Aquifer Mechanics - FORS(GEOL) 8730

Todd Rasmussen (2-4300) John Dowd (2-2383)

February 11, 2004

Course Summary

This course focuses on understanding the mechanics of flow through subsurface me-

dia. Theories of flow include: confined homogeneous aquifers, leaky aquifers, delayed

yield, effects of partially penetrating wells, unconfined (water table) aquifers, effects of

boundaries and multiple wells, dual porosity media, fractured rock aquifers. Applications

focus on using pump test data to identify the relative importance of flow and transport

processes and to estimate hydraulic properties of aquifers.

Textbook

Kruseman GP, de Ridder NA, 1991, Analysis and Evaluation of Pumping Test Data,

Second Edition, International Institute for Land Reclamation and Improvement, P.O. Box

45, 6700 AA Wageningen, The Netherlands

1

Contents

1 Introduction to Aquifer Testing 3

1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Definitions of Hydraulic Properties . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Common Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Aquifer Test Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Ground-Water Flow Geometries 13

2.1 Flow Parallel to Bedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Flow in Series Through Bedding . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Flow at an Angle to the Bedding . . . . . . . . . . . . . . . . . . . . . . . 14

3 The Ground-Water Flow Equation in Radial Coordinates 16

4 Flow in Confined Aquifers 18

4.1 Unsteady Flow in Confined Aquifers (Theis Method) . . . . . . . . . . . . 18

4.2 Unsteady Flow Problem Example . . . . . . . . . . . . . . . . . . . . . . . 18

4.3 Estimating the Theis Parameters . . . . . . . . . . . . . . . . . . . . . . . 19

4.4 Confined Aquifer Parameter Sensitivity Coefficients . . . . . . . . . . . . . 21

4.5 Aquifer Derivative Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Flow in Leaky Aquifers 23

5.1 Steady Flow in a Leaky Aquifer . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Unsteady, Leaky Flow (Hantush-Jacob Method) . . . . . . . . . . . . . . . 24

