Fluid Displacement in a 2D DFN Fracture: Time Integration of the Interface Position

A crucial part in a two-dimensional discrete fracture network (DFN) model of two-phase flow in a naturally fractured rock is the updating algorithm for the fluid interface positions inside the fractures. In this study, a single fracture situated in an impermeable rock matrix is considered. Displacements of a Newtonian fluid by another Newtonian fluid, by a power-law fluid, or by a Bingham fluid are analysed. It is shown that a significant improvement in accuracy and computational efficiency in all three types of displacement can be achieved by updating the interface position with predictor–corrector scheme instead of forward Euler scheme currently used in such simulators. In order to achieve the same accuracy in the breakthrough time as with the forward Euler method, predictor–corrector can be run with a timestep that is 3 to 50 times larger. The gain in accuracy (or, equivalently, increase of the adjustable timestep) achieved with predictor–corrector is shown to outweigh the extra computational cost of this scheme. It is also shown that further increase in accuracy, e.g. by using the Runge–Kutta 4 scheme, does not pay off. In order to have control over the error in the interface positions, a viable strategy is to choose the timestep equal to a given fraction of the shortest single-fracture breakthrough time in the network. The value of this fraction can be chosen based on the required accuracy in the breakthrough time. In the case of Newtonian/Newtonian displacement, the optimal ratio of timestep to the single-fracture breakthrough time is found to depend only on the mobility ratio and the prescribed accuracy in the numerically computed breakthrough time. In case of a single interface in the fracture, numerical experimentation yields simple equations for recommended timestep in Newtonian/Newtonian displacement based on the fluid mobility ratio and the maximum allowed error in the breakthrough time.