Microfluidics

Microfluidics is the study and design of the control or transport of small volumes of fluid flow through porous material or narrow channels for a variety of applications (e.g. mixing, separations). Capillary pressure is one of many geometry-related characteristics that can be altered in a microfluidic device to optimize a certain process. For instance, as the capillary pressure increases, a wettable surface in a channel will pull the liquid through the conduit. This eliminates the need for a pump in the system, and can make the desired process completely autonomous. Capillary pressure can also be utilized to block fluid flow in a microfluidic device.

A schematic of fluid flowing through a microfluidic device by capillary action (refer to image of capillary rise of water for left and right contact angles in microfluidic channels)

The capillary pressure in a microchannel can be described as:

${\displaystyle p_{c}=-\gamma \left({\frac {cos\theta _{b}+cos\theta _{t}}{d}}+{\frac {cos\theta _{l}+cos\theta _{r}}{w}}\right)}$

where:

${\displaystyle {\gamma }}$ is the surface tension of the liquid

${\displaystyle {\theta _{b}}}$ is the contact angle at the bottom

${\displaystyle {\theta _{t}}}$ is the contact angle at the top

${\displaystyle {\theta _{l}}}$ is the contact angle at the left side of the channel

${\displaystyle {\theta _{r}}}$ is the contact angles at the right side of the channel

${\displaystyle {d}}$ is the depth

${\displaystyle {w}}$ is the width

Thus, the capillary pressure can be altered by changing the surface tension of the fluid, contact angles of the fluid, or the depth and width of the device channels. To change the surface tension, one can apply a surfactant to the capillary walls. The contact angles vary by sudden expansion or contraction within the device channels. A positive capillary pressure represents a valve on the fluid flow while a negative pressure represents the fluid being pulled into the microchannel.^[3]

Petrochemical industry

Capillary pressure plays a vital role in extracting sub-surface hydrocarbons (such as petroleum or natural gas) from underneath porous reservoir rocks. Its measurements are utilized to predict reservoir fluid saturations and cap-rock seal capacity, and for assessing relative permeability (the ability of a fluid to be transported in the presence of a second immiscible fluid) data.^[7] Additionally, capillary pressure in porous rocks has been shown to affect phase behavior of the reservoir fluids, thus influencing extraction methods and recovery.^[8] It is crucial to understand these geological properties of the reservoir for its development, production, and management (e.g. how easy it is to extract the hydrocarbons).

Deepwater Horizon offshore drilling unit on fire 2010

The Deepwater Horizon oil spill is an example of why capillary pressure is significant to the petrochemical industry. It is believed that upon the Deepwater Horizon oil rig’s explosion in the Gulf of Mexico in 2010, methane gas had broken through a recently implemented seal, and expanded up and out of the rig. Although capillary pressure studies (or potentially a lack thereof) do not necessarily sit at the root of this particular oil spill, capillary pressure measurements yield crucial information for understanding reservoir properties that could have influenced the engineering decisions made in the Deepwater Horizon event.^[9]

Capillary pressure, as seen in petroleum engineering, is often modeled in a laboratory where it is recorded as the pressure required to displace some wetting phase by a non-wetting phase to establish equilibrium.^[10] For reference, capillary pressures between air and brine (which is a significant system in the petrochemical industry) have been shown to range between 0.67 and 9.5 MPa.^[11] There are various ways to predict, measure, or calculate capillary pressure relationships in the oil and gas industry. These include the following:^[7]

Mercury Injection

Flow diagram of the Mercury Injection Method for measuring capillary pressure: 1. Dried sample evacuated, 2. Mercury added, 3. System opened up to atmospheric pressure, mercury level drops, 4. Pressure increased drastically so that mercury enters the sample pores

Capillary pressure curve from the mercury injection method.

This method is well suited to irregular rock samples (e.g. those found in drill cuttings) and is typically used to understand the relationship between capillary pressure and the porous structure of the sample.^[12] In this method, the pores of the sample rock are evacuated, followed by mercury filling the pores with increasing pressure. Meanwhile, the volume of mercury at each given pressure is recorded and given as a pore size distribution, or converted to relevant oil/gas data. One pitfall to this method is that it does not account for fluid-surface interactions. However, the entire process of injecting mercury and collecting data occurs rapidly in comparison to other methods.^[7]

Centrifuge Method

The centrifuge method relies on the following relationship between capillary pressure and gravity:^[7]

A simplified diagram of a centrifuge set-up for measuring capillary pressure of a brine-oil system

${\displaystyle p_{c}=hg(\rho _{w}-\rho _{nw})}$

where:

${\displaystyle h}$ is the height of the capillary rise

${\displaystyle g}$ is gravity

${\displaystyle \rho _{w}}$ is the density of the wetting phase

${\displaystyle \rho _{nw}}$ is the density of the non-wetting phase

The centrifugal force essentially serves as an applied capillary pressure for small test plugs, often composed of brine and oil. During the centrifugation process, a given amount of brine is expelled from the plug at certain centrifugal rates of rotation. A glass vial measures the amount of fluid as it is being expelled, and these readings result in a curve that relates rotation speeds with drainage amounts. The rotation speed is correlated to capillary pressure by the following equation:

${\displaystyle p_{c}=7.9e^{-8}(\rho _{1}-\rho _{2})\omega ^{2}(r_{b}^{2}-r_{t}^{2})}$

where:

