Computing Pressure Front Propagation Using the Diffusive Time of Flight inStructured and Unstructured Grid Systems Via the Fast Marching Method

摘要

Diffusive-Time-of-Flight(DTOF),representing the travel time of pressure front propagation,has foundmany applications in unconventional reservoir performance analysis.The computation of DTOF typicallyinvolves 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 achallenge.In this paper,we present a novel and robust method for solving the Eikonal equation using finitevolume discretization and the FMM.The finite volume form of the Eikonal equation is derived by differential manipulation,volumeintegration and the divergence theorem.Using product rule,the differential term is first converted to thedivergence form.Then volume integrals that contain divergence terms are converted to surface integralsusing the divergence theorem.Consequently,the spatial coordinates are replaced by cell volumes andtransmissibilities 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 modelswith homogeneous reservoir properties.Consistent DTOF distributions are obtained between the proposedapproach and the analytical solutions.Next,the implementation is applied to unconventional reservoirs withhydraulic 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 illustrativepurposes,we present applications of the proposed method to embedded discrete fracture models(EDFM),dual-porosity dual-permeability models,and unstructured PEBI grids with heterogeneous reservoirproperties.Visualization of the DTOF provides flow diagnostics such as evolution of the drainage volumeof the wells and well interactions.The novelty of the proposed approach is its broad applicability to arbitrary grid systems and easeof implementation in commercial reservoir simulators.This makes the approach well-suited for fieldapplications with complex grid geometry and complex well architecture.