Final Exam

• May 10, 5 – 7:30 pm, ESS 081

Energy Transformation

• 1 Caloria of heat = energy necessary to raise the temperature of one gram of pure water from 14.5 – 15.5oC

• Latent Heat of vaporization

Hv = 597.3 – 0.564T (Cal./g)

• Latent Heat of condensation

Energy Transformation, Cont.

• Latent heat of fusion – Hf – 1 g of ice at 0oC => ~80 cal of heat must be added to melt ice. Resulting water has same temperature.

• Sublimation – Water passes directly from a solid state to a vapor state. Energy = Hf + Hv => 677 cal/g at 0oC.

• Hv > 6Hf > 5 x amt. to warm water from 0oC -> 100oC

Hydrologic Equation

• Inflow = outflow +/- Changes in storage

• Equation is simple statement of mass conservation

Condensation

• Condensation occurs when air mass can no longer hold all of its humidity.

• Temperature drops => saturation humidity drops.

• If absolute humidity remains constant => relative humidity rises.

• Relative humidity reaches 100% => condensation => Dew point temperature.

Cool, moist Cool, moistWarm, dry

Limited soil-moisture storage

All infiltrate

All infiltrate

some water always on the surface

Puddles and overland flow

Q0

Determining ground water recharge from baseflow (1)

• Meyboom method (Seasonal recession method): utilizes stream hydrographs from two or more consecutive years.

• Assumptions: the catchment area has no dams or other method of streamflow regulation; snowmelt contributes little to the runoff.

Determining ground water recharge from baseflow (2)

• Rorabaugh method (Recession curve displacement method): utilizes stream hydrograph during one season.

Aquifer

• Properties: Porosity, specific yield, specific retention.

• Potential: Transmissivity, storativity.

• Types: confined, unconfined.

• Hydraulic conductivity, Physical Laws controlling water transport.

d60

d60

d10

d10

Sediment Classification

• Sediments are classified on basis of size of individual grains

• Grain size distribution curve• Uniformity coefficient Cu = d60/d10

• d60 = grain size that is 60% finer by weight.• d10 = grain size that is 10% finer by weight.• Cu = 4 => well sorted; Cu > 6 => poorly

sorted.

Specific Yield and Retention

• Specific yield – Sy: ratio of volume of water that drains from a saturated rock owing to the attraction of gravity to the total volume of the rock.

• Specific retention – Sr: ratio of the volume of water in a rock can retain against gravity drainage to the total volume of the rock.

• n = Sy + Sr.• Sr increases with decreasing grain size.

Darcy’s Law

• Q = -KA(dh/dl).

• dh/dl = Hydraulic gradient.

• dh = change in head between two points separated by small distance dl.

Laminar flow (Small R < 10)

Turbulent flow (Large R)

Flow lines

Flow lines

Darcy’s Law: Yes

Darcy’s Law: No

Hydraulic conductivity

• K = hydraulic conductivity (L/T).

• K is also referred to as the coefficient of permeability.

• K = -Q[A(dh/dl)] [ L3/T/[L2(L/L)] = L/T]

• V = Q/A = -K(dh/dl) = specific discharge or Darcian velocity.

Intrinsic Permeability

• Intrinsic permeability Ki = Cd2 (L2).• K = Ki (γ/μ) or K = Ki (ρg/ μ)• Petroleum industry 1 Darcy = unit of intrinsic

permeability Ki

• 1 darcy = 1 cP x 1 cm3/s / (1 atm/ 1 cm). cP – centipoise - 0.01 dyn s/cm2

atm – atmospheric pressure – 1.0132 x 1016 dyn/cm2

• 1 darcy = 9.87 x 10-9 cm2 ~ 10-8 cm2

Aquifer

• Aquifer – geologic unit that can store and transmit water at rates fast enough to supply amounts to wells. Usually, intrinsic permeability > 10-2 Darcy.

• Confining layer – unit with little or no permeability … < 10-2 Darcy.

aquifuge – absolutely impermeable unit. aquitard - a unit can store and transmit water

slowly. Also called leaky confining layer. Raritan formation on Long Island.

-- all these definitions are in a relative sense.

Transmissivity

• The amount of water that can be transmitted horizontally through a unit width by the full saturated thickness of the aquifer under a hydraulic gradient of 1.

• T = bK• T = transmissivity.• b = saturated thickness.• K = hydraulic conductivity.• Multilayer => T1 + T2 + … + Tn

Specific Storage

• Specific storage Ss = amount of water per unit volume stored or expelled owing to compressibility of mineral skeleton and pore water per unit change in head (1/L).

• Ss = ρwg(α+nβ)• α = compressibiliy of aquifer skeleton.• n = porosity.• β = compressibility of water.

Storativity of confined Unit

S = b Ss

• Ss = specific storage.

• b = aquifer thickness.

• All water released in confined, saturated aquifer comes from compressibility of mineral skeleton and pore water.

Storativity in Unconfined Unit

• Changes in saturation associated with changes in storage.

• Storage or release depends on specific yield Sy and specific storage Ss.

• S = Sy + b Ss

Volume of water drained from aquifer

• Vw = SAdh

• Vw = volume of water drained.

• S = storativity (dimensionless).

• A = area overlying drained aquifer.

• dh = average decline in head.

Hydraulic head, h

• Hydraulic head is energy per unit weight.

• h = v2/2g + z + P/gρ. [L].

• Unit: (L; ft or m).

• v ~ 10-6 m/s or 30 m/y for ground water flows.

• v2/2g ~ 10-12 m2/s2 / (2 x 9.8 m/s2) ~ 10-13 m.

• h = z + P/gρ. [L].

Flow lines and flow nets

• A flow line is an imaginary line that traces the path that a particle of ground water would flow as it flows through an aquifer.

