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**Unit 04 : Advanced Hydrogeology**

Hydraulic Testing

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Well Hydraulics A water well is a hydraulic structure that is designed and constructed to permit economic withdrawal of water from an aquifer Water well construction includes: Selection of appropriate drilling methods Selection of appropriate completion materials Analysis and interpretation of well and aquifer performance

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**Pumping Well Terminology**

Static Water Level [SWL] (ho) is the equilibrium water level before pumping commences Pumping Water Level [PWL] (h) is the water level during pumping Drawdown (s = ho - h) is the difference between SWL and PWL Well Yield (Q) is the volume of water pumped per unit time Specific Capacity (Q/s) is the yield per unit drawdown ho h s Q

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**Cone of Depression High Kh aquifer Low Kh aquifer Kh Kv**

A zone of low pressure is created centred on the pumping well Drawdown is a maximum at the well and reduces radially Head gradient decreases away from the well and the pattern resembles an inverted cone called the cone of depression The cone expands over time until the inflows (from various boundaries) match the well extraction The shape of the equilibrium cone is controlled by hydraulic conductivity

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**Aquifer Characteristics**

Pump tests allow estimation of transmission and storage characteristics of aquifers Transmissivity (T = Kb) is the rate of flow through a vertical strip of aquifer (thickness b) of unit width under a unit hydraulic gradient Storage Coefficient (S = Sy + Ssb) is storage change per unit volume of aquifer per unit change in head Radius of Influence (R) for a well is the maximum horizontal extent of the cone of depression when the well is in equilibrium with inflows

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**Steady Radial Confined Flow**

ho h s Q r b Assumptions Isotropic, homogeneous, infinite aquifer, 2-D radial flow Initial Conditions h(r,0) = ho for all r Boundary Conditions h(R,t) = ho for all t Darcy’s Law Q = -2prbKh/r Rearranging h = - Q r 2pKb r Integrating h = - Q ln(r) + c 2pKb BC specifies h = ho at r = R Using BC ho = - Q ln(R) + c 2pKb Eliminating constant (c) gives s = ho – h = Q ln(r/R) This is the Thiem Equation

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**Steady Unconfined Radial Flow**

ho h s Q r Assumptions Isotropic, homogeneous, infinite aquifer, 2-D radial flow Initial Conditions h(r,0) = ho for all r Boundary Conditions h(R,t) = ho for all t Darcy’s Law Q = -2prhKh/r Rearranging hh = - Q r 2pK r Integrating h2 = - Q ln(r) + c pK BC specifies h = ho at r = R Using BC ho2 = - Q ln(R) + c pK Eliminating constant (c) gives ho2 – h2 = Q ln(r/R) This is the Thiem Equation

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**Thiem Equation Assumptions**

The equation for unconfined flow can be rearranged to give: s = ho – h = Q ln(r/R) 2pK (ho+ h)/2 Compare this with the confined equation: 2pKb It is clear than the only difference is that the aquifer thickness b is replaced by (ho + h)/2 The implicit assumption in the derivation (2-D flow) implies that s is small compared with ho and that the (ho + h)/2 does not deviate significantly from ho This assumption may not be valid in the immediate vicinity of a pumping well in an unconfined aquifer

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**Thiem Equation Applications**

The equation for unconfined flow can be rearranged to give: K = Q ln(r/R) p (ho2 - h2) Similarly with the confined equation: 2pb (ho – h) The radius of influence R is hard to estimate but any two wells at different radial distance can be used in the equations K = Q ln(r2/r1) and K = Q ln(r2/r1) p (h22 – h12) pb (h2 – h1) This means that for a well producing at a steady rate (Q) with a steady drawdown, any pair of observation points at different radial distances can be used to estimate K

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Specific Capacity For a confined well producing at a steady rate (Q) the specific capacity is given by: Q = 2pKb sw ln(rw/R) This means that for a confined well producing at a steady rate (Q) the specific capacity is constant. The equation for unconfined flow can be rearranged: Q = pK (ho + hw) Writing hw = ho - sw gives: Q = - pK (sw + 2ho) sw ln(rw/R) For an unconfined well producing at a steady rate (Q) the specific capacity reduces with increasing drawdown. The maximum specific capacity for an unconfined well is given by: Q = 2pKho

