Examining the Significance of Advective Acceleration to Single-Phase Flow Through Heterogeneous Porous Media
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Date
2006-11-15
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Abstract
Practically every groundwater flow simulator in use today implements Darcy's law to model saturated flow [1, 4, 64, 65]. Darcy's law is a linear relation derived experimentally in the mid 1800's by Henry Darcy. Since that time it has been used as an approximation to momentum conservation, thus simplifying the model of saturated flow through porous media.
In spite of its widespread appeal, Darcy's law does have limitations. Particularly in the case of higher velocities (i.e. Reynolds numbers much larger than one), Darcy's law is no longer a sufficient model for saturated flow through porous media. In this document, we propose a model for momentum balance which includes two terms that are nonlinear in velocity. One term, based upon the early 1900s work of Forchheimer, incorporates the magnitude of the fluid velocity. The other, known as the advective acceleration term in this document, models how the momentum is advected by the pore velocity and includes velocity gradients.
While the advective acceleration term is a core component of the exact momentum conservation equation, it is neglected in the groundwater flow literature due to the assumption that groundwater flow is generally slow. However, to our knowledge no one has performed numerical studies to examine the magnitude of the error introduced by this simplifying assumption. In this document, we focus on two-dimensional regional flow through heterogeneous porous media and examine the significance of advective acceleration in both the Darcy and Forchheimer flow regimes.
The non-Darcy flow simulator was written using the finite element simulation framework, Sundance 2.0. While the Raviart-Thomas finite elements are the appropriate element combination for solving the non-Darcy flow system, they were not available in Sundance 2.0. Therefore, a multiscale residual-based stabilization approach was implemented [50, 73, 78, 89]. We have proven that the modified equation set is stable. Moreover, we have shown that the discrete equations conserve mass both locally and globally and conserve momentum globally.
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heterogeneous porous media, stabilized finite elements, non-Darcy groundwater flow
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PhD
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Computational Mathematics
