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Unit 04 : Advanced Hydrogeology Hydraulic Testing.

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1 Unit 04 : Advanced Hydrogeology Hydraulic Testing

2 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

3 Pumping Well Terminology Static Water Level [SWL] (h o ) is the equilibrium water level before pumping commences Pumping Water Level [PWL] (h) is the water level during pumping Drawdown (s = h o - 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 hoho h s Q

4 Cone of Depression 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 Low K h aquifer High K h aquifer K h K v

5 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 = S y + S s b) 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

6 Steady Radial Confined Flow Assumptions Isotropic, homogeneous, infinite aquifer, 2-D radial flow Initial Conditions h(r,0) = h o for all r Boundary Conditions h(R,t) = h o for all t Darcys LawQ = -2 rbK h/ r Rearranging h = - Q r 2 Kb r Integrating h = - Q ln(r) + c 2 Kb BC specifies h = h o at r = R Using BC h o = - Q ln(R) + c 2 Kb Eliminating constant (c) gives s = h o – h = Q ln(r/R) 2 Kb This is the Thiem Equation hoho h s Q r b

7 Steady Unconfined Radial Flow Assumptions Isotropic, homogeneous, infinite aquifer, 2-D radial flow Initial Conditions h(r,0) = h o for all r Boundary Conditions h(R,t) = h o for all t hoho h s Q r Darcys LawQ = -2 rhK h/ r Rearranging h h = - Q r 2 K r Integrating h 2 = - Q ln(r) + c 2 2 K BC specifies h = h o at r = R Using BC h o 2 = - Q ln(R) + c K Eliminating constant (c) gives h o 2 – h 2 = Q ln(r/R) K This is the Thiem Equation

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

9 Thiem Equation Applications The equation for unconfined flow can be rearranged to give: K = Q ln(r/R) (h o 2 - h 2 ) Similarly with the confined equation: K = Q ln(r/R) 2 b (h o – 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(r 2 /r 1 ) and K = Q ln(r 2 /r 1 ) (h 2 2 – h 1 2 ) 2 b (h 2 – h 1 ) 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

10 Specific Capacity For a confined well producing at a steady rate (Q) the specific capacity is given by: Q = 2 Kb s w ln(r w /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 = K (h o + h w ) s w ln(r w /R) Writing h w = h o - s w gives: Q = - K (s w + 2h o ) s w ln(r w /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 = 2 Kh o s w ln(r w /R)

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

12 Theis PDE to ODE Let = S/T (to simplify notation where is called the hydraulic diffusivity) PDE 1 (r h ) = h r r r t u = r 2 S = r 2 4Tt 4t So u = r = 2u r 2t r And u = - r 2 = -u t 4t 2 t Rewriting the PDE in terms of u: 1 d (r u dh ) u = u dh r du r du r t du Rewriting partial derivatives in terms of u: 1 d (r 2u dh ) 2u = - u dh r du r du r t du Rearranging and cancelling: d (u dh ) = - r 2 dh = -u dh du du 4t 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

13 Theis Integration The resulting ODE is: dh = -(1 + 1) du h u Integrating: ln(h) = -u – ln(u) + c Simplifying: ln(hu) = c – u Inverting: hu = exp(c).exp(-u) To eliminate exp(c), use Darcys Law: lim r h = -Q = lim 2hu r 0 r 2 Kb u 0 Remember r h = rh u = h r 2 = 2hu r r 2t lim hu = -Q = exp(c) u 0 4 Kb Simplifying: h = -Q exp(-u) 4 T u Recall h = dh/du dh = -Q exp(-u).du 4 T u Integrating: h = -Q exp(-u) du + C 4 T u u Finally, using h(,t) = h o to eliminate C: h o - h = Q exp(-u) du 4 T u u The integral is called the exponential integral but is often written as the Theis well function W(u) s = h o - h = Q W(u) 4 T

