after Fetter Unconfined Aquifer Water Table: Subdued replica of the topography Hal Levin demonstration

Confined Aquifer: aquifer between two aquitards. = Artesian aquifer if the water level in a well rises above aquifer = Flowing Artesian aquifer if the well level rises above the ground surface. e.g., Dakota Sandstone: east dipping K sst, from Black Hills- artesian) Unconfined Aquifer: aquifer in which the water table forms upper boundary. “Water table aquifer” Head h = z P = 1 atm e.g., Missouri, Mississippi & Meramec River valleys Hi yields, good quality Ogalalla Aquifer (High Plains aquifer): CO KS NE NM OK SD QT Sands & gravels, alluvial apron off Rocky Mts. Perched Aquifer: unconfined aquifer above main water table; Generally above a lens of low-k material. Note- there also is an "inverted" water table along bottom! Hydrostratigraphic Unit: e.g. MO, IL C-Ord sequence of dolostone & sandstone capped by Maquoketa shale Aquifer Types

Cambrian-Ordovician aquifer Dissolved Solids mg/l

USGS

Typical Yields of Wells in the principal aquifers of the three principal groundwater provinces USGS 1967 Alluvial Valleys & SE Lowland Osage & Till Plains Springfield Plateau Ozark Aquifer St. Francois Aquifer <--Maquoketa Shale <--Davis/Derby-Doe Run gpm

Ss = specific storage Units: 1/length = Volume H 2 O released from storage /unit vol. aquifer /unit head drop (F&C p. 58) Ss =  g  B  where  aquifer compressibility ~ /m for sandy gravel  = water compressibility  = porosity S y = Specific yield Units: dimensionless = storativity for an unconfined aquifer "unconfined storativity" = Vol of H 2 O drained from storage/total volume rock (D&S, p. 116) = Vol of H 2 O released (grav. drained) from storage/unit area aquifer/unit head drop S y = V wd /V T Typically, S y = 0.01 to 0.30 F&C, p. 61 Specific retention: Sr =  = Sy + Sr + unconnected porosity

Storativity S Units: dimensionless S = Volume water/unit area/unit head drop = "Storage Coefficient" S = m Ss confined aquifer S = Sy + m Ss unconfined; note Sy >> mSs For confined aquifers, typically S = to Transmissivity T = K*m m = aquifer thickness Units m 2 /sec = Rate of flow of water thru unit -wide vertical strip of aquifer under a unit hyd. Gradient T ≥ m 2 /s in a good aquifer

HYDRAULIC DIFFUSIVITY (D): Freeze & Cherry p. 61 D = T/S Transmissivity T /Storativity S = K/S s Hydraulic Conductivity K/ Specific Storage S s

FUNDAMENTAL CONCEPTS AND PARTIAL DERIVATIVES Scalars: Indicate scale (e.g., mass, Temp, size,...) Have a magnitude Vectors: Directed line segment, Have both direction and magnitude; e.g., velocity, force...) v = f i + g j + h k where i, j, k are unit vectors Two types of vector products: Dot Product (scalar product): a. b = b. a = |a| |b| cos  commutative Cross Product (vector product): a x b = - b x a = |a| |b| sin  anticommutative i. i = 1 j. j = 1 k. k = 1 i. j = i. k = j. k = 0 Scalar Field: Assign some magnitude to each point in space; e.g. Temp Vector Field: Assign some vector to each point in space; e.g. Velocity

FUNCTIONS OF TWO OR MORE VARIABLES Thomas, p. 495 There are many instances in science and engineering where a quantity is determined by many parameters. Scalar function w = f(x,y) e.g., Let w be the temperature, defined at every point in space Can make a contour map of a scalar function in the xy plane. Can take the derivative of the function in any desired direction with vector calculus (= directional derivative). Can take the partial derivatives, which tell how the function varies wrt changes in only one of its controlling variables. In x direction, define: In y direction, define:

FUNCTIONS OF TWO OR MORE VARIABLES Thomas, p. 495 There are many instances in science and engineering where a quantity is determined by many parameters. Scalar function w = f(x,y) e.g., Let w be the temperature, defined at every point in space Define the Gradient: “del operator” The gradient of a scalar function w is a vector whose direction gives the surface normal and the direction of maximum change. The magnitude of the gradient is the maximum value of this directional derivative. The direction and magnitude of the gradient are independent of the particular choice of the coordinate system.

