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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)

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Partially Penetrating Wells

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  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 sympolsb,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; • 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.

  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 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.

  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) • 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

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