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Partially Penetrating Wells By: Lauren Cameron

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Introduction Partially penetrating wells: aquifer is so thick that a fully penetrating well is impractical Increase velocity close to well and the affect is inversely related to distance from well (unless the aquifer has obvious anisotropy) Anisotropic aquifers The affect is negligible at distances r > 2D sqrt(Kb/Kv) *standard methods cannot be used at r < 2D sqrt(Kb/Kv) unless allowances are made Assumptions Violated: Well is fully penetrating Flow is horizontal

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Corrections Different types of aquifers require different modifications Confined and Leaky (steady-state)- Huisman method: Observed drawdowns can be corrected for partial penetration Confined (unsteady-state)- Hantush method: Modification of Theis Method or Jacob Method Leaky (unsteady-state)-Weeks method: Based on Walton and Hantush curve-fitting methods for horizontal flow Unconfined (unsteady-state)- Streltsova curve-fitting or Neuman curve-fitting method Fit data to curves

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Confined aquifers (steady-state) Huisman's correction method I Equation used to correct steady-state drawdown in piezometer at r < 2D (Sm)partially - (Sm)fully = (Q/2∏KD) * (2D/∏d) ∑ (1/n) {sin(n∏b/D)-sin(n∏Zw/D)}cos(n∏Zw/D)K0(n∏r/D) Where (Sm)partially = observed steady-statedrawdown (Sm)fully = steady state drawdown that would have occuarred if the wellhad been fully penetrating Zw= distance from the bottom of the well screen to the underlying b= distance from the top of the well screen to the underlying aquiclude Z = distance from the middle of the piezometer screen to the underlying aquiclude D = length of the well screen

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Re: Confined aquifers (steady-state) Assumptions: The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by: The well does not penetrate the entire thickness of the aquifer. The following conditions are added: The flow to the well is in steady state; r > rew Remarks Cannot be applied in the immediate vicinity of well where Huisman’s correction method II must be used A few terms of series behind the ∑-sign will generally suffice

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Huisman’s Correction Method II Huisman’s correction method- applied in the immediate vicinity of well Expressed by: (Swm)partially – (Swm)fully = (Q/2∏D)(1-P/P)ln(εd/rew) Where: P = d/D = the penetration ratio d = length of the well screen e =l/d = amount of eccentricity I = distance between the middle of the well screen and the middle of the aquifer ε = function of P and e rew = effective radius of the pumped well

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Huisman’s Correction method II Assumptions: The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by: The well does not penetrate the entire thickness of the aquifer. The following conditions are added: The flow to the well is in a steady state; r = rew.

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Confined Aquifers (unsteady-state): Modified Hantush’s Method Hantush’s modification of Theis method For a relatively short period of pumping {t < {(2D-b-a)2(S,)}/20K, the drawdown in a piezometer at r from a partially penetrating well is S = (Q/8 ∏K(b-d)) E(u,(b/r),(d/r),(a/r)) Where E(u,(b/r),(d/r),(a/r)) = M(u,B1) – M(u,B2) + M(u,B3) – M(u,B4) U = (R^2 Ss/4Kt) Ss = S/D = specific storage of aquifer B1 = (b+a)/r (for sympols b,d, and a) B2 = (d+a)/r B3 = (b-a)/r B4 = (d-a)/r

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Re: Confined Aquifers (unsteady-state): Modified Hantush’s Method Assumptions: - The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by: The well does not penetrate the entire thickness of the aquifer. The following conditions are added: The flow to the well is in an unsteady state; The time of pumping is relatively short: t < {(2D-b-a)*(Ss)}/20K.

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Confined Aquifers (unsteady-state): Modified Jacob’s Method Hantush’s modification of the Jacob method can be used if the following assumptions and conditions are satisfied: The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by: The well does not penetrate the entire thickness of the aquifer. The following conditions are added: The flow to the well is in an unsteady state; The time of pumping is relatively long: t > D2(Ss)/2K.

