# A Practical Guide to Fluid Flow in Porous Media

In porous media, fluids move for the same reason that they do in ducts, namely, there must be a continuous pressure gradient. The source of the pressure (or vacuum) is ultimately the wetting of some portion of the porous network. To describe fluid transport in conventional lateral flow strips, Darcy’s Law for one-dimensional flow in porous media applies, *Q* α *D**P*/*d*, where the volumetric flow rate *Q* is proportional to a pressure difference *D**P* across the fluid column of length *d *(Figure 1). In the case of wet-out flow, Darcy’s Law reduces to the well-known Washburn relation, *d*^{2 }= *W**×**t,* where the square of the distance *d* that the fluid front has traveled into the strip is proportional to the time *t*. The proportionality constant *W* depends directly on an effective surface tension (that includes any contact angle dependence) and the pore diameter, and inversely on viscosity. During wet-out flow, surface tension pulls the fluid further into the dry paper. Counteracting this is the viscous resistance, which is proportional to the fluid flow velocity with a coefficient that increases as the fluid column lengthens. The result is the decrease in flow velocity of the fluid front as the fluid penetrates the porous media. Deviations from Washburn flow are attributable to violation of one or more of the many assumptions of Washburn flow, including a non-limiting source, flow in an isotropic homogeneous medium, flow through constant cross-sectional area, and flow of a perfectly wetting liquid.

##### Figure 1. Schematic of the wet-out (top) and fully-wetted (bottom) flow regimes in a strip of porous material. Fluid moves with volumetric flow rate *Q* through the rectangular strip toward a wicking pad on the right.

In the case of fully-wetted flow in the strip, i.e., when the fluid front is located in the wicking pad, the velocity of fluid flow in the strip is *approximately* steady, *d*α *t*. The source of the pressure (or vacuum) is now the wet-out of the wicking pad. Darcy’s Law tells us that there must be a pressure difference between any two points for flow to occur between those points. Therefore, if the fluid is moving, the portion of the network toward which the fluid is moving is less wetted than that from which the fluid flows. Thus, the phrase “fully-wetted flow” does not imply that the entire network is wetted, and instead refers only to a portion of the network that is well behind the fluid front in the wicking pad.

##### Figure 2. Schematic of the wetted fraction of pores for the case of wet-out (top) and fully-wetted (bottom) cases for strips of porous material.

In the following examples, we’ve used our flow visualization methods to measure the flow rate in “fully-wetted” porous materials. Note that the measurement in the example of Figure 2 indicates constant velocity across a minute and a centimeter, so the degree of wetting difference across the strip must be small. In the example of Figure 3, one can observe a decrease in the speed of successive pulses, likely due to being either near depletion of the source or near saturation of the wicking pad.

##### Figure 3. Measurement of constant flow rate across a centimeter of the strip using the fluorescence-based marking method.

##### Figure 4. Measurement of decreasing flow velocity for successive pulses using the electrochemical marking method. The decrease in speed of successive pulses is likely due to being near depletion of the source pad or near saturation of the wicking pad.