How boundary slip controls emergent Darcy flow of liquids in tortuous and in capillary pores

Fundamental investigations of how boundary slip relative to the no-slip condition for liquid flow in a set of two distinct idealized pore geometries, i.e., a diverging-converging tortuous pore, in contrast to a straight tube capillary pore, contribute to emergent Darcy flow and flow enhancement are presented. Using steady-state solutions to Navier-Stokes equations, a sensitivity study investigates the role of (a) a large variation in boundary slip reported in the literature, and (b) a large variation in pore-throat sizes found in geologic porous media. Results show that both the pore geometry and their pore-throat sizes contribute to differences over several orders of magnitude in the emergent Darcy flow behavior and the flow enhancement. Tortuous pores contribute to a lower flow enhancement relative to the capillary pores, and while the larger pore throats (i.e., $\ge 10\mu \mathrm{m}$) negligibly enhance flow, it increasingly becomes significant for the micron-size pore throats. From capillary pores, flow enhancement is found to increases linearly in an unlimited manner with an increment in boundary slip relative to the no-slip condition. In contrast, flow enhancement from diverging-converging tortuous pores is found to get limited defined by an asymptote for flows with a larger boundary slip. Capillary pores offer no change in resistance to flow due to boundary slip. In contrast, the very nature of diverging-converging tortuous pore geometry offers growth in drag forces and energy dissipation rate, i.e., an increase in resistance to flow, which contributes to the asymptote or the limited flow enhancement. A set of theoretical models are presented, which can be used to predict the flow enhancement as a function of boundary slip and spatial-scale of pore throats. This study may have implications for predicting flow enhancement and pressure loss during fluid injection or recovery from low permeability geologic reservoirs, and relevant to other engineering applications, e.g., hydraulics in corrugated channels or design of carbon nanotube membranes for desalinization purposes.

Figure 1

A cut section from a 3D rendition of an axis symmetric periodic tortuous pores composed of periodic array of three pore units (red dashed square highlights a single pore unit), which are formed around staggered arrangement of sediment grains of size $s$, (b) 2D section of a single tortuous pore unit showing anisotropic pore throats of diameter ($d$) and boundary conditions, and (c) A cut section from a 3D rendition of an axis symmetric capillary pore.

Figure 2

Orthogonal cross-sectional slices of computed velocity fields from a tortuous pore ($d=1\mu \mathrm{m}$) showing evolution of velocity within a pore as a results of no-slip to slip boundary conditions (a)–(c), and (d) computed velocity profiles from a half section of an axis symmetric capillary pore ($d=1\mu \mathrm{m}$) as a result of different slip lengths ($b$).

Figure 3

Average velocity ($U_a$) and permeability ($k$) of various size of pore throats ($d$) as a function of boundary slip, i.e., slip length, ($b$), from capillary (a) and tortuous (b) pores.

Figure 4

Average velocity, $U_a$ and permeability, ($k$) as a function of boundary slip, ($b$) for various size pore throats ($d$), of capillary (a) and tortuous (b) pore domains.

Figure 5

2D maps of the flow enhancement factor, $E_1$ as a function of pore-throat size, $d$ and slip length, $b$ from capillary (a) and tortuous (b) pores.

Figure 6

3D maps of the flow enhancement factor, $E_1$ showing a planar or linear flow enhancement from capillary pores (a) relative to the limiting flow enhancement behavior found from the tortuous pores (b).

Figure 7

Flow enhancement computed data ($E_1$ and $E_2$) from straight tube capillary pores and the plots of theoretical Eqs. (10) and (11) which can predict flow enhancement due to boundary slip for various size ($d$) pore throats as a function of slip length $b$ (a) and (b), and as a function of nondimensional parameter $b/d$ (c) and (d). Note: the frequency of total number of domains with various pore-throat sizes have been subsampled to present clarity in figures.

Figure 8

Flow enhancement computed data ($E_1$ and $E_2$) from diverging-converging tortuous pores and the plots of theoretical Eqs. (14) and (15), which can predict flow enhancement due to boundary slip for various size ($d$) pore throats as a function of slip length $b$ (a) and (b), and as a function of nondimensional parameter $b/d$ (c) and (d). Note: the frequency of total number of domains with various pore-throat sizes have been subsampled to present clarity in figures.

Figure 9

Net changes in form drag, ${\mathrm{F}}_{\mathrm{N}}\left(\mathrm{N}\right)$ and friction drag, ${\mathrm{F}}_{\tau }\left(\mathrm{N}\right)$ due to an increase in slip length $b$ at boundaries of tortuous pore domains of various pore-throat sizes $d$. Note, for a better display, (a) and (c) are semilog plots of forces also included in (b) and (d), respectively. Only (c) shows friction drag from the capillary pore.

Figure 10

(a)–(h) Energy dissipation rate $\varepsilon \left(\mathrm{W}{\mathrm{m}}^{-3}\right)$ showing an increase in energy dissipation from 2D half sections of tortuous pores of $d=1\mu \mathrm{m}$ (notice its location highlighted by dashed red rectangle in the inset), in comparison to no change in energy dissipation found from the capillary pore (i)–(l).