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ECMOR I  1st European Conference on the Mathematics of Oil Recovery
 Conference date: 14 Jul 1989  16 Jul 1989
 Location: Cambridge, UK
 ISBN: 9789462821347
 Published: 01 July 1989
1  20 of 46 results


A Review of Reservoir Simulation Techniques and Considerations on Future Developments
More LessReservoir simulation technology has been developed ever since the first computers were built. By the late 1980’ies it is now a considerable industry in itself with a large userbase. The advancements of computing technology together with enhancements in numerical techniques have given the industry a sophisticated tool for prediction of reservoir behaviour. Black oil and compositional reservoir simulation has to a large extent driven this development especially through the l970’ies where also the engineering methodology was conceived to do reservoir simulation studies properly with pseudofunctions etc. In the early 1980’ies the concept of integrated studies became to a larger extent a vital part of any reservoir simulation studies. The use of geological modelling with a computer has during the last years also become an important part of reservoir analysis for certain types of fields to give a better reservoir input definition. The historical trends in the technological development are reviewed, and consideration of future technological developments is given. Reservoir geological modelling is foreseen to be a growth area. The standard black oil reservoir simulation will be continuously refined and compositional formulations will gradually be more and more used. The use of dimensionless analysis and numbers should be advanced.



Simulation of Waterflood Fracture Growth with Coupled Fluid Flow, Temperature and Rock Elasticity
More LessMethods used to simulate thermal fracturing of water injection wells in three dimensions are described. The calculation combines a finite difference model of fluid and heat flow within the reservoir, with determination of the poro and thermoelastic stress field, and with a boundary element fracture mechanics model to compute fracture growth. The components of the calculation are closely linked, but display behaviour over a range of timescales. Methods of coupling this complex problem with adequate efficiency and stability are discussed. Solutions are presented for a number of problems derived from field well behaviour. It is shown that behaviour can be governed by changes in the vertical extent of fracturing, and that different types of solution are derived for constant rate and constant pressure wells.



Corner Point Geometry in Reservoir Simulation
More LessHydrocarbon reservoirs commonly possess a geometry which incorporates both dip and faulting. In order to distinguish depth variations due to these two effects, it is useful to specify a simulation cell through the positions of its eight corner points. The faces of such a cell may be bilinear surfaces, and may form part of a distorted grid. Such a grid may be chosen such that cell boundaries lie along faults, which may be vertical or sloping. Normal and fault connections may be treated on an equivalent basis. A system of ‘coordinate lines’ and corner point depths used to construct such a model is described. An exact analytic expression is obtained for the cell volume. Transmissibility values are calculated in terms of threevector mutual interface areas, which automatically incorporate corrections for dip and inclined flow. The application of calculated transmissibilities to a simple five point model can lead to an inconsistent finite difference scheme, and to significant errors if the grid is highly distorted. A method of avoiding such errors is presented, which more accurately reflects the interaction between pressures and cell geometry to produce flows. This is derived from basic finite element type principles, and results are presented for test cases, comparing the corrected scheme with a simple five point model.



A Boundary Element Solution to the Transient Pressure Response of Multiply Fractured Horizontal Wells
Authors C. P. J. W. van Kruysdijk and G. M. DullaertOne of the advantages of a horizontal well over a vertical one is that it can be fractured at a number of positions along its horizontal section. This technique is particularly useful in tight reservoirs where economic production can not be achieved by conventional means. As this type of reservoir will produce in transient state for a considerable part of its producing life, accurate transient inflow models are required, not just for well testing purposes, but also for production forecasting. We constructed a model that yields the transient pressure response of a horizontal well intersected by several fractures. The properties of the fracture, length, conductivity, position and orientation can vary from fracture to fracture. By using the Boundary Element Method in combination with the Laplace transform, we succeeded in minimising the discretisation errors for the early time transients without requiring massive amounts of CPU time.



