InterPore2021

From the non-linear Darcy law for immiscible two-phase flow in porous media to constitutive equations for each fluid species

2 Jun 2021, 19:50

15m

Oral Presentation (MS6-A) Physics of multi-phase flow in diverse porous media MS6-A

Speaker

Alex Hansen (NTNU)

Description

There is growing evidence that the flow velocity $\vec v_p$ of an immiscible fluid mixture flowing in a porous medium depends on the local pressure gradient to a power in the range 1.5 to 2 when capillary and viscous forces compete [1]. The relative permeability equations relate the flow velocity of each immiscible fluid species, $\vec v_w$ and $\vec v_n$, to a gradient in the corresponding pressure field. These equations allow the mapping $(\vec v_w,\vec v_n) \to \vec v_p$. However, the opposite mapping, $\vec v_p \to (\vec v_w,\vec v_n)$ is not unique. Hence, attempts at generalizing the relative permeability equations to account for the non-linear behavior of $\vec v_p$ cannot use $\vec v_p$ as a starting point. Hansen et al. [2] have defined a co-moving velocity $\vec v_m$ which is related to but not equal to the velocity difference between the two fluid species and provided a transformation $(\vec v_p,\vec v_m) \to \vec (v_w,\vec v_n)$, making it possible relate the non-linear behavior of $\vec v_p$ to non-linearities in the behavior of $\vec v_w$ and $\vec v_n$. We use a dynamic network model [3] and relative permeability data from the literature to explore this mapping and what it means.

References

[1] Tallakstad, K. T., Knudsen, H. A., Ramstad, T., Løvoll, G., Måløy, K. J., Toussaint, R., and Flekkøy, E. G., Phys. Rev. Lett., 102, 074502 (2009); Tallakstad, K. T., Løvoll, G., Knudsen, H. A., Ramstad, T., Flekkøy, E. G., and Måløy, K. J., Phys. Rev. E, 80, 036308 (2009); Sinha, S. and Hansen, A., EPL, 99, 44004 (2012); Roy, S., Hansen, A. and Sinha, S., Front. Phys. 7, 92 (2020).Aursj{\o}, O., Erpelding, M., Tallakstad, K. T., Flekkøy, E. G., Hansen, A., and Måløy, K. J., Front. Phys. 2, 63 (2014); Sinha, S., Bender, A.T., Danczyk, M., Keepseagle, K., Prather, C.A., Bray, J.M., Thrane, L.W., Seymour, J.D., Codd, S.L. and Hansen, A., Transport Por. Media, 119, 77 (2017); Gao, Y., Lin, Q., Bijeljic, B. and Blunt, M. J., Water Resources Research, 53, 10274 (2017); Gao, Y., Lin, Q., Bijeljic, B. and Blunt, M. J., Phys. Rev. Fluids, 5, 013801 (2020); Zhang, Y., Bijeljic, B., Gao, Y., Lin, Q. and Blunt, M. J., eartharXiv, https://doi.org/10.31223/osf.io/2rxbn (2020).
[3] Hansen, A., Sinha, S., Bedeaux, D., Kjelstrup, S., Gjennestad, M. A., and Vassvik, M. Transp. Porous Media, 125, 565 (2018).}
[4] Sinha, S., Gjennestad, M. Aa., Vassvik, M. and Hansen, A., arXiv:1907.12842 (2019).