2 Darcy’s Law as a basis for measuring groundwater velocity

Darcy’s Law is a disarmingly simple relationship between the rate of groundwater discharge (volume per time) through a specified area of an aquifer (A = y z in Figure 3, measured perpendicular to the flow direction) to quantities that can be readily measured, i.e., hydraulic conductivity (K) and the hydraulic gradient (denoted by i, and calculated as the difference in head between two locations, ΔH, divided by the distance between the locations, Δℓ, i = ΔH/Δℓ in Figure 3). The volume per time, Q, is the product of K i A or (KiA). Darcy’s Law is commonly applied at various scales, with useful insights resulting. However, it should be kept in mind that by utilizing Darcy’s Law in this fashion, the hydrogeologist is treating an aquifer as a simple, homogeneous porous medium over the scale being tested. As discussed below, and throughout this book, aquifers are neither simple nor homogeneous, and any measurement made to characterize them will likely vary with the scale of test employed. With this caveat understood, Darcy’s Law is worthy of further examination since it has been an underpinning of hydrogeology for over a century and continues to be at the heart of both field and modeling methods for aquifer characterization, resource development, and remediation.

The quantity, K, is a measure of the aquifer’s ability to conduct water flow and is obtained from any of a variety of field or laboratory tests. Field tests conducted in situ are generally considered to produce the most representative values of K. The details of these tests can be found elsewhere. For the purposes of this discussion, it is enough to recognize that the values of K obtained from any tests can vary spatially, depending on the scale of the test — which can range from the sub-meter scale to tens of meters (Butler and Healy, 1998). Moreover, in aquifers with active chemical or microbiological processes occurring, K may also vary locally in time (Schillig et al., 2011). For these reasons, K is difficult to pin down in the characterization of an aquifer and is commonly credited as the greatest source of error in Darcy’s Law calculations (Bright et al., 2002).

The hydraulic gradient, i, is generally considered a more reliably measurable quantity than K, but it is sometimes also subject to high levels of uncertainty (Devlin and McElwee, 2007). The hydraulic gradient between two locations is obtained by dividing the difference in hydraulic head at the two locations by the distance between them (hydraulic head is expressed as a water level elevation measured from a common datum). Given the accepted practice of ascribing errors in flow calculations primarily to imperfect knowledge of K, the issue of error in hydraulic gradient values can be overlooked. Notable errors in i may arise from a variety of causes, including a) measurements of water levels in closely spaced wells with nearly identical hydraulic heads, b) measurements in highly permeable sediments, again with small differences in hydraulic head, c) measurements in wells with differing screen lengths that may intersect geologic units in poor hydraulic connection, d) measurements in wells that are not hydraulically connected to each other due to either geological barriers or clogged well screens, e) measurements in wells intersecting zones containing waters of different density (perhaps due to different amounts of dissolved solids), as might occur in deep groundwater systems or near coastlines where seawater intrudes into aquifers (Post and Asmuth, 2013).

The Darcy equation is concerned with the volume of water that passes through a specified area in a given time, i.e., a discharge. It does not make any direct pronouncements on the speed at which the water is moving through that area, only the volume per time. Although at times people speak of a Darcy flux, q, which is the product of hydraulic conductivity and gradient (Ki) (note: this quantity is also known as the specific discharge because it can also be calculated by dividing the discharge rate by the area through which the water flows) and has units of distance per time, this quantity is not the same as the seepage velocity of a parcel of water as it would be measured in linear distance per time on a map. The distinction might at first seem lost in subtlety. The difference can be easily visualized in the case of water discharging from a common garden hose, held in a horizontal attitude (Figure 4a). If the hose outlet is unobstructed, water will stream out at what might be perceived as a normal rate — the stream moves a horizontal distance of only a few centimeters from the outlet before falling into a 4-liter bucket, for example, which it fills in a minute.

Now consider the same hose, with the same four liter per minute flow rate (discharge), but with a thumb partially obstructing the outlet (Figure 4b). The thumb causes the stream of water to exit the hose in a jet of water that may travel horizontally several meters before falling to the ground. The speed of the water in these two scenarios is notably different, the first being slower and the second faster, even though it still fills the bucket in a minute. Therefore, the discharge rate is the same in both scenarios. Since the area of the hose perpendicular to flow is unchanging, the specific discharge is also the same in the two scenarios. The Darcy equation is concerned with estimating the discharge or the Darcy flux (specific discharge); the speed can only be obtained if, as in the hose example, the obstruction presented by the thumb i.e., the fraction of the hose area open to flow, is considered.

In this example, the thumb does not reduce the flow rate of water from the hose. The water speed increases because the area available for flow from the hose outlet is diminished. Consequently, pressure builds behind the thumb and drives the resulting water jet. In an aquifer, the obstruction to flow results not from a thumb, but from the solid matrix of the aquifer, generally sediment grains or rock, which contains pores spaces between the grains, analogous to variable diameter tubes. These ‘tubes’, or pores, are available for water flow, i.e., they are the space not blocked by the thumb in the hose example. The ratio of open space to total space in a volume of aquifer is called porosity, n. For a specified flow rate, the smaller the porosity the higher the backpressure (i.e., up-gradient pressure head) must be to maintain that flow, and the faster water must move through the aquifer. In porous material, some pores are dead ends or not connected to the other pores and do not participate in flow. Only the connected pores that contain flowing water are considered when calculating the seepage velocity. The porosity based on these openings is therefore less than n, and is called the effective porosity, n_e. The revised equation for seepage velocity becomes: v = Ki/n_e = q/n_e. The direction of the water movement is obtained from the hydraulic gradient term in Darcy’s Law; as a first approximation, water flows in the direction given by the steepest descent of hydraulic head.

The above discussion of groundwater velocity is the basis for the majority of field estimates of v and is common hydrogeological practice. As a first pass, low cost, method of aquifer characterization, the method has proven very effective. However, this simplified approach implicitly assumes a homogeneous (K is the same everywhere) and isotropic aquifer (K is the same no matter what direction water moves through the material — essential for the assumption that water flows in the direction of steepest descent of hydraulic head), as well as field measurements that are not subject to scale-related biases. Where significant heterogeneities are present that can channelize flow, or where in situ remediation activities require better knowledge of flow behavior on a small scale, greater fidelity than can currently be provided by a field-based Darcy approach may be required. Several specific issues that may demand higher levels of velocity characterization than Darcy’s Law calculations typically afford are discussed in the section “The importance of knowing groundwater velocity” later in this book. The search for a reliable alternative — or more likely, reliable complimentary technologies — has produced some promising prospects, and is still underway. Among the issues that must be resolved before such alternatives can gain wide acceptance are 1) reasonably low measurement costs, 2) levels of training to conduct the measurements that do not greatly exceed those currently obtained by college-level hydrogeologists, 3) a measurement scale that makes both theoretical and practical sense, and 4) underpinning all of the above there must be a clear understanding of what the estimated velocity physically represents, otherwise interpretations of the data could be erroneous. This wish list may seem far off presently. However, it continues to drive innovation.