A Convective Segregation Model for Predicting Reservoir Fluid Compositional Distribution
Authors F. Montel, J. P. Caltagirone and L. PebayleConvective rolls resulting from a geothermal gradient coupled with thermodiffusion and gravity effects may induce large spatial changes of fluid composition in hydrocarbon reservoirs. This paper presents a theoretical modelling of these phenomena based on the equations of natural convection in porous media and of mass transfer. The numerical model aims at predicting the fluid property distribution from a single representative sample. The therniodynamical properties of the fluid are computed by the PENGROBINSON equation of state. Viscosity, diffusion and ther modiffusion coefficients of the various components are functions of temperature, pressure and total fluid composition. The porous medium can be anisotropic and heterogeneous. The equations are solved by a control volume method and the numerical integration scheme is discussed. An example is given for a typical reservoir. The distributions of the components and of the corresponding fluid properties are computed in different cases. A simple criterion is proposed to define the application range of the model.



Effects of Heterogeneities on Phase Behaviour in Enhanced Oil Recovery
Authors F. J. Fayers, J. W. Barker and T. M. J. NewleyThe effects that heterogeneities can create on multiphase flows and phase behaviour are discussed. A statistical indicator simulation method is used to generate twodimensional heterogeneity descriptions for two hypothetical Representative Elements of Volume (REV’s A and B). The REV’s are used to investigate how the heterogeneities will influence the recovery of residual oil by several EOR processes. Finely gridded comparisons are made between homogeneous and heterogeneous cases in a compositional simulator, demonstrating the potential impact of heterogeneities.



On the Strict Hyperbolicity of the BuckleyLeverett Equations for ThreePhase Flow
By Lars HoldenIt is proved that the standard assumptions on the BuckleyLeverett equations for threephase flow imply that the equation system is not strictly hyperbolic. Therefore, the solution of the BuckleyLeverett equations for threephase flow is very complicated. We also discuss four different models for the relative permeability. It is stated that Stone’s model almost always gives (an) elliptic region(s). Furthermore, it is proved that Marchesin’s model is hyperbolic under very weak assumptions. The triangular model is hyperbolic and the solution is welldefined and depends L1continuously upon the initial values in the Riemann problem.



Application of FractionalFlow Theory To 3Phase, 1Dimensional Surfactant Flooding
More LessSurfactants are used in enhanced oil recovery to decrease the interfacial tension (IFT) between oil and water, and thus the residual oil saturation. The lowest IFT values are obtained in a Type III phase environment, where a 3component system of oil, water and surfactant may appear as 1, 2, or 3 phases depending on overall composition. An analytical solution of the Riemann problem describing continuous injection of an aqueous surfactant solution into a core containing oil and water is presented for a constant salinity, Type III system. Special consideration is given to the problem of determining how the composition path crosses the phase boundaries.



Averaging of Relative Permeability in Heterogeneous Reservoirs
Authors Steinar Ekrann and Magnar Dalebeen increasingly used to capture the effects of small scale heterogeneities, in coarse grid reservoir simulation. This mode of use suggests a close relationship to the concept of effective properties, e.g. effective relative permeabilities. The two concepts are reviewed in some detail in the paper, and the relationship between them is explored. It is shown that fundamental differences exist. Effective properties can be defined only on a much larger spatial scale than that of the heterogeneities themselves. Effective properties become universally valid in the sense that they appear as parameters in the ensuing partial differential equations for the large scale variables. Dynamic pseudos are constructed from one single (approximate) solution of the governing equations. No scale restrictions apply to this “parent” solution. If properly constructed, dynamic pseudos allow exact reproduction, in a subsequent coarse grid simulation, of the parent solution. Dynamic pseudos generally become dependent on the particular parent solution chosen, and on the numerical grid (and numerical method) with which they are later to be employed.



Equations for TwoPhase Flow in Porous Media Derived from Space Averaging
By D. PavoneWe start from mass and momentum balance equations which are valid at the pore level. Then, space averaging is used to define macroscopic variables and to derive equations that link these variables. That is the way a macroscopic mass balance equation and a macroscopic momentum balance equation are derived. But, the mometum balance equation fails to produce the generalised Darcy equation. Hence, thermodynamics of irreversible processes is used to generate two phenomenological laws. One is a generalised Darcylike equation, the other is a capillary pressure equation. The genera.lised Darcylike equation is standard except that it includes viscous and temperature couplings. The derived capillary pressure equation takes macroscopic fluid/fluid and fluid/solid interfacial areas into account plus a dynamic term. Finally, an explicit calculation of the derived capillary pressure equation in a conical capillary is given. It validates the derived equation and indicates a new way to compute capillary pressure in porous media.