14 Theis Plot : 1/u vs W(u)

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

16 [1,1] Type Curve s=0.17m t=51s

17 Theis Analysis 1.Overlay type-curve on data-curve keeping axes parallel 2.Select a point on the type-curve (any will do but [1,1] is simplest) 3.Read off the corresponding co-ordinates on the data-curve [t d,s d ] 4.For [1,1] on the type curve corresponding to [t d,s d ], T = Q/4 s d and S = 4Tt d /r 2 = Qt d / r 2 s d 5.For the example, Q = 32 L/s or m 3 /s; r = 120 m; t d = 51 s and s d = 0.17 m 6.T = (0.032)/(12.56 x 0.17) = m 2 /s = 1300 m 2 /d 7.S = (0.032 x 51)/(3.14 x 120 x 120 x 0.17) = 2.1 x 10 -4

18 Cooper-Jacob Cooper and Jacob (1946) pointed out that the series expansion of the exponential integral or W(u) is: W(u) = – - ln(u) + u - u 2 + u 3 - u 4 +..… 1.1! 2.2! 3.3! 4.4! where is Eulers constant (0.5772) For u<< 1, say u < 0.05 the series can be truncated: W(u) – ln(e - ln(u) = - ln(e u) = -ln(1.78u) Thus: s = h o - h = - Q ln(1.78u) = - Q ln(1.78r 2 S) = Q ln( 4Tt ) 4 T 4 T 4Tt 4 T 1.78r 2 S s = h o - h = Q ln( 2.25Tt ) = 2.3 Q log( 2.25Tt ) 4 T r 2 S 4 T r 2 S The Cooper-Jacob simplification expresses drawdown (s) as a linear function of ln(t) or log(t).

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

20 t o = 84s s =0.39 m

21 Cooper-Jacob Analysis 1.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 2.Determine intercept on the time axis for s=0 3.Determine drawdown increment ( s) for one log-cycle 4.For straight-line fit, T = 2.3Q/4 s and S = 2.25Tt o /r 2 = 2.3Qt o /1.78 r 2 s 5.For the example, Q = 32 L/s or m 3 /s; r = 120 m; t o = 84 s and s = 0.39 m 6.T = (2.3 x 0.032)/(12.56 x 0.39) = m 2 /s = 1300 m 2 /d 7.S = (2.3 x x 84)/(1.78 x 3.14 x 120 x 120 x 0.39) = 1.9 x 10 -4

22 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

23 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).

24 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.

25 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.

26 Recharge Effect : Leakage Rate 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. High Leakage Low Leakage

27 Hantush Type Curves Theis Curve Data are fitted in a manner similar to the Theis curve. The parameter r/B = r( {K v / b} / {K h b} ) ½ increases with the amount of leakage.

28 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.

29 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.

30 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

31 Schafer (1978) suggests that an estimate of the critical time to exhaust casing storage can be made more easily: t c = 3.75 (d c 2 – d p 2 ) / (Q/s) = 15 V a /Q where t c is the critical time (d); d c is the inside casing diameter (m); d p is the outside diameter of the rising main (m); Q/s is the specific capacity of the well (m 3 /d/m) V a 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 t c in wells in low-K HSUs Casing Storage s dcdc dpdp Q

32 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) 4 T r 2 S 4 T S 4 T So the log(distance)-drawdown curve can be used to estimate aquifer characteristics by measuring s for one log-cycle and the r o intercept on the distance- axis. T = 4.6Q and S = 2.25Tt 4 s r o 2

33 Distance-Drawdown Graph

34 Aquifer Characteristics For the example: t = 0.35 days and Q = 1100 m 3 /d T = x 1100 / 3.8 = 106 m 2 /d S = 2.25 x 106 x 0.35 / (126 x 126) = 5.3 x 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.

35 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.

36 Theoretical Pumped Well Drawdown

37 Unconfined Well Correction The adjusted drawdown for an unconfined well is given by: s c = (1 - s a ) s a 2b where b is the initial saturated thickness; s a is the measured drawdown; and s c is the corrected drawdown For example, if b = 20 m; s a = 6 m; then the corrected drawdown s c = 0.85s a = 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 s c / s a.

38 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

39 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.