5.3 The Universal Confining Layer (by P. Stone) . . . . . . . . . . . . . . . . . 25

2

1 Introduction to Aquifer Testing

1.1 Notation

• Physical Properties

– Aquifer thickness, b

– Aquitard thickness, b′

– Aquifer porosity, n = Vv

Vb= ne + ni

∗ Volume of voids, Vv

∗ Bulk volume, Vb

∗ Aquifer effective porosity, ne

∗ Immobile zone porosity, ni

– Bulk density, ρb = (1− n) ρs

– Skeletal density, ρs = 2.65 g/cm3

– Water content, θ = Vw

Vv

– Relative saturation, Θ = thetan

= Vw

Vb

• Conductance Parameters

– Aquifer hydraulic conductivity, K = k γµ

∗ Intrinsic permeability, k ∝ d2

∗ Pore diameter, d

∗ Fluid specific weight, γ = ρw g = 9807 Pa/m

∗ Fluid density, ρw = 1 g/cm3

∗ Fluid dynamic viscosity, µ = 0.001 Pa · s

– Aquifer transmissivity, T = Kh b

– Horizontal component of hydraulic conductivity, Kh

3

– Vertical component of hydraulic conductivity, Kv or K ′

– Anisotropy ratio, m = Kh

Kv= K

K′

– Hydraulic conductance, C = Kv

b′= K′

b′

– Hydraulic resistance, c = 1C

= b′

K′

– Aquitard leakance, L =√

T c =√

TC

=√

K b b′

K′ =√

m b b′

– Hydraulic conductivity tensor, K =

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

• Flow Parameters

– Hydraulic head, h = z + pγ

+ v2

2 g

∗ Elevation, z

∗ Fluid pressure, p

∗ Fluid specific weight, γ = ρ g

∗ Fluid density, rho

∗ Gravitational constant, g = 9.807m/s2

– Hydraulic gradient vector, ı = [ix, iy, iz] = gradh =[

∂h∂x

, ∂h∂y

, ∂h∂z

]– Fluid flux vector, q = Q

A= [qx, qy, qz] = −K ∇h

∗ Fluid flow, Q = ∆Vw

∆t

∗ Cross sectional area, A

– Richards’ equation, q = −D ∇θ

– Aquitard flux, q′ = −K ′∆hb′

= −L ∆h

• Storage Parameters

– Specific storativity, Ss = − ∂θ∂h

= −∂

[VwVb

]∂h

= Vw

V 2b

∂Vb

∂h− 1

Vb

∂Vw

∂h= Vw

Vb

[1Vb

∂Vb

∂h− 1

Vw

∂Vw

∂h

]=

γ n (βw − βs) = γ (α + n βw)

4

– Water compressibility, βw = − 1Vw

∂Vw

∂p= 4.4× 10−10 Pa−1 = 5× 10−6 m−1

– Aquifer compressibility, α = − 1Vb

∂Vb

∂p

– Effective stress, p

– Bulk volume, Vb

– Skeletal compressibility, βs

– Barometric efficiency, BE = n/Ss

– Tidal efficiency, TE = 1−BE

– Aquifer storativity, S = b Ss

– Aquifer hydraulic diffusivity, D = K/Ss =T/S

– Specific yield, Sy = b ∂θ∂h

= b ∂θ∂h

• Transport Parameters

– Solute effective dispersion coefficient, De = Do + α v

– Molecular diffusion coefficient, Do

– Solute dispersivity, α

– Average fluid pore velocity, v

– Retardation factor, R = 1 + ρb

nKd

– Distribution coeficient, Kd

– Cation exchange capacity, CEC

– Anion exchange capacity, AEC

– Organic carbon content, OC

– Specific surface area, Sa

– Sorption kinetic parameter, k

• Additional Aquifer Parameters

5

– Specific capacity, Cp = Q/s

– Pumping rate, Q

– Aquifer drawdown, s

– Recharge rate, R

– Deep percolation rate, DP

1.2 Definitions of Hydraulic Properties

• Anisotropic: hydraulic conductivity is differ in different directions. We must repre-

sent the hydraulic conductivity, K, using a tensor

• Anisotropy ratio: Ratio of hydraulic conductivity in one principal direction to the

hydraulic conductivity in another principal direction.

• Aquiclude: A geologic formation in which negligible fluid flow is possible

• Aquifer: A geologic formation that transmits appreciable quantities of water to a

well

• Aquifer Compressibility: reciprocal of bulk modulus of elasticity of aquifer where is

the effective stress. ranges from 10−6Pa−1 for clays, to 10−10Pa−1 for rock

• Aquitard or confining layer: geologic formation that resists the movement of water

between two aquifers. The layer has material properties K’ and b’ to distinguish

them from aquifer properties. Flow through a confining layer can be assumed to be

vertical:

• Conductance: hydraulic conductivity of resistive layer (K ′) per unit thickness of the

resisting layer (b’):

• Darcian flux: see flux

• Darcy’s Law: relates hydraulic flux to hydraulic gradient and hydraulic conductivity

6

• Dimension of Flow:

– Cartesian Flow: flux is different in each of the three cartesian directions

– Planar Flow: flux is different in two directions, but zero in one cartesian direc-

tion

– Linear Flow: flux is different in one cartesian direction, but zero in the other

two

– Radial or Cylindrical Flow: flux is planar, radiating inward or outward from a

central axis.

– Spherical Flow: flux is radiating uniformly in all three directions from a point

• Discharge: Water moving from the subsurface to the surface across the earth’s

surface

• Drawdown: water level decline due to pumping from well, where hi is baseline

(initial) water level before pumping and h is observed water level during aquifer

test.

