A Semi-Analytical Solution for Countercurrent Spontaneous Imbibition in Water-Wet Fractured Reservoirs

Countercurrent spontaneous imbibition (SI) is an important flow mechanism for oil recovery in fractured reservoirs during waterflooding. SI plays a key role in the mobilization of oil in the matrix because it facilitates water infiltration by capillarity even when the matrix permeability is low, which limits fluid transport by advection. However, the modeling of SI in fractured media under dynamic conditions has been insufficiently studied. Most imbibition models assume conventional exponential functions and empirical constants based on experimental results of oil recovery under static conditions. Thus, the modeling of water distribution in the fracture and the matrix has been ignored which may lead to incorrect estimates of the efficiency of countercurrent SI to recover oil. Using the classic fractional flow equation, we present a semi-analytical solution to model countercurrent SI in a water-wet fractured medium by including a transfer function to account for continuous fluid exchange between the fracture and the matrix. The model is numerically solved using finite differences by including an effective imbibition time as a function of water advance in the fracture, which overcomes the difficulty of solving iterative numerical summations as shown in other imbibition models. The novelty of the presented solution is that it enables the modeling of water infiltration in the matrix under dynamic conditions. We verified the semi-analytical model against 2D numerical simulations and validated against experimental data to demonstrate that the model accurately predicts oil recovery and water infiltration in the matrix.