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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
48 results
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Stochastic Simulation of Lithofacies: an Improved Sequential Indicator Approach
Authors V. Suro-Pérez and A. G. JournelReservoir characterization demands the estimation or simulation of important reservoir parameters such as permeability and porosity. Often, reservoir geology is accounted for via continuous values of permeability or porosity, without explicit reference to the reservoir facies. The approach proposed starts by modeling the spatial distribution of the reservoir facies and, only then, conditions the generation of permeability/porosity values to the simulated facies geometry. It allows accounting for spatial relationships between different lithologies (covariances and crosscovariances), and uses such relations to estimate (or simulate) the most probable lithology at any specific location. The ARCO data set is used to build stochastic simulations of six different lithofacies over a particular vertical section, considering only three conditioning wells out of ten actually available. Each stochastic simulation is a lithological reservoir image which reproduces the patterns of continuity of the lithofacies considered and honor the data values at the conditioning well locations. Repeated generation of such lithofacies images, allows assesment of the geological heterogeneities impact on the oil recovery.
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Combining Geology, Geostatistics and Multiphase Fluid Flow for 3D Reservoir Studies
Authors A. Galli, D. Guérillot and C. RavenneA geostatistical model is used to generate a 3D lithofacies model between wells. Then fluid flow simulations are performed in this 3D geometry. The geological investigations and the main characteristics of the geostatistical model are recalled and comments on the simulation and the fluid flow behavior are made.
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Development of Geostatistical Methods Dealing with the Boundary Conditions Problem Encountered in Fluid Mechanics of Porous Media
Authors A. Dong, S. Ahmed and G. de MarsilyUntil recently, partial differential equations (pde) and geostatistics were seldom associated. Yet, in earth sciences, examples of phenomena controlled by these equations are numerous: fluid flow or heat transfer...The presence of a pde implies three main points:
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Large-Scale Barriers in Extensively Drilled Reservoirs
Authors J. Høiberg, H. Omre and H. TjelmelandShale and carbonate semented barriers in petroleum reservoirs are of major concern due to its influence on flow characteristics. A stochastic model for shale distribution in a sandstone reservoir is established. A Markov field model combined with marked point processes are applied. Based on the model and constrained by the observations in wells, several realizations of the reservoir description are generated. The description is homogenized and transfered to a format suitable as input to ECLIPSE. Reservoir production simulations are performed on the realizations, hence the uncertainty in the production profiles can be evaluated.
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Accurate Calibration of Empirical Viscous Fingering Models
Authors F. J. Fayers, M. J. Blunt and M. A. ChristieWe review the use and calibration of empirical models for viscous fingering. The choice of parameters for the three principal approaches (Koval, Todd and Longstaff, and Fayers methods) is outlined. The methods all give similar levels of accuracy when compared with linear ex-periments, but differ in performance in two-dimensional applications. This arises from differences in the formulation of the total mobility terms. The superiority of the Todd and Longstaff and Fayers methods is demonstrated for two-dimensional and gravity influenced flows by comparison with experiments and high resolution simulation. The use of high resolution simulation to calibrate empirical models in a systematic manner is described. Results from detailed simulation demonstrate the sensitivity of empirical model parameters to viscous to gravity ratio, recovery process (secondary, tertiary or WAG), and geological heterogeneity. It is shown that for large amplitude heterogeneities with short correlation lengths, the accuracy of the empirical models is not satisfactory, but is improved by the addition of a diffusive term.
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Effective Permeability of Heterogeneous Reservoir Regions
Authors L. J. Durlofsky and E. Y. ChungMost reservoir heterogeneities, such as detailed crossbedding, are on a scale that is too fine to be directly in cluded in reservoir simulation or reservoir engineering calculations. Therefore, ‘averaging’ procedures are required to scale up permeabilities from the fine scale, over which heterogeneities occur, to the coarser scales appropriate for reservoir engineering computation. The purpose of this paper is to describe the numerical implementation of such an averaging technique and to apply this technique to the scale up of a representative region of a crossbedded eolian depositional system. Toward this end, a triangle based nonconforming finite element approximation of the fine scale pressure equation, subject to appropriately formulated periodic boundary conditions, is salved to give the effective coarse scale permeability. It will be seen that the magnitudes of the cross terms of the effective permeability tensor are significant for the eolian system considered and that the accurate determination of these terms is essential for modeling flow through crossbedded systems.
