# 4.5 Applicability of Darcy’s Law

Darcy’s law is a macroscopic law applied using average values of parameters for a representative portion of the porous media, the REV as discussed in Section 3 of this book. Darcy’s law is applied to the flow of fluids through porous media. It describes a linear relationship between specific discharge and the hydraulic gradient. This relationship is valid for most all groundwater conditions. However, as the flow rate approaches zero or when high rates of flow occur in high hydraulic conductivity material like fractures or karst features, flow may not be linear and thus, Darcy’s law does not appropriately represent groundwater flow.

It has been suggested that a more general formulation of the law may be posed as Equation 32.

 (32)

where:

 m = coefficient (dimensionless), if m = 1, as in all the common situations, the flow law is linear, if m is not equal to 1 then the equation describes non-linear conditions and should not be called Darcy’s law. (Freeze and Cherry, 1979)

Darcy’s law is strictly applicable for laminar flow of an incompressible fluid in a solid matrix (non-deforming) of porous medium in which the gradient of mechanical energy is the only driving force (Figure 29). It is applicable under steady state or transient conditions. Laminar flow occurs when a fluid flows in parallel lines with no disturbance between the lines. Laminar flow is by definition “not turbulent” (Figure 29). Under turbulent flow conditions packets of water exhibit chaotic changes in velocity (for example, flow in a white-water stream).

Turbulent conditions begin to develop as flow velocity increases. The Reynolds number that characterizes the ratio of internal forces to viscous forces acting on fluid elements is often used to test for laminar or turbulent conditions (Equation 33).

 (33)

where:

 Re = Reynolds number characterizes the ratio of internal to viscous forces acting on fluid elements (dimensionless) q = specific discharge (L/T) d = characteristic length (L) ρ = density (M/L3) μ = dynamic viscosity (M/(LT))

The characteristic length term attempts to provide some information on the pore diameters available for flow. Authors suggest using an effective grain size (e.g., d10 finer than) (Todd and Mays, 2004), the mean pore dimension, mean grain diameter, or some function of the square root of the intrinsic permeability k (Freeze and Cherry, 1979). Bear (1972) states “Darcy’s law is valid as long as the Reynolds number, based on average grain diameter, does not exceed some value between 1 and 10 (page 126)”. Most authors on the topic agree that when Re is less than one, laminar flow occurs, and Darcy’s law is valid.

Flow rates that exceed the upper limit of Darcy’s law have been noted to occur in caverns and cavities of karstic limestones and dolomites, cavernous volcanic rocks (e.g., lava tubes), and some open framework boulder dominated deposits (in short, materials with large interconnected pores and extremely high hydraulic conductivities). Under most natural conditions, groundwater flows are laminar and Darcy’s law is valid (Freeze and Cherry, 1979). However, turbulent flow may occur in a portion of a groundwater system when flow rates accelerate in the vicinity of high yield pumping wells and drains. Turbulent flow may also occur in fractures with large apertures when the system is stressed by pumping. The lower limit of Darcy’s law is of little concern to most hydrogeologists as flow rates are extremely small.

Darcy’s law as presented here is not valid for compressible fluids. Fortunately, water has a relatively low compressibility. So, although water is not completely incompressible, this requirement can be relaxed to accommodate use of Darcy’s law for the small compressibility of water. However, Darcy’s law is not applicable if the density of the fluid varies due to differing pressure, temperature, and/or high concentrations of dissolved constituents; it is not applicable if there are substantial differences in density from location to location within a flow system; and it is not applicable if thermal, chemical, or electrical gradients drive fluid flow. However, forms of flux equations based on Darcy’s Law have been developed to accommodate density variations due to compressibility, non-uniform solute concentrations and non-uniform temperature. These equations are usually referred as representing Darcy’s Law.

Again, fortunately, the density and viscosity changes of water are usually trivial at the low pressures and narrow range of temperatures occurring in most of the Earth’s shallow subsurface making Darcy’s law applicable.

The lengthy presentation of Darcy’s Law in this section is purposeful. Darcy’s Law is the key to understanding and interpreting groundwater flow in porous media. It provides the foundational relationships between head, gradients and hydraulic conductivities that hydrogeologists used every day to develop conceptual models of how natural groundwater systems work, generate groundwater budgets, identify source areas of groundwater contamination, manage groundwater supplies and quantify exchanges between surface water and groundwater systems. In the authors’ experience, whenever a problem seems overwhelming or unclear, check Darcy’s Law and be sure basic principles are being correctly applied. This advice has been valuable in resolving issues encountered by students attacking assigned problems as well as professional hydrogeologists and engineers managing large complex groundwater sites.