Advanced Program
Hanoi University of Mining and Geology, HUMG
ECH 155B, July 2016
Chemical Engineering Laboratory

Radial Capillary Penetration into Fibrous Media

Goals of the Experiment
In this experiment we will measured the rate of capillary penetration into fibrous media (i.e., the wicking velocity of liquid into paper) to determine the effective permeability of the paper. Relative permeability and capillary pressures are very important in crude oil extraction and reservoir engineering.
The fundamental goals of the experiment are to provide hands-on experience in determining relative permeabilities,data collection, modeling, error analysis, andmemo report writing.

Background
Radial capillary penetration of liquids in thin porous paper filters is of specific interest because of its applictions in the paper and textile industries including printing technologies as well as in protein and drug chromatography. Capillary transport also works under variable gravity conditions and is therefore of interest in space applications for fuel transport. More generally, capillary pressures and relative permeability find a host of applications in any system with small, substrate porocity and multiphase flow. In the petrochemical industry, enhanced fluid recovery fluids such as salt water with or without additives are frequently to displace crude oil from sandstone for enhanced oil recovery.
In this laboratory experiment, relatively simple experimental equipment and measurements can be used to determine effective permeability and model the process of wicking of a fluid in a porous media. Indeed, the process of capillary penetration of fluid wicking is very common in every day life. Consider what happens when you dip just the corner of a paper towel into a glass of water: you will observe the water spontaneously propagate (i.e., “wick”) into the paper towel. If you wait long enough, eventually the entire paper towel will be wet. This process is referred to as “capillary imbibition,” since it is driven by the so-called capillary pressure difference across the air/liquid interface at the leading edge of the propagating front. We ask the question: how fast will the water wick into the paper?
As you can see in the image above, paper is a porous medium (in many oil fields, crude oil is trapped in sandstone and must be pushed out for recovery). If you think back to ECH 155A, you had a similar system of a porous media with fluid flow. Analogously to column draining, the velocity of the fluid in our porous filter paper media can be modeled with Darcy’s law,
𝒖−𝜅∇P
𝜇
(1)
is the “permeability” of the medium, 𝜇 is the fluid viscosity, and ∇𝑃 is the pressure gradient. If the fluid is incompressible, then by conservation of mass the fluid velocity must obey the continuity equation,
∇∙𝒖=0 (2)

Combination of eqs. (1) and (2) yields a governing equation for the pressure,


2
𝑃=0 (3)
Predicting the velocity thus reduces down to knowing the pressure distribution. Often one uses a pump to apply a certain pressure gradient to move fluid through a porous medium (as in crude oil recovery), but in wicking the pressure difference is provided by capillarity. A key physical idea to remember is that there is always a pressure difference across a curved liquid/liquid interface as shown in the diagram below.
/
For example, in a capillary tube shown above the pressure difference is given by


𝑃
𝑙𝑖𝑞
𝑃
𝑎𝑡𝑚
2𝛾𝑐𝑜𝑠𝜃
𝑎
(4)

is the air/liquid interfacial tension, 𝑎 is the tube radius, and 𝜃 is the contact angle of the liquid/solid interface (see diagram below).


Thus, for fluids that wet the solid phase (with the magnitude of 𝜃< 90°), the pressure on the liquid side of the interface will be lower than on the gas side. This pressure difference drives fluid motion.

Although the actual paper medium is tortuous and disordered, as an approximation we will model it as being composed primarily of tubes of radius 𝑎. In this experiment, we will be lowering a horizontal piece of paper onto a vertical glass capillary tube with radius connected to a liquid reservoir. As soon as the paper contacts the tube, the liquid will begin wicking radially outward in the paper. If we denote the moving liquid/air interface position as (𝑡), then the pressure in the liquid phase at 𝑅(𝑡), for all times 𝑡>0, is given by the capillary pressure in eq. 4. We also assume that the pressure in the paper right above the capillary tube, at 𝑟=, is simply atmospheric. In other words, we need to solve the system of equations

Governing Equation:
1
𝑟
𝑑
𝑑𝑟
𝑟
𝑑𝑃
𝑑𝑟=0 (5)

Boundary Condition 1:
𝑃
𝑙𝑖𝑞
𝑃
𝑎𝑡𝑚
2𝛾𝑐𝑜𝑠𝜃
𝑎, 𝑟=𝑅(𝑡) (6)

Boundary Condition 2: P