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Application of Analytical Methods in Predicting Waterflood Performance of Reservoirs with Stochastic Sand Bodies
Authors O.B. Abu-elbashar, T.S. Daltaban, C.G. Wall and J.S. ArcherDue to the complex architecture of fluvial reservoirs composed of uncorrelatable (stochastic) sand bodies, straightforward application of simple one dimensional analytical methods to predict their waterflood performance is not possible. This is due to the inadequacy of the existing techniques to represent the complex connectivities between flow elements in one dimensional models. In this paper, a new stochastic sand modelling method is presented. It simplifies sand description in grid blocks for simulation, and for averaging of the reservoir parameters such as permeability and porosity. Although the method is conditioned by the available seismic and well data which are extremely sparse, it employs only modest use of statistical methods. Also, contrary to many current techniques which are restricted to two dimensional realisation, the current method extends the realisation into three dimensions. This is carried out by using the special stochastic sand conditioning technique in which the generated flow elements are matched to a prescribed girded domain.
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Numerical Simulation and Homogenization of Diphasic Flow in Heterogeneous Reservoir
Authors B. Amaziane, A. Bourgeat and J. V. KoebbeBy mean of the so called homogenization theory, see for instance [16], we derive mathematically rigorous “effective” reservoir equations from exact local equations of incompressible two-phase flow (miscible or immiscible) in a heterogenous reservoir. The main result is that “effective” equations are exactly of the same type as the original ones. In general cases the effective permeability tensor is given only as a mathematical limit. In some special cases where there is some additional knowledge on the heterogeneities repartition as for instance a spatial periodic repartition, we may really compute the effective parameters and then numerically compare both behaviour in a heterogeneous or in a homogenized reservoir. In [1] and [12], we have presented some simulation of stratified medium; in [2] and [12] we have presented several simulations on spatially periodic heterogeneities. Herein we are presenting only one of such a simulation to illustrate our results.
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Stochastic Characterization of Grid-Block Permeabilities: from Point Values to Block Tensors
Authors J. J. Gomez-Hernandez and A. G. JournelPermeability is generally measured at the core scale and only at a few locations. By using geostatistical techniques, one can generate alternative high resolution images of the reservoir at the measurement scale. These images will honor both the data values at the data locations and some prior structural information as modeled, for instance, by a covariance function. However, these images must be averaged to the scale of the flow simulation grid-blocks.
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Large-Scale Properties for Flow through a Stratified Medium: A Discussion of Various Approaches
Authors A. Ahmadi, A. Labastie and M. QuintardThe determination of effective properties or pseudo-functions for two-phase flow through heterogeneous porous media is a problem of central importance in petroleum engineering (reservoir simulation). Our purpose was to compare published theories by performing test case studies. The test case corresponded to the waterflooding of a two-strata heterogeneous medium. Generalized Darcy’s equations were solved by using a black-oil reservoir simulator. Several simulations were performed by varying the filtration velocity and the relative importance of gravity effects versus capillary and viscous effects. Based on an extensive literature survey, two fundamental methods were essentially tested: the large-scale averaging method and an application of the fine-grid to coarse-grid method.
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The Use of Second-Order Godunov-Type Methods for Simulating EOR Processes in Realistic Reservoir Models
Authors K. Holing, J. Alvestad and J. A. TrangensteinEnhanced oil recovery using polymer or chemical flooding involves the displacement of oil by slugs with dynamic behavior critical to the response and the efficiency of the process. The numerical modelling of these processes is very difficult, since the dynamics of the slug flow lead to concentration fronts that are not self-sharpening. Conventional upstream-weighted finite differences lead to both substantial grid orientation errors and to smearing of sharp fronts. This paper reports the formulation and performance of a second-order Godunov-type finite difference method for modelling of two-dimensional (2-D), three-component incompressible polymer floods. The scheme successfully handles realistic applications and significantly reduces both the spreading of fronts and grid orientation effects, when compared to the standard upstream-weighted finite difference schemes. Polymer slug injection examples will be shown for both areal and cross sectional models involving various reservoir heterogeneities and realistic reservoir models. Also, a one-dimensional (1-D) example is given.
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Modelling Flow through Heterogeneous Porous Media with Boundary Integrals using Higher-Order Surface Singularities
Authors D. W. Wong, J. S. Archer and J. M. R. GrahamNumerical modelling of heterogeneous porous media has received attention at a variety of scales from pore dimensions to those of hundreds of meters. Most of the approaches to predict the behaviour of flow have focussed on enhancements to established finite difference schemes (e.g., local grid refinement, multi-grid methods). Although finite difference schemes have many advantages, in fine grids they are difficult to apply properly to curved surfaces with Neumann boundary conditions, which occur at the boundary between media of different permeability and they appear very difficult to use with multiple heterogeneities. In such circumstances boundary conditions can often be expressed using boundary integral equations, and solutions by powerful boundary integral equation methods can be applied.