${\displaystyle r_{b}}$ is the radius of rotation of the bottom of the core sample

${\displaystyle r_{t}}$ is the radius of rotation of the top of the core sample

${\displaystyle \omega }$ is the rotational speed

The primary benefits to this method are that it's rapid (producing curves in a matter of hours) and is not restricted to being performed at certain temperatures.^[13]

Other methods include the Vapor Pressure Method, Gravity-Equilibrium Method, Dynamic Method, Semi-dynamic Method, and the Transient Method.

averaging primary drainage capillary pressure vs. normalized saturation data

${\displaystyle {\text{average Pc}}={\frac {\sum _{i=1}^{n}\left({\phi V_{\mathit {b}}P_{\mathit {c}}}\right)_{i}}{\sum _{i=1}^{n}\left({\phi V_{\mathit {b}}}\right)_{i}}}}$

in which ${\displaystyle n}$ is the number of core samples, ${\displaystyle \phi }$ is the effective porosity, ${\displaystyle Vb}$ is the bulk volume of sample, and ${\displaystyle Pc}$ is the primary drainage capillary pressure data vs. normalized water saturation.

averaging imbibition and secondary drainage capillary pressure vs. normalized saturation data

${\displaystyle {\text{average Pc}}={\frac {\sum _{i=1}^{n}\left({\frac {P_{\mathit {c}}{\sqrt {k/\phi }}}{\gamma }}\right)_{i}}{\sum _{i=1}^{n}\left({\frac {\sqrt {k/\phi }}{\gamma }}\right)_{i}}}}$

in which ${\displaystyle n}$ is the number of core samples, ${\displaystyle \phi }$ is the effective porosity, ${\displaystyle k}$ is the absolute permeability, ${\displaystyle \gamma }$ is the interfacial tension or IFT, and ${\displaystyle Pc}$ is the imbibition or secondary drainage capillary pressure data vs. normalized water saturation.

Needle ice

Image of needle ice

In addition to being manipulated for medical and energy applications, capillary pressure is the cause behind various natural phenomena as well. For example, needle ice, seen in cold soil, occurs via capillary action. The first major contributions to the study of needle ice, or simply, frost heaving were made by Stephen Taber (1929) and Gunnar Beskow (1935), who independently aimed to understand soil freezing. Taber’s initial work was related to understanding how the size of pores within the ground influenced the amount of frost heave. He also discovered that frost heave is favorable for crystal growth and that a gradient of soil moisture tension drives water upward toward the freezing front near the top of the ground.^[17] In Beskow’s studies, he defined this soil moisture tension as “capillary pressure” (and soil water as “capillary water”). Beskow determined that the soil type and effective stress on the soil particles influenced frost heave, where effective stress is the sum of pressure from above ground and the capillary pressure.^[18]

In 1961, D.H. Everett elaborated on Taber and Beskow’s studies to understand why pore spaces filled with ice continue to experience ice growth. He utilized thermodynamic equilibrium principles, a piston cylinder model for ice growth and the following equation to understand the freezing of water in porous media (directly applicable to the formation of needle ice):

Piston cylinder model for ice growth

${\displaystyle P_{s}-P_{l}=\Psi _{sl}{\frac {dA_{r}}{dV}}=\Psi _{sl}{\tilde {K}}}$

where:

${\displaystyle {P_{s}}}$ is the pressure of the solid crystal

${\displaystyle {P_{l}}}$ is the pressure in the surrounding liquid

${\displaystyle {\Psi _{sl}}}$ is the interfacial tension between the solid and the liquid

${\displaystyle {A_{r}}}$ is the surface area of the phase boundary

${\displaystyle {V}}$ is the volume of the crystal

${\displaystyle {\tilde {K}}}$ is the mean curvature of the solid/liquid interface

With this equation and model, Everett noted the behavior of water and ice given different pressure conditions at the solid-liquid interface. Everett determined that if the pressure of the ice is equal to the pressure of the liquid underneath the surface, ice growth is unable to continue into the capillary. Thus, with additional heat loss, it is most favorable for water to travel up the capillary and freeze in the top cylinder (as needle ice continues to grow atop itself above the soil surface). As the pressure of the ice increases, a curved interface between the solid and liquid arises and the ice will either melt, or equilibrium will be reestablished so that further heat loss again leads to ice formation. Overall, Everett determined that frost heaving (analogous to the development of needle ice) occurs as a function of the pore size in the soil and the energy at the interface of ice and water. Unfortunately, the downside to Everett's model is that he did not consider soil particle effects on the surface.^[19][20]

Circulatory system

Capillaries in the circulatory system are vital to providing nutrients and excreting waste throughout the body. There exist pressure gradients (due to hydrostatic and oncotic pressures) in the capillaries that control blood flow at the capillary level, and ultimately influence the capillary exchange processes (e.g. fluid flux).^[21] Due to limitations in technology and bodily structure, most studies of capillary activity are done in the retina, lip and skin, historically through cannulation or a servo-nulling system. Capillaroscopy has been used to visualize capillaries in the skin in 2D, and has been reported to observe an average range of capillary pressure of 10.5 to 22.5 mmHg in humans, and an increase in pressure among people with type 1 diabetes and hypertension. Relative to other components of the circulatory system, capillary pressure is low, as to avoid rupturing, but sufficient for facilitating capillary functions.^[22]