4.3 Poiseuille’s Law
When differences in soil water potential occur, water flows from regions of higher potential to regions of lower potential, unless those regions are separated by an impermeable layer. This knowledge alone is enough for us to determine the direction of soil water flow in any situation where we know or can measure the soil water potentials. But, if we want to estimate the rate of water flow through the soil, we need to know the relationship between the differences in water potential and the flow rate. To understand that relationship, let’s begin by considering Poiseuille’s Law for laminar flow through a tube:
![\[ \varrho=\frac{\pi r^4}{8\eta}\frac{\Delta p}{L} \]](https://open.library.okstate.edu/app/uploads/quicklatex/quicklatex.com-94a4136a5b0250d628f217bb737e9fa7_l3.png "Rendered by QuickLaTeX.com")
(Eq. 4-1)
where Q is the volume of flow per unit time (m3 s-1), r is the radius of the tube (m), Δp is the pressure difference from one end of the tube to the other (Pa), η is the dynamic viscosity of the fluid (Pa×s), and L is the length of the tube (m). Recall that soil water potentials can also be expressed as pressures, and you can see that the term Δp is analogous to the difference in water potential from one point in the soil to another. The term Δp/L defines the hydraulic gradient, the ratio of the difference in hydraulic pressure to the distance over which that difference occurs. So, Poiseuille’s Law tells us that the flow rate through a tube is proportional to the hydraulic gradient. If the hydraulic gradient decreases, the flow rate also decreases. We will see that the same principle holds true for soil water flow.
We can gain a second key insight from Poiseuille’s Law. Often it is useful to consider not only the volumetric flow rate, Q, but also the flux, q, which is the volumetric flow rate per unit area (m s-1). By inspecting Poiseuille’s Law and recalling that the cross-sectional area of a cylindrical tube is πr2, we can see that the flux through a tube is given by:
![\[ q=\frac{r^2}{8 \eta}\frac{\Delta p}{L} \]](https://open.library.okstate.edu/app/uploads/quicklatex/quicklatex.com-f611bdad3ca2ad5f358aa9ab83f287cc_l3.png "Rendered by QuickLaTeX.com")
(Eq. 4-2)
The term r2 shows that the magnitude of the flux for a particular hydraulic gradient depends strongly on the radius of the tube. If r is larger by a factor of 10 then q is larger by a factor of 100. We will see that a similar principle holds true for soil water flow; the water flux through the soil depends strongly on the size of the soil pores.
A third key insight we can gain from Poiseuille’s Law is that properties of the fluid also influence the flow. The term η-1 shows that as the viscosity of the fluid increases the magnitude of the flux decreases. The viscosities of water and air, the two fluids most commonly found in soil, increase as temperature decreases. Thus, from Poiseuille’s Law, we can infer that flow rates for these fluids through soil will decrease as the temperature decreases, and they do.
Although Poiseuille’s Law, published in 1841, allows us to gain some helpful insights about soil water flow, it is of little practical value for solving soil water flow problems. Soil is not a smooth straight tube, nor is it a bundle of smooth straight tubes. We usually cannot identify any meaningful “radius” for the soil pore network which would allow us to directly apply Poiseuille’s Law to the soil. We will need another approach.