If the function is a vector (v) rather than a scalar, there are two different types of differential operations, somewhat analogous to the two ways of multiplying two vectors together {i.e. the cross (vector) and dot (scalar) products}: Type 1: the curl of v is a vector: Type 2: the divergence of v is a scalar: So: Great utility for fluxes & material balance

dx dy dz Overall Difference Rate of Gain in box Significance of Divergence Measure of stuff in - stuff out

Laplacian: Gauss Divergence Theorem: where u n is the surface normal

Continuity Equation (Mass conservation): A = source or sink term;  = flow porosity No sources or sinks  = constant Steady Flow Steady, Incompressible Flow Because the Mass Flux q m :

Continuity Equation (Mass conservation): A = source or sink term;  = flow porosity So, “Diffusion Equation” where Ss = specific storage

Cartesian Coordinates Cylindrical Coordinates Cylindrical Coordinates, Radial Symmetry ∂h/∂  = 0 Cylindrical Coordinates, Purely Radial Flow ∂h/∂  = 0 ∂h/∂z = 0 “Diffusion Equation”

Derivative of Integrals: CRC Handbook Thomas p. 539

Gradient: “del operator” Divergence: Diffusion Equation:

Darcy's Law : Hubbert (1940; J. Geol. 48, p ) where: q v  Darcy Velocity, Specific Discharge or Fluid volumetric flux vector (cm/sec) k  permeability (cm 2 ) K = kg/  hydraulic conductivity (cm/sec)  Kinematic viscosity, cm 2 /sec  = (k/  [force/unit mass]

Gravitational Potential  g

If fdx +gdy+hdz is an “exact differential” (= du), then it is easy to integrate, and the line integral is independent of the path: Condition for exactness: Exact differential: If true: => Curl u = 0

Suppose that force F = fi +gj + hk acts on a line segment dl = idx+jdy+kdz : If fdx + gdy + hdz is exact, then the work integral is independent of the path, and F represents a conservative force field that is given by the gradient of a scalar function u (= potential function). 1. Conservative forces are the gradients of some potential function. 2. The curl of a gradient field is zero i.e., Curl (grad u) = 0 In general: Conservative Forces

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

Fetter No Flow

Flow Net Rules: No Flow boundaries are perpendicular to equipotential lines Flowlines are tangent to such boundaries (// flow) Constant head boundaries are parallel to and equal to the equipotential surface Flow is perpendicular to constant head boundary

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

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

MK Hubbert (1940)

Fetter, after 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)

after Freeze and Witherspoon Effect of Topography on Regional Groundwater Flow

for unconfined flow

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

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

Craig et al 1996 Union Beach, NJ Water Level & Chlorinity

Air Fresh Water   =1.00 hfhf Fresh Water-Salt Water Interface? Sea level  Salt Water  =1.025 ? ? ?

Salt Water Fresh Water hfhf z Ghyben-Herzberg P Sea level z interface 

Ghyben-Herzberg Analysis Hydrostatic Condition  P -  g = 0 No horizontal P gradients Note: z = depth  fw = 1.00  sw = 1.025

Salt Water Fresh Water hfhf z Ghyben-Herzberg P Sea level z interface 

Physical Effects Tend to have a rather sharp interface, only diffuse in detail e.g., Halocline in coastal caves Get fresh water lens on saline water Islands: FW to 1000’s ft below sea level; e.g., Hawaii Re-entrants in the interface near coastal springs, FLA Interesting implications: 1) If  is 10’ ASL, then interface is 400’ BSL 2) If  decreases 5’ ASL, then interface rises 200’ BSL 3) Slope of interface ~ 40 x slope of water table

