Fig. 5.2 Ring permeameter shown in (a) three-dimensional oblique view and (b) schematic cross-section. (Reproduced from Chappell and Ternan (1997) with permission of Wiley-Blackwell.)

(observed within the water supply tube; see Section 6.2) to fall from level 1 (hj) to level 2 (h2) (BS 1377-5, 1990).

Other permeameters are available to allow undisturbed soil cores to be tested in the field. One such method is ring permeametry, which applies a constant-head to relatively large, undisturbed cores (7000 cm3) to derive the saturated hydraulic conductivity directly (Fig. 5.2). The large size of the cores has the advantage that error due to leakage between the soil core and metal ring can be minimised (Chappell and Lancaster, 2007). With ring permeametry, the soil core is first lifted from the ground and tested on a stand with atmospheric pressure being maintained at the base of the core. If the metal ring is inserted into the ground (using a press or hammer) and the test undertaken with the core still in the ground, the strata beneath the core, if less permeable that that within the core, will affect (reduce) the value obtained.

Inserting a ring into the ground surface (to a depth of 10 cm), and then inserting a larger outer ring (e.g. 53-cm diameter) to encompass the inner test ring

(e.g. 28-cm diameter) gives a double-ring infiltrometer (Fig. 5.3). By flooding the outer ring (as well as the inner ring) with water, the flow out of the base of the inner ring is primarily vertical. Note: with small-diameter infiltrometer rings, predominantly vertical flows cannot be maintained and so should not be used. Once the test has saturated the soil within and beneath the rings, the rate of infiltration will reduce to a constant value; this constant rate is called the infiltration capacity. The infiltration capacity determined by the standard method (using large diameter rings) is equivalent to the saturated hydraulic conductivity of the ground surface or topsoil (where permeameter cores have been tested with the core in the ground).

Tests can be undertaken using individual piezometers (see Section 6.2.1) to derive a saturated hydraulic conductivity. One such test was developed by Hvorslev (1951) and remains a standard test within the UK (BS 6316, 1992). Within this test the original water level within a piezometer is raised or lowered artificially, and the return to the original level, which occurs at an exponential rate, is dependent on the saturated hydraulic conductivity. Consequently, if the height to which the water level is raised at the start of the test is ho and the height of the water level above the original water level is h after time t, then a semi-logarithmic plot of the ratio h/ho versus time should yield a straight line.

If the length of the piezometer screen, L, is more than eight times the radius of the filter pack (borehole), R, then the saturated hydraulic conductivity can be found from, r2ln(L/R) s 2LT„

where r is the piezometer radius and To is the time taken for the water level to rise to 37 per cent of the initial change. As the water level within the piezometer can be raised or lowered by introducing a solid object (or slug), such tests are often called slug tests. Further details of such tests can be found within Butler (1997).

Tests on observation wells (see Section 6.2.2) and wells used for procuring groundwater is an essential prerequisite to any exploitation of well field for public water supply. These tests form part of a hydrological sub-discipline known as well hydraulics. Different approaches apply depending upon whether the aquifer is confined (by an overlying impeding layer) or unconfined and whether the conditions are unsteady or in steady state.

When the well fully penetrates a horizontal confined aquifer (Fig. 5.4), flow to the well is also horizontal from all directions (i.e. radial two-dimensional flow). To ensure a steady flow, there must be continuous recharge to the aquifer from sources distant to the well. Assuming also that the aquifer is homogeneous and isotropic and is not affected by compression in dewatering, the flow to the well at any radius r can be expressed by Darcy's law, dh

Q = 2n Ksb—h ^ s dr where Q is the pumping rate. Integrating over the radius distance rw to r1 and hw to h1 gives,

where T is the transmissivity of the aquifer, or the product of the saturated hydraulic conductivity and saturated zone thickness, b. This is called the Thiem equation.

By pumping the well at a steady rate and waiting until the well level hw is constant, observation of the drawdown level hi at an observation well at a known distance ri, from the pumped well, allows estimation of the transmissivity of the aquifer. In practice, the observations from two or more observation wells at different radii are more useful since head losses in the well, caused by friction in the well casing, can then be allowed for.

