Unconfined aquifer with constant thickness

Consider a simple unconfined aquifer in which recharge (R) is spatially uniform, the aquifer has a constant thickness (L), and the porosity (q) is spatially uniform. A flow net for this system is shown in Fig. 6.1. The distri-bution of travel times (i.e. age) in this system was shown by Vogel (1967) to be:

(6.1a)

It is interesting to note that the travel time given by Eq. (6.1a) is independent of the horizontal location in the aquifer. Thus, contours of travel time are horizontal lines as shown in Fig. 6.1. An important consequence of the

t L R

L

= L z -Ê ËÁ

ˆ

¯˜

qln

age distribution in this system is that large differences in age can occur over relatively short vertical distances when the recharge rate is small. For example, if the aquifer thickness, porosity and recharge rate are 30 m, 0.3 m and 0.02 m/a, respectively, then the difference in age from the water table to a depth of 1 m is more than 15 years. It is also worth noting that for cases in which z << L (i.e.

sampling a thick aquifer near the water table), then the travel time is approxi-mately given by:

(6.1b)

In this case, the travel time distribution is a linear function of depth.

The horizontal groundwater velocity in this system can be derived by considering a fluid mass balance and is given by:

FIG. 6.1. Unconfined aquifer of constant thickness (L) and uniform recharge (R). The upper diagram is a flow net for this system; the solid lines represent flow paths and the broken lines represent equipotential lines. The lower diagram shows contours of travel time, with age increasing exponentially with depth. The contour interval for travel times is arbitrary, as it depends on the values for recharge and porosity.

t z ª Rq

(6.2)

The horizontal velocity in this model increases with increasing distance in the direction of flow (x) because the total input of water from recharge increases continuously in the direction of flow.

The vertical velocity for this system can be obtained by differentiating Eq. (6.1a) with respect to z and is given by:

(6.3)

Like the travel time, the vertical velocity in this model is not a function of the horizontal location. Also note that at z = 0, Eq. (6.3) reduces to simply R/q.

If it is possible to obtain an integrated sample from all flow tubes, then this unconfined aquifer can be evaluated using a groundwater discharge model.

For example, if all of the flow tubes discharge into a spring or river where they are effectively mixed (i.e. where water of different ages is mixed together), then the mean age (also known as the mean transit time or the residence time) of this mixed sample is given by:

(6.4)

It is interesting to note that Eq. (6.4) is not a function of horizontal location. In other words, if at a given horizontal location in the system all flow paths are effectively integrated, then the residence time will be the same if one examines the flow system at some other location provided again that all flow paths are integrated (or mixed together). This concept can be seen more clearly by examining two horizontal locations in the flow system as illustrated by Fig. 6.2. There are three flow tubes that cross the line A-A’. The travel time in these flow tubes is about 114, 41.6 and 11.2 years, respectively (for a porosity of 0.3 and a recharge rate of 1 m/a). Of the total discharge, 33.3% comes from each tube. Multiplying the travel times in each tube by 0.33 and then summing, we obtain a flow-weighted mean travel time of 55.5 years. Now consider flow that crosses the line B-B’. We now have a total of six flow tubes, which includes the three tubes that crossed line A-A’. In this case, each tube contributes 16.7%

of the total flow. If we multiply the travel times in each flow tube (155, 78.6, 48.6, 31, 15.7, and 3.71 years) by 0.167 and then sum them, we arrive at a mean

v Rx

x =L q

v R L z

z = (L- ) q

t q

= L R

travel time of 55.5 years, the same as for line A-A’. Even though the travel time in tube 1 increases from 114 to 155 years as water moves from the A-A’ line to B-B’, this increase in age is diluted by the addition of younger water resulting from recharge to the right of the line A-A’.

The independence of the mean transit time to horizontal position is a fundamental attribute of an unconfined aquifer that receives uniform recharge and has a constant thickness. Haitjema (1995) showed how this attribute leads to a mean transit time in the base flow of an idealized watershed that is independent of scale. In other words, where recharge to a watershed is spatially uniform, and the aquifer is of constant thickness, the integrated travel time of water discharging from the system will be the same everywhere, and is related to the aquifer thickness (L), recharge rate (R) and porosity (q) as given by Eq. (6.4). The practical significance of this concept is that two of the most fundamental properties of a subsurface flow system (the flow of water per unit area into the system, R, and the volume of water in storage per unit area, Lq) can be, in theory, evaluated by collecting flow integrated samples of discharge anywhere in the watershed.

The discharge model described above can be (and historically was) developed without considering the details of the aquifer geometry, but treating FIG. 6.2. An illustration of how the mean transit time in an unconfined aquifer of constant thickness and uniform recharge is independent of the horizontal scale. Numbers inside flow tubes represent travel times in years for a recharge rate of 1.0 m/a. At the line A-A’ the mean travel time is 55.5 years. At the line B-B’ the mean travel time (i.e. average of the travel time in each flow tube) is also 55.5 years. Although water in flow tubes 1, 2 and 3 has increased in age from A-A’ to B-B’, the addition of younger water in tubes 4, 5 and 6 exactly offsets this increase to produce a mean travel time that is independent of horizontal scale. See text for more details.

the system as a ‘black box’. The basis for the mathematical development of groundwater discharge models is the ‘convolution integral’:

(6.5)

where C_in and C_out are the input and output concentrations, respectively, of some tracer, g(t) is a weight function that describes the age frequency distri-bution of the sample, t_i is the time of observation, t is the transit time, and the exponential term accounts for first-order decay in the case of a radioactive tracer. For CFCs, the exponential term could also be used to describe biodegra-dation in some systems. The weighting function, g(t), for an unconfined aquifer of constant thickness and uniform recharge is given by:

(6.6)

Because of the exponential term in Eq. (6.6) that describes the frequency distribution of age, this model is known as the ‘exponential’ model.

Equation (6.5), together with Eq. (6.6), allows calculation of tracer concentra-tions in integrated aquifer outflow.

6.3. UNCONFINED AQUIFER WITH LINEARLY