• Flux: Volume of water (Vw) flowing per unit area (A) per unit time (t)

• Homogeneous Media: K is a constant is space

• Hydraulic Resistance: the reciprocal of the conductance (C)

• Hydraulic conductivity: Measure of ability of geologic media to transmit water,

related in a general way to pore size and shape:

– C = constant of proportionality

– d = median pore or grain size

– γ specific weight of fluid used to measure total head

– µ = water dynamic viscosity

7

• Hydraulic Diffusivity: ratio of hydraulic conductivity to specific storage, or, equiv-

alently, of transmissivity to the storage coefficient

• Hydraulic Gradient: Change in total head per unit distance (h)

• Infiltration: Water passing across the earth’s surface into the subsurface

• Isotropic: hydraulic conductivity is the same in all directions. It can be represented

as a scalar

• Leakance, or Leakage factor

• Microporosity or matrix porosity: pores too small to see, such as voids between

mineral grains or clay platelets.

• Macroporosity: visible pores, such as fractures, voids or vugs

• Nonlinear: K is not constant for all values of total head or fluid potentials

• Percolation: Water moving through the unsaturated zone

• Porosity: volume of voids (Vv) per unit bulk volume (Vb), ranging from 10−5 for

dense granites to over 0.70 for volcanic tuffs.

• Precipitation: Water falling on the surface of the earth

• Recharge: Water moving across the water table from the unsaturated zone into the

saturated zone

• Relative Saturation: Water content of material relative to saturated water content:

• Richards’ Equation: relates hydraulic flux to change in volume of water in storage

• Specific Capacity: water discharge from well divided by amount of drawdown

• Specific Storage: A measure of the volume of water released (or added) to storage,

per unit volume of aquifer, per unit increase (or decrease) in fluid pressure. It

8

is the sum of the aquifer plus water compressibilities: where is the fluid specific

weight, is the bulk compressibility, is the compressibility of water, and = - n is the

compressibility of the mineral skeleton.

• Specific Yield: volume of water released (or added) from an unconfined aquifer per

unit area of aquifer per unit decline (rise) in water table position. Similar to the

storage coefficient for confined aquifers.

• Storage Coefficient: water released from storage for a given thickness of aquifer (b).

• Storativity: Identical to the storage coefficient

• Total Head: sum of elevation, pressure, velocity, osmotic and other potentials

• Transmissivity: Total hydraulic conductivity for a given aquifer thickness

• Water Compressibility: reciprocal of bulk modulus of elasticity of water where p is

the water pressure.

• Water Content: volume of water (Vw) per bulk volume of aquifer (Vb)

1.3 Common Errors

• DANGER #1: q and v are not the same!

– q is specific flux (units of velocity)

– v is groundwater velocity (units of velocity)

• DANGER #2: Darcy’s Law is not always valid!

– Darcy’s law is only appropriate for laminar, and not turbulent flow, turbulent

conditions arise in large voids (fractures, caves, etc.), and where velocities are

high, such as near boreholes.

9

– The Reynolds number helps to define the conditions for laminar flow d is void

opening, is kinematic viscosity (approximately 10−6 m2/s), R is the ratio of

inertial to viscous forces.

– When the inertial forces are too great, such as when R > 1, then flow can

become turbulent.

– For turbulent conditions, the flow rate varies with the square root of the gra-

dient.

1.4 Aquifer Test Preliminaries

• Definition of Test Objectives

– scientific vs engineering objectives

∗ identification of system behavior

∗ estimation of system parameters

– scale of interest

∗ local vs regional test

∗ short-term vs long-term effects

– hydraulic vs transport effects

∗ average or extreme behavior

∗ fluid vs solute migration

• Definition of Conceptual Model

– Define state of knowledge of area

∗ Type of aquifer

∗ Type of geologic media, homogeneous/heterogeneous, isotropic/anisotropic

∗ Geometries of aquifers and confining units

∗ Recharge and discharge areas

10

– Inventory existing wells in area for baseline (reference) data

∗ Find local and regional flow gradients, horizontally and vertically

∗ Use the ground and surface water chemical characteristics to indicate of

flow system

• Physical Installation

– Pumping well:

∗ Type of gravel or sand pack

∗ Depth of well

∗ Well and borehole diameters

∗ Length of screened interval and screen integrity

∗ Depth of pump, pumping capacity, and previous pumping records

∗ Measurement of pumped water rate and duration

∗ Location of discharge

– Observation wells:

