Applied Hydrogeology VI Прикладная Гидрогеология Yoram Eckstein, Ph.D. Fulbright Professor 2013/2014 Tomsk Polytechnic University Tomsk, Russian Federation Spring Semester 2014
Useful links • http://www.onlineconversion.com/ • http://www.digitaldutch.com/unitconverter/ • http://water.usgs.gov/ogw/basics.html • http://water.usgs.gov/ogw/pubs.html • http://ga.water.usgs.gov/edu/earthgwaquifer.html • http://water.usgs.gov/ogw/techniques.html • http://water.usgs.gov/ogw/CRT/
Applied Hydrogeology VI. Groundwater Flow to Wells
Type of Water Wells • Production Wells • Injection Wells • Remediation Wells • Pumping wells • Injection wells
Ground-water flow to wells • Extract water • Remove contaminated water • Lower water table for constructions • Relieve pressures under dams • Injections – recharges • Control salt-water intrusion
Our purpose of well studies • Compute the decline in the water level, or drawdown, around a pumping well whose hydraulic properties are known. • Determine the hydraulic properties of an aquifer by performing an aquifer test in which a well is pumped at a constant rate and either the stabilized drawdown or the change in drawdown over time is measured.
Wells in Confined and Unconfined Aquifers • In unconfined aquifers, pumping will result in drawdown of the water table • In confined aquifers, pumping will cause drawdown of the potentiometric surface • All pores in the confined aquifer will still remain saturated
Wells in Confined and Unconfined Aquifers • All pores in the confined aquifer remain fully saturated
Wells in Confined and Unconfined Aquifers • The pores within the cone of depression in an unconfined aquifer are dewaterd
Wells in Confined and Unconfined Aquifers http://dli.taftcollege.edu/streams/geography/Animations/ConeDepression.html
Wells in Confined and Unconfined Aquifers If qr is the radial groundwater flux at a distance r from the well, it follows from the mass balance equation that the total radial flow towards the well should be equal to the pumping rate: Q = −2πrbqr
Wells in Confined and Unconfined Aquifers Using Darcy’s law to express the groundwater flux becomes Q = −2πrbqr Q = 2rb b·K = T
Wells in Confined and Unconfined Aquifers Q = 2rbhence: and where c is an integration constant and therefore:
Wells in Confined and Unconfined Aquifers or Thiem equation for steady-state flow of water into a well in confined aquifer
Wells in Confined and Unconfined Aquifers In the case of an unconfined aquifer thickness (Ho) equals the elevation of the water table (ho) above the aquifer bottom. However, within the cone of depression H < Ho
Wells in Confined and Unconfined Aquifers The mass balance equation is: using Darcy’s law to express the groundwater flux and assuming that s<<Ho this becomes:
Wells in Confined and Unconfined Aquifers • further integration yields:
Wells in Confined and Unconfined Aquifers or: which is • Dupuitequation for steady-state flow of water into a well in unconfined aquifer
Steady Flow in an Unconfined Aquifer Integrate both sides of the equation Dupuit Equation L is the flow length Steady flow through an unconfined aquifer resting on a horizontal impervious surface
Steady Flow in an Unconfined Aquifer • Flow lines are assumed to be horizontal and parallel to impermeable layer • The hydraulic gradient of flow is equal to the slope of water. (slope very small)
Wells in Our Considerations • Fully penetrate the aquifer • The flow is radial symmetric • The aquifer is homogeneous and isotropic
Basic Assumptions • The aquifer is bounded on the bottom by a confining layer. • All geologic formations are horizontal and have infinite horizontal extent. • The potentiometric surface of the aquifer is horizontal prior to the start of the pumping.
Basic Assumptions (cont.) • The potentiometric surface of the aquifer is not recharging with time prior to the start of the pumping. • All changes in the position of the potentiometric surface are due to the effect of the pumping alone.
Basic Assumptions (cont.) • The aquifer is homogeneous and isotropic. • All flow is radial toward the well. • Ground water flow is horizontal. • Darcy’s law is valid.