• A flow net is a network of equipotential lines and associated flow lines.

Boundary conditions

• No-flow boundary – flow line – parallel to the boundary. Equipotential line - intersect at right angle.• Constant-head boundary – flow line – intersect at right angle. Equipotential line - parallel to the boundary.• Water-table boundary – flow line – depends. Equipotential line - depends.

Estimate the quantity of water from flow net

• q’ = Kph/f.• q’ – total volume discharge per unit width of aquifer

(L3/T; ft3/d or m3/d).• K – hydraulic conductivity (L/T; ft/d or m/d).• p – number of flowtubes bounded by adjacent pairs of

flow lines.• h – total head loss over the length of flow lines (L; ft

or m).• f - number of squares bounded by any two adjacent

flow lines and covering the entire length of flow.

Water table

• Water table = undulating surface at which pressure in fluid in pores = atmospheric pressure.

Water table

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.

Drawdown

• T = Q/ 4(h0-h)G(u)

• G(u) =

W(u) - completely confined.

W(u,r/B) – leaky, confined, no storage.

H(u,) – leaky, confined, with storage.

W(uA,uB,) - unconfined.

Aquifer test

• Steady-state conditions.

Cone of depression stabilizes.

• Nonequilibrium flow conditions.

Cone of depression changes.

Needs a pumping well and at least one observational well.

Aquifer tests

• T = Q/ 4(h0-h)G(u)

• G(u) =

W(u) - completely confined.

W(u,r/B) – leaky, confined, no storage.

H(u,) – leaky, confined, with storage.

W(uA,uB,) - unconfined.

Slug test

• Overdamped

– water level recovers to the initial static level in a smooth manner that is approximately exponential.

• Underdamped

– water level oscillates about the static water level with the magnitude of oscillation decreasing with time until the oscillations cease.

Cooper-Bredehoeft-Papadopulos Method (confined aquifer)

• H/H0 = F(,)

• H – head at time t.

• H0 – head at time t = 0.

= T t/rc2

= rs2S/rc

2

Underdamped Response Slug Test

• Van der Kamp Method – confined aquifer and well fully penetrating.

• H(t) = H0 e-t cos t

H(t) - hydraulic head (L) at time t (T)

H0 - the instantaneous change in head (L)

- damping constant (T-1)

- an angular frequency (T-1)

= ln[H(t1)/H(t2)]/ (t2 – t1)

= 2/(t2-t1)

Underdamped Response Slug Test (cont.)

• T = c + a ln T

c = -a ln[0.79 rs2S(g/L)1/2]

a = [rc2(g/L)1/2] / (8d)

d = /(g/L)1/2

L = g / (2 + 2)

Mass transport of solutes

• Diffusion – both ionic and molecular species dissolved in water move from area of higher concentration (chemical activity) to areas of lower concentration.

• Advection – moving water carries it dissolved solutes.

Diffusion – Fick’s laws

• Fick’s first law F = -D dC/dx F = mass flux of solute per unit area per unit time. D = diffusion coefficient (area/time) C = solute concentration (mass/volume) dC/dx = concentration gradient

(mass/volume/distance).• D ranges from 1 x 10-9 to 2 x 10-9 m2/s, for the

major cations and anions.

Diffusion – Fick’s laws (cont.)

• Fick’s second law

C/t = D 2C/x2

D = diffusion coefficient (area/time)

C = solute concentration (mass/volume)

t = time

Advection

• Advecting contaminants travel at the same rate as the average linear velocity of ground water

vx = -(K/ne) dh/dl

vx = average linear velocity K = hydraulic conductivity

ne = effective porosity dh/dl = hydraulic gradient

Mechanical Dispersion

• Longitudinal dispersion: if the mixing occurs along the pathway of fluid flow

- it moves faster through the center of the pore;

- some of the fluid will travel in longer pathways;

- fluid travels faster through larger pore.• Transverse or lateral dispersion: if the mixing

occurs normal to the pathway of fluid flow.

- flow paths can split and branch out to the side.

Hydrodynamic Dispersion

• Hydrodynamic dispersion:

DL = D* + aLvx

DL = longitudinal coefficient of hydrodynamic dispersion

D* = effective molecular diffusion coefficient

aL = dynamic dispersivity

vx = average linear ground-water velocity

Advection-dispersion Equation

• DL2C/x2 – vxC/x = C/t

DL2C/x2 – dispersion (diffusion + dispersivity).

vxC/x – Advection

Solute Transport by Advection-Dispersion

• C = C0/2{erfc[(L-vxt)/2(DLt)1/2] + exp(vxL/DL)erfc[(L-vxt)/2(DLt)1/2] }

C = solute concentration (M/L3, mg/L)

C0 = initial concentration (M/L3, mg/L)

L = flow path length (L; ft/m)

vx = average ground velocity (L/T)

t = time since release of the solute (T)

DL = longitudinal dispersion coefficient (L2/T)

Retardation Factor

• Retardation factor = 1 + (b/)(Kd)

b = dry bulk mass density of the soil (M/L3; gm/cm3)

= volumetric moisture content of the soil (dimensionless).

Kd = distribution coefficient for solute with the soil (L3/M; mL/g)

Solute Movement with Retardation

• vc = vx/[1+ (b/)(Kd)]

vc = velocity of the solute front. In one-dimensional column the solute concentration is one-half of the original value (L/T; ft/day or m/day).

vx = average linear velocity (L/T; ft/day or m/day).

x = -y/tan(2Kbiy/Q)

Q - pumping rateK - conductivityb – initial thicknessi – initial h gradient

x0 = -Q/(2Kbi)

ymax = Q/(2Kbi)

Confined

Static fresh and slat water

Ghyben-Herzberg principle