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:

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)

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)

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

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