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

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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.

From the experiments, he obtained the following data:

The data describes how total discharge (flux) \(Q\) (\(\frac{L^3}{time}\)) 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 (or head), ie it moves from h_{2} to h_{1 }(left to right) if h_{1}>h_{2}. If h_{1}<h_{2} then the fluid moves from h_{2} to h_{1} (right to left) then the value would be positive. Of course, all sign conventions depend on where you define your datum, so be careful in this respect.

The more consolidated the material, the lower its permeability. Thus "loose" materials like gravel have high permeability. 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 cm^{2}, and it is then represented as \(k\). The full range of values for hydraulic conductivity is 1-10^{13} \(\frac{m}{s}\). Experiments like Darcy's are used to measure K in real materials.

We can rewrite the equation as

\[\begin{align} q&=\frac{Q}{A} \\[4pt] &=-K\frac{dh}{dx} \end{align}\]

\(q\) is Darcy flux with units of \(\frac{m}{s}\). It is important to note that Darcy flux does **not** equal the fluid velocity, even though it is in units that indicate velocity. The true fluid velocity is found through the average linear velocity (\(v_x\)) (averaging the velocity of all possible fluid paths through the porous media), but finding this value is outside the scope of this class.

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\(\frac{m}{min}\).

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.

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

Returning to Darcy flux,

\[\begin{align} q&=-K\frac{dh}{dx} \label{Darcy1}\\[4pt] &=\frac{-k}{\mu}\frac{dp}{dx} \end{align}\]

where \(\mu\) is the fluid viscosity in Pa·s and \(k\) is the permeability in m^{2}.

We can rewrite Darcy's expression to include pressure. Pressure: \(\rho\)gh_{1}=h_{1} and \(\rho\)gh_{2}=h_{2 }in a non level pipe.

\[\begin{align} q &=\frac{-k}{\mu}\frac{(\rho gh_1-\rho gh_2)}{dx} \\[4pt] &=\frac{-k\rho g}{\mu}\frac{dh}{dx} \label{Darcy2} \end{align}\]

Equating Equations \ref{Darcy1} and \ref{Darcy2} results in

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

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.

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

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

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

\[v\approx .9\frac{m}{yr}\]

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.

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

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

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). A 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.

Darcy's law is the main equation that governs flow in aquifers and wells. Without it, we would not be able to drill wells or understand water movement in aquifers, especially in California's central valley, where both these ideas are crucial to both agricultural productivity and groundwater subsidence.