EAS44600 Groundwater Hydrology Lecture 8: Storage Properties of Aquifers

Dr. Pengfei Zhang Moisture Content Recall that the porosity (n) of earth materials is the ratio of the volume of the voids (Vv) to the total volume (VT) of the sample. The volume of the voids may be further divided into the volume of the water (Vw) and the volume of the air (Va). The volumetric moisture content is defined as:

T

w

VV= (8-1)

For saturated zones Vw = Vv and = n; for unsaturated zones Vw < Vv and < n. Water Table A water table is defined as the surface on which the fluid pressure in the pores of a porous medium is exactly atmospheric. We commonly think that the water table is the boundary between the unsaturated zone and the saturated zone. However, a saturated capillary fringe often existed above the water table (Figure 8-1). The capillary fringe is different from the typical saturated zone in that its pressure is less than one atmosphere. Therefore, a hydrologic system is often divided into three zones: an unsaturated zone, a capillary fringe (sometimes also called tension-saturated zone), and a saturated zone (Figure 8-1).

Figure 8-1. A hydrologic system consists of an unsaturated zone, a capillary fringe and a saturated zone (Freeze and Cherry). If one places a well a few feet or so below the water table, the water in that well will rise to the elevation of the water table at the location of the well (Table 8-1). The pressure hydraulic head

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(hp) on the water table is zero since the pressure on the water table is exactly atmospheric. Recall that the total hydraulic head (h) is the sum of the elevation head (z) and the pressure head (hp). Therefore, the total hydraulic head at any point on the water table must be equal to the elevation head (i.e., h = z). On figures the position of the water table is often indicated by a small inverted triangle, as in Figure 8-1. The water table often follows topography, although the relief of the water table is less than the topography (Figure 8-2). Groundwater generally flows from topographic highs to topographic lows, and discharges at the topographic low spots.

Figure 8-2. Unconfined, or water table aquifer (Fetter).

Aquifer An aquifer is a geologic unit that can store and transmit water at sufficient rates to supply wells. This requires an intrinsic permeability of 10-2 darcy and above. A confining layer is a geologic unit that has low to no intrinsic permeability (e.g., 10-2 darcy or less), such as in clays and till. Confining layers are subdivided into aquifuges (absolutely impermeable) and aquitards (impermeable relative to the adjacent units). Water table aquifers, those with no confining layer above, are called unconfined aquifers (Figure 8-2). Aquifers overlain by a confining layer are called confined aquifers (Figure 8-3). When you think about it, the aquifer inside the pipe used in Darcys experiment was a confined aquifer. The pipe itself was a confining layer. Because this aquifer was confined, any water that was introduced to the pipe faster than it was released caused pressure in the pipe. Natural confined aquifers act the same way. As water recharges the confined aquifer, pressure may build up in the aquifer. If one were to drill a well in a pressured confined aquifer, the water

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level in the well (piezometer) might rise far above the aquifer. An artesian well is a well in which the water rises above the top of the aquifer. In some cases the water level may rise above the ground surface, in which case the well is called as a flowing well (Figure 8-4).

Recharge Confining Layer

Confined Aquifer

Figure 8-3. Confined aquifer (Fetter).

Figure 8-4. Artesian and flowing well in confined aquifer (Fetter).

Where do aquifers get their water? In the case of an unconfined aquifer, the water largely comes directly from precipitation since there is no obstruction preventing infiltrating water from reaching the water table. You can imagine then that the water table may fluctuate significantly through the seasons because of changes in precipitation and evapotranspiration that serve to add and remove water from the subsurface. The case of the confined aquifer is less straightforward, since there is a confining layer atop the aquifer; its recharge must come from means other than from direct infiltration. An important means of recharge to confined aquifers is by infiltration of precipitation into portions of the aquifer that outcrop at the surface (Figure 8-3). For example,

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confined aquifers in intermontane basins may rise laterally with the topography, and actually outcrop in the adjacent foothills. Hence, the outcrops in the foothills catch precipitation and then transmit the water to lower elevation in the aquifer where it becomes confined. Potentiometric surface If the water levels in wells tapped in a confined aquifer are plotted on a map and contoured, the resulting surface, which is actually a map of the hydraulic head in the aquifer, is referred to as a potentiometric surface (Figures 8-5 and 8-6). If the aquifer is unconfined, the contour map is referred to as the map of the water table. Groundwater will flow in the general direction that the potentiometric surface is sloping, i.e., from higher hydraulic head to lower hydraulic head (Figure 8-6).