Example. A well in a confined aquifer was pumped at a steady rate of 0.0311 m3 s-1. When the well level remained constant at 85.48 m, the observation well level at a distance of 10.4 m was 86.52 m. Calculate the transmissivity,

Qln(Vrw)

5.1.3.2 Steady flow in an unconfined aquifer

The groundwater flow to a well in an unconfined aquifer may be complicated by the downward movement of recharge water from ground surface infiltration, but here only distant sources in the aquifer are assumed to maintain the steady-state flow. In Fig. 5.5, the water table intersects the well at hr with the water level in the well at hw. Between the two levels is a seepage zone. However, in estimating the flow to the well, the Dupuit-Forchheimer assumption of horizontal flow may be used, thus, dh

Q _ 2n rKh — dr with h the height of the water table replacing b the thickness of the confined aquifer. Integrating over the radius distance r1 to r2 with corresponding values h and h2 then:

from which K can also be evaluated. In applying this equation to an unconfined aquifer, the effects of the seepage from the water table into the well (hw and rw) are negligible if the distance of the nearest observation well r1 is greater than 1.5 times the original water table height ho.

The steady-state groundwater flow situations analysed above, and consequent steady-state positions for the piezometric surface or water table, are only achieved after what may be very long periods of time after the start of pumping. During such periods, even though the pumping rate may be steady, the piezometric surface or water table will be falling with time until the steady-state position is reached. Theoretically, if the aquifer were infinite in extent (with no vertical recharge), a steady state would never be completely achieved. Consequently, an ability to analyse such non-steady groundwater flow situations for wells being pumped at a constant rate, is required.

In confined aquifers, the relief of pressure as a piezometric surface falls introduces two compressibility effects: the pore water expands owing to the smaller water pressure, and simultaneously the pore space contracts owing to a greater mechanical stress from the overburden as the reduced water pressure takes less of the load. Thus water is released from storage over the aquifer to make up the pumped abstraction; no dewatering of the pore spaces occurs.

In unconfined aquifers, on the other hand, compressibility effects are usually negligible, but dewatering of the pore spaces does occur as the water table falls. The water being released from the whole aquifer integrates to equal the constant pumping abstraction.

In both types of aquifer, the head, h, varies both with distance and time after the start of pumping. Even with a constant pumping rate, the situation is described as being 'non-steady' flow!

Assuming negligible recharge and no head gradient in the vertical, the horizontal, two-dimensional, non-steady flow can be described by, x ( ™ + dih )=s, -

For a homogeneous, isotropic, confined aquifer of thickness b, this can be written, d 2h d 2h _ S d h dx2 dy2 T d t with T and S(S = Ssb, T = Kb), the transmissivity and storativity, respectively. For non-steady flow to a well, this equation may be transformed to radial coordinates. Thus with r = *J(x2 + y2 ):

The solution of this equation yields h(r, t) (Fig. 5.6), the hydraulic head at distance r from a well at time t after the commencement of steady pumping at a rate Q.

For practical purposes, what is usually required is (h0 — h(r, t)}, which is the drawdown, s(r, t), from the initial rest level head, ho. The solution of the equation is,

4n T Ju

(known as the Theis equation), where Q is the steady pumping rate and u= r2S/4Tt. Expansion of the integral gives,

U2 U3

The expression within the brackets is usually denoted by W(u) known as the well function so that:

Values of the well function, W(u), for a range of u values are given in Table 5.2. Knowing the 'formation constants' of the aquifer, S and T, and for a given Q, the drawdown s(r, t) can be estimated directly for any radius r and time t.

Example. To calculate the drawdown in a confined aquifer at r |
= 25 m after | |

6 hours of pumping. Water with a constant discharge of 0.0311 |
m3 s-1. The | |

aquifer constants are S = 0.005 and T = 0.0092 m2 s-1. |
4Tt | |

252 x 0.005 | ||

4 x 0.0092(6 x 3600) | ||

3.125 | ||

794.88 | ||

= 3.93 x 10-3 | ||

So that, | ||

s(r, t) = -QTw (u) | ||

0.0311 = x 4.97 4 x n x 0.0092 | ||

= 1.337m |

The well function can also be used to derive values of S and T by matching measurements of drawdown against time at an observation well against the well function (see e.g. Fetter, 2001).

The Theis solution of the non-steady radial flow equation has, for certain conditions, a simplified form due to Jacob. If u is small (< 0.01) only the first two terms in the series of the equation steady flow in an unconfined aquifer need be used (to within 1 per cent of the full series) so that:

4n T

u |
1.0 |
2.0 |
3.0 |
4.0 |
5.0 |
6.0 |
7.0 |
8.0 |
9.0 | |

XI |
0.2I9 |
0.049 |
0.0I3 |
0.004 |
0.00I | |||||

XI0- |
■ I |
1.82 |
I .22 |
0.9I |
0.70 |
0.56 |
0.45 |
0.37 |
0.3I |
0.26 |

XI0- |
-2 |
4.04 |
3.35 |
2.96 |
2.68 |
2.47 |
2.30 |
2.I5 |
2.03 |
1.92 |

XI0- |
-3 |
6.33 |
5.64 |
5.23 |
4.95 |
4.73 |
4.54 |
4.39 |
4.26 |
4.I4 |

XI0- |
-4 |
8.63 |
7.94 |
7.53 |
7.25 |
7.02 |
6.84 |
6.69 |
6.55 |
6.44 |

XI0- |
-5 |
I0.94 |
I0.24 |
9.84 |
9.55 |
9.33 |
9.I4 |
8.99 |
8.86 |
8.74 |

XI0- |
-6 |
I3.24 |
I2.55 |
I2.I4 |
II.85 |
II.63 |
II .45 |
II .29 |
II.I6 |
II.04 |

XI0- |
-7 |
I5.54 |
I4.85 |
I4.44 |
I4.I5 |
I3.93 |
I3.75 |
I3.60 |
I3.46 |
I3.34 |

XI0- |
-8 |
I7.84 |
I7.I5 |
I6.74 |
I6.46 |
I6.23 |
I6.05 |
I5.90 |
I5.76 |
I5.65 |

XI0- |
-9 |
20. I5 |
I9.45 |
I9.05 |
I8.76 |
I8.54 |
I8.35 |
I8.20 |
I8.07 |
I7.95 |

XI0- |
I0 |
22.45 |
2I .76 |
2I .35 |
2I .06 |
20.84 |
20.66 |
20.50 |
20.37 |
20.25 |

XI0- |
II |
24.75 |
24.06 |
23.65 |
23.36 |
23. I4 |
22.96 |
22.8I |
22.67 |
22.55 |

XI0- |
I2 |
27.05 |
26.36 |
25.96 |
25.67 |
25.44 |
25.26 |
25. II |
24.97 |
24.86 |

XI0- |
I3 |
29.36 |
28.66 |
28.26 |
27.97 |
27.75 |
27.56 |
27.4I |
27.28 |
27. I6 |

XI0- |
I4 |
3I.66 |
30.97 |
30.56 |
30.27 |
30.05 |
29.87 |
29.7I |
29.58 |
29.46 |

XI0- |
I5 |
33.96 |
33.27 |
32.86 |
32.58 |
32.35 |
32. I7 |
32.02 |
3I .88 |
3I.76 |

4nT \ r2S J 4nT Intercept Slope

A semi-log plot of s(r, t) versus ln(t/r2) allows T and S to be estimated from the slope and intercept of a best-fit straight line, drawn given greater regard to the points at longer times (Fig. 5.7).

Then, given the slope of the fitted straight line, m

4n m

2.25Tc

Drawdown sfcf)

Slope | |

Intercept (c) Fig. 5.7 Fitted straight line to drawdown versus log time data for the case of non-steady flow in a confined aquifer. (Adapted from a figure prepared by Andrew Binley). ## 5.1.3.5 Non-steady flow in an unconfined aquiferIn an unconfined aquifer, it has been seen that steady flow, based on the Dupuit-Forchheimer assumptions, becomes somewhat unrealistic close to the well when the slope of the water table becomes steep. In assessing non-steady flow, dewatering of the medium results in changes in transmissivity and, when the water table is lowered, the storativity is decreased. Thus the application of the non-steady flow equations becomes difficult. However, for small drawdowns over short time periods, the Theis equation may be used for rough estimates of s(r, t). Various methods have been derived to deal with the problems of analysis of observations in more complex flow situations, including non-steady flow in unconfined aquifers, and for these the reader is referred to the specialist texts on groundwater (e.g. Fetter, 2001; Todd and Mays, 2005), which also deal with the added difficulties encountered with non-homogeneous and anisotropic aquifers, more complex boundary conditions in aquifers of limited extent, and the rather common situation where leaky aquifers receive recharge or lose water through a low permeability aquitard. ## 5.2 Unsaturated hydraulic conductivity curveDarcy's law describing subsurface flow within the saturated zone can be extended to describe flow with unsaturated soil, regolith and rock by replacing the saturated hydraulic conductivity with the unsaturated hydraulic conductivity curve (K(h)). For small pressure heads of up to about -30cm H2O below atmospheric pressure (Fig. 5.8), this curve can be measured directly using a tension infiltrome-ter (Fig. 5.9). For larger negative pressure heads, the unsaturated hydraulic conductivity curve is normally derived by modelling the data within the moisture release curve (Section 6.3). The models of Brooks and Corey (1964) and van Genuchten (1980) are most commonly used. |

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