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**Unsteady Radial Confined Flow**

ho h s Q r b Assumptions Isotropic, homogeneous, infinite aquifer, 2-D radial flow Initial Conditions h(r,0) = ho for all r Boundary Conditions h(,t) = ho for all t PDE 1 (rh ) = S h r r r T t Solution is more complex than steady-state Change the dependent variable by letting u = r2S 4Tt The ultimate solution is: ho- h = Q exp(-u) du 4pT u u where the integral is called the exponential integral written as the well function W(u) This is the Theis Equation

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Theis PDE to ODE Let a = S/T (to simplify notation where a is called the hydraulic diffusivity) PDE 1 (rh ) = a h r r r t u = r2S = ar2 4Tt 4t So u = ar = 2u r 2t r And u = -ar2 = -u t t t Rewriting the PDE in terms of u: 1 d (r u dh ) u = a u dh r du r du r t du Rewriting partial derivatives in terms of u: 1 d (r 2u dh ) 2u = -a u dh r du r du r t du Rearranging and cancelling: d (u dh ) = -ar2 dh = -u dh du du t du du Expanding the LHS derivative: u d (dh ) + dh = -u dh du du du du d (dh ) = -(u + 1) dh du du u du Writing dh/du as h’ gives an ODE: dh’ = -(u + 1) h’ or dh’ = -(1 + 1) du du u h’ u

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**Theis Integration The resulting ODE is: dh’ = -(1 + 1) du h’ u**

Integrating: ln(h’) = -u – ln(u) + c Simplifying: ln(h’u) = c – u Inverting: h’u = exp(c).exp(-u) To eliminate exp(c), use Darcy’s Law: lim rh = -Q = lim 2h’u r 0 r 2pKb u 0 Remember rh = rh’u = h’ar2 = 2h’u r r t lim h’u = -Q = exp(c) u 0 4pKb Simplifying: h’ = -Q exp(-u) 4pT u Recall h’ = dh/du dh = -Q exp(-u).du 4pT u Integrating: h = -Q exp(-u) du + C 4pTu u Finally, using h(,t) = ho to eliminate C: ho - h = Q exp(-u) du The integral is called the exponential integral but is often written as the Theis well function W(u) s = ho - h = Q W(u) 4pT

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Theis Plot : 1/u vs W(u)

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**Theis Plot : Log(t) vs Log(s)**

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**Theis Plot : Log(t) vs Log(s)**

s=0.17m [1,1] Type Curve t=51s

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**Theis Analysis Overlay type-curve on data-curve keeping axes parallel**

Select a point on the type-curve (any will do but [1,1] is simplest) Read off the corresponding co-ordinates on the data-curve [td,sd] For [1,1] on the type curve corresponding to [td,sd], T = Q/4psd and S = 4Ttd/r2 = Qtd/pr2sd For the example, Q = 32 L/s or m3/s; r = 120 m; td = 51 s and sd = 0.17 m T = (0.032)/(12.56 x 0.17) = m2/s = 1300 m2/d S = (0.032 x 51)/(3.14 x 120 x 120 x 0.17) = 2.1 x 10-4

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**where g is Euler’s constant (0.5772)**

Cooper-Jacob Cooper and Jacob (1946) pointed out that the series expansion of the exponential integral or W(u) is: W(u) = – g - ln(u) + u - u2 + u3 - u4 + ..… 1.1! 2.2! 3.3! 4.4! where g is Euler’s constant (0.5772) For u<< 1 , say u < 0.05 the series can be truncated: W(u) – ln(eg) - ln(u) = - ln(egu) = -ln(1.78u) Thus: s = ho - h = - Q ln(1.78u) = - Q ln(1.78r2S) = Q ln( 4Tt ) 4pT pT Tt pT r2S s = ho - h = Q ln( 2.25Tt ) = 2.3 Q log( 2.25Tt ) 4pT r2S pT r2S The Cooper-Jacob simplification expresses drawdown (s) as a linear function of ln(t) or log(t).