∗ Number and location (distance from well and depth):

∗ Barometric pressure

∗ Precipitation

∗ Nearby surface water levels

∗ Reference ground water levels

– Aquifer test data:

∗ Discharge sampling (flow meter, weir, bucket, etc) and control (valves,

voltage)

∗ Manual (steel tape, electrical sounder) vs automated (pressure transducer,

float)

∗ Analog (strip charts) vs digital (computer)

11

∗ Water level sampling rates and duration

∗ Well effects (cascading water, water quality changes)

• Aquifer test data interpretation

– Removing background effects

∗ Loading effects due to barometric pressure, rainfall, oceanic and earth tides

∗ Trends (seasonal or short-term)

∗ Fluid column density effects (salinity, temperature, dissolved gasses)

– Identifying aquifers type

∗ confined aquifer (leaky, double porosity, fractured rock)

∗ unconfined aquifer (delayed yield)

– Identifying other effects

∗ well bore storage (and excessive pumping rate)

∗ partial penetration

∗ boundaries

– Standard plots

∗ horizontal-axis: log t or log(t/r2)

∗ vertical-axis: s or log s

– Derivative plots

∗ horizontal-axis: log t or log(t/r2)

∗ vertical-axis: ∆s/∆t or ∆s/(∆ log t)

12

2 Ground-Water Flow Geometries

2.1 Flow Parallel to Bedding

Let us assume that two permeable layers overlie each other with horizontal flow in each

layer, and no flow between each layer. The flow in each layer is:

Q1 = −K1 A1ha−hb

xa−xb

Q2 = −K2 A2ha−hb

xa−xb

(1)

where A1 = w b1 and A2 = w b2. We sum the amount of flow in each layer:

Qh = Q1 + Q2

Qh = −Kh A ha−hb

xa−xb

(2)

where A = A1 + A2 = w (b1 + b2) = w b, and Kh is the effective horizontal hydraulic

conductivity. Substitution results in:

Kh A ha−hb

xa−xb= K1 A1

ha−hb

xa−xb+ K2 A2

ha−hb

xa−xb

Kh A = K1 A1 + K2 A2

Kh = K1A1

A+ K2

A2

A

Kh = K1b1b

+ K2b2b

(3)

We can also write this in terms of transmissivity:

Te = T1 + T2 (4)

because Te = Ke b, T1 = K1 b1, and T2 = K2 b2.

For multiple layers, we have:

Te =∑n

i=1 TiKe =∑n

i=1bi Ki∑n

i=1bi

(5)

2.2 Flow in Series Through Bedding

In this case the flow is perpendicular to the beds:

Q1 = −K1 A ha−hb

za−zb

Q2 = −K2 A hb−hc

zb−zc

(6)

13

where A is the same for each bed. In this case, the flows are equal to each other:

Qv = Q1 = Q2

−Kv A ha−hc

b= −K1 A ha−hb

b1= −K2 A hb−hc

b2

Kvha−hc

b= K1

ha−hb

b1= K2

hb−hc

b2

(7)

where b = za − zc, b1 = za − zb, b2 = zb − zc. Solving for the intervening head, hb, yields:

hb =K1

b1ha + K2

b2hc

K1

b1+ K2

b2

(8)

Substituting into the previous equation yields:

Kv

bha−hc

=K1

b1

[ha −

K1b1

ha+K2b2

hc

K1b1

+K2b2

]Kv

b= 1

b1K1

+b2K2

bKv

= b1K1

+ b2K2

(9)

or, by using the hydraulic resistance, c = Kb, we have:

cv = c1 + c2 (10)

Thus, like transmissivity, the hydraulic resistance adds. For multiple aquifers this is:

cv =n∑

i=1

ci (11)

Kv =

∑ni=1 bi∑ni=1

bi

Ki

(12)

