Abstract
The accuracy and the limits of validity of the discontinuous pressure model, which describes fluid flow inside a fracture using a subgrid scale approach, is assessed by comparing simulation results with those from direct simulation using Stokes flow. While the subgrid scale approach assumes a unidirectional flow, the Stokes model includes both velocity components. This is at the cost of meshing the interior of the fracture, which is here achieved through a spline-based mesh generation scheme. This scheme explicitly couples the spline representing the discontinuity to the fracture mesh and thereby alleviates the (re)meshing requirements for the interior of the fracture. The subgrid model and the direct simulation of Stokes flow approaches are compared by simulating a typical case containing a pressurised fracture, highlighting the advantages of using a subgrid model for the range in which its assumptions are valid, and showing its capabilities to accurately include the influence of the fracture on the porous material even outside this range.
Highlights
Fluid flow inside fractures is commonly modelled using the cubic law, allowing the two- or three-dimensional interior of the fracture to be reduced to a line or a plane, respectively
By introducing an interface permeability term [14, 24] governing the pressure drop between the fracture and the porous material, resulting in a discontinuous pressure model [30, 37]. Comparisons between these models indicate that all three model variations provide the same results when the interface permeability is sufficiently high [29] and their results are independent of the used discretisation method [16]. While these extended models allow for more physical phenomena to be included, and provide more information about the fluid behaviour inside the fracture, they still maintain the same assumptions as the cubic law, i.e. the fluid flow is solely dependent on the tangential pressure gradient, and a constant pressure for the complete fracture height
While the vertical velocity component is clearly present in the Stokes flow simulation, this component is not included in the discontinuous pressure model as it assumes a unidirectional flow in the interior of the fracture
Summary
Fluid flow inside fractures is commonly modelled using the cubic law, allowing the two- or three-dimensional interior of the fracture to be reduced to a line or a plane, respectively. An extension of the cubic law is the continuous pressure model [32, 41], which directly imposes the fluid flowing into the porous material based on changes in the velocity profile instead of modelling the total fluid transport inside the fracture. The interior of the fracture can be considered a separate domain, allowing the fluid flow inside the fracture to be described through the Stokes equations [1, 2, 5, 8, 27] This requires a separate mesh to be generated for the interior of the fracture, and this mesh needs to deform and extend to account for further fracture opening and propagation, respectively. Three cases will be assessed, namely a stationary fracture, a propagating fracture with a realistic opening height, and a case in which this height has been increased beyond the assumptions normally valid for the discontinuous pressure model
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