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**Cooper-Jacob Plot : Log(t) vs s**

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**Cooper-Jacob Plot : Log(t) vs s**

to = 84s Ds =0.39 m

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**Cooper-Jacob Analysis**

Fit straight-line to data (excluding early and late times if necessary): – at early times the Cooper-Jacob approximation may not be valid – at late times boundaries may significantly influence drawdown Determine intercept on the time axis for s=0 Determine drawdown increment (Ds) for one log-cycle For straight-line fit, T = 2.3Q/4pDs and S = 2.25Tto/r2 = 2.3Qto/1.78pr2Ds For the example, Q = 32 L/s or m3/s; r = 120 m; to = 84 s and Ds = 0.39 m T = (2.3 x 0.032)/(12.56 x 0.39) = m2/s = 1300 m2/d S = (2.3 x x 84)/(1.78 x 3.14 x 120 x 120 x 0.39) = 1.9 x 10-4

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**Theis-Cooper-Jacob Assumptions**

Real aquifers rarely conform to the assumptions made for Theis-Cooper-Jacob non-equilibrium analysis Isotropic, homogeneous, uniform thickness Fully penetrating well Laminar flow Flat potentiometric surface Infinite areal extent No recharge The failure of some or all of these assumptions leads to “non-ideal” behaviour and deviations from the Theis and Cooper-Jacob analytical solutions for radial unsteady flow

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**Recharge Effect : Recharge > Well Yield**

Recharge causes the slope of the log(time) vs drawdown curve to flatten as the recharge in the zone of influence of the well matches the discharge. The gradient and intercept can still be used to estimate the aquifer characteristics (T,S).

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**Recharge Effect : Recharge < Well Yield**

If the recharge is insufficient to match the discharge, the log(time) vs drawdown curve flattens but does not become horizontal and drawdown continues to increase at a reduced rate. T and S can be estimated from the first leg of the curve.

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Sources of Recharge Various sources of recharge may cause deviation from the ideal Theis behaviour. Surface water: river, stream or lake boundaries may provide a source of recharge, halting the expansion of the cone of depression. Vertical seepage from an overlying aquifer, through an intervening aquitard, as a result of vertical gradients created by pumping, can also provide a source of recharge. Where the cone of depression extends over large areas, leakage from aquitards may provide sufficient recharge.

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**Recharge Effect : Leakage Rate**

High Leakage Low Leakage Recharge by vertical leakage from overlying (or underlying beds) can be quantified using analytical solutions developed by Jacob (1946). The analysis assumes a single uniform leaky bed.

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**Hantush Type Curves Theis Curve**

Data are fitted in a manner similar to the Theis curve. The parameter r/B = r( {K’v / b’} / {Khb} )½ increases with the amount of leakage.

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**Barrier Effect : No Flow Boundary**

Steepening of the log(time) vs. drawdown curve indicates an aquifer limited by a barrier boundary of some kind. Aquifer characteristics (T,S) can be estimated from the first leg.

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**Potential Flow Barriers**

Various flow barriers may cause deviation from the ideal Theis behaviour. Fault truncations against low permeability aquitards. Lenticular pinchouts and lateral facies changes associated with reduced permeability. Groundwater divides associated with scarp slopes. Spring lines with discharge captured by wells. Artificial barriers such as grout curtains and slurry walls.

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Casing Storage It has been known for many decades that early time data can give erroneous results because of removal of water stored in the well casing. When pumping begins, this water is removed and the amount drawn from the aquifer is consequently reduced. The true aquifer response is masked until the casing storage is exhausted. Analytical solutions accounting for casing storage were developed by Papadopulos and Cooper (1967) and Ramey et al (1973) Unfortunately, these solutions require prior knowledge of well efficiencies and aquifer characteristics

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**tc = 3.75p(dc2 – dp2) / (Q/s) = 15 Va /Q**

Casing Storage Q Schafer (1978) suggests that an estimate of the critical time to exhaust casing storage can be made more easily: tc = 3.75p(dc2 – dp2) / (Q/s) = 15 Va /Q where tc is the critical time (d); dc is the inside casing diameter (m); dp is the outside diameter of the rising main (m); Q/s is the specific capacity of the well (m3/d/m) Va is the volume of water removed from the annulus between casing and rising main. Note: It is safest to ignore data from pumped wells earlier than time tc in wells in low-K HSU’s s dp dc

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**s = 2.3 Q log( 2.25Tt ) = 2.3 Q log( 2.25Tt ) – 4.6Q log(r)**