Hubbert’s (1940) Analysis Hydrodynamic condition with immiscible fluid interface 1) If hydrostatic conditions existed: All FW would have drained out Water sea level, everywhere w/ SW below 2) G-H analysis underestimates the depth to the interface Assume interface between two immiscible fluids Each fluid has its own potential h everywhere, even where that fluid is not present! FW potentials are horizontal in static SW and air zones, where heads for latter phases are constant

Ford & Williams 1989 ….…...

after Ford & Williams 1989 ….…... Fresh Water Equipotentials  Fresh Water Equipotentials 

For any two fluids, two head conditions: P sw =  sw g (h sw + z) and P fw =  fw g (h fw + z) On the mutual interface, P sw = P fw so: ∂z/∂x gives slope of interface ~ 40x slope of water table Also, 40 = spacing of horizontal FW equipotentials in the SW region Take ∂/∂z and ∂/∂x on the interface, noting that h sw is a constant as SW is not in motion

after USGS WSP 2250 Saline ground water Fresh Water Lens on Island Saline ground water 0

Confined Unconfined Fetter

Saltwater Intrusion Mostly a problem in coastal areas: GA NY FL Los Angeles From above analysis, if lower  by 5’ ASL by pumping, then interface rises 200’ BSL! Abandonment of freshwater wells- e.g., Union Beach, NJ Can attempt to correct with artificial recharge- e.g., Orange Co Los Angeles, Orange, Ventura Counties; Salinas & Pajaro Valleys; Water level have dropped as much as 200' since Correct with artificial recharge Also, possible upconing of underlying brines in Central Valley FLA- now using reverse osmosis to treat saline GW >17 MGD Problems include overpumping; upconing due to wetlands drainage (Everglades) Marco Island- Hawthorn 540’: Cl to 4800 mg/l (cf. 250 mg/l Cl drinking water std)

Possible Solutions Artificial Recharge (most common) Reduced Pumping Pumping trough Artificial pressure ridge Subsurface Barrier

End

USGS WSP 2250

Potentiometric Surface defines direction of GW flow: Flow at rt angle to equipotential lines (isotropic case) If spacing between equipotential lines is const, then K is constant In general K 1 A 1 /L 1 = K 2 A 2 /L 2 where A = x-sect thickness of aquifer; L = distance between equipotential lines For layer of const thickness, K 1 /L 1 = K 2 /L 2 (eg. 3.35; D&S p. 79)

FLUID DYNAMICS Consider flow of homogeneous fluid of constant density Fluid transport in the Earth's crust is dominated by Viscous, laminar flow, thru minute cracks and openings, Slow enough that inertial effects are negligible. What drives flow within a porous medium? Down hill? Down Pressure? Down Head? Consider: Case 1: Artesian well- fluid flows uphill. Case 2: Swimming pool- large vertical P gradient, but no flow. Case3: Convective gyre w/i Swimming pool- ascending fluid moves from hi to lo P descending fluid moves from low to hi P Case 4: Metamorphic rocks and magmatic systems.

after Toth (1963)

Potentiometric Surface ("Piezometric sfc) Map of the hydraulic head = Imaginary surface representing level to whic water would rise in a well cased to the aquifer. Vertical planes assumed; no vertical flow Concept rigorously valid only for horizontal flow w/i horizontal aquifer Measure w/ Piezometers- small dia well w. short screen- can measure hydraulic head at a point (Fetter, p. 134) Potentiometric Surface defines direction of GW flow: Flow at rt angle to equipotential lines (isotropic case) If spacing between equipotential lines is const, then K is constant In general K1/L1 = K2/L2 L = distance between equipotential lines (eg. 3.35; D&S p. 79) For confined aquifers, get large changes in pressure (head) with virtually no change in the thickness of the saturated column. Potentiometric sfc remains above unit