A Dynamic Network Model for Imbibition
Authors: Nguyen Viet Hoai , Sheppard Adrian P. , Knackstedt Mark A. , Pinczewski W. Val 
Published: December 2007

Abstract
A dynamic network model for imbibition based on a physically realistic and mathematically rigorous treatment of the complex dynamics of frontal displacements, film swelling, snap-off and snap-off nucleated displacements is described. Many of the previously reported observations in micro-model displacements, X-ray CT measured saturation profiles in core floods and the competition between frontal displacements and snap-off all arise as a natural consequence of the dynamics of the model.
The model is used to study the effect of displacement rate on imbibition patterns, relative permeabilities and residual saturations. The model shows that at very low displacement rates the displacement is dominated by film swelling and snap-off with a displacement pattern similar to bond percolation. Trapping is extensive and relative permeabilities are low. At high displacement rates snap-off is completely suppressed and the displacement pattern is similar to invasion percolation. Trapping is low and relative permeabilities are high.

Introduction
Quasi-static network models based on realistic rock topology, geometry and pore-scale physics have been used to predict relative permeabilities and residual saturations for displacements in simple porous media1,2,3,4,5. A major limitation of these models in imbibition displacements is the treatment of wetting film or corner flow and snap-off.
In imbibition the wetting fluid advances by piston- like displacements of pores and throats from a connected invasion front and by film swelling and snap-off ahead of the connected front. Snap-off is the dominant trapping mechanism and the competition between snap-off and frontal piston-like pore displacements determines the displacement pattern, residual saturation and relative permeability. Quasi-static models assume that the displacement is dominated by capillary forces and neglect viscous pressure gradients in bulk fluid and in films. Neglecting viscous losses in bulk fluid is a reasonable assumption for capillary numbers Ca&lt;10−3-10−4 (Dullien6), where the capillary number is defined asEquation 1
and µ is the viscosity of the displacing fluid, v is the displacement velocity and s is the interfacial tension. However, this is not true for film flow where the small conductivity of films can result in significant rate effects at capillary numbers as low as 10−6-10−8 (Blunt and Scher7 and Hughes and Blunt8). Displacement rate has a profound effect on the competition between piston-like displacements and snap-off and therefore on the displacement pattern, residual saturation and relative permeability.
A number of dynamic network models have been proposed to account for rate effects in imbibition7,8,9,10,11,12,13,14,15. All of these models involve the computation of pressure fields based on simplifying assumptions regarding the nature of wetting films and film flow (constant film conductivity, steady-state flow, lumped hydraulic resistances for film and bulk flow). The computed pressure fields are used to modify the order of pore and throat displacements. None of the models fully capture the complex spatial and temporal dependence of film swelling, snap-off and snap-off nucleated pore and throat displacements ahead of the connected front.