Mathematics in Oil Production

The performance of a surfactant flooding system is critically dependent on its phase behaviour. Thus simulations of surfactant floods must be able to represent the phase behaviour of the system adequately. This paper describes the package which has been developed to represent phase behaviour in a major three-dimensional chemical enhanced oil recovery simulator (SCORPIO). Put simply the task of the package is to take an overall composition and to determine how much material is adsorbed and how the rest is divided into up to three phases of different composition. The concept of the model assumes that it is possible to describe the phase behaviour in terms of a series of pseudo-ternary slices through the quaternary space. The methodology employed deliberately avoids the usual analytical description of the pseudo-ternary phase behaviour in favour of a numerical approach. This enables experimental data on phase behaviour to be used directly. The model also permits the adsorption of two surface active species to be described on the phase diagram. This treatment reflects the thermodynamics of the situation which requires chemical equilibrium between components which are adsorbed and in solution. Again this treatment avoids having to force the adsorption into an analytical form. Similarly interfacial.tensions and phase viscosities are represented numerically thus enabling direct application of experimental results.

A reservoir in the subsurface is in itself intrinsically deterministic. It is there, it has potentially measurable, deterministic properties and features at all scales (if we could only excavate every part of it) and it is the end product of many complex processes (sedimentation, erosion, burial, compaction, diagenesis, ...) which operated over millions of years to form it. So why do we still speak of the stochastic nature of reservoirs if they are all actually deterministic? The word stochastic has its origin in the Greek adjective stochastikos which means skilful at aiming or guessing. As we shall see later, reservoir description ultimately is a combination of observations (the deterministic component) and formalized, educated aiming (geology, sedimentology) or guessing (the stochastic component). Today we think of a stochastic phenomenon or a stochastic parameter as something which is characterized by the property that its observation under a given set of circumstances does not always lead to the same observed outcome (so that there is no deterministic regularity) but rather to different outcomes in such a way that there is statistical regularity.

There is a need in the numerical simulation of reservoir performance to use average permeability values for the grid blocks. The permeability distributions to be averaged over are based on samples taken from cores and from logs using correlations between permeabilities and porosities and from other sources. It is necessary to use a suitable “effective” value determined from this sample. The effective value is a single value for an equivalent homogeneous block. Conventionally this effective value has been determined from detailed numerical simulation or use of a simple estimate such as the geometric mean. If the permeability fluctuations are small then perturbation theory or effective medium theory (EMT) give reliable estimates of the effective permeability. However, for systems with a more severe permeability variation or for those with a finite fraction of non-reservoir rock all the simple estimates are invalid as well as EMT and perturbation theory. This paper describes a real-space renormalization technique which leads to better estimates and is able to resolve details on a much finer scale than conventional numerical simulation. This method involves averaging over small regions of the reservoir first to form a new ‘averaged permeability’ distribution with a lower variance than the original. This pre-averaging may be repeated until a stable estimate is found. Examples are given to show that this is in excellemt agreement with computationally more expensive numerical simulation but
significantly different from simple estimates such as the geometric mean.

Reservoir simulation is a major consumer of computer resources in the petroleum industry where vector processors such as Crays and IBM3O9O5 are used for large field studies. Finite difference techniques are used to solve the differential equations describing the flow of oil, gas and water through the porous rock of a reservoir. Recently ECL obtained the Queen’s Award for technology for the development of the Eclipse reservoir simulator which is used at over 100 sites worldwide. This paper reviews the methods used in Eclipse such as active cell addressing, vector run addressing, memory management, nested factorization and orthomin and compares these with alternatives such as ICCGO and biconjugate gradients. This paper introduces several new concepts not previously reported in the literature. In particular it shows that the workcount for Nested Factorization can be reduced from 25 to 18 leading to a substantial increase in computing efficiency. A parallel version of Nested Factorization is described. A new highly parallel hybrid algorithm is described for the first time in the oil literature. A new efficient disking technique for memory management is introduced.

The spatial discretisation of finite difference numerical reservoir simulation is reviewed with reference to both areal and vertical description of the reservoir. The complexities of several special fluid flow environments are compared with the standard approach and new discretisation techniques suggested. Actual examples are used to illustrate more complex discretisations and realistic simulation of fluid flow. During the 1960’s, the availability of high speed digital computers permitted the development of the first practical numerical petroleum reservoir models (1,2,3). During this decade, reservoir models increased in complexity so that a large model numbered some hundreds of space points over which the equations were evaluated. The initial techniques used, in terms of finite difference formulation and solution of linearised equations, were far less sophisticated than at the present time.

The first part of this paper reviews some of the processes involved in extracting oil from underground reservoirs. Typical magnitudes of the quantities relevant to the fluid mechanics are given. The second part of the paper describes recent studies which aim at generalising Darcy’s law to the flow of two immiscible fluids through a porous medium.

We consider the problem of numerical simulation of hydrodynamic dispersion in porous media, in cases where the medium includes regions where the fluid velocity is unusually slow. This situation can occur at the microscopic level due to sufficient randomness in the pore space, or macroscopically in the presence of broad permeability distributions. The conventional numerical simulation technique, particle tracking, is shown to be incorrect in the presence of slow regions. An alternative and efficient method, “probability propagation”, is introduced to deal with these difficulties, and illustrative results for microscopic disorder are presented. The research described herein was done in collaboration with S. Redner and D. Wilkinson.