Distance-Drawdown Simultaneous drawdown data from at least three observation wells, each at different radial distances, can be used to plot a log(distance)-drawdown graph. The Cooper-Jacob equation, for fixed t, has the form: s = 2.3 Q log( 2.25Tt ) = 2.3 Q log( 2.25Tt ) – 4.6Q log(r) 4pT r2S pT S pT So the log(distance)-drawdown curve can be used to estimate aquifer characteristics by measuring Ds for one log-cycle and the ro intercept on the distance-axis. T = 4.6Q and S = 2.25Tt 4pDs ro2

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**Distance-Drawdown Graph**

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**Aquifer Characteristics**

For the example: t = 0.35 days and Q = 1100 m3/d T = x 1100 / 3.8 = 106 m2/d S = 2.25 x 106 x 0.35 / (126 x 126) = 5.3 x 10-3 The estimates of T and S from log(time)-drawdown and log(distance)-drawdown plots are independent of one another and so are recommended as a check for consistency in data derived from pump tests. Ideally 4 or 5 observation wells are needed for the distance-drawdown graph and it is recommended that T and S are computed for several different times.

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Well Efficiency The efficiency of a pumped well can be evaluated using distance-drawdown graphs. The distance-drawdown graph is extended to the outer radius of the pumped well (including any filter pack) to estimate the theoretical drawdown for a 100% efficient well. This analysis assumes the well is fully-penetrating and the entire saturated thickness is screened. The theoretical drawdown (estimated) divided by the actual well drawdown (observed) is a measure of well efficiency. A correction is necessary for unconfined wells to allow for the reduction in saturated thickness as a result of drawdown.

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**Theoretical Pumped Well Drawdown**

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**Unconfined Well Correction**

The adjusted drawdown for an unconfined well is given by: sc = (1 - sa ) sa 2b where b is the initial saturated thickness; sa is the measured drawdown; and sc is the corrected drawdown For example, if b = 20 m; sa = 6 m; then the corrected drawdown sc = 0.85sa = 5.1 m If the drawdown is not corrected, the Jacob and Theis analysis underestimates the true transmissivity under saturated conditions by a factor of sc/ sa.

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**Causes of Well Inefficiency**

Factors contributing to well inefficiency (excess head loss) fall into two groups: Design factors Insufficient open area of screen Poor distribution of open area Insufficient length of screen Improperly designed filter pack Construction factors Inadequate development, residual drilling fluids Improper placement of screen relative to aquifer interval

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Radius of Influence The radius of influence of a well can be determined from a distance-drawdown plot. For all practical purposes, a useful comparative index is the intercept of the distance-drawdown graph on the distance axis. Radius of influence can be used as a guide for well spacing to avoid interference. Since radius of influence depends on the balance between aquifer recharge and well discharge, the radius may vary from year to year. For unconfined wells in productive aquifers, the radius of influence is typically a few hundred metres. For confined wells may have a radius of influence extending several kilometres.

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Determining ro Unconfined Well Confined Well

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Unconfined Aquifers Most analytical solutions assume isotropic, homogeneous, confined aquifers or assume drawdowns are small for the unconfined case. There are three distinct parts to the time drawdown curve in an unconfined aquifer: early time response follows Theis equation with the confined “elastic” storage corresponding to storativity (bSs) intermediate times respond as a leaky aquifer with vertical flow in the vicinity of the pumped well with storage release controlled by the aquifer Kh/Kv ratio late time response follows Theis equation with gravity drainage providing storage corresponding to the specific yield (Sy)

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**Delayed-Yield Response**

Unconfined response is complex with theory developed by Boulton, Dagan, Steltsova, Rushton and Neuman. uA uB h Neuman (1975) defines a well function W(uA,uB,h) where each parameter corresponds to a different time phase: early-time response is controlled by uA = r2S/4Tt intermediate-times are controlled by h = r2Kv/Khb2 late-time response is controlled by uB = r2Sy/4Tt The Hantush leakage parameter (r/B) is closely related to h (r/B)2 = r2Kv’/Khbb’ where the Kv’ and b’ parameters refer to the leaky bed. If these leaky-bed parameters become aquifer parameters (r/B)2 = h.

