Darcy Lab: Describe Apparatus Q = K A ∂h/∂x cm 3 /sec = cm/sec cm 2 cm/cm

after Domenico & Schwartz (1990) Flow toward Pumping Well, next to river = line source = constant head boundary Plan view River Channel Line Source

Pathlines ≠ Flowlines for transient flow Flowlines | to Equipotential surface if K is isotropic Can be conceptualized in 3D Flow Nets: Set of intersecting Equipotential lines and Flowlines Flowlines  Streamlines  Instantaneous flow directions Pathlines  Actual particle path

Flow Net Rules: Flowlines are perpendicular to equipotential lines (isotropic case) Spacing between equipotential lines L: If spacing between lines is constant, then K is constant In general K 1 m 1 /L 1 = K 2 m 2 /L 2 where m = x-sect thickness of aquifer; L = distance between equipotential lines For layer of const thickness, K 1 /L 1 ~ K 2 /L 2 No Flow Boundaries Equipotential lines meet No Flow boundaries at right angles Flowlines are tangent to such boundaries (// flow) Constant Head Boundaries Equipotential lines are parallel to constant head boundaries Flow is perpendicular to constant head boundary

Impermeble Boundary Constant Head Boundary Water Table Boundary after Freeze & Cherry FLOW NETS

MK Hubbert

MK Hubbert (1940)

MK Hubbert (1940) Consider piezometers emplaced near hilltop & near valley

Fetter, after Hubbert (1940)

Cedar Bog, OH

Piezometer Cedar Bog, Ohio

Topographic Highs tend to be Recharge Zones h decreases with depth Water tends to move downward => recharge zone Topographic Lows tend to be Discharge Zones h increases with depth Water will tend to move upward => discharge zone It is possible to have flowing well in such areas, if case the well to depth where h > sfc. Hinge Line: Separates recharge (downward flow) & discharge areas (upward flow). Can separate zones of soil moisture deficiency & surplus (e.g., waterlogging). Topographic Divides constitute Drainage Basin Divides for Surface water e.g., continental divide Topographic Divides may or may not be GW Divides

Bluegrass Spring Criss

MK Hubbert (1940)

Equipotential Lines Lines of constant head. Contours on potentiometric surface or on water table  map => Equipotential Surface in 3D Potentiometric Surface: ("Piezometric sfc") Map of the hydraulic head; Contours are equipotential lines Imaginary surface representing the level to which water would rise in a nonpumping well cased to an aquifer, representing vertical projection of equipotential surface to land sfc. Vertical planes assumed; no vertical flow: 2D representation of a 3D phenomenon Concept rigorously valid only for horizontal flow w/i horizontal aquifer Measure w/ Piezometers  small dia non-pumping well with short screen- can measure hydraulic head at a point (Fetter, p. 134)

How do we know basic flownet picture is correct?

Mathematical solutions (Toth, 1962, 1963) Numerical Simulations Data

Basin Geometry: Sinusoidal water table on a regional topo slope Toth (1962, 1963) h(x, z 0 ) = z 0 + Bx/L + b sin (2  x/ ) constant + regional slope + local relief B

Basin Geometry: Sinusoidal water table on a regional topo slope Toth (1962, 1963) h(x, z 0 ) = z 0 + Bx/L + b sin (2  x/ ) constant + regional slope + local relief Solve Laplace’s equation Simulate nested set of flow systems e.g., D&S How do we get q?

Regional flow pattern in an area of sloping topography and water table. Fetter, after Toth (1962) JGR 67, No Flow Discharge Recharge

after Toth 1963Australian Government Local Flow Systems Intermediate Flow System Regional Flow System

Conclusions General slope causes regional GW flow system, If too small, get only local systems If the regional slope and relief are both significant, get regional, intermediate, and local GW flow systems. Local relief causes local systems. The greater the amplitude of the relief, the greater the proportion of the water in the local system If the regional slope and relief are both negligible, get flat water table often with waterlogged areas mostly discharged by ET For a given water table, the deeper the basin, the more important the regional flow High relief & deep basins promote deep circulation into hi T zones

End 24 Begin 25

Hubbert (1940) MK Hubbert Equipotential Line Flow Line FLOW NETS AIP

How do we know basic flownet picture is correct? Data Mathematical solutions (Toth, 1962, 1963) Numerical Simulations

Piezometer Cedar Bog, Ohio

Regional flow pattern in an area of sloping topography and water table. Fetter, after Toth (1962) JGR 67, No Flow Discharge Recharge Pierre Simon Laplace

