### 10. Partially penetrating wells

```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
Assumptions Violated:

Well is fully penetrating

Flow is horizontal
Corrections

Different types of aquifers require different modifications

Confined and Leaky (steady-state)- Huisman method:


Confined (unsteady-state)- Hantush method:


Modification of Theis Method or Jacob Method
Leaky (unsteady-state)-Weeks method:


Observed drawdowns can be corrected for partial penetration
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)

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 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
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

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
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 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

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

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;

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.
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)



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
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

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.
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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