Accurate simulation of hydraulic fracturing requires the modeling and computing of fracturing fluids leakoff from fractures to the porous formation, as well as flowback in the reverse direction. For practical applications in complex fracture networks, the pressure drop from fracture to formation varies dramatically in time and space and can be negative, inducing flowback. Leakoff flow is also affected by filter-cakes that can be created on fracture faces by the filtration of fracturing fluids. Leakoff models incorporating filter-cake effects exist in the literature, assuming a constant, positive pressure drop and a single fluid. This work proposes a model that can include the filter cake, leakoff, and flowback, allowing for varying pressure drop. This new feature is vitally important when injection pressures vary in time and when solid mechanics models affect the fracture aperture and consequently the fracture pressure.
The classic Carter leakoff model is based on an analytic solution of slightly compressible Darcy flow in the formation, assuming a positive, constant pressure drop. This analytic solution is also used in the filtration with linear-invasion and crossflow (FLIC) model (McGowan et al. ), which further incorporates filter-cake effects. Here, a more general analytic solution is introduced that is valid for any temporally varying pressure drop when a single fluid is used. For efficient computation of the model, a numerical discretization of the formation Darcy flow is alternatively formulated. Multiphase flow can also be modeled in the formation and solved numerically. The model also includes a compressible filter-cake model and an invaded region having incompressible flow with Carreau viscosity, as in the FLIC model. In contrast to McGowan et al. , the model uses a mathematical formulation allowing for negative pressure drops and velocities (i.e., flowback).
An efficient method for solving the proposed model numerically is presented, as well as equations for coupling to a fluid dynamics model inside the fractures. Simulation results demonstrate the wide applicability of the model to operations in complex fracture networks with highly varying pressure drops.