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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
1 - 20 of 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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