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A Finite Element Method for Calculating Transmissibilities in N-point Difference Equations using a Non-Diagonal Permeability Tensor
By P. SamierPetroleum reservoir are always heterogeneous. Averaging techniques consist mostly in defining an equivalent homogeneous permeability tensor for a given heterogeneous porous medium whose absolute permeability is a space dependent function. The equivalent permeability tensor is generally symmetric but non-diagonal: three unusual off-diagonal terms Kxy, Kxz and Kyz are to be considered in addition to the standard diagonal permeability terms in x, y and z direction. Non-diagonal tensors arise also in a mesh whose axes are not the principal directions of the permeability tensor: an application for horizontal well simulation is briefly presented.
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Implicit Flux Limiting Schemes for Petroleum Reservoir Simulation
More LessExplicit total variation decreasing (TVD) numerical methods have been used in the past to give convergent, high order accurate solutions to hyperbolic conservation equations, such as those governing flow in oil reservoirs. To ensure stability there is a restriction on the size of time step that can be used. Many petroleum reservoir simulation problems have regions of fast flow away from sharp fronts, which means that this time step limitation makes explicit schemes less efficient than the best implicit methods. This work extends the theory of TVD schemes to both fully implicit and partially implicit methods. We use our theoretical results to construct schemes which are stable even for very large time steps. In general these schemes are only first order accurate in time over all, but locally may achieve second order time accuracy. Results are presented for a one dimensional Buckley Leverett problem, which demonstrate that these methods are more accurate than conventional implicit algorithms and more efficient than explicit methods, where smaller time steps must be used. Results from black oil and compositional simulators are presented.
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The Use of Boundary Element Method in Front Tracking for Composite Reservoirs
Authors J. Kikani and R. N. HorneThis paper demonstrates a new approach using the Boundary Element Method (BEM) to solve for pressure transient behavior in composite and sectionally homogeneous reservoirs. A boundary element solution is proposed in Laplace space to a piecewise homogeneous reservoir with arbitrary geometry of each region. Any number of such regions with different rock and fluid properties can be included in the solution procedure. This formulation can solve fluid injection problems which show composite behavior (as in steam injection and CO2 flooding). In addition, impermeable barriers of any shape and orientation as well as large pressure support sources (aquifers) can be included.
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Control Volume Method to Model Fluid Flow on 2D Irregular Meshing
By I. FailleModelling fluid flow in porous media requires the approximation of an elliptic partial differential operator such as div which appears when substituting Darcy’s law m mass conservation equation. When the medium is heterogeneous, the permeability tensor K is discontinuous and depends on the lithologic nature of the medium. In order to take this dependence well into account, the grid used to discretize the set of equations is such that cell boundaries are aligned with geological discontinuities.
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Heterogeneous Porous Media and Domain Decomposition Methods
Authors M. S. Espedal, R. Hansen, P. Langlo, O. Saevareid and R. E. EwingThe physics governing flow in porous media are character ized by localized phenomena taking place on scales that are small compared to the overall size of the reservoir. To obtain proper resolution, local refinement, both in space and time, seems to be promising. The approach is especially attractive within the framework of domain-decomposition, where the global solution is constructed from local computations on sub-domains where the resolution and even the entire solution strategy can be specially designed to match local properties of the model. The strategy is well suited for parallel computer architectures as well as integration into existing large scale simulators. In the present paper we will focus on heterogeneous rock properties. Core-samples, well-tests and production data provide information on widely different scales. To be useful in simulations, all the data have to be brought to the scale of the discrete blocks of the simulator. For this purpose, a stochastic approach based on assuming a multivariate lognormal distribution of permeability data will be investigated in the context of local grid refinement.
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Parallel Simulation of Petroleum Reservoirs
More LessThe parallelisation of an oil reservoir simulator is de scribed. The target machine architecture is an array of transputers. The reservoir simulator is described. Since the most time consuming part of the reservoir simulation is the solution of the resultant systems of equations the parallelisation of this is discussed. The scaling properties and perfonnance of a key routine are reported.
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Comprehensive Mathematical Modeling of Horizontal Wells
By M. R. IslamRecently, most technological advances in the petroleum industry have been in the areas of horizontal wells. Horizontal wells have been reported to produce up to 10 times more oil than that produced by vertical wells. Even though recent advances in horizontal well technology have led to a large drop in drilling and operation costs, very little has been done to advance mathematical modeling of a horizontal well in order to achieve more realistic prediction capabilities. This paper describes a reservoir simulator capable of dealing with important features of a horizontal wells, including weilbore dynamics and near weilbore radial flow.
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Curvilinear Grid Generation Techniques
Authors C. L. Farmer and D. E. HeathAlgebraic and numerical techniques for constructing curvilinear simulation grids are reviewed. Attention is given to the control of smoothness and orthogonality while deforming grids around geological structures, well patterns, faults and reservoir boundaries. A new method for constructing grids using global minimisation is outlined.