Figure 8-5. Unconfined aquifer and its water table; confined aquifer and its potentiometric surface (Fetter).

Figure 8-6. Potentiometric surface of the Dakota Sandstone (Darton, 1909).

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Compressibility of Water Fluids are compressible, e.g., an increase in pressure dp will lead to a decrease in the volume of a given mass of water (Vw). The compressibility of water () is defined as:

dpVdV ww /= (8-2)

where dVw is the change in the volume of the water, Vw is the original volume of the water, and dp is the change in pressure. The negative sign is necessary to ensure a positive . Effective Stress The weight of overlying rock and water creates a downward stress (T) on a saturated aquifer (Figure 8-7). This stress is borne by the granular skeleton of the porous medium and the fluid pressure p of the water in the pore spaces (Figure 8-7). Mathematically speaking, we have: peT += (8-3) where e is the effective stress, the portion of the total stress that is borne by the granular skeleton. In terms of the changes in these parameters, we have:

dpdd eT += (8-4) where dT is the change in total stress, de is the change in effective stress, and dp is the change in fluid pressure.

Total StressT

eEffective Stress

pFluid Pressure

Total StressT

eEffective Stress

pFluid Pressure

Figure 8-7. Total stress, effective stress, and fluid pressure on an arbitrary plane through a saturated porous medium (Freeze and Cherry).

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At a given point of the saturated porous medium, the weight of the overlying rock and water remains essentially the same over time. In such cases, the change in the total stress dT = 0, and dpd e = (8-5) In plain words, equation 8-5 states that if the fluid pressure increases, the effective stress decreases by an equal amount; and if the fluid pressure decreases, the effective stress increases by an equal amount. Recall that pghp =

h, where is the density of the fluid and g is the

gravitational constant. Also recall that zhp += , where h is the total hydraulic head and z is the elevation head. Therefore, the fluid pressure can be expressed in terms of the hydraulic head as: )( zhgghp p == (8-6) Since z is a constant, differentiating equation 8-6 gives gdhzhgdgdhdp p === )( (8-7) Substituting equation 8-7 into equation 8-5 yields: d gdhe = (8-8) Equation 8-8 states that the change in the effective stress (de) at a given point in a saturated aquifer is governed by the change in the hydraulic head at that point. Compressibility of a Porous Medium The compressibility of a porous medium, , is defined as

e

TT

dVdV

/= (8-9)

where VT is the total volume of the porous medium, dVT is the change in the volume of the porous medium, and de is the change in effective stress. Recall that VT = Vs + Vv, where Vs is the volume of the solids and Vv is the volume of the water-saturated voids. An increase in effective stress de leads to a reduction dVT in the total volume of the porous medium. In granular materials the reduction in the total volume of the porous medium is almost entirely due to grain rearrangement. In general, vsT dVdVdV += ; but the volume change for individual grains due to the change in effective stress is negligible (in other words, individual grains are almost incompressible). Therefore, we can assume dV vT dV= (8-10)

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Equation 8-10 states that the change in the total volume of a porous medium is equal to the change in the volume of voids (or porosity) due to the rearrangement of the grains under increased effective stress. Aquifer Compressibility Pumping from a well will reduce the pressure head in a saturated aquifer, leading to increased effective stress (equation 8-8). The aquifer skeleton may consolidate or compact due to this increased effective stress by the rearrangement of the grains (Figure 8-8). Aquifer compressibility is defined as

edbdb

/= (8-11)

where is the aquifer compressibility, db is the change in aquifer thickness, b is the original aquifer thickness, and de is the change in effective stress (Figure 8-8). The negative sign indicates that the aquifer thickness reduces as the effective stress increases.

Figure 8-8. Aquifer compaction caused by groundwater pumping (Freeze and Cherry).