Statistical Physics and Fractal Displacement Patterns in Porous Media
More LessIt is well known from early experiments that in the process of displacing a high viscosity by a low viscosity fluid in a porous medium, one creates irregular and highly ramified fluidfluid interfaces which are fractals. Such processes and their resulting structures are the central and motivating theme for this commencing talk where stochastic modelling techniques from statistical physics and computer simulations will be extensively applied.



A Simple Analytical Model of the Growth of Viscous Fingers in Heterogeneous Porous Media
Authors J. N. M. van Wunnik and K. WitFingering of unstable immiscible displacements in moderately heteroge neous reservoirs has been modelled analytically. The displacements are assumed to take place in the horizontal plane. Stabilising mechanisms such as capillary smear out or lateral dispersion are not incorporated. An accurate expression for the viscous growth of an initially small disturbance (finger) in a displacement front is derived first. The porous medium is assumed homogeneous and the saturation of the invading liquid in the fingers constant. For a constant finger width, the derived expression shows exponential growth for small fingers and linear growth for large fingers. Next, this expression is merged with an expression which describes the development of a marginally stable displacement (mobility ratio M=1) in a heterogeneous porous medium. The results of this merging show that the more widespread and the more profound the permeability variations are, the faster the fingers initiate and the faster they show linear growth. The results of the analytical model are compared with the results of a revised version of the numerical technique, diffusionlimited aggregation (DLA). The applied version of such a DLA simulator is summarised and its application justified. The results of the analytical model match the results of the numerical simulator, if in the latter only interface movements in the main flow direction (longitudinal) are included.



Simple Renormalization Schemes for Calculating Effective Properties of Heterogeneous Reservoirs
More LessThis paper describes real space renormalization schemes for determining effective flow properties of heterogeneous reservoirs, focusing on effective permeabilities. Effective properties are calculated in a hierarchical fashion by dividing the system up into cells, consisting of a number of gridblocks, and calculating the effective properties of each of these cells. Each cell is then treated as a single gridblock in a larger cell and so on. This paper discusses two types of scheme: smallcell methods, which are very simple to implement and computationally very efficient, and largecell methods, which provide the greater accuracy required to deal with strongly anisotropic systems. The largecell method has advantages in determining the effective per meability of crosssectional models, where discontinuous impermeable shales often cause difficulties for smallcell schemes with the locally imposed cell boundary conditions. Each of the large cells contains many gridblocks and the effective permeability is calculated by replacing each cell by its equivalent resistor network. In 2D these resistor networks can be reduced to single equivalent resistors by a systematic sequence of exact resistor transformations mirroring the ‘differential real space renormalization’ technique developed in statistical physics. This large cell method appears more attractive for anisotropic systems with a finite fraction of impermeable material and can be extended to allow for mesh refinement. A test problem using outcrop data illustrates the capabilities of the method.



Multigrid Methods in Modelling Porous Media Flow
Authors F. Brakhagen and T. W. FogwellA new multigrid method is presented for the solution of the unsymmetric system of differential equations



Local Grid Refinement
Authors I. M. Cheshire and A. RenriquezLocal grid refinement is a technique which aims to provide improved accuracy in subregions of a reservoir simulation study. HEINEMANN et al (1983) discuss dynamic grid refinement to improve the resolution at flood fronts. However, most papers on the subject published since MROSOVSKY et al (1973) focus on enhanced accuracy near wells. The objective is to develop efficient techniques for field scale studies with full resolution of awkward coning effects near troublesome production wells. Figure 1 illustrates typical radial and Cartesian refinements required by reservoir engineers. Growing interest in the application of horizontal wells gives added impetus to the technology. Recent work in local grid refinement has been reviewed by EWING et al (1989).



Analysis of a Model and Sequential Numerical Method for Thermal Reservoir Simulation
More LessIn this paper we present and analyze a model for twocomponent, threephase, nonisothermal fluid flow in a onedimensional fluid reservoir, We discuss the thermodynamic principles that constrain the model functions, and analyze the effect of these thermodynamic principles on the flow equations. This analysis allows us to formulate a sequential approach to steam flooding: first a parabolic equation is solved to find the pressure and total fluid velocity, then a system of hyperbolic conservation laws is solved to update the fluid composition and energy. The thermodynamic principles allow us to compute the characteristic speeds and directions in the component/energy conservation equations, and to use these in a secondorder Godunov method.



