Chapter 3: Confined Aquifers Matt Thomas WASR 8730 1/11/2018

Overview Review of confined aquifers Thiem’s method – (transient) Steady-state Theis’s method – Unsteady-state Jacob’s method – Unsteady-state

Confined aquifers Aquifer unit bounded above and below by low permeability confining units Water is generally pressurized, causing well water levels to often rise above the upper confining unit Investigate piezometric surface as opposed to the water table (for unconfined aquifers)

Confined aquifers Important quantities Distance from well to piezometer i (ri) Drawdown in piezometer i (si) Depth of aquifer (D) Discharge from pumping well (Q)

Assumptions Aquifer is confined Infinite areal extent Homogenous, isotropic, constant thickness Piezometric surface is horizontal prior to pumping Aquifer pumped at constant rate Well is fully penetrating and screened over entire thickness of aquifer Water removed from storage is discharged instantaneously with head decline (unsteady-state) Well diameter is small (unsteady-state)

Note In a confined aquifer, water must come from storage In theory, this means must always have unsteady-state flow In practice, if Δs is small can treat conditions as steady-state

Study area We’ll use data from a pump test in the Oude Korendijk polder in The Netherlands Aquifer unit is from 18 m to 25 m depth

Thiem’s method (1906) Well discharge may be expressed as More commonly expressed in terms of drawdown (sm1 – sm2) instead of head Assumptions: 1-6, steady-state flow

Thiem’s method (1906) – one piezometer If drawdown data is only available from one piezometer and the well itself, can use Where smw is drawdown in the well Not an ideal method; always drill multiple piezometers if possible

Thiem’s method (1906) – procedures Plot time-drawdown curves on semi-log plot (time on log scale) Check that late-times are approx. parallel for all curves This means steady-state assumption is met Plug drawdown data for a given late-time into Thiem equation in terms of drawdown and solve for KD (recall T = KD) Repeat for all combinations of piezometers and calculate the mean KD

Thiem’s method (1906) - procedures Plot distance-drawdown curve on semi-log plot (dist. on log scale) Calculate slope of line as drawdown per log-cycle r (Δsm) Plug into Thiem equation as , solve for KD

Thiem’s method (1906) - notes Can use Thiem’s method with either time-drawdown or distance-drawdown data Often times the cone of depression will continue to deepen over whole pumping period Thiem’s method still valid as long as rate is constant in all piezometers This is called transient steady-state flow Usually quicker, easier, and cheaper to achieve in a pump test

Theis’s method (1935) For unsteady-state flow Includes time and storativity Recall: Storativity is a function of specific storage and the area of the aquifer Specfic storage is in turn a function of the elasticities of the mineral grains and water, as well as the porosity of the material General idea: discharge can be calculated as the product of rate of head decline and storativity summed over the area of effect Assumptions: 1-8, unsteady-state flow

Theis’s method (1935) Mathematically,

Theis’s method (1935) – well function W(u) is called the “well function of u” W(u) is called “dimensionless drawdown”, u is called “dimensionless time” Can generally calculate u as in the previous slide, then look up the value of W(u) for a given u

Theis’s method (1935) – curve fitting If have s, r, and t values then can determine S and KD Problem: have two unknowns and an exponential integral This prevents a traditional explicit solution Must use the curve-fitting method to solve

Theis’s method (1935) – curve fitting Rewrite as Rewrite as Note that the first term in each equation is constant Plot s vs. t/r2 and W(u) vs. 1/u on same log-log plot The two curves will be of the same shape Offset horizontally by Q/4πKD and vertically by 4KD/S See pg. 63 for more info

Theis’s method (1935) - notes Don’t consider early-time data Assumptions involved in this and many other models requiring curve-fitting are not valid in early-times If plot of data produces a flat curve, the solution is indeterminant

Jacob’s method (1946) For unsteady-state flow Assumptions: 1-8, unsteady-state, u < 0.1 Theis: As t increases and r decreases, u decreases When t sufficiently large enough and r sufficiently small enough, all terms after ln(u) become negligible For u < 0.1, can approximate Theis equation as

Jacob’s method (1946) From here, can go three different directions r = constant t = constant Use t/r2 All follow same general pattern: Plot X-drawdown data, where X is the non-constant variable and on log scale Fit a straight line to the data Find the x-intercept (X0) and slope (Δs per log-cycle X) Plug into relevant equations and solve for KD and S Be sure to verify that u < 0.1 with the calculated values!

Jacob’s method (1946) Data Procedure in text Equations Notes r = constant 3.4 Use piezometers at moderate distances Should be close agreement among KD and S values t = constant 3.5 Need data from at least three piezometers Repeat for several t Should have agreement among calculated values s vs. t/r2 3.6

Summary Three methods shown generally agree with each other

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