40 Determining r o Unconfined Well Confined Well

41 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 (bS s ) –intermediate times respond as a leaky aquifer with vertical flow in the vicinity of the pumped well with storage release controlled by the aquifer K h /K v ratio –late time response follows Theis equation with gravity drainage providing storage corresponding to the specific yield (S y )

42 Delayed-Yield Response Neuman (1975) defines a well function W(u A,u B, ) where each parameter corresponds to a different time phase: early-time response is controlled by u A = r 2 S/4Tt intermediate-times are controlled by = r 2 K v /K h b 2 late-time response is controlled by u B = r 2 S y /4Tt The Hantush leakage parameter (r/B) is closely related to (r/B) 2 = r 2 K v /K h bb where the Kv and b parameters refer to the leaky bed. If these leaky-bed parameters become aquifer parameters (r/B) 2 = Unconfined response is complex with theory developed by Boulton, Dagan, Steltsova, Rushton and Neuman. uAuA uBuB

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

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

45 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).

46 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( L) ( r )} b 2b 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 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.

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

48 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(0.5 /2) (1/200)} = 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(0.5 /2) (1/40)} = 0.89 This is clearly a much more efficient well completion.

49 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.

50 Recovery Curve 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. Drawdown 10 m Recovery 10 m Pumping Stopped

51 Superposition The drawdown (s) for a well pumping at a constant rate (Q) for a period (t) is given by: s = h o - h = Q W(u) where u = r 2 S 4 T 4Tt 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 (s r ) for a well recharged at a constant rate (-Q) for a period (t = t - t r ) starting at time t r is given by: s r = - Q W(u) where u = r 2 S 4 T 4Tt The total drawdown for t > t r is: s = s + s r = Q (W(u) - W(u)) 4 T

52 Residual Drawdown and Recovery The total drawdown for t > t r is: s = s + s r = Q (W(u) - W(u)) 4 T The Cooper-Jacob approximation can be applied giving: s = s + s r = Q (ln(2.25Tt) - ln(2.25Tt)) 4 T r 2 S r 2 S Simplification gives the residual drawdown equation: s = s + s r = Q ln(t) 4 T t The equation predicting the recovery is: s r = - Q ln(2.25Tt) 4 T r 2 S For t > t r, the recovery s r is the difference between the observed drawdown s and the extrapolated pumping drawdown (s).

53 Time-Recovery Graph 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) s r = 4.6 m t o = 0.12 hrs

54 Time-Recovery Analysis For a constant rate of pumping (Q), the recovery any time (t) after pumping stops: T = Q = - Q = Q 4 (s - s) - 4 s r 4 s r For the example, s r = 4.6 m and Q = 1100 m 3 /d so: T = 1100 / (12.56 x 4.6) = 19 m 2 /d The storage coefficient can be estimated for an observation well (r = 30 m) using: S = 4Tt o r 2 For the example, t o = 0.12 and Q = 1100 m 3 /d so: S = 4 x 19 x 0.12 / (24 x 30 x 30) = 4.3 x It is necessary to use an observation well for this calculation because well bore storage effects render any calculation based on r w potentially subject to huge errors.

55 Time-Residual Drawdown Graph 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. s = 5.2 m

56 Time-Residual Drawdown Analysis For a constant rate of pumping (Q), the recovery any time (t) after pumping stops: T = Q 4 s For the example, s r = 5.2 m and Q = 1100 m 3 /d so: T = 1100 / (12.56 x 5.2) = 17 m 2 /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 - t r where t r is the elapsed pumping time before recovery starts.

57 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 << 1indicates incomplete recovery of initial head - small aquifer volume

58 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. Packer Perforated Section Gauge Valve Drill Stem

59 DST Analysis Recall that the final form of the recovery equation is: h o - h = s = 2.3Q log(t) 4 T t For a DST, the pressure (rather than head) is measured p o - p = 2.3Q log(t) 4 kb t Remembering that p = h,T = Kb and K = k / The Horner-plot has an intercept p o 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 = 2.3Q 4 b p t / t p (kPa) popo p

60 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 Displaced Head Initial Head Displacer

61 Slug Analysis r a is access tube internal radius r w is perforated section external radius L is length of perforated section h o is initial head, t = t o h(t) is head after recovery time t Ais the tube or casing csa = r a 2 Fis a shape factor = 2 L / ln(L/r w ) 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) = r a 2 ln(L) ln(h) F(t - t o ) h o 2L(t - t o ) r w h o Tube or Casing 2r a 2r w L