Operator Splitting and Domain Decomposition Techniques for Reservoir Flow Problems
Authors Helge K. Dahle, Magne S. Espedal, Øystein Pettersen and Ove SaevareidAn algorithm for solving the (near) hyperbolic saturation equation of reservoir flow dynamics numerically is described. The procedure is split into several subproblems, namely operator splitting, tracking of discontinuous modes, and diffusion correction. To resolve rapid variations and reduce processor time, grid refinement, domain decomposition, and parallel computations have been used.



Simulation of Compositional Reservoir Phenomena on a Hypercube
More LessTechniques to improve the efficiency of numerical simulation for the prediction of compositional reservoir processes are described. In particular, parallel computational approaches to the numerical nndell ing of the cycling of a gas condensate reservoir have been investigated on a hypercube parallel computer ( INTEL IPSC/2 ). A commercial reservoir simulation model was used as the initial basis for the seven component simulation. Parallelization of the coefficient and saturation function routines was accomplished using a technique to minimize the overhead of messages in this distributed memory environment. For the linear equation solution a donm.in decomposition approach was implemented along with multigrid techniques to obtain more robust parallel solutions. The initial goal of the research to develop efficient parallel numerical models for the prediction of compositional processes was achieved. Greater than ninetyeight per cent of the CPU time of the compositional model was parallelized. Future work to achieve further parallelization will involve both the linear equation solution and the coefficient routines.



The Parallelisation of Bosim, Shell’s Black/Volatile Oil Reservoir Simulator
Authors D. T. van Daalen, P. J. Hoogerbrugge, J. A. Meijerink and R. J. A. ZeestratenIt is reported how the IMPEC version of Bosim, the black/volatile oil variant of Shell’s Multisim family of reservoir simulators has been adapted to run on a mediumscale (10100 processors) Meiko Computing Surface, a localmemory parallel MIMD computer based on Inmos transputers. Bosim is widely used and currently takes up considerable amounts of CPU time on vector supercomputers and large mainframes. Since most of its computations are inherently parallel, reservoir simulation makes a good candidate for parafleisation, now that parallel processing has become a practical reality. The strategy chosen is coarse grain paralleisation using 2D domain decomposition: the reservoir is divided among a 2D grid of processors. First we have parallelised the local (e.g. PVT) and semilocal (e.g. matrix set up) computations. Afterwards we have handled the linear solver and the well computations (both requiring global communication). Parallel Bosim is in Fortran 77. The communication between the Fortran programs running on different processors is provided by a library of communication subroutines in cooperation with a small Occam ‘harness”. A source code analyser has helped us generate the calls to the communication routines that had to be inserted. Currently over 98% (for a small model) of the timestep computations run in parallel, and on 60 processors the time steps run at about half the Cray speed at a fraction of the cost.



Monte Carlo Simulation of Lithology from Seismic Data in a ChannelSand Reservoir
Authors P. Doyen, T. Guidish and M. de BuylIn areas of rapid lithologic variations, the areal extent of sand and shale units usually cannot be inferred from sparse well data alone. Seismically derived interval velocity data can be used to help predict lithologic variations away from wells. However, in general, the overlap of the velocity ranges for sands and shales is such that seismic discrimination of lithology is ambiguous. We present a Monte Carlo technique for numerically simulating the spatial arrangement of sand/shale units. This technique accoimts for the ambiguous nature of the seismic velocity information. Rather than calculating a unique sand/shale model, the Monte Carlo method provides a family of alternative lithologic images, all of which are consistent with the data. The range of models reflects the uncertainty of the lithologic classification and is used to assess risk in reservoir development. The sand/shale simulation technique is illustrated using a data set from an oilproducing channelsand reservoir. Sand/shale crosssectional simulations are generated along a seismic traverse that intersects three wells. The simulated models reproduce the logderived lithologic sequences at the wells; they are conditioned by interval velocities inverted from the seismic amplitude data and are consistent with the spatial autocorrelation and crosscorrelation structures of the seismic and well data. The seismically derived lithologic models of the reservoir are better spatially constrained than models solely conditioned by well data. However, in keeping with the inherent ambiguity of the seismic information, the exact location of the lateral truncation of the channel sand is not precisely defined in the Monte Carlo lithologic simulations.
