Fluid Absorption in Porous Media
Capillarity is the driving force that plays an important role in the displacement of one fluid by another at the microscale on a solid wall. Understanding and quantifying capillary forces is crucial in designing products and applications that deal with immiscible fluids. In context of fluid transport in fibrous materials, the rate of fluid absorption is often predicted using the Lucas–Washburn (LW) equation. Typical LW models are developed on the basis of treating a porous medium as bundles of 1D parallel capillary tubes with a given singlevalue contact angle. A more accurate (but more rigorous), model for fluid transport is the Richards equation, which requires mathematical relationships for the capillary pressure p_{c}=p_{c}(S) and relative permeability K(S) of the porous media as a function of its instantaneous fluid saturation. 
Richards equation can be solved numerically to predict fluid saturation as a function of time and space S=S(x,y,z,t) in a fluid absorbing (or fluid releasing) porous material. Instantaneous saturation is needed to predict the rate of fluid absorption into (or release from) the material [Jaganathan et al., 2009]. 
Capillary pressure is the pressure required to force a nonwetting fluid into a porous media (e.g., forcing air into a hydrophilic media initially filled with water) as can be seen in the figure below. 
2D numerical simulation of water drainage from a fibrous media comprised of fibers with different degrees of hydrophilicity. The red and black fibers have a YoungLaplace contact angle (YLCA) of 65 and 35 degrees, respectively. It can be seen that fluid saturation percentage decreases (from left to right) by increasing the intrusion (capillary) pressure [Bucher and Tafreshi 2014]. 
Such simulations can be used to produce a mathematical relationship between capillary pressure and fluid saturation in single or multicomponent (comprised of fibers with different fiber diameter of contact angles) as can be seen in the figure below. 
Effects of reducing the mass fraction of hydrophilic fibers (from A to E) on capillary pressure of bicomponent media comprised of hydrophilic (black) and relatively hydrophobic (red) fibers. All media have the same porosity and fiber diameters and are under the same intrusion pressure of 7 kPa [Bucher and Tafreshi 2014]. 
A fast and flexible method to estimate the capillary pressure of a 3D fibrous media is the Full Morphology (FM) method implemented in GeoDict software. The FM method uses a spherecaging algorithm to produce a range of intrusion pressures that would have been needed for a nonwetting fluid to enter a 3D structure if it was made up of a series of fictitious cylindrical pores with different diameters. By marking the fraction of the total void space that is made up of “pores” of certain diameter, the FM method obtains a capillary pressure–saturation relationship for the whole media using the Laplace equation written for a cylindrical pore (see figure below). 
Morphological simulation of nonwetting phase penetration (that is wetting phase drainage) from a fibrous media under an increasing intrusion pressure [Jaganathan et al., 2009]. 
Despite the advances in computational methods, capillary pressuresaturation relationship for media comprised of fluidstoring swellingcontracting porous fibers (e.g., Rayon or cotton fibers) can only be found experimentally. Figure below shows an example of such an experiment [Jaganathan et al., 2009]. 
Fluid height rise experiment is conducted to obtain a capillary pressuresaturation relationship for nonwoven fabrics made of Rayon (swelling) fibers [Ashari et al., 2010]. 
Relative permeability is the inverse of the resistance that a partiallywetted media shows against fluid flow. The relative permeability tensor can be obtained by solving the Stokes equations numerically at each saturation levels and post processing the resulting data using Darcy’s law. It is important to note that relative permeability (unlike capillary pressure) is a tensorial property. This means that fluid absorption in the x,y and z directions can be quite different in a fibrous media dependent on the x,y, and z orientation of the fibers. This also means that fibrous media with different overall fiber orientations (but identical solid volume fractions and fiber diameters) can absorb (or release) fluids at different rates [Jaganathan et al. 2008]. 
Relative permeability tensor can be obtained by solving the Stokes equations for flow of a wetting fluid in a fibrous media under different saturation levels. Figure to the right is an example of relative permeability calculation performed for the 2D macroscale simulations shown in the next figure below [Ashari et al. 2010]. 
With relative permeability and capillary pressure obtained from the above 3D microscale (fiberlevel) simulations in terms of wetting saturation, we solved the Richards equation on macroscale (scales comparable with dimesions of the media) using a finite element PDE solver [Jaganathan et al., 2009]. Figures below show an example of such dualscale calculations conducted for fluid absorption in fibrous sheets with identical parameters but different inplane fiber orientations. 
Contour plots of saturation at t = 0.41 s planar media with different inplane fiber orientations. Different colors from red to blue represent different saturation values from one to zero, respectively. Coordinates are normalized by the actual sample dimensions. The volume of the absorbed liquid in these media is plotted versus time. The figure to the left shows that the rate of fluid spread increases with increasing the inplane alignment of the fibers. [Ashari et al. 2010]. 
We also used our dualscale modeling approach to simulate the rate of fluid release from moving partially saturated nonwoven sheets in contact with a solid surface. This is a challenging task as the release rate depends on many parameters, some of which are difficult to quantify. In this concern, we developed a diffusioncontrolled boundary treatment to simulate fluid release from partially saturated porous materials onto surfaces with different hydrophilicy. Our numerical simulations were accompanied with experimental data obtained for lotionsaturated Rayon and PEY nonwoven wipes using a custommade test rig. 
The figure on the left is a schematic illustration of our experimental rig built to test nonwoven wet wipes. The figure on the right compares wipes’ average saturation obtained from our simulations with those obtained experimentally for Rayon wipes at at different speeds [Ashari et al., 2011]. 
Representative Publications:

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