Since dpd e = (equation 8-5) and gdhdp = (equation 8-7), equation 8-11 can also be written as

gdh

bdbdp

bdb

// == (8-12) Specific Storage The specific storage (Ss) of a saturated aquifer is defined as the volume of water released from the storage per unit volume of the aquifer per unit decline in hydraulic head. As discussed earlier, a decrease in hydraulic head h will lead to a decrease in fluid pressure p and an increase in effective stress e. A decrease in fluid pressure will cause the fluid to expand; and an increase in effective stress will cause the compaction of the aquifer. Therefore, the water released from

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the storage due to a decrease in h is produced by two mechanisms: 1) the expansion of the water caused by decreasing p, and 2) the compaction of the aquifer caused by increasing e. The volume of water produced (dVw) by the expansion of the water can be derived from equation 8-2: dpVdV ww = (8-13) Recall that for a saturated aquifer Vw = Vv = nVT, where n is the porosity. With VT = 1 (unit volume of aquifer), dp gdh= (equation 8-7), and dh = -1 (unit decline in hydraulic head), equation 8-13 becomes dV gngdhnVTw == ))(( (8-14) The volume of water expelled (dVw) from the unit volume of aquifer during compaction is equal to the reduction in volume of the aquifer (dVT). From equation 8-9, we have: eTT dVdV = (8-15) Recall that dVT = dVv (equation 8-10), and Vw = Vv for a saturated aquifer. Therefore,

Tw dVdV = (8-16) The negative sign is added since the volumetric reduction dVT is negative, but the amount of water produced dVw is positive. Combining equations 8-15 and 8-16, we have:

eTTw dVdVdV == (8-17) For VT = 1 (unit volume of aquifer), dh = -1 (unit decline in hydraulic head), and gdhd e = (equation 8-8), equation 8-16 becomes: dV gw = (8-18) The specific storage Ss is the sum of the two terms given by equations 8-14 and 8-18: )( ngSs += (8-19) Transmissivity and Storativity of a Confined Aquifer We have mentioned transmissivity (T) and storativity (S) earlier in the course. For a confined aquifer of thickness b, the transmissivity T is defined as T bK= (8-20)

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where K is the hydraulic conductivity, and the storativity (or storage coefficient) S is defined as S bSs= (8-21) Substituting equation 8-19 into equation 8-21 gives: )( ngbS += (8-22) In plain words, the storativity of a confined aquifer of thickness b is the volume of water released from the storage per unit surface area of the aquifer per unit decline in hydraulic head or potentiometric surface (Figure 8-9).

Figure 8-9. Diagrams illustrating the concept of storativity in (a) an unconfined aquifer and (b) a confined aquifer (Domenico and Schwartz). Transmissivity and Specific Yield in an Unconfined Aquifer In an unconfined aquifer, the transmissivity is defined by the same equation (8-20) but b is the saturated thickness of the aquifer or the height of the water table above the top of the underlying confining aquifer. In an unconfined aquifer, the release of water from storage is primary due to the dewatering of the pore spaces. This dewatering is directly related to the specific yield Sy of the aquifer materials. Water may also be released in an unconfined aquifer due to the secondary effects of water expansion and aquifer compaction. Therefore, the storativity of an unconfined aquifer is defined as S bSS sy += (8-23)

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For an unconfined aquifer the value of Sy is typically several orders of magnitude greater than the value of Ssb, and the storativity is often taken to be equal to Sy. The volume of water drained from an aquifer due to the drop in hydraulic head can be estimated from the formula V hSAw = (8-24) where Vw is the volume of water drained, S is the storativity, A is the surface area overlying the drained aquifer, and h is the decline in hydraulic head. Storativity values for confined aquifers range from 0.00005 to 0.005; storativity values for unconfined aquifers are much higher, ranging from 0.02 to 0.30. Therefore, for the same decline in hydraulic head, the volume of water released from an unconfined aquifer will be much greater than the volume of water released from a confined aquifer.

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Moisture ContentWater TableAquiferPotentiometric surfaceCompressibility of WaterEffective StressCompressibility of a Porous MediumAquifer CompressibilitySpecific StorageTransmissivity and Storativity of a Confined AquiferTransmissivity and Specific Yield in an Unconfined Aquifer