Non equilibrium equation for unsteady radial flow


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Non equilibrium equation for unsteady radial flow

  1. 1. Aquifer and Non Equilibrium equation for unsteady radial flow Presented by: AbhiShek Gupta
  2. 2. Aquifer • A saturated, permeable, geologic unit that can transmit a significant amount of groundwater under an ordinary gradient.
  3. 3. Unsteady flow in confined aquifer • Assumptions  The aquifer is confined  The aquifer has infinite aerial extent  The aquifer is homogeneous, isotropic and of uniform thickness  The piezometric surface is horizontal prior to pumping  The aquifer is pumped at a constant discharge rate  The well penetrates the full thickness of the aquifer and thus receives water by horizontal flow
  4. 4. principle of continuity equation of flow, Inflow, Outflow,
  5. 5. • Change in volume, S (Storage coefficient) is the volume of water released per unit surface area per unit change in head normal to the surface. • In this equation, h is head, r is radial distance from the well, S is storage coefficient, T is transmissivity, and t is the time since the beginning of pumping.
  6. 6. The Theis Method (Curve Matching Method) • Theis assumed that the well is replaced by a mathematical sink of constant strength and imposing the boundary conditions h = h0 for t = 0, and h → h0 as r →∞ for t ≥0, the solution, Where W(u) is the well function and u is given by,
  7. 7. After taking log on both equations, therefore, a graph of log s against log t should be the same shape as a graph of log (W(u)) against log (1/u) constants
  8. 8. Theis curve Filed plot on logarithmic paper
  9. 9. Match of field data plot to Theis Type curve
  10. 10. Cooper-Jacob Method (Time-Drawdown) This is also based upon the Theis analysis Base 10 Straight line
  11. 11. If and are drawdown at time and then, Transmissivity is calculated by above equation. When s=0 (storativity) Semi –log plot
  12. 12. Cooper-Jacob Method (Distance-Drawdown) When observation of 3 or more wells are to be made. transmissivity, per one logarithmic cycle When s=0, Therefore,
  13. 13. Unsteady Radial Flow in an Unconfined Aquifer Equation of flow of water (Neuman’s equation), where, h is the saturated thickness of the aquifer (m) r is radial distance from the pumping well (m) z is elevation above the base of the aquifer (m) is specific storage (1/m) is radial hydraulic conductivity (m/day) is vertical hydraulic conductivity (m/day) T is time (day)
  14. 14. Three phases of drawdown First phase: • pressure drops • specific storage as a major contribution behaves as an artesian aquifer •flow is horizontal • time-drawdown follows Theis curve S - the elastic storativity.
  15. 15. Second phase Third phase • water table declines • specific yield as a major contribution • flow is both horizontal and vertical • time-drawdown is a function of Kv/Kh r, b • rate of drawdown decreases • flow is again horizontal • time-drawdown again follows Theis curve S - the specific yield.
  16. 16. Neuman’ assumptions • The aquifer is unconfined. • The vadose zone has no influence on the drawdown. • Water initially pumped comes from the instantaneous release of water from elastic storage. • Eventually water comes from storage due to gravity drainage of interconnected pores. • The drawdown is negligible compared with the saturated aquifer thickness. • The specific yield is at least 10 times the elastic storativity. • The aquifer may be- but does not have to be- anisotropic with the radial hydraulic conductivity different than the vertical hydraulic conductivity.
  17. 17. Neuman’s solution, Where, is the well function of water-table aquifer For early time, and For late time, and and Parameters can be found by Penman method
  18. 18. Penmen method to find parameters • Two sets of type curves are used and plotted on log-log paper (Theoretical curve vs 1/u). • Superpose the early (t − s) data on Type-A curve. • The data analysis is done by matching the observed data to the type curve. • From the match point of Type-A curve, determine the values for and the value of • Use the previous equations to determine T and S • The latest (s − t) data are then superposed on Type-B Curve for the Γ - values of previously matched Type-A curve, from the match point of Type-B curve, determine the values for • By using the previous equations, the T and S can be determined. .
  19. 19. Type curves for unconfined aquifers
  20. 20. Unsteady Radial Flow in a Leaky Aquifer Equation for Unsteady radial flow for leaky aquifer, Where, r is the radial distance from a pumping well (m) e is the rate of vertical leakage (m/day)
  21. 21. Hantush-Jacob Method Assumptions: • The aquifer is leaky and has an "apparent" infinite extent, • The aquifer and the confining layer are homogeneous, isotropic, and of uniform thickness, over the area influenced by pumping, • The potentiometric surface was horizontal prior to pumping, • The well is pumped at a constant rate, • The well is fully penetrating, • Water removed from storage is discharged instantaneously with decline in head, • The well diameter is small so that well storage is negligible, • Leakage through the aquitard layer is vertical.
  22. 22. Hantush and Jacob solution for leaky aquifer, Where, where, is the well function for leaky confined aquifer B is the leakage factor given as where, b' is thickness of the aquitard (m) K' is hydraulic conductivity of the aquitard (m/day)
  23. 23. Walton Graphical Solution Log-log plot
  24. 24. Procedure • Field data are plotted on drawdown vs. time on full logarithmic scale. • Field data should match one of the type curves for r/B (interpolation if between two lines) • From a match point, the following are known values • Substitute in Hantush-Jacob equation: (From match) r = distance between pumping well and observation well B = leakage factor
  25. 25. Thank you