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Neuman Type Curves The Neuman type curves are fitted to data in a manner similar to that for Theis curves. Higher values of h indicate more rapid gravity drainage.

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Partial Penetration Partial penetration effects occur when the intake of the well is less than the full thickness of the aquifer

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**Effects of Partial Penetration**

The flow is not strictly horizontal and radial. Flow-lines curve upwards and downwards as they approach the intake and flow-paths are consequently longer. The convergence of flow-lines and the longer flow-paths result in greater head-loss than predicted by the analytical equations. For a given yield (Q), the drawdown of a partially penetrating well is more than that for a fully penetrating well. The analysis of the partially penetrating case is difficult but Kozeny (1933) provides a practical method to estimate the change in specific capacity (Q/s).

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Q/s Reduction Factors Kozeny (1933) gives the following approximate reduction factor to correct specific capacity (Q/s) for partial penetration effects: F = L {1 + 7 cos(pL) ( r )} b b 2L where b is the total aquifer thickness (m); r is the well radius (m); and L is screen length (m). The equation is valid for L/b < 0.5 and L/r > 30 For a 300 mm dia. well with an aquifer thickness of 30 m and a screen length of 15 m, L/b = 0.5 and 2L/r = 200 the reduction factor is: F = 0.5 x {1 + 7 x (1/200)} = 0.67 Other factor are provided by Muskat (1937), Hantush (1964), Huisman (1964), Neuman (1974) but they are harder to use.

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**Partial Penetration Alternative**

Multiple screened sections distributed over the entire saturated thickness functions more efficiently for the same open area.

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Screen Design 300 mm dia. well with single screened interval of 15 m in aquifer of thickness 30 m. L/b = 0.5 and 2L/r = 200 F = 0.5 x {1 + 7 x cos(0.5p/2) (1/200)} = 0.67 300 mm dia. well with 5 x 3 m solid sections alternating with 5 x 3m screened sections, in an aquifer of thickness 30 m. There effectively are five aquifers. L/b = 0.5 and 2L/r = 40 F = 0.5 x {1 + 7 x cos(0.5p/2) (1/40)} = 0.89 This is clearly a much more efficient well completion.

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Recovery Data When pumping is halted, water levels rise towards their pre-pumping levels. The rate of recovery provides a second method for calculating aquifer characteristics. Monitoring recovery heads is an important part of the well-testing process. Observation well data (from multiple wells) is preferable to that gathered from pumped wells. Pumped well recovery records are less useful but can be used in a more limited way to provide information on aquifer properties.

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Recovery Curve Drawdown 10 m Recovery 10 m Pumping Stopped The recovery curve on a linear scale appears as an inverted image of the drawdown curve. The dotted line represent the continuation of the drawdown curve.

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Superposition The drawdown (s) for a well pumping at a constant rate (Q) for a period (t) is given by: s = ho - h = Q W(u) where u = r2S 4pT Tt The effects of well recovery can be calculated by adding the effects of a pumping well to those of a recharge well using the superposition theorem. The drawdown (sr) for a well recharged at a constant rate (-Q) for a period (t’ = t - tr) starting at time tr is given by: sr = - Q W(u’) where u’ = r2S 4pT Tt’ The total drawdown for t > tr is: s’ = s + sr = Q (W(u) - W(u’)) 4pT

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**Residual Drawdown and Recovery**

The total drawdown for t > tr is: s’ = s + sr = Q (W(u) - W(u’)) 4pT The Cooper-Jacob approximation can be applied giving: s’ = s + sr = Q (ln(2.25Tt) - ln(2.25Tt’)) 4pT r2S r2S Simplification gives the residual drawdown equation: s’ = s + sr = Q ln(t) 4pT t’ The equation predicting the recovery is: sr = - Q ln(2.25Tt’) 4pT r2S For t > tr, the recovery sr is the difference between the observed drawdown s’ and the extrapolated pumping drawdown (s).