2.3 Flow at an Angle to the Bedding

In general, we can define the horizontal and vertical components of flux using:

q =Q

A= [qh, qv] = − [Kh ih, Kv iv] (13)

where ih = ∂h∂x

and iv = ∂h∂z

The magnitude of the hydraulic gradient, |i|, and flux |q|, are:

|i|2 = i2h + i2v

|q|2 = q2h + q2

v

(14)

14

The flux components can be determined from the direction, φ and magnitude, |q|, of

flow:

qh = |q| cos φ

qv = |q| sin φ(15)

Substitution provides:

(Ke |i|)2 = (Kh |ih|)2 + (Kv |iv|)2 (16)

Simplification yields:

Ke =

√i2h

|i|2 K2h + i2v

|i|2 K2v√

cos2 φ K2h + sin2 φ K2

v

Kh

√cos2 φ + m sin2 φ

(17)

where m = Kv/Kh is the anisotropy ratio.

If the hydraulic gradient is at an angle of 45 to the bedding planes, we have:

Ke =

√K2

h + K2v

2= Kx

√1 + m2

2(18)

If the flux is at an angle of 45 to the bedding planes, we have qh = qv, so that

Kh ih = Kv iv, and m = ix/iv, resulting in:

Ke = Kh

√2 m2

1 + m2(19)

15

3 The Ground-Water Flow Equation in Radial Co-

ordinates

We derive the well flow equation by first defining a representative cylindrical volume with

fixed height, b, an inner radius, r1, and an outer radius, r2 = r1 + r. Flow across the

inner surface, Q1, and outer surface, Q2, are:

Q1 = q1 A1

Q2 = q2 A2

(20)

with Ai = 2 π ri b and qi = −K(

∆h∆r

)i, and where A is the cross-sectional area of the

cylinder perpendicular to the flow direction, q is the darcian flux across the cylindrical

face, K is the hydraulic conductivity of the medium within the cylinder, i = ∂h∂r

is the

hydraulic gradient, h is the hydraulic head, and r is the radial distance from the center

axis of the cylinder. Combining terms yields:

Qi = −2 π b K ri

(∆h

∆r

)i

(21)

We now specify the mass balance equation as:

∆Q = Q1 −Q2 =∆Vw

∆t(22)

where Vw is the change in volume of water within the cylinder needed to balance the

inflows with the outflows, and T is time. The change in the volume of water within the

cylinder is related to the change in the water content, ∆θ, within the cylinder using:

∆θ =∆Vw

∆Vb

(23)

where Vb is the volume of the hollow cylinder, equal to:

∆Vb = V2 − V1 = b π[r22 − r2

1

]= b π ∆r2 (24)

We can relate the hydraulic head to the water content using:

Ss =∆θ

∆h(25)

16

where Ss is the specific storage coefficient. Combining equations yields:

2 π b K

[r2

(∆h

∆r

)2

− r1

(∆h

∆r

)1

]= π b Ss ∆r2 ∆h∆t (26)

which is the same as:

2 K ∆

[r

(∆h

∆r

)]= Ss ∆r2 ∆h

∆t(27)

2 K∆[r(

∆h∆r

)]∆r2

= Ss∆h

∆t(28)

2 D∂[r(

∂h∂r

)]∂r2

= Ss ∂h∂t (29)

(30)

where D = TS

is the hydraulic diffusivity, and the partials are introduced to denote

infinitesimal changes.

We can simplify this further by noting that ∂r2

∂r= 2 r:

D∂[r(

∂h∂r

)]r ∂r

= ∂h∂t (31)

D

[∂2h

∂r2+ ∂1r

∂h

∂r

]= ∂h∂t (32)

∂2h

∂r2+ ∂1r

∂h

∂r=

1

D∂h∂t (33)

(34)

17

4 Flow in Confined Aquifers

4.1 Unsteady Flow in Confined Aquifers (Theis Method)

We start with the ground-water flow equation:

T

[∂2h

∂r2+

1

r∂h∂r

]= S

∂h

∂t(35)

D

[∂2h

∂r2+

1

r∂h∂r

]=

∂h

∂t(36)

