Diffusive-Time-of-Flight (DTOF), representing the travel time of pressure front propagation, has found many applications in unconventional reservoir performance analysis. The computation of DTOF typically involves upwind finite difference of the Eikonal equation and solution using the fast marching method (FMM). However, application of the finite difference based FMM to irregular grid systems remains a challenge. In this paper, we present a novel and robust method for solving the Eikonal equation using finite volume discretization and the FMM.
The finite volume form of the Eikonal equation is derived by differential manipulation, volume integration and the divergence theorem. Using product rule, the differential term is first converted to the divergence form. Then volume integrals that contain divergence terms are converted to surface integrals using the divergence theorem. Consequently, the spatial coordinates are replaced by cell volumes and transmissibilities which are universal for both structured and unstructured grids in finite volume simulators. When applied with the upstream scheme, the finite volume form evolves into a set of quadratic equations, and fast marching method is implemented to solve these equations.
The implementation is first validated with analytical solutions using isotropic and anisotropic models with homogeneous reservoir properties. Consistent DTOF distributions are obtained between the proposed approach and the analytical solutions. Next, the implementation is applied to unconventional reservoirs with hydraulic and natural fractures. Our approach relies on cell volumes and connections (transmissibilities) rather than the grid geometry, and thus can be easily applied to complex grid systems. For illustrative purposes, we present applications of the proposed method to embedded discrete fracture models (EDFM), dual-porosity dual-permeability models, and unstructured PEBI grids with heterogeneous reservoir properties. Visualization of the DTOF provides flow diagnostics such as evolution of the drainage volume of the wells and well interactions.
The novelty of the proposed approach is its broad applicability to arbitrary grid systems and ease of implementation in commercial reservoir simulators. This makes the approach well-suited for field applications with complex grid geometry and complex well architecture.