62 Hvorslev Analysis K = r a 2 ln(L) ln(h) 2L(t - t o ) r w h o Plot time against log (h/h o ) Measure basic time lag T o when ln(h 0 /h) = 1 K = r a 2 ln (L) 2LT o r w Time lag T o occurs when: h = e -1 h o = 0.37h o If T o = 1000 secs for a 50 mm dia. x 1 m length Casagrande piezometer with 38 mm dia access tube K = 2 x m/s Tube or Casing 2r a 2r w L Time, t - t o hhohho ToTo

63 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)

64 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 r r r

65 General Solution The general solution for adding image wells to a real pumping well can be written: s = s p s i = Q [W(u) W(u i )] 4 T where u p = r p 2 S and u i = r i 2 S 4Tt 4Tt and r p,r i are the distances from the pumping and image wells respectively. r r rprp riri For a barrier boundary, for all points on the boundary r p = r i and the drawdown is doubled. For a recharge boundary, for all points on the boundary r p = r i and the drawdown is zero.

66 Specific Solutions For the recharge boundary case: s = Q [W(u) - W( u)] 4 T where = (r i /r p ) 2 and 0< <1 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(r i 2 u)] = Q [W(u) W( u)] 4 T r p 2 4 T For the barrier boundary case: s = Q [W(u) + W( u)] 4 T where = (r i /r p ) 2 and 0< <1

67 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. r2r2 r1r1 r2r2 r1r1

68 Parallel Boundaries A parallel recharge boundary and a barrier boundary (or any pattern with parallel boundaries) requires an infinite array of image wells. r1r1 r2r2

69 Boundary Location For an observation well at distance r 1, measure off the same drawdown (s), before and after the dog leg on a log(time) vs. drawdown plot. Find the times t 1 and t 2. Assuming that the dog leg is created by an image well at distance r 2, if the drawdowns are identical then W(u 1 ) = W(u 2 ) so u 1 = u 2. Thus: r 1 2 S/4Tt 1 = r 2 2 S/4Tt 2 So r 1 2 t 2 = r 2 2 t 1 and r 2 = r 1 (t 2 / t 1 ) ½ The distance r 2 the radial distance from the observation point to the boundary. Repeating for additional observation wells may help locate the boundary. t1t1 t2t2 s s

70 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: s w = CQ 2

71 Well Efficiency The total drawdown at a pumping well is given by: s t = s + s w = Q W(u) + CQ 2 = BQ + CQ 2 4 T The ratio of the aquifer loss and total drawdown (s/s t ) is known as the well efficiency. s = W(u) = B. s t W(u) + 4 TCQ 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 = 1 = 1. s t TCQ / [ln (2.25Tt /S) - 2 ln(r w )] 1 + CQ/B(r w ) Written in this form it is clear that well efficiency reduces with pumping rate (Q) and increases with well radius (r w ), where B is inversely related to well radius. The specific capacity is given by: Q = 1. s t B + CQ

72 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 Q 1 < Q 2 < Q 3 < Q 4 < Q 5 takes place for 1-2 hours ( t) and for 5 to 8 steps. The entire test usually takes place in one day. Equal pumping times ( t) simplifies the analysis. At the end of each step, the pumping rate (Q) and drawdown (s) is recorded. Time, t Drawdown, s s1s1 s2s2 s3s3 s4s4 s5s5

73 Step-Drawdown Test Analysis Step-drawdown tests are analysed by plotting the reciprocal of specific capacity (s/Q) against the pumping rate (Q). Q (L/s) s/Q (m/m 3 /d) The intercept of the graph at Q=0 is B = W(u)/4 T 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 m 3 /d and s = 33.3 m the B = m/m 3 /d If C = 4 x 10 -5, then CQ 2 = 18.2 m The well efficiency is 33.3/( ) = 65% B C

74 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 m 3 /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.

75 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

76 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

77 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

78 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

79 Measuring Drawdown Pumped 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) h(t) increases with time so a longer test is required – boundary and other effects can swamp aquifer response

80 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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