∂2h

∂r2+

1

r∂h∂r =

1

D

∂h

∂t(37)

(38)

4.2 Unsteady Flow Problem Example

Find the distance from a pumping well where the drawdown is 1 foot, given:

• Radius of pumping well, ro = 2” = 0.1667 ft

• Hydraulic conductivity, K = 2× 10−4 ft/min

• Aquifer saturated thickness, b = 10 ft

• Specific yield, Sy = 0.1 ft3/ft3/ft

• Pumping duration, t = 3 days = 72 hrs = 4320 min

• Drawdown at pumped well, so = 10 ft

Derived Parameters: Aquifer Transmissivity, Storage Coefficient, and Diffusivity are:

T = Kb = (2× 10−4 ft/min)× (10 ft) = 2× 10−3 ft2/min

S = Sy b = [0.1 (ft3 water)/(ft3 soil)/(ft drawdown)]× (10 ft soil) = 1 ft3/ft2/ft

D = T/S= (K b)/(Sy b) = K/Sy = 2× 10−4 ft/min/0.1 ft3/ft3/ft = 2× 10−3 ft2/min

(39)

The well function argument is:

uo =r2o

4Dt=

(0.1667 ft)2

4× (2× 10−3 ft2/min)× (4320 min)= 8× 10−4 (40)

18

This satisfies the Jacob condition: u < 0.01, so the well function at the pumped well is:

W (uo) = −0.5772− ln u = 6.55 (41)

The pumping rate must be:

Q =4Tso

W (uo)= 4× (2× 10−3 ft2/min)× 10 ft

6.55= 0.0384 ft3/min = 0.287 gpm (42)

The well function at the observation well is simply:

W (u1) = W (uo)s1

so

= 20.37× 1 ft

10 ft= 0.655 (43)

The well function argument at the observation well is:

u1 = exp(−0.5772−W (u1) = 0.292 (44)

The distance at which the drawdown is 1 foot is:

r =√

4 D t u1 =√

4× (2× 10−3 ft2/min)× (4320 min)× 0.292 = 3.18 feet (45)

4.3 Estimating the Theis Parameters

The following data are normally provided for an aquifer test:

• Q, aquifer pumping rate (constant)

• r, distance of observation well from pumping well (constant)

• s, drawdown vector in observation well

• t, time vector at which drawdowns are measured

We can remove the constants Q and r from consideration by defining two normalized

variables, a normalized drawdown, w, and a normalized time, tau :

wi =4 π si

Qτi =

4 tir2

(46)

so that the Theis Solution becomes:

wi =W (ui)

Tui =

1

D τi

(47)

Our objective is to determine the following aquifer parameters:

19

• D, aquifer diffusivity

• T , aquifer transmissivity

• S = T/D, aquifer storage coefficient

It is clear that there are two parameters to determine, T and D. The storage coefficient

is determined once D and T are known.

Two unknowns require two equations, which we select to be the observed drawdowns

at two separate times:

w1 =W (u1)

T(48)

w2 =W (u2)

T(49)

u1 =1

D τ1

(50)

u2 =1

D τ2

(51)

(52)

We can remove T from these equations by taking the ratio of the normalized drawdown

at two different times:

w1

w2

=W (u1)

Wu2

(53)

w1

w2

W (u2)

Wu1

= 1 (54)

ln

[w1

w2

W (u2)

Wu1

]= 0 (55)

(56)

which is now only a function of the unknown diffusivity, D. We solve for this unknown

at each time step using Newton’s method:

Dj+1 = Dj −F (Dj)

f(Dk)(57)

The functions, F (D) and f(D), are defined as:

F (Dj) = ln

[w1

w2

W (u2)

Wu1

](58)

20

f(Dj) =dF (D)

dD(59)

(60)

Solving for f(D) is assisted by use of the chain rule:

d [ln y]

dD=

d [ln y]

dy

∂y

∂u

∂u

∂D(61)

We can show that:

f(D) =1

D

[exp−u2

W (u2)− exp−u1

W (u1)

