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2.3: Darcy's Law - Flow in a Porous Medium

  • Page ID
    3528
  • Darcy's law is crucial to understanding many branches of geology, especially hydrogeology. Before we look at the law and what it can tell us, let's look at how it was developed. Darcy's law is named after Henry Darcy, a 19th century French engineer who developed an underground pressurized pipe system to deliver water around the city of Dijon. The system, which also provided water to the famous Dijon fountains, revolutionized city water and sewage systems. The system required no pumps and was driven purely by gravity. During the process of developing the new system, Darcy conducted a series of experiments where he tried to move water solely using gravity.

    FIGURE Darcy experiment

    From the experiments, he obtained the following data:

    FIGURE DARCY data

    The data describes how total discharge (flux) \(Q\) changes based on a variety of variables. Darcy found that

    \[Q\propto\frac{dh}{dx}\]

    and then that

    \[Q=-KA\frac{dh}{dx}\]

    A is the cross sectional area, K is the hydraulic conductivity, and \(\frac{dh}{dx}\) is the hydraulic gradient. \(Q\), the total discharge rate, has units of \(\frac{m^3}{s}\), the volume of water per time. You might now be wondering why there is a minus sign in Darcy's law. This is because as we saw in the diffusion definition, fluid flows from high to low pressure, ie it moves from h1>h2

    FIGURE Hydraulic Conductivity

    The more consolidated the material, the lower its permeability. Thus "loose" materials like gravel have high permeabilities. The highlighted column in the figure is K, the hydraulic conductivity in \(\frac{m}{s}\). These are the units that we will be using, but the conductivity can also have units of darcys or cm2, and it is then represented as \(k\). The full range of values for hydraulic conductivity is 1-1013 \(\frac{m}{s}\). Experiments like Darcy's are used to measure K in real materials.

    We can rewrite the equation as 

    \[q=\frac{Q}{A}\]

    \[=-K\frac{dh}{dx}\]

    \(q\) is Darcy flux with units of \(\frac{m}{s}\). It is important to note that Darcy flux does not equal the fluid velocity.

    We can also say that

    \[v=\frac{q}{\varphi}\]

    where \(\varphi\) is the porosity and v is the fluid velocity. \(\varphi\) is calculated as \(\varphi=\frac{V_{void}}{V_{tot}}\), and is usually expressed as a fraction between 0-1 or a percent. If \(\varphi\leq\)1 ⇒ v>q.

    Fluid Velocity

    Let's do a basic example. We are given that \(q=2\frac{m}{min}\) and \(\varphi\) is 25% and want to find the fluid velocity. We first convert 25% to .25. Then, \(v=\frac{2}{.25}\frac{m}{min}\) and v=8\(\frac{m}{min}\). 

    FIGURE q with pebbles and dye

    From the figure we can see that \(v=\frac{L}{t}\). Adding the blue pebbles decreases the pore space, and thus decreases the porosity. With the new longer path created by adding the smaller pebbles, the fluid velocity will increase due to the larger L component. The net motion is still to the right. 

    FIGURE Pa Pb pipe.

    What is actually driving the flow in the figure? ⇒ Gravity.

    Returning to Darcy flux,

    \[q=-K\frac{dh}{dx}\]

    \[=\frac{-k}{\mu}\frac{dp}{dx}\]

     \(\mu\) is the fluid viscosity in Pa·s and \(k\) is the permeability in m2.

    We can rewrite Darcy's expression to include pressure. Pressure:\(\rho\)gh1=h1 and \(\rho\)gh2=hin a non level pipe.

    \[q=\frac{-k}{\mu}\frac{(\rho gh_1-\rho gh_2)}{dx}\]

    \[=\frac{-k\rho g}{\mu}\frac{dh}{dx}\]

    \[K=\frac{-k\rho g}{\mu}\]

    When \(\mu\uparrow\), then K\(\downarrow\), because \(\mu\) resists flow and slows things down. 

    Now that we have these relationships, we can write Darcy's law in two ways:

    \[q=-K\frac{dh}{dx}\;\;\; or\;\;\; q=\frac{-k}{\mu}\frac{dp}{dx}\]

    The flow is driven by "pressure" but often expressed as just "height" because \(\rho g\) are hiding in \(K\) (implicit).

    Water Entering an Aquifer

    Let's now do an example calculating the fluid velocity of water entering an aquifer.

    FIGURE stream/top of aquifer

    We can write the relationship between the variables as:

    \[\frac{dh}{dx}=\frac{h_x-h_s}{L}\]

    If dh=10 m, L=100 m, K=10-6\(\frac{m}{s}\), and \(\varphi\)=30%, calculate the fluid velocity. 

    Darcy flux \[q=10^{-6}(\frac{10}{100})\]

    \[=10^{-7}\frac{m}{s}\]

    Fluid velocity \[v=\frac{q}{\varphi}\]

    \[=\frac{10^{-7}}{.3}\frac{m}{s}\]

    \[=3\: 10^{-8}\frac{m}{s}\cdot 3.15\: e^7\frac{s}{yr}\]

    \[v\approx .9\frac{m}{yr}\]

     

    FIGURE laminar flow RE=1.54

    FIGURE turbulent flow RE=2000

    We have just looked at how fluid can be transported, and how it moves on a large scale. Now let's take a closer look at the characteristics of the flow itself, and examine the Reynold's number. The Reynold's number determines the type of flow the fluid has, either laminar or turbulent. Laminar flow has a Reynold's number, Re<1-10. There are several basic assumptions of laminar flow. 

    \[\rho u\frac{du}{dx}\;\;\;and\;\;\;\mu\frac{d^2u}{dx^2}\;\;\ must\;balance\]

    The first term is the inertial force per unit volume, which drives the flow. The second term is the viscous force per unit volume, which resists flow. Since both terms must balance, we can set them equal and do some dimensional analysis. 

    \[\rho\frac{u_ou_o}{L}=\mu\frac{u_o}{L^2}\]

    \[Re=\frac{\rho u_oL}{\mu}\;\;\;Reynold's\;number\]

    Where the variables for Reynold's number (Re) are \(u_o\) flow velocity, L fluid depth, and the variables we have seen earlier, \(\rho\) fluid density, and \(\mu\) fluid viscosity. To have Darcy flow, the Re<1-10 (laminar) or Re>1-10 (non-linear/not-laminar). An Re>1-10 may not be laminar flow, but it is not turbulent either. To have turbulent flow, the Re>2000.

    Now let's look at how different variables affect the Re. A denser fluid that carries large grains and has a fast velocity will destabilize the flow and cause the Re to become more turbulent. In contrast, a fluid such as ice that has a high viscosity and moves very slowly will have a low Re and will likely have laminar flow. The high viscosity stabilizes the flow. As we can see, flow type depends on a lot more than just velocity. Two fluids could both be moving at \(u_o\), but it is \(\frac{\rho L}{\mu}\) that controls if the flows are laminar or not. 

    FIGURE turbulent flow RE=10,000

    Darcy's law is the main equation that governs aquifers and wells. Without it, we would not be able to drill wells or understand how waters flows in aquifers, especially in California's central valley, where both these ideas are crucial. 

    FIGURE aquifers and wells