical Analysis of a Hyperbolic System Modeling Solvent Flooding
Authors T. Johansen and R. WintherThe fluid system under considerations in this paper consists of three chemical components. The phase properties of this system depends on its composition and we assume that a maximum of two phases can be formed. The mathematical model governs the purely convective transport of this fluid system through a one dimensional homogeneous porous medium. The model is discussed through mathematical analysis and numerical experiments.
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Mixed Methods, Operator Splitting, and Local Refinement Techniques for Simulation on Irregular Grids
Authors M. S. Espedal, R. E. Ewing and T. F. RussellThe partial differential equations used to model multiphase and multicomponent fluid flows are convection-dominated, with important local properties. Operator-splitting techniques have been defined to address these different phenomena. Convection is treated by time stepping along the characteristics of the associated pure convection problem and diffusion is modeled via a Galerkin method for miscible displacement and a Petrov Galerkin method for immiscible displacement. These ideas have been generalized to Eulerian-Lagrangian Localized Adjoint Method (ELLAM) formulations which conserve mass and allow more accurate treatment of boundary conditions. Accurate approximations of the fluid velocities needed in the characteristic time stepping are obtained by mixed finite-element methods.
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Domain Decomposition Methods in Reservoir Simulation Coupling Well and Full Field Models
Authors O. Gosselin and J. M. ThomasIn industrial reservoir simulators wells are usually with just a few large discrete cells and simplified source terms. The complex flow mechanisms that arise around wells are thus not accurately represented. This can have serious consequences on results. A natural idea to obviate these defects would be to use a finer grid mesh around the wells. But such local grid refinements intoduces mesh irregularities with an excessive contrast in the mesh sizes between the grids. Conventional numerical schemes and conventional solvers to handle such irregularities are often inadequate and considerbly degrade the computational performance of codes. This papers considers an other approach by decomposing the above problem over two overlapping or non overlapping subdomains: reservoir and wells. For each time step, we solve the differential equations in separate mesh resolutions and iterate between subdomains until convergence is reached at the internal boundary. The boundary conditions are provided by results of the adjacent domain (pressures, saturations and fluxes). We present some techniques of decomposed modelling applied to model equations of diphasic immiscible flows. We use overlapping subdomains or alternately non overlapping domains with relaxation of interface conditions to achieve convergence. These algorithms have been developped for a threedimensional Dead-Oil model with slight compressibility, under “fully implicit” formulation. We compare different strategies related to imposed boundary conditions on the interface of subdomains (Dirichlet/Neumann) and their influence on number of global iterations.
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A Characteristic Finite Element Method for Solving Non-Linear Convection-Diffusion Equations on Locally Refined Grids
More LessA method for solving nonlinear convection-diffusion equations in two or three space dimensions is described. These equations play an important role in the numerical simulation of immiscible, two-phase flow through porous media. All computations are performed on locally refined and dynamically adapted grids. This increases efficiency and ensures an optimal representation of shock fronts. Operator-splitting is used to decouple convection and diffusion, which reduces the problem to an alternating sequence of hyperbolic and elliptic equations. An accurate characteristic method deals with the hyperbolic equations. Nonlinearities in the convection term are treated by solving Riemann problems along stream lines. Elliptic equations are discretised by mixed finite elements and solved by multi-grid. Gravity effects are included by a spatial splitting of the convection term. The method induces almost no numerical diffusion. It also permits to use large time steps and it conserves mass exactly. Numerical results are presented which demonstrate the performance of the method for some multi-dimensional test problems.
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A Coordinate System for Local Grid Refinement Close to Wells
By S. EkrannThe construction of an orthogonal curvilinear grid is described, suitable for local refinement close to wells. The grid is obtained by conformal mapping. It is approximately polar close to the well, and provides for a smooth transition to a surrounding cartesian grid. It is shown that this grid has several advantages over competing grids. Examples illustrate that strongly improved accuracy, over coarse grid simulations, is obtainable with relatively few extra grid blocks.
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Data Structure and Algorithms for Adaptive Mesh Refinement
Authors T. Hermitte and D. GuérillotFluid flow in an oil and gas reservoir is governed by a system of nonlinear partial-differential equations with different types of boundary conditions. To compute an approximate solution to this evolving problem, the integration domain (geometry of the reservoir) is discretized. Given the large size of this three-dimensional geometry (several kilometers in areal extent and sometimes up to 100 meters thick), and the cost of fluid flow simulation depending on the number of unknowns for each grid block, the time spans evolved (several years of production) and the features of current computers, the mesh used for solving these partial-differential equations is not refined enough.
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Two Dimensional Stochastic Modelling of Flow in Non-Uniform Confined Aquifers. Correction of the Systematic Bias Introduced by Numerical Models when they are used Stochastically
Authors P. Lachassagne, E. Ledoux and G. de Marsily1 Averaging Permeabilities / 2 Use Of A Two Dimensional Numerical Model With Probabilistic Theory
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An Estimator for the Effective Permeability
Authors L. Holden, J. Høiberg and O. LiaAn estimator for the effective permeability, based on one-phase incompressible flow, is presented. The method gives accurate estimates for afl types of heterogeneous blocks. It is considerably faster than a full simulation and also provides a measure of the error involved.