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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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Constant-Time Step Deconvolution Model for Variable-Rate Well Test Pressure Data
Authors S. Buitrago, G. Gedler and R. ManzanillaThe transient pressure well test analysis is a technique which allows the petroleum engineer to determine reservoir properties, such as permeability, porosity, the drainage volume of the reservoir, static pressure and, in general, to characterize or describe the reservoir-well system in order to indicate damage or stimulation of well, fracturing or not of well, the existence of faults or flow barriers, the approximate shape of the drainage area of the reservoir or the change of the reservoir lithological properties.
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Inverse Modeling for Compressible Flow. Application to Gas Reservoirs
Authors B. Bréfort and V. PelcéClassical methods to determine permeability distribution in a gas reservoir are sometimes inadequate: for a gas field, only core measurements are available and scale problems always appear when applying these values to the whole reservoir. An inverse modeling procedure is proposed to determine automatically the permeability field for a dry-gas reservoir. This method consists in the minimization of the difference between a historical record of real pressures and pressures computed from discretized equations of monophasic gas flow in porous medium. A Lagrangian method combined with geostatistical information is used. Two applications are presented: a history matching of a gas field giving a permeability distribution then used for prediction and a transient draw-down test on a gas well which gives an estimate of the local permeability in the surrounding matrix.
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Computer Geological Simulation in Oil Recovery
More LessAt the Siberian Scientific-Research Institute for the Oil Industry we have developed some methods of solving geological problems. They are realized in the form of a computer system GEOPAK-2. The system can solve three classes of problems providing construction of a producing formation geological model according to the type of a filtration model used: a detailed geological correlation of well profiles; an estimation of the object geological structure complexity parameters; geometrization of pools and differentiated calculation of reserves.The first of them is ba sic for solving the others.The re sults of solving the second one are used in oil recovery estima tion procedures based on one—di mensional statistical models of filtration.The solution of the third-class problems is necessary when using two- and three-dimensional determined models of filtration.
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A New Formulation for Generalised Compositional Simulation
Authors R. E. Mott and C. L. FarmerAn important recent trend in reservoir simulation has been the development of multiple application simulators which can model both black oil and compositional processes within a single program. A number of different mathematical formulations have been proposed for solving the fluid flow equations in these simulators. With an IMPES solution scheme there are two basic approaches; Newton-Raphson methods, such as that of Young and Stephenson (1983), and the volume balance method of Watts (1986). Various fully implicit schemes have also been described. Although these methods all solve the same underlying equations, the relationship between them is not easily understood.
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P3D Modeling of Vertical Hydraulic Fracture Growth
More LessRecent short cut design procedures for hydraulic fracturing treatment are based on two dimensional geometric description: either a penny shape or constant height is supposed. Unless the minimal horizontal stress is constant along the vertical direction or a high confining stress acts in the overburden and underburden these assumptions cannot be justified.
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Irregular Averaging of Filtration Transfer Processes in Heterogeneous Media
More LessThe modification of the asymptotic method of averaging operators with rapid ly oscillating coefficients (Sanchez-Palencia, 1980, Bakhvalov et al., 1984) is suggested for the media with heavy heterogeneity (great differences in permeability of separate areas). Several new models of the filtration processes generalizing a known phenomenological simulator (Barenblatt et al., 1960) are constructed with the aid of this method. The composite has been selected as a physical model, the components of which are a connected high-conductive system and periodically scattered isolated compact blocks. The system period l is muchless than macroscale L, so ε=1/L<<1.
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History Matching Problems of Filtration Theory. Complex Adaptative Geological Models of Fields
Authors B. Palatnik, I. Zakirov and G. AgaevTo choose improved oil or gas field development its necessary to create algorithm providing the possibility to calculate a field development performance during all the period of production for the different controlled influences on the bed. The simulation reliability and therefore the validity of the accepted project decisions depends on the degree of the adequacy of the mathematical models used to real physical processes as well as on the degree of the truthfulness of the initial geological-field information used at the simulation. It is the important peculiarity of the large oil and gas fields that they have common hydrodynamical connection between processes which take place in an aquifer, in a bed, in wells and fluid gathering, compressing, cleaning and treating and field pipeline transferring systems. Therefore to carry out long-term forecast of field development performance it is necessary to create the complex adaptive mathematical model which unites both reservoir and ground technological equipment (GTE) models.
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Multilevel Methods in Porous Media Flow
Authors R. Teigland and G. E. FladmarkThe purpose of this paper is to describe the development and application of a cell-centered multilevel method applied to multiphase flow in porous media. The model that we consider is two-phase, 2-D compressible flow in a porous media. In section 2 we set up the coupled set of non linear partial differential equations to be solved and discretize them using cell-centered differencing. A fully implicit scheme with water saturation and oil pressure as unknowns is used in the solution of the system.
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