J. Phys. A: Math. Gen. 23 (1990) 5089-5094. Printed in the UK COMMENT Numerical simulations of radial displacement of a wetting fluid by a non-wetting fluid in a porous medium D G Kiriakidis, G H Neale and E Mitsoulis Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario, KIN 6N5. Canada Received 26 June 1990, in final form 14 August 1990 Abstract. An approach to simulate the radial displacement of a wetting fluid by a non- wetting fluid in a porous medium is described. The computer algorithm is based on the DLA, anti-DLA and invasion percolation models as well as on the notion of the phase diagram. The transition from DLA and anti-mA to invasion percolation is made according to a transition probability. The numerical results are in very good qualitative agreement with numerical and experimental results available in the literature. The three types of immiscible displacement of a wetting fluid by a non-wetting fluid in a porous medium, namely, viscous fingering, capillary fingering and stable displace- ment are described by the phase diagram [l, 21. Each type corresponds to a region within the phase diagram having as its axes the viscosity ratio, M(=pnw/pw), and the capillary number, Ca( = Vp,,/ 7). The boundaries of each region are calculated in terms of the viscosity ratio, the capillary number and the geometrical properties of the porous medium. Three distinct statistical models have been developed in order to describe the above regions: (a) the DLA model (diffusion-limited aggregation) [3,4] for viscous fingering at low viscosity ratios, (b) the anti-DLA model [4] for stable displacement, and (c) the invasion percolation model [5] at very low capillary numbers. Both the DLA and anti-DLA models solve the Laplace equation by letting random walkers wander in the displaced and displacing phases, respectively, and stick upon contact with the interface. The absence of walkers from one phase implies negligible pressure gradients in that phase. According to the invasion percolation model, in the case of drainage the interface moves along the paths of least resistance which are present in the largest channels, since they provide the lowest capillary pressure. An algorithm has been developed by Leclerc and Neale [6] in order to describe radial, immiscible displacement of a wetting fluid in a porous medium represented by a network of interconnected capillaries. By using the DLA and the anti-DLA models, as well as the notion of open bonds for percolation, they described the transition from viscous fingering to capillary fingering at low viscosity ratios and the transition from stable displacement to capillary fingering at high viscosity ratios. A similar method has been employed by Kiriakidis et a1 [73 to simulate linear displacement of a wetting fluid by a non-wetting one. Although Leclerc and Neale’s algorithm describes successfully the intermediate regions it fails to describe the capillary fingering region at very low capillary numbers. According to their approach one expects an almost complete recovery of the wetting 0305-4470/90/215089 + 06%03.50 0 1990 IOP Publishing Ltd 5089