# 4.1 Limitations of Darcy’s Law for Application to Karst Aquifers

Darcy’s law was derived empirically (Figure 33). Laboratory studies undertaken by Darcy (1856) for one-dimensional laminar flow through a known cross-sectional area, A, of porous media, indicated a constant of proportionality between discharge, Q, and the hydraulic gradient, , for water flow through a material comprised of small pores (less than 10 mm). The constant of proportionality is called hydraulic conductivity, K, as presented in Equation 1.

Figure 33  Schematic diagram for experimental permeameter determination of hydraulic conductivity based on Darcy’s Law. The porous material (yellow) is within a circular tube of cross-sectional area, A, of known length Δl. The volumetric rate of flow through the porous material, Q, is measured as head in the water reservoirs remain constant (hin and hout) in order to have a constant head gradient, . The reservoir elevations are modified, and a new equilibrium established and a new flow, Q, and new gradient, , recorded. This is repeated several times to determine the slope of a line fit through the points created with Q/A values on the x-axis and on the y-axis. The slope of that line is the hydraulic conductivity, K.

 and rearranging, produces: (1)

where:

 Q = volumetric flow through the system (L3T−1) K = hydraulic conductivity of the porous medium (LT−1) A = cross-sectional area perpendicular to flow (L2) Δh = measured head difference (L) Δl = length over which the head difference is measured (L) q = specific discharge (LT−1)

Darcy’s law is valid for laminar flow conditions. Laminar flow occurs when water particles flow in smooth, parallel path lines no matter the shape of the conveyance. For laminar flow, the velocity along a path line is constant. If the conveyance shape changes, obstacles are present in the flow channel, or wall roughness changes, the smooth path lines may begin to cross each other. Initially, it is the change in geometry of the flow path that causes flow lines to cross, then the fluid behavior begins to transition away from Darcian flow, however, this initial deviation from Darcian flow has nothing to do with turbulence, instead it is the flow path geometry that causes the water to change direction even at relatively low velocities. In the 1970s, researchers studying flow through rock fractures (Sharp, 1970; Maini, 1971) reported that when they observed the onset of non-Darcian flow in experiments (that is, larger head change per unit increase in flow), flow was still laminar based on the linear behavior of dye injected into the flow stream. They termed this initial deviation from Darcian flow as nonlinear laminar flow. As velocity increases, the viscous forces are gradually overcome by increasing inertial forces. At full turbulence, path lines cross each other, eddies form, the average forward velocity along the direction of the conveyance is relatively constant and flow is less orderly. For most porous media, water cannot move through the rock fast enough for turbulence to occur because of the large gradient required to reach the velocity that results in turbulence. Figure 34a shows path lines under laminar flow and at the onset of turbulent flow in a smooth straight pipe. In a smooth circular pipe, the velocity is zero at the wall and is maximum at the center, forming a three-dimensional, parabolic, cone-shaped profile (Figure 34a). For a given pipe diameter, a smoother and straighter pipe can support higher velocities before flow transitions to turbulent conditions (Figure 34b). Figure 35a shows laminar flow (Figure 35a) and turbulent steady flow (Figure 35b) from a spring in Alabama, USA.

Figure 34  Water path lines in a pipe showing a) laminar flow where the velocity profile is parabolic and the average velocity is 50 percent of the maximum velocity in the center; and b) turbulent flow where the velocity profile across the pipe is constant and equal to the average velocity.

Figure 35  Photographs of spring discharge from Dry Spring cave in Jackson County, Alabama, USA: a) discharge on April 2, 2006 showing laminar flow conditions; and b) discharge on April 8, 2006 showing turbulent steady flow conditions. Photographs by Alan Cressler (2006), used with permission.

For most porous media, water cannot move through the rock fast enough for turbulence to occur. With typical porous media, the pore size is generally so small that the gradient required to cause turbulent flow is extremely large thus flow tends to be slow and remain laminar. However, laboratory flow experiments through optically smooth channels with a separation distance of 200 to 500 mm used a water pump capable of producing 250 pounds per square inch (250 psi = 1,700,000 N/m2) of pressure and these achieved the entire flow range from laminar through turbulent (Acosta et al., 1985). An exception to the condition of laminar flow in porous media occurs in rare situations in aquifers near the wall of a well that is pumped at an extremely high rate because a large rate of flow converges on a small area as defined by the surface of the cylindrical wellbore. This turbulence can occur in any aquifer type if the hydraulic conductivity is large enough that the high pumping rate can be maintained.

The main difference between karst aquifers and most other aquifers is that the rocks have dissolved along fractures creating large water conveyances of high hydraulic conductivity and turbulent flow occurs during major recharge events, such as large storms or near areas of focused recharge such as sinking streams. In any rock type with interconnected pores greater than approximately 10 mm, hydraulic conductivity can be extremely large and turbulent flow can occur.

Exercise 9 invites the reader to consider what other aquifer types may have extremely large pores and high hydraulic conductivity, where flow can be laminar or turbulent. A second part of the exercise invites the reader to consider converging flow to a production well for different radial distances and aquifer types.