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Leaky Aquifers (steady-state) The effect of partial penetration is, as a rule, independent of vertical replenishment; therefore, Huisman correction methods I and II can also be applied to leaky aquifers if assumptions are satisfied…

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Leaky Aquifers (unsteady-state): Weeks’s modification of Walton and Hantush curve-fitting method Pump times (t > DS/2K): Effects of partial penetration reach max value and then remain constant Drawdown equation: S = (Q/4 ∏KD){W(u,r/D) + Fs((r/D),(b/D),(d/D),(a/D)} OR S = (Q/4 ∏KD){W(u,β) + Fs((r/D),(b/D),(d/D),(a/D)} Where W(u,r/L) = Walton's well function for unsteady-state flow in fully penetrated leaky aquifers confined by incompressible aquitard(s) (Equation 4.6, Section 4.2.1) βW(u,) = Hantush's well function for unsteady-state flow in fully penetrated leaky aquifers confined by compressible aquitard(s) (Equation 4.15, Section 4.2.3) r,b,d,a = geometrical parameters given in Figure 10.2.

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Re:Leaky Aquifers (unsteady-state): Weeks’s modification of Walton and Hantush curve-fitting methods The value of f, is constant for a particular well/piezometer configuration and can be determined from Annex 8.1. With the value of Fs, known, a family of type curves of {W(u,r/L) + fs} or {W(u,p) + f,} versus I/u can be drawn for different values of r/L or p. These can then be matched with the data curve for t > DS/2K to obtain the hydraulic characteristics of the aquifer.

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Re:Leaky Aquifers (unsteady-state): Weeks’s modification of Walton and Hantush curve-fitting methods Assumptions: The Walton curve-fitting method (Section 4.2.1) can be used if: The assumptions and conditions in Section 4.2.1 are satisfied; A corrected family of type curves (W(u,r/L + fs} is used instead of W(u,r/L); Equation 10.12 is used instead of Equation 4.6. The Hantush curve-fitting method (Section 4.2.3) can be used if: T > DS/2K The assumptions and conditions in Section 4.2.3 are satisfied; A corrected family of type curves (W(u,p) + fs} is used instead of W(u,p); Equation 10.13 is used instead of Equation 4.15.

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Unconfined Anisotropic Aquifers (unsteady-state):Streltsova’s curve-fitting method Early-time drawdown S = (Q/4∏KhD(b1/D))W(Ua,β,b1/D,b2/D) Where Ua = (r^2Sa)/ (4KhDt) Sa = storativity of the aquifer Β = (r^2/D^2)(Kv/Kh) Late-time drawdown S = (Q/4∏KhD(b1/D))W(Ub,β,b1/D,b2/D) Where Ub = (r^2 * Sy)/(4KhDt) Sy = Specific yield Values of both functions are given in Annex 10.3 and Annex 10.4 for a selected range of parameter values, from these values a family of type A and b curves can be drawn

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Re: Unconfined Anisotropic Aquifers (unsteady-state):Streltsova’s curve-fitting method Assumptions: The Streltsova curve-fitting method can be used if the following assumptions and conditionsare satisfied: The assumptions listed at the beginning of Chapter 3, with the exception of the first, third, sixth and seventh assumptions, which are replaced by The aquifer is homogeneous, anisotropic, and of uniform thickness over the area influenced by the pumping test The well does not penetrate the entire thickness of the aquifer; The aquifer is unconfined and shows delayed watertable response. The following conditions are added: The flow to the well is in an unsteady state; SY/SA > 10.

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Unconfined Anisotropic Aquifers (unsteady-state):Neuman’s curve-fitting method Drawdown eqn: S = (Q/4∏KhD)W{Ua,(or Ub),β,Sa/Sy,b/D,d/D,z/D) Where Ua = (r^2Sa/4KhDt) Ub = (r^2Sy/4KhDt) Β = (r/D)^2 * (Kv/Kh) Eqn is expressed in terms of six dimensionless parameters, which makes it possible to present a sufficient number of type A and B curves to cover the range needed for field application More widely applicable Both limited by same assumptions and conditions

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