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Partially Penetrating Wells

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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)
- 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

- Anisotropic aquifers

- Assumptions Violated:
- Well is fully penetrating
- Flow is horizontal

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 and Leaky (steady-state)- Huisman method:

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)

- 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 assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by:
- 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- 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

- Where:

- (Swm)partially – (Swm)fully = (Q/2∏D)(1-P/P)ln(εd/rew)

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.

- The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by:

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

- 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

- 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 assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by:
- 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)

- 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 S = (Q/4 ∏KD){W(u,β) + Fs((r/D),(b/D),(d/D),(a/D)}

- 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,r/D) + 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

- 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 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.

- The Walton curve-fitting method (Section 4.2.1) can be used if:

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)

- Where

- S = (Q/4∏KhD(b1/D))W(Ua,β,b1/D,b2/D)
- 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

- Where

- S = (Q/4∏KhD(b1/D))W(Ub,β,b1/D,b2/D)
- 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

- 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.

- The assumptions listed at the beginning of Chapter 3, with the exception of the first, third, sixth and seventh assumptions, which are replaced by

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

- 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

- Where

- S = (Q/4∏KhD)W{Ua,(or Ub),β,Sa/Sy,b/D,d/D,z/D)

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