Numerical Simulation Of Displacements In Systems Containing A Lens Heterogeneity
Authors: McKean H.C. , Dawe R.A. 
Published: March 2010

Abstract
Miscible fluid displacement through a lens heterogeneity has been studied. Flow displacement and effluent profiles have been obtained from beadpack experiments and used to validate the results obtained from numerical simulation. Subsequently the effects of permeability contrast, lens size and fluid viscosity ratio have been examined with the simulator. The effect of the lens heterogeneity on the displacement profiles is emphasized, especially the significance of premature breakthrough. Its neglect will cause incorrect reservoir engineering interpretation,' for example that of core floods for relative permeability determinations. Koval's theory on representing heterogeneity in porous media has been applied and, although not in agreement with the results obtained here, such an approach appears promising in representing well-defined heterogeneity, such as lenses, for simulation purposes.

Introduction
The understanding of how fluids flow in response to heterogeneity is fundamental to the study of displacement in reservoir systems. Heterogeneity in the form of permeability contrast within a porous system causes distortion of fluid streamlines and deviation from the production profiles of equivalent homogeneous systems.
Layered systems have been extensively studied for both miscible and immiscible displacements(1,8) but systems containing lenses have not yet been fully examined(1,2,8,9) Lenses are non-continuous heterogeneities(1,10). Parameters determined from core tests (e.g. absolute permeability, relative permeability, residual saturation) normally assume homogeneous core properties, but an unknown lens of differing permeability within the core will affect the fluid flow and hence the reliability of these parameters. Huppler(9) examined theoretically the effect of lenses on the determination of relative permeabilities in core samples and showed that they can seriously affect the determination of displacement profiles and residual oil saturation within the core. It is also necessary to know how reservoir sweep patterns are affected by lens heterogeneity for recovery profile predictions at the reservoir scale.
This study has investigated modelling of displacement through systems containing lenses. Pertinent parameters for study included permeability contrast (C = permeability of lens/permeability of surrounding medium), length and width ratios (L = length of heterogeneity/length of system, W = width of heterogeneity/width of system) and position of the lens, as well as viscosity ratio (M = viscosity displaced/viscosity displacing). Numerical simulation results have been validated by some experimental measurements, and the simulator used to investigate systematically the parameters C, L, W and M. The production profiles were then examined using Koval's theory in an attempt to incorporate heterogeneity effects into a simulator grid block.

Experimental
Two-dimensional glass bead models (20 cm by 10 cm by 0.6 cm) containing a rectangular centrally placed lens structure or dimension 6.7 cm by 3 cm were used (Fig. 1)(1,2,8) Permeability contrasts were produced by using glass beads of differing sizes, here a permeability ratio of 2.5 was used (actually 360: 144 Darcy). The porosity was constant at 40% throughout the system. Suitable plumbing allowed linear displacements to be achieved. Constant rate displacements were used with appropriate rates (0.5-1 ml/ minute i.e. relatively fast compared to field rates) to minimize dispersion effects (examined in separate experiments.(11,12).