ECMOR I - 1st European Conference on the Mathematics of Oil Recovery

Methods 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.

One 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.

The effects that heterogeneities can create on multi-phase 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.

Surfactants 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 3-component 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.

We 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 Darcy-like equation, the other is a capillary pressure equation. The genera.lised Darcy-like 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.

Fingering 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 wide-spread 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, diffusion-limited 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.

A new multigrid method is presented for the solution of the unsymmetric system of differential equations

In this paper we present and analyze a model for two-component, three-phase, non-isothermal fluid flow in a one-dimensional 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 second-order Godunov method.

Techniques 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 ninety-eight 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.

In 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 oil-producing channel-sand reservoir. Sand/shale cross-sectional simulations are generated along a seismic traverse that intersects three wells. The simulated models reproduce the log-derived 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.