The Use Of Finite Element Models In Simulating North Sea Waterfloods
Authors: Wilson David C. , Casinder Prins C. 
Published: December 2007

Abstract
This paper presents the results of Finite Element simulation of multiphase immiscible displacement for a wide range of data. The formulation is based on Galerkin's method and the cubic Hermite space is employed to represent the two main variables namely pressure and saturation.
While there is little doubt that the Finite Element method is better suited. than Finite Difference formulations, to the generation of sharp fronts, several problems have been encountered. The results obtained show that three factors strongly influence the final answer. For high stability it is necessary to use a device known as capacity lumping. For correct positioning of the front upstream weighting and a somewhat modified fw curve had to be employed. The form of upstream weighting employed is related to the rate of advance of each saturation and is therefore relatively easy to apply in a multi-dimensional case. The modified fw curve recognizes the impossibility (using reservoir oil and water) of measuring values of unsteady state relative permeability at less than the flood front saturation. It is believed that when injection projects in the North Sea have advanced to the point of giving sufficient history use of finite element simulators such as described in this paper will enable the progress of the 'front' to be monitored much more accurately than at present.
Intense interest is being displayed in possible tertiary recovery techniques. A natural consequence of the finite element technique described in this work should be improved reservoir description. This is essential if any of the tertiary techniques are to be successful.

1. INTRODUCTION
The simultaneous flow of two fluids through porous media has always been a subject of considerable importance to the Petroleum Industry. It has now become standard practice to commence water injection fairly early in the life of an undersaturated oil reservoir. so that the maintenance of reservoir pressure, together with the consequent suppression of gas liberation, will result in quick, efficient recovery of oil.
In the case of offshore oil production, such as in the North Sea, the cost of the platform and equipment far outweighs any other item of expenditure. Moreover, orders for platforms have to be placed well in advance of field development. It is essential to be able to predict as accurately as possible from the scant data available from exploration wells, the behaviour of the reservoir under various water injection schemes. This enables management to take decisions on the feasibility of different projects, select locations for platforms and wells. and also delineate those unfavourable parts of the reservoir which contribute to low oil recovery and/or premature water breakthrough.
Numerical methods have proved to be a powerful tool for this purpose. and waterflood models in particular have, as their objective, the obtaining of accurate pressure and saturation profiles across the reservoir, so that the progress of the flood front with time can be monitored. Conventionally, this has been achieved by the use of Finite Difference Methods (FDM) in which the reservoir is divided into rectangular blocks. and average values of pressure and saturation are calculated for each block. FDM yield reasonably good results for such problems, but their principal disadvantage lies in their 'step-wise' averaging nature which leads to significant numerical dispersion. This means that unless a fine mesh is used. the saturation values may be 'smeared out', and sometimes a flood front can hardly be detected at all. With this in mind the search for alternative techniques in recent years has led to variational or Finite Element Methods (FEM) being introduced into reservoir simulators.