Abstract
We develop a computational geometry-based upscaling approach to accurately capture the dynamic processes occurring within and between subsurface fracture networks and the surrounding porous matrix named the Upscaled Discrete Fracture Matrix model (UDFM). Fracture attributes (orientations and apertures) are upscaled and combined with matrix attributes (permeability and porosity) into properties of an unstructured spatially variable Delaunay tetrahedral continuum mesh. The resolution of the mesh depends on proximity to the fractures to preserve the geometric and topological integrity of the underlying fracture network as well as increasing the accuracy of gradients in the dynamics between the fractures and matrix. The UDFM Delaunay mesh and its dual Voronoi mesh can be used in existing multiphysics simulators for flow, solute/heat mass transport, and geomechanics, thereby eliminating the need for the additional development of numerical methods to couple processes in fracture/matrix systems. The model is verified by comparing flow, transport, and coupled heat-mass flow simulations on the provided meshes against analytical and numerical benchmarks. We also provide an additional example to demonstrate the applicability of the UDFM to complex heteregenous fractured media. Overall, the UDFM is accurate in all of the cases considered here and presents an attractive alternative to other modeling strategies that require novel numerical methods or multi-dimensional meshes.
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Acknowledgments
We would like to thank an anonymous reviewer and the editor for helpful comments that greatly improved this manuscript.
Funding
Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). M.R.S. and J.D.H. acknowledge support from the LANL LDRD program office Grant Number # 20180621ECR. M.R.S. acknowledges support from the Center for Nonlinear Studies. J.D.H. also thanks the partial support of U.S. Department of Energy’s Office of Science Basic Energy Sciences E3W1. P.H.S. was funded through the National Nuclear Security Administration Office of Defense Nuclear Nonproliferation Research and Development. S.K. thanks U.S. Department of Energy’s Office of Fossil Energy and National Energy Technology Laboratory’s Strategic Center for Natural Gas and Oil for support.
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Appendices
Appendix 1: Verification of permeability correction factor
Appendix 2: Lauwerier (1955) analytical 784 solution
Lauwerier [35] derived a mathematical model for the injection of hot water into an oil bearing layer. In this section, we present the equation and solution, which is valid for heat flow in a fracture surrounded by impermeable rock by assuming the sand and oil have equivalent thermal properties. For the complete derivation and solution see the original text. The following symbols are used in Lauwerier [35],
Symbol | Description |
---|---|
ρ w c w | Specific heat of the water per m3 |
v w | Linear water velocity in m/s |
λ 2 | Thermal conductivity of the oil saturated |
sand surrounding rock | |
f | Porosity of the sand |
ρ s c s | Specific heat of the sand per m3 |
ρ 0 c 0 | Specific heat of the oil per m3 |
s r | Residual oil saturation in the water layer |
s 0 | Connate-water saturation of the oil layer |
2b | Aperture of water layer (fracture) in m |
T 2 | Temperature of oil sand |
T 1 | Temperature of the water layer |
T 0 | Temperature of injected water |
Lauwerier [35] introduces the dimensionless variables ξ, η, τ, and 𝜃 defined by
where
and
such that the equations governing heat in the water layer (fracture) are: For |η| > 1,
For |η| = 1,
For τ = 0,
The temperature in the water layer (fracture) is therefore
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Sweeney, M.R., Gable, C.W., Karra, S. et al. Upscaled discrete fracture matrix model (UDFM): an octree-refined continuum representation of fractured porous media. Comput Geosci 24, 293–310 (2020). https://doi.org/10.1007/s10596-019-09921-9
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DOI: https://doi.org/10.1007/s10596-019-09921-9