Early and Late Time Analytical Solutions for Co-Current Spontaneous Imbibition and Generalized Scaling
Authors: Andersen Østebø Pål 
Published: June 2020

Abstract
Cocurrent spontaneous imbibition is an important driving mechanism for oil (and gas) production in naturally fractured reservoirs, especially when matrix blocks are partially covered by both a wetting and a non-wetting phase (assumedly water and oil in this work). A 1D model is considered where water covers one side (inlet) and oil the other (outlet). Water then imbibes and displaces oil, mainly co-currently towards the outlet, spontaneously driven by capillary forces, but also to some extent counter-current production takes place at the inlet.
The behavior of this system is described using (1) an advection-capillary diffusion transport equation combined with (2) a pressure equation. The pressure equation is solved to continuously update the total velocity in the advection term of the first equation. This system is tightly coupled and must be solved simultaneously to get solutions of pressures and saturations vs distance and time. Experimental and numerical works have indicated that the saturation profile is comparable with a Buckley-Leverett solution (obtained for forced displacement in absence of capillary forces). The aim of this work is to use the Buckley-Leverett profile explicitly to solve the pressure equation. This, combined with the boundary conditions will provide an analytical solution for recovery as function of time until the saturation front reaches the outlet. A solution is also suggested after the outlet is reached which corrects the Buckley- Leverett solution to maintain the imbibed water inside the system in agreement with the co-current spontaneous imbibition process and preserve continuity in recovery and spatial saturation profiles. For early times a numerical calculation is required based on the Buckley-Leverett profile to generate an effective total mobility and an effective capillary pressure. The solution can then be calculated explicitly. At late times an ordinary differential equation must be solved and the mentioned parameters change with time.
The suggested solution is compared against numerical simulations. The solution provides a direct and accurate estimate of the time scale for the water front to reach the outlet and shapes of the recovery profile and was demonstrated to scale cocurrent imbibition recovery. It is shown that imbibition rate can increase, decrease and stay constant with time based on a derived effective mobility ratio which also can be used for evaluating effectiveness of displacement as it incorporates the entire saturation functions. Square root of time recovery is a special case only seen for very high oil mobility. It is demonstrated that co-current imbibition scales with the square of length both at early and late times.
To our knowledge, previous analytical solutions have only considered infinite-acting systems, are limited to piston-like displacement assumptions or have focused only on the period before the outlet boundary is reached. They are also often based on implicit formulations that do not provide much more insight than numerical simulations. In addition to scaling recovery time, more understanding is given to the period after the outlet boundary is reached.