](62)

so that:

Dj+1 = Dj

1− ln[

w1

w2

W (u2)W (u1)

]e−u2

W (u2)− e−u1

W (u1)

(63)

Iteration proceeds until the aquifer diffusivity converges. Upon convergence, we calculate

the aquifer transmissivity and storage coefficient for each time step using:

Ti =W (ui)

wi

(64)

Si =Ti

Di

(65)

4.4 Confined Aquifer Parameter Sensitivity Coefficients

Confined aquifer sensitivity coefficients are used to relate changes in drawdown (raw, U ,

and normalized, U∗) to each of the coefficients that determine drawdown.

s = Q4 π T

W (u)

Ur = ∂s∂r

= − Q4 π T

2 e−u

r

Ut = ∂s∂t

= Q4 π T

e−u

t

UT = ∂s∂T

= Q4 π T

e−u−W (u)T

US = ∂s∂S

= − Q4 π T

e−u

S

UD = ∂s∂D

= Q4 π T

e−u

T

(66)

The sensitivity between T and D is:

∂D

∂T=

UT

US

=e−u −W (u)

S e−u(67)

21

The sensitivity between T and S is:

∂S

∂T=

UT

US

=W (u)− e−u

D e−u(68)

Note that T and S are independent of each other when:

∂S

∂T= 0 (69)

which occurs when W (u) = e−u. T and D are independent for these same conditions. In

fact, drawdowns are insensitive to transmissivity at this point, because UT = 0 for this

condition. This condition is satisfied at u = 0.4348.

4.5 Aquifer Derivative Curves

We calculate derivative curves using observed drawdowns:

s′ = Ut =∂s

∂t=

Q

4 π T

e−u

t(70)

For normalized variables, this is:

w′ = U∗t =

e−u

T τ(71)

w =4 π s

Q=

W (u)

T(72)

τ =4 t

r2(73)

u =1

D τ(74)

(75)

The maximum of this function is found by setting the derivative of w′ to zero:

w′′ =∂2w

∂τ 2=

e−u (u2 − 1)

T τ 2= 0 (76)

which occurs when u = 1. This implies that at the time of the maximum, t∗, and the

value of drawdown at the maximum, w∗, can be used to determine aquifer parameters,

D = 1/t∗, because:

T = W (1)w∗ = 0.2194

w∗

w = W (u)T

S = TD

= 0.2194 τ∗

w∗

(77)

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5 Flow in Leaky Aquifers

5.1 Steady Flow in a Leaky Aquifer

For a leaky aquifer, we have:

T

[∂2s

∂r2+

1

r∂s∂r

]− q = 0 (78)

We account for leakage using:

q =K ′

b′s =

s

c(79)

Dividing by the aquifer transmissivity yields the equation:

∂2s

∂r2+

1

r∂s∂r − s

L2= 0 (80)

where L2 = T c.

This form is similar to the Modified Bessel Function Differential Equation (Abramowitz

and Stegun, 9.6.1):

z2 ∂2w

∂z2+ z ∂w∂z −

(z2 + v2

)w = 0 (81)

for the conditions:

v = 0 (82)

s = w (83)

z =r

L(84)

(85)

The Modified Bessel Function solution for v = 0 is:

w = Ko(z) =∫ ∞

o

cos zt√t2 + 1

dt =∫ ∞

ocos(z sinh t) dt (86)

for boundary conditions:

limz→∞ w = 0

limz→0 z ∂w∂z

= 1(87)

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Our boundary conditions are:

limr→∞ s = 0

limr→0 r ∂s∂r

= Q2 π T

(88)

which yields the solution:

s =Q

2 π TKo

(r

L

)(89)

Leakage into aquifers during well tests often result in steady flow conditions. This

attribute is quantified using the specific capacity coefficient, Cp:

Cp =Q

s=

2 π T

Ko

(rL

) (90)

5.2 Unsteady, Leaky Flow (Hantush-Jacob Method)

For unsteady flow conditions we have:

T

[∂2s

∂r2+

1

r∂s∂r

]− S

∂s

∂t− q = 0 (91)

We again account for leakage using:

q =K ′

b′s =

s

c(92)

Dividing by the aquifer transmissivity yields the equation:

∂2s

∂r2+

1

r∂s∂r − 1

D

∂s

∂t− s

L2= 0 (93)

where L2 = T c.

