In this paper, we propose and numerically solve a new model considering confined flow in dual-porosity media coupled with free flow in embedded macrofractures and conduits. Such situation arises, for example, for fluid flows in hydraulic fractured tight/shale oil/gas reservoirs. The flow in dual-porosity media, which consists of both matrix and microfractures, is described by a dual-porosity model. And the flow in the macrofractures and conduits is governed by the Stokes equation. Then the two models are coupled through four physically valid interface conditions on the interface between dual-porosity media and macrofractures/conduits, which play a key role in a physically faithful simulation with high accuracy. All the four interface conditions are constructed based on fundamental properties of the traditional dual-porosity model and the well-known Stokes-Darcy model. The weak formulation is derived for the proposed model, and the well-posedness of the model is analyzed. A finite element semidiscretization in space is presented based on the weak formulation, and four different schemes are then utilized for the full discretization. The convergence of the full discretization with the backward Euler scheme is analyzed. Four numerical experiments are presented to validate the proposed model and demonstrate the features of both the model and the numerical method, such as the optimal convergence rate of the numerical solution, the detail flow characteristics around macrofractures and conduits, and the applicability to the real world problems.


Mathematics and Statistics

Second Department

Geosciences and Geological and Petroleum Engineering

Research Center/Lab(s)

Center for High Performance Computing Research

Keywords and Phrases

Convergence of Numerical Methods; Flow of Fluids; Horizontal Wells; Hydraulic Fracturing; Navier Stokes Equations; Numerical Methods; Oil Wells; Petroleum Reservoir Engineering; Porosity; Backward Euler Scheme; Dual Porosity Model; Flow Charac-Teristics; Fundamental Properties; Interface Conditions; Numerical Experiments; Stokes Equations; Wellbore; Finite Element Method

International Standard Serial Number (ISSN)


Document Type

Article - Journal

Document Version

Final Version

File Type





© 2016 Society for Industrial and Applied Mathematics (SIAM), All rights reserved.

Publication Date

01 Oct 2016