Numerical Simulations Basically reproduce Toth’s patterns High K layers act as “pirating agents Refraction of flow lines tends to align flow parallel to hi K layer, and perpendicular to low K layers

after Freeze and Witherspoon Effect of Topography on Regional Groundwater Flow Isotropic Systems Regular slope Sinusoidal slope

Isotropic Aquifer Anisotropic Aquifer K x : K z = 10:1 after Freeze *& Witherspoon 1967

Layered Aquifers

after Freeze *& Witherspoon 1967 Confined Aquifers Sloping Confining Layer Horizontal Confining Layer

Conclusions General slope causes regional GW flow system, If too small, get only local systems Local relief causes local systems. The greater the amplitude of the relief, the greater the proportion of the water in the local system If the regional slope and relief are both negligible, get flat water table often with waterlogged areas mostly discharged by ET If the regional slope and relief are both significant, get regional, intermediate, and local GW flow systems. For a given water table, the deeper the basin, the more important the regional flow High relief & deep basins promote deep circulation into hi T zones

Flow in a Horizontal Layers Case 1: Steady Flow in a Horizontal Confined Aquifer Flow/ unit width: Darcy Velocity q: Typically have equally-spaced equipotential lines

Case 2: Steady Flow in a Horizontal, Unconfined Aquifer Flow/ unit width: m 2 /s Dupuit (1863) Assumptions: Grad h = slope of the water table Equipotential lines (planes) are vertical Streamlines are horizontal Q’dx = -K h dh Dupuit Equation Fetter p. 164

Impervious Base h cf. Fetter p. 164 Steady flow No sources or sinks

cf. Fetter p. 167 F&C 189 Q’ = -K h dh/dx dQ’/dx = 0 continuity equation So: More generally, for an Unconfined Aquifer: Steady flow with source term: Poisson Eq in h 2 where w = recharge cm/sec Steady flow: No sources or sinks Laplace’s equation in h 2 Better Approach for one dimensional flow

Steady unconfined flow: with a source term Poisson Eq in h 2 1-D Solution: Boundary x= 0 h= h 1 x= L h= h 2 cf. Fetter p. 167 F&C 189

cf. Fetter p. 167 F&C 189 w Unconfined flow with recharge w = m/s K = x=0 h 1 = x=1000m h 2 = 10m

Finally, for unsteady unconfined flow: Boussinesq Eq. Sy is specific yield Fetter p For small drawdown compared to saturated thickness b: Linearized Boussinesq Eq. (Bear p ) Laplace’s Equation Steady flow Poisson’s Equation Steady Flow with Source or Sink Diffusion Equation

End Part II

Pierre Simon Laplace Dibner Lib.

MK Hubbert

wikimedia.org Leonhard Euler

Charles V. Theis 19-19

for unconfined flow

After Toth 1983

after Johnson 1975

Radial flow Initial Condition & Boundary conditions: Transient flow, Confined Aquifer, No recharge Constant pumping rate Q

Radial flow Initial Condition & Boundary conditions: and where Solution: “Theis equation” or “Non-equilibrium Eq.” where

Approximation for t >> 0 D&S p. 151

USGS Circ 1186 Pumping of Confined Aquifer Not GW “level” Potentiometric sfc!

USGS Circ 1186 Pumping of Unconfined Aquifer

USGS Circ 1186 Santa Cruz River Martinez Hill, South of Tucson AZ 1989 >100’ GW drop 1942 Cottonwoods, Mesquite

for unconfined flow

USGS Circ 1186 rate Q 1 (note divide) Initial Condition rate Q 2 >Q 1

Domenico & Schwartz (1990) Flow beneath Dam Vertical x-section Flow toward Pumping Well, next to river = line source = constant head boundary Plan view River Channel

after Toth

after Toth 1963Australian Government

PROBLEMS OF GROUNDWATER USE Saltwater Intrusion Mostly a problem in coastal areas: GA NY FL Los Angeles Abandonment of freshwater wells; e.g., Union Beach, NJ Los Angeles & Orange Ventura Co; Salinas & Pajaro Valleys; Fremont Water level have dropped as much as 200' since Correct with artificial recharge Upconing of underlying brines in Central Valley

Saltwater Intrusion Saltwater-Freshwater Interface: Sharp gradient in water quality Seawater Salinity = 35‰ = 35,000 ppm = 35 g/l NaCl type water  sw = Freshwater < 500 ppm (MCL), mostly Chemically variable; commonly Na Ca HCO 3 water  fw = Nonlinear Mixing Effect: Dissolution of mixing zone of fw & sw Possible example: Lower Floridan Aquifer: mostly 1500’ thick Very Hi T ~ 10 7 ft 2 /day in “Boulder Zone” near base,  ~30% paleokarst? Cave spongework

Clarence King 1 st Director of USGS