Systematic Modelling of Unstable Displacement

Modelling unstable displacement is a challenge which may lead to large errors in reservoir simulations. Field scale coarse grid simulations therefore need to be anchored to more reliable fine grid models which capture fluid displacement instabilities in a physically correct manner. In this paper, a recently developed approach for accurately modelling viscous fingering has been applied to various types of unstable displacement. The method involves estimation of dispersivity of the porous medium and length scale of the model to determine the required size of the simulation grid cell. Fractional flow theory is then applied to obtain the correct saturation of the injected phase in the unstable fingers formed due to the adverse mobility ratio. Unstable displacement experiments have been history matched using 2D-imaging of in-situ saturation as a calibration of our method, before carrying out sensitivity calculations on the effect of fluid viscosity, and rock heterogeneity. Our modelling approach allows us to carry out simulations using a conventional numerical simulator using elementary numerical methods (e.g. single-point upstreaming). The methods used to model instability (Sorbie et al, 2020) was originally developed for immiscible water/oil systems. The current paper now presents new results applying this approach to unstable gas displacements, where adverse viscosity ratios may be even higher than in water/oil systems. The displacement with injected gas is shown to be influenced by mass exchanges between the gas and oil as the alternating fluids (water and gas) are injected in WAG processes. Swelling of fingers delay the gas front and WAG processes divert the injected gas and improve sweep efficiency. We have also modelled water-oil displacement at adverse mobility and shown the benefit which is obtained by reducing the instability by adding polymers to viscosify the injected water. The impact of rock heterogeneity has different effect depending on buoyancy forces and the degree of crossflow into the high permeable zones. This paper extends our novel approach to modelling the fine scale distribution of the injected fluids in adverse mobility ratio displacements. This approach has now been applied to both, gas/oil and water/oil systems where viscous fingering is present, either at extremely adverse mobility ratios and/or for reservoirs where the permeability field is very heterogeneous.

A Finite-Element Method for Reservoir Simulation

Several previous studies have applied finite-element methods to reservoir simulation problems. Accurate solutions have been demonstrated with these methods; however, competitiveness with finite difference has not been established for most nonlinear reservoir simulation problems. In this study a more efficient finite-element procedures is presented and tested. The method is Galerkin-based, and improved efficiency is obtained by combining Lagrange trial functions with Lobatto quadrature in a particular way. The simulation of tracer performance, ion exchange preflush performance, and adverse mobility ratio miscible displacements is considered. For the problems considered, the method is shown to yield accurate solutions with less computing expense than finite differences or previously proposed finite-element techniques. For the special case of linear trial functions, the method reduces to a five-point central difference approximation. In contrast to previously reported results, this approximation is found to simulate adverse mobility ratio displacements without grid orientation sensitivity, provided a sufficiently fine grid is used. Introduction In the past few years several studies have investigated the use of finite-element methods in reservoir simulation. These include single-phase two-component simulations in one1–3 and two4,5 spatial dimensions and two-phase immiscible calculations in both one6–8 and two9,10 dimensions. These studies have demonstrated that the method is capable of giving accurate solutions, particularly for small slug problems and adverse mobility ratio displacements. All these studies used what we term conventional Galerkin finite-element techniques,1 and, unfortunately, these methods have not proved to be cost competitive with finite differences for most nonlinear reservoir simulation problems. A reduction in computing requirements is, therefore, necessary to make finite-element methods truly useful for reservoir simulation. Relative to finite differences, the increased computing requirements of conventional Galerkin-based methods are due to the following.The approximation of time-derivative terms involves the same number of surrounding grid points as the approximation of flow terms; thus, implicit-pressure/explicit-saturation (IMPES) techniques are not possible (see Ref. 12, Chap. 7).The matrices which result from the approximation of flow terms are not nearly so sparse as in finite differences; thus, the solution of matrix problems requires more computation.The computational work required to generate matrix coefficients is considerably greater than with finite differences due to the number of numerical integrations which must be performed.