Theoretical Investigation of Unconditionally Stable Sequential Algorithms of Coupled Flow and Geomechanics for Hydraulically Fractured Systems

2021 ◽  
Author(s):  
Jihoon Kim
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
A Priori ◽  
Coupled Systems ◽  
Stability And Convergence ◽  
Sequential Algorithms ◽  
Coupled Flow ◽  
Unconditionally Stable ◽  
Fracture Stiffness ◽  
Element Method

Abstract We investigate unconditionally stable sequential algorithms for coupled hydraulically fractured geomechanics and flow systems, which can account for poromechanics behavior within the fractures. We focus on modifying the concepts of the fixed stress and undrained sequential methods properly for the coupled systems by taking appropriate stabilization terms for stability and convergence with energy analyses. Specifically, an apparent fracture stiffness is used for for numerical stabilization. Because this fracture stiffness depends on the fracture length, the stabilization term needs to be updated dynamically, different from the drained bulk modulus used for typical poromechanics problems. For numerical tests, we take the extended finite element method for geomechanics while the piecewise constant finite element method is used for flow within an existing hydraulic fracture. The numerical results support a priori stability analyses.

scholarly journals On convergence of schemes of finite element method of high accuracy for the equation of heat and moisture transfer

2021 ◽  
Vol 102 (2) ◽  
pp. 129-141
Author(s):  
D. Utebaev ◽  
◽  
G.Kh. G.Kh. Utepbergenova ◽  
K.O. Tleuov ◽  
◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Moisture Transfer ◽  
A Priori Estimates ◽  
A Priori ◽  
Numerical Algorithms ◽  
Stability And Convergence ◽  
Differential Problem ◽  
Order Of Accuracy ◽  
Element Method

In this paper difference schemes of the finite element method of a high order of accuracy for the non-stationary equation of moisture transfer of Aller are constructed and investigated. The increased order of accuracy is achieved through special sampling of temporal and spatial variables. The stability and convergence of the constructed numerical algorithms are proved, the corresponding a priori estimates are obtained in various norms, which are used later to obtain estimates of the accuracy of the scheme under weak assumptions on the smoothness of solutions to the differential problem.

scholarly journals Adaptive modelling of variably saturated seepage problems

2021 ◽  
Author(s):  
B Ashby ◽  
C Bortolozo ◽  
A Lukyanov ◽  
T Pryer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Water Flux ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
Element Method

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

Sign in / Sign up

Export Citation Format

Share Document