Basic Assumptions (cont.) • Ground water has a constant density and viscosity. • The pumping well and the observation wells are fully penetrating. • The pumping well has an infinitesimal diameter and is 100% efficient.
A completely confined aquifer • Addition assumptions: • The aquifer is confined top and bottom. • The is no source of recharge to the aquifer. • The aquifer is compressible. • Water is released instantaneously. • Constant pumping rate of the well.
Theis’ Solution to Transient Flow Equation Two-dimensional flow with no vertical components: thus
Theis’ Solution to Transient Flow Equation Two-dimensional flow with no vertical components:
Theis’ Solution to Transient Flow Equation h0 = initial hydraulic head (L; m or ft) h = hydraulic head (L; m or ft) s = h0– h = drawdown (L; m or ft) Q = constant pumping rate (L3/T; m3/d or ft3/d) W(u) = well function. T = transmissivity (L2/T; m2/d or ft2/d)
Theis’ Solution to Transient Flow Equation S = storativity (dimensionless) t = time since pumping began (T; d) r = radial distance from the pumping well (L; m or ft)
Purpose of Pumping Well Tests • Determine the hydraulic properties of an aquifer by performing an aquifer test in which a well is pumped at a constant rate and either the stabilized drawdown or the change in drawdown over time is measured. • Determine the hydraulic properties of a water well by performing variable-rate production test.
Static Water Level [SWL] (ho) is the equilibrium water level before pumping commences Pumping Water Level [PWL] (h) is the water level during pumping Drawdown (s = ho - h) is the difference between SWL and PWL Well Yield (Q) is the volume of water pumped per unit time Specific Capacity (Q/s) is the yield per unit drawdown Q s ho h Pumping Well Terminology
Cone of Depression High Kh aquifer • A zone of low pressure is created centred on the pumping well • Drawdown is a maximum at the well and reduces radially • Head gradient decreases away from the well and the pattern resembles an inverted cone called the cone of depression • The cone expands over time until the inflows (from various boundaries) match the well extraction • The shape of the equilibrium cone is controlled by hydraulic conductivity Kh Kv Low Kh aquifer
Known values: Q – pumping rate r - distance from the pumping well to the observation well (can be rw) rw – pumping well radius
Cooper-Jacob simplification • Cooper and Jacob (1946) pointed out that the series expansion of the exponential integral or W(u) is: where γ is Euler’s constant = 0.5772 • For u<< 1 , say u < 0.05 the series can be truncated: W(u) – ln(eγ)- ln(u) = - ln(eγu) = -ln(1.78u)
Cooper-Jacob simplification • For u<< 1 , say u < 0.05 the series can be truncated: W(u) -ln(1.78u) Thus: • The Cooper-Jacob simplification expresses drawdown (s) as a linear function of ln(t) or log(t).
Cooper-Jacob Plot: Log(t) vs. Drawdown (s) to = 84sec. Ds =0.40 m
Cooper-Jacob Analysis • Fit straight-line to data (excluding early and late times if necessary): – at early times the Cooper-Jacob approximation may not be valid – at late times boundaries may significantly influence drawdown • Determine intercept on the time axis for s=0 • Determine drawdown increment (Ds) for one log-cycle
Cooper-Jacob Analysis (cont.) • For straight-line fit, and S = • For the example, Q = 32 L/s or 0.032 m3/s; r = 120 m; to = 84 s and Ds = 0.40 m 7.
Recharge Effect : Recharge > Well Yield Recharge causes the slope of the log(time) vs drawdown curve to flatten as the recharge in the zone of influence of the well matches the discharge. The gradient and intercept can still be used to estimate the aquifer characteristics (T,S).
Recharge Effect : Recharge < Well Yield If the recharge is insufficient to match the discharge, the log(time) vs drawdown curve flattens but does not become horizontal and drawdown continues to increase at a reduced rate. T and S can be estimated from the first leg of the curve.