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Partially Penetrating Wells. By: Lauren Cameron. 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)

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partially penetrating wells

Partially Penetrating Wells

By: Lauren Cameron

introduction
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
corrections
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
confined aquifers steady state
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
re confined aquifers steady state
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
huisman s correction method ii
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
huisman s correction method ii1
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.
confined aquifers unsteady state modified hantush s method
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 sympolsb,d, and a)
      • B2 = (d+a)/r
      • B3 = (b-a)/r
      • B4 = (d-a)/r
re confined aquifers unsteady state modified hantush s method
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.
confined aquifers unsteady state modified jacob s method
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.
leaky aquifers steady state
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…
leaky aquifers unsteady state weeks s modification of walton and hantush curve fitting method
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.
re leaky aquifers unsteady state weeks s modification of walton and hantush curve fitting methods
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.
re leaky aquifers unsteady state weeks s modification of walton and hantush curve fitting methods1
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;
      • Acorrected 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;
      • Acorrected 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.
unconfined anisotropic aquifers unsteady state streltsova s curve fitting method
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
re unconfined anisotropic aquifers unsteady state streltsova s curve fitting method
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 conditionsaresatisfied:
    • 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.
unconfined anisotropic aquifers unsteady state neuman s curve fitting method
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)
          • Eqnis 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