Using the same approach as the solution for the confined (Theis) equation:

u

[∂2s

∂u2+ ∂s∂u

]+

∂s

∂u− r2 s

4 u L2= 0 (94)

which is equivalent to:

u2 ∂2s

∂u2+ u [u + 1] ∂s∂u +

∂s

∂u− r2 s

4 u L2= 0 (95)

We can find an approximate solution for Jacob conditions, i.e., u << 1:

u2 ∂2s

∂u2+ u ∂s∂u +

∂s

∂u−[u2 + v2

]s = 0 (96)

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where:

v =

√r2

4 L2− u2 (97)

yielding:

s =Q

4 π TKv(u) (98)

which holds as long as u < 0.01 and r2 L

> u

Hantush and Jacob found a solution for all u:

s =Q

4 π T

∫ ∞

u

exp[−x− r2

4 r L2

]x

dx (99)

which is also known as the Walton solution.

5.3 The Universal Confining Layer (by P. Stone)

An unrecognized protector of our ground water

There is a unique geological unit – economically critical, hydrologically essential, ex-

tremely widespread (ubiquitous among some investigated environments) – that amazingly

has not been formally recognized and described as an interrelated unit, and not named

according to the rules for stratigraphic nomenclature.

This unit, whose lithology and thickness varies among sites but is always highly imper-

meable, even when very thin, is found to lie atop the first ”usable” drinking-water aquifer

apparently virtually everywhere. Or, perhaps one should say, it is found in virtually every

place where there has been some shallow stratigraphic and geo-hydrologic investigation

that could reveal it; these of course being at sites where actual, or suspected, or potential,

ground-water contamination has prompted such investigation. Surprisingly, this unit al-

most always perches some completely unconnected, sometimes ephemeral, always minor

and inconsequential, zone or ground water atop it, that being the contaminated zone

where present. Or else no water table whatsoever reportedly exists above the confining

layer for the underlying aquifer in hydraulic nature even though it often occurs at an

uncommonly shallow depth for such an aquifer.

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This formation and confining layer is unique not only because of its near ubiquity

among the widely scattered investigated sites, but because it encompasses various differ-

ent geologic formations and, astoundingly, completely different geologic provinces also.

What other geologic formation in the shallow subsurface encompasses both Appalachian

Piedmont and Atlantic Coastal-Plain provinces, of vastly different ages and especially

origins? Host materials range from soft sediments to residua from hard crystalline rocks

(curiously, the same condition, if not the layer itself, is commonly reported for fractured

hard-rock aquifers also). One might conjecture that this layer relates somehow to alluvial

deposition, stream valleys certainly being one of the few depositional environments to

span both Piedmont and Coastal-Plain geologic provinces. Or perhaps it is some unrec-

ognized sub- ”C” soil horizon or phenomenon, formed in-place. Will close examination

of contamination-site investigations from other geologic provinces, in other parts of the

country, reveal this same formation? Is it found still further afield, at least at sites where

there is some exploratory interest able to find it?

The economic importance of this stratum cannot be overestimated. By universally

isolating the uppermost ”true” aquifer from the host of dangerous materials at the ground

surface above, or especially those dissolved in the shallowest ”perched” or ”ephemeral”

ground water so nearby above, the environmental Cleanup costs that are avoided, or

deemed unnecessary, are astronomical in their cumulative total, and represent some very

considerable sum at any given site also.

Other distinctive distributional characteristics include (I thank a colleague for this

observation) the apparent tendency, judged from reported contaminant-plume geometry,

for the confining unit to rise near property boundaries. Incidentally, I hope the identifi-

cation of this physical mechanism lays to rest the suspicious sometimes voiced