1. Partially Penetrating Wells By: Lauren Cameron

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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.

8. 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

9. 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.

10. 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.

11. 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…

12. 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.

13. 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.

14. 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.

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

16. 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.

17. 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