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Numerical Simulation of Hydraulic Fracturing in a Discrete Element System
Authors S. Thallak, L. Rothenburg, M. Dusseault and R. BathurstHydraulically induced fracture in an assembly of cohesionless discs is numerically simulated using a discrete element model, consisting of discrete particles coupled with an interLvoid fluid flow model. Grains are represented by circular discs; to simulate flow, a geometrically coupled channel network is created by assigning nodes to pores, and flow channels to pore throats. Flow rates in channels are assumed to be proportional to the pressure gradient according to the Hagen-Poiseuille equation. The paper describes the main features of the model and explains the fracture initiation due to fluid injection and the propagation process at a grain level.
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A 3-D Network Simulating Two Phase Immiscible Displacements in Porous Media
Authors D. Zhou and E. H. StenbyAs a continuation of the work on simulation of multiphase flow in porous media, associated with Enhanced Oil Recovery, the previously developed two dimensional network model ( Thou and Stenby, 1989) was extended to a three dimensional version. The model can simulate immiscible two phase displacement in porous media using invasion percolation theory. In this paper, the structure of the 3D model, the stability, and the boundary conditions at the outlet of the medium will be discussed briefly. The simulated results are compared with experimental data with respect to the capillray and effects, viscosity ratio and the wettability effects on the displacement process.
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An Analytical and Numerical Study of the Three-Phase Surfactant Displacement Problem
By J. W. BarkerNumerical simulation has been used to investigate the effects of middle phase mobility, dispersion, and a salinity gradient, on the performance of a tertiary surfactant flood involving Type III (three phase) phase behaviour. The BPOPE simulation model’ was used. The results indicate that the relative mobility of the middle phase is an important consideration in the design of a surfactant flood. This mobility should be low compared to the mobility of the oil phase. Unfortunately, the effects of dispersion on the low mobility flood are severe, and “self-sharpening” behaviour must be introduced, for example by means of a salinity gradient. In the salinity gradient flood, the middle phase mobility again has a strong influence on the solution, and the salinity is not necessarily optimal at any point within the surfactant bank. In a future paper, we shall demonstrate that inclusion of an alcohol in the surfactant slug can also produce self-sharpening behaviour, provided the properties of the alcohol are chosen correctly. No additional benefit is gained by varying the alcohol concentration within the chemical slug. Varying the composition of the sur factant within the surfactant slug does not produce selfsharpening behaviour.
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A Triangular Model for Three-Phase Flow
More LessIn this paper some numerical experiments with a recently proposed model for the flow of water, oil and gas in a porous medium are discussed. We consider the case of incompressible flow and neglect cappilary effects. In many oil reservoars the three phases are in contact with each other. The most important parameters describing the flow are the three-phase relative permeabilities. Unfortunately, these data are typkally very difficult to measure and is often not available.
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A Practical Front Tracking Technique for Control of Numerical Diffusion
Authors M. Halilu and R. I. IssaA front tracking scheme capable of drastically reducing the numerical diffusion which inhibits resolution of sharp contact fronts is developed and implemented in a multi-dimensional simulation technique. The most attractive feature of the scheme is its ease of incorporation into standard reservoir simulation methods, whereby only the finite difference equations in the vicinity of the contact front need to be altered. Implementation of the scheme in conjunction with an IMPES method is outlined here, and the results of one and two dimensional flow applications are presented. These show marked improvements in the ability to capture sharp fronts.
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FAG Method for Reservoir Simulation
Authors R. Boyer, B. Martinet and K. SaïkoukThe simulation of a bidimensional two-phase immiscible flow is considered using a static composite grid (locally refined). This composite grid consists of a global, rectangular coarse grid and a set of locally refined patches around wells. For the spatial discretisation, we use cell centered finite volume approximation and for time stepping, the IMPES scheme with upstream weighting. In order to solve the pressure equation on composite grid efficiently, we use the FAC (Fast adaptive composite) method proposed by Mc. CORMICK [6]. This technique has analogies with BEPS method used by EWING and al. [2], [3]. A numerical application (oil-water flow) is developed to illustrate the FAC method.
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Mixed Finite Elements for Multiphase Flow in Porous Media Consisting of Different Rock Types
Authors Ø. Bøe and G. E. FladmarkModelling of heterogeneous reservoirs has received increased attention during the last years. In high permeable zones, the flow could be dominated by advection, while processes such as imbibition or drainage might dominate in other parts. To illustrate the ideas presented in this paper, we consider two phase incompressible, immiscible flow in two spatial dimensions. The ideas could be generalized to more than two phases. The relative permeabilities kr l = w, nw (w: wetting phase, nw: nonwetting phase) and capillary pressure data Pc are highly dependent upon the rock type. Indeed, we shall define a rock type as a porous medium for which such data are dependent upon the saturations only. Let the reservoir domain Ω concist of Nr rock types.
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Radial Transport in Porous Media with Dispersion and Adsorption
More LessA transport equation characterizing dispersion Darcy’s velocity varies inversely with the radius: and absorption of a chemical solution for radial porous flow can be derived by a mass balance as follows:
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