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**Time-Recovery Graph to’ = 0.12 hrs Dsr = 4.6 m**

Aquifer characteristics can be calculated from a log(time)-recovery plot but the drawdown (s) curve for the pumping phase must be extrapolated to estimate recovery (s - s’)

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**Time-Recovery Analysis**

For a constant rate of pumping (Q), the recovery any time (t’) after pumping stops: T = Q = Q = Q 4pD(s - s’) pDsr 4pDsr For the example, Dsr = 4.6 m and Q = 1100 m3/d so: T = 1100 / (12.56 x 4.6) = 19 m2/d The storage coefficient can be estimated for an observation well (r = 30 m) using: S = 4Tto’ r2 For the example, to’ = and Q = 1100 m3/d so: S = 4 x 19 x 0.12 / (24 x 30 x 30) = 4.3 x 10-4 It is necessary to use an observation well for this calculation because well bore storage effects render any calculation based on rw potentially subject to huge errors.

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**Time-Residual Drawdown Graph**

Ds’ = 5.2 m Transmissivity can be calculated from a log(time ratio)-residual drawdown (s’) graph by determining the gradient. For such cases, the x-axis is log(t/t’) and thus is a ratio.

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**Time-Residual Drawdown Analysis**

For a constant rate of pumping (Q), the recovery any time (t’) after pumping stops: T = Q 4pDs’ For the example, Dsr = 5.2 m and Q = 1100 m3/d so: T = 1100 / (12.56 x 5.2) = 17 m2/d Notice that the graph plots t/t’ so the points on the LHS represent long recovery times and those on the RHS short recovery times. The storage coefficient cannot be estimated for the residual drawdown plot because the intercept t / t’ 1 as t’ . This more obvious, remembering t’ = t - tr where tr is the elapsed pumping time before recovery starts.

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**Residual Drawdown for Real Aquifers**

Theoretical intercept is 1 >> 1 indicates a recharge effect >1 may indicate greater S for pumping than recovery ?consolidation < 1 indicates incomplete recovery of initial head - finite aquifer volume << 1 indicates incomplete recovery of initial head - small aquifer volume

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DST The drill stem test, used widely in petroleum engineering, is a recovery test. Packers are used to isolate the HSU of interest which has been flowing for some time. Initially the bypass valve is open allowing free circulation. When the bypass valve is closed, the formation pressure is “shut-in” and begins to recover towards the static value. The Horner plot is a direct analogue of the residual drawdown plot. Drill Stem Valve Packer Perforated Section Packer Gauge

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**DST Analysis Recall that the final form of the recovery equation is:**

ho - h = s’ = 2.3Q log(t) 4pT t’ For a DST, the pressure (rather than head) is measured po - p = 2.3Qm log(t) 4pkb t’ Remembering that p = gh,T = Kb and K = kg/m The Horner-plot has an intercept po when t / t’ = 1 This intercept is taken to be the static formation pressure. K can be estimated from the gradient of the graph: k = 2.3Qm 4pbDp t / t’ po p (kPa) Dp

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Slug Test The recovery test in a borehole after withdrawal or injection (or displacement) of a known volume of water is called a slug test. The slug test is a rapid field method for estimation of moderate to low K-values in a single well. The procedure is: initial head is noted the slug is removed, added or displaced instantaneously (displacement is best in this respect) head recovery is monitored (usually with a submerged pressure logging device) typical head changes are 2-3 m in mm dia. piezometers so the volume of the slug is typically only 1-10 litres Displacer Displaced Head Initial Head

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**Slug Analysis ra is access tube internal radius**

rw is perforated section external radius L is length of perforated section ho is initial head, t = to h(t) is head after recovery time t A is the tube or casing csa = pra2 F is a shape factor = 2pL / ln(L/rw) Analysis methods include: Hvorslev (1951) Cooper et al (1967) The Cooper analysis considers storage but the Hvorslev analysis is more widely used. K = A ln (h) = ra2 ln(L) ln(h) F(t - to) ho 2L(t - to) rw ho Tube or Casing 2ra L 2rw

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**Hvorslev Analysis K = ra2 ln(L) ln(h) 2L(t - to) rw ho**

Plot time against log (h/ho) Measure basic time lag To when ln(h0/h) = 1 K = ra2 ln (L) 2LTo rw Time lag To occurs when: h = e-1ho = 0.37ho If To = 1000 secs for a 50 mm dia. x 1 m length Casagrande piezometer with 38 mm dia access tube K = 2 x 10-6 m/s Tube or Casing 2ra 2rw L 0.6 0.5 0.4 0.3 0.2 0.1 Time, t - to h ho To

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Bounded Aquifers Superposition was used to calculate well recovery by adding the effects of a pumping and recharge well starting at different times. Superposition can also be used to simulate the effects of aquifer boundaries by adding wells at different positions. For boundaries, the wells that create the same effect as a boundary are called image wells. This relatively simple application of superposition for analysis of aquifer boundaries was for described by Ferris (1959)

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Image Wells Recharge boundaries at distance (r) are simulated by a recharge image well at an equal distance (r) across the boundary. Barrier boundaries at distance (r) are simulated by a pumping image well at an equal distance (r) across the boundary. r

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**s = sp si = Q [W(u) W(ui)]**

General Solution The general solution for adding image wells to a real pumping well can be written: s = sp si = Q [W(u) W(ui)] 4pT where up = rp2S and ui = ri2S 4Tt Tt and rp,ri are the distances from the pumping and image wells respectively. r rp ri For a barrier boundary, for all points on the boundary rp = ri and the drawdown is doubled. For a recharge boundary, for all points on the boundary rp = ri and the drawdown is zero.

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Specific Solutions Using the Cooper-Jacob approximation is only possible for large values of to ensure that u < 0.05 for all r so the Theis well function is used: s = Q [W(u) W(ri2u)] = Q [W(u) W(au)] 4pT rp pT For the barrier boundary case: s = Q [W(u) + W(au)] 4pT where a = (ri/rp)2 and 0<a<1 For the recharge boundary case: s = Q [W(u) - W(au)] 4pT where a = (ri/rp)2 and 0<a<1

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Multiple Boundaries A recharge boundary and a barrier boundary at right angles can be generated by two pairs of pumping and recharge wells. Two barrier boundaries at right angles can be generated by superposition of an array of four pumping wells. r2 r2 r1 r1

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Parallel Boundaries A parallel recharge boundary and a barrier boundary (or any pattern with parallel boundaries) requires an infinite array of image wells. r1 r2

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Boundary Location For an observation well at distance r1, measure off the same drawdown (s), before and after the “dog leg” on a log(time) vs. drawdown plot. Find the times t1 and t2. Assuming that the “dog leg” is created by an image well at distance r2 , if the drawdowns are identical then W(u1) = W(u2) so u1 = u2. Thus: r12S/4Tt1 = r22S/4Tt2 So r12t2 = r22t1 and r2 = r1(t2 / t1)½ The distance r2 the radial distance from the observation point to the boundary. Repeating for additional observation wells may help locate the boundary. t1 t2 s

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Pumping Wells The drawdown observed in a pumping well has two component parts: aquifer loss drawdown due to laminar flow in the aquifer well loss drawdown due to turbulent flow in the immediate vicinity of the well through the screen and/or gravel pack Well loss is usually assumed to be proportional to the square of the pumping rate: sw = CQ2

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**st = s + sw = Q W(u) + CQ2 = BQ + CQ2**

Well Efficiency The total drawdown at a pumping well is given by: st = s + sw = Q W(u) + CQ2 = BQ + CQ2 4pT The ratio of the aquifer loss and total drawdown (s/st) is known as the well efficiency. s = W(u) = B . st W(u) + 4pTCQ B + CQ Mogg (1968) defines well efficiency at a fixed time (t = 24 hrs). Thus, writing W(u) as the Cooper-Jacob approximation gives: s = = st pTCQ / [ln (2.25Tt /S) - 2 ln(rw)] CQ/B(rw) Written in this form it is clear that well efficiency reduces with pumping rate (Q) and increases with well radius (rw), where B is inversely related to well radius. The specific capacity is given by: Q = st B + CQ

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Step-Drawdown Test Step-drawdown tests are tests at different pumping rates (Q) designed to determine well efficiency. Normally pumping at each successively greater rate Q1 < Q2 < Q3 < Q4 < Q5 takes place for 1-2 hours (Dt) and for 5 to 8 steps. The entire test usually takes place in one day. Equal pumping times (Dt) simplifies the analysis. At the end of each step, the pumping rate (Q) and drawdown (s) is recorded. s1 s2 s3 Drawdown, s s4 s5 Time, t

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**Step-Drawdown Test Analysis**

Step-drawdown tests are analysed by plotting the reciprocal of specific capacity (s/Q) against the pumping rate (Q). The intercept of the graph at Q=0 is B = W(u)/4pT and the slope is the well loss coefficient, C. B can also be obtained independently from a Theis or Cooper-Jacob analysis of a pump test. For Q = 2700 m3/d and s = 33.3 m the B = m/m3/d If C = 4 x 10-5, then CQ2 = 18.2 m The well efficiency is 33.3/( ) = 65% C s/Q (m/m3/d) B Q (L/s)

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Well Yield The chart is used to select casing sizes for a particular yield. The main constraint is pumping equipment. For example, if the well is designed to deliver 4,000 m3/d, the optimum casing dia. is 360 mm (2 nom. sizes > pump dia.) and the minimum 300 mm. The drilled well diameter would have to be 410 to 510 mm to provide at least a 50 mm grout/cement annulus.

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Pump Test Planning Pump tests will not produce satisfactory estimates of either aquifer properties or well performance unless the data collection system is carefully and QA/QC is addressed in the design. Several preliminary estimates are needed to design a successful test: Estimate the maximum drawdown at the pumped well Estimate the maximum pumping rate Evaluate the best method to measure the pumped volumes Plan discharge of pumped volumes distant from the well Estimate drawdowns at observation wells Simulate the test before it is conducted Measure all initial heads several times to ensure that steady-conditions prevail Survey elevations of all well measurement reference points

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**Number of Observation Wells**

Number depends on test objectives and available resources for test program. Single well can give aquifer characteristics (T and S). Reliability of estimates increases with additional observation points. Three wells at different distances are needed for time-distance analysis No maximum number because anisotropy, homogeneity, and boundaries can be deduced from response

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**Pump Test Measurements**

The accuracy of drawdown data and the results of subsequent analysis depends on: maintaining a constant pumping rate measuring drawdown at several (>2) observation wells at different radial distances taking drawdowns at appropriate time intervals at least every min (1-15 mins); (every 5 mins) mins; (every 30 mins) 1-5 hrs; (every 60 mins) 5-12 hrs; (every 8 hrs) >12 hrs measuring barometric pressure, stream levels, tidal oscillations as necessary over the test period measuring both pumping and recovery data continuing tests for no less than 24 hours for a confined aquifers and 72 hours for unconfined aquifers in constant rate tests collecting data over a 24 hour period for 5 or 6 pumping rates for step-drawdown tests

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**Measuring Pumping Rates**

Control of pumping is normally required as head and pump rpm changes. Frequent flow rate measurements are needed to maintain constant rate. Lower rates periodic measurements of time to fill a container of known volume “v” notch weir - measure head (sensitive at low flows) Higher rates impellor driven water meter - measure velocity (insensitive) circular orifice weir - measure head v=(2gh)½ rectangular notch weir - measure head free-flow Parshall flume (drop in floor) - measure head cutthroat flume (flat floor) – measure head

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**Measuring Drawdown Pumped wells Observation wells**

heads are hard to measure due to turbulence and pulsing. data cannot reliably estimate storage. Observation wells smallest possible diameter involves least time lag screens usually 1-2 m; longer is better but not critical should be at same depth as centre of production section if too close (< 3 to 5 x aquifer thickness) can be strongly influenced by anisotropy (stratification) if too far away (>200 m unconfined) Dh(t) increases with time so a longer test is required – boundary and other effects can swamp aquifer response

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**Drawdown Instrumentation**

Dipmeters let cable hang to remove kinks rely on light or buzzer, have spare batteries Steel tapes read wetted part for water level (chalking helps) hard to use where high-frequency readings are needed Pressure gauges measure head above reference point – need drawdown estimates to set gauge depth Pressure tranducers/data loggers – hang in well and record at predetermined interval can be rented (cheaply) for tests remote sites (no personnel) and closest wells (frequency)

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ESS 454 Hydrogeology Module 4 Flow to Wells Preliminaries, Radial Flow and Well Function Non-dimensional Variables, Theis “Type” curve, and Cooper-Jacob.

ESS 454 Hydrogeology Module 4 Flow to Wells Preliminaries, Radial Flow and Well Function Non-dimensional Variables, Theis “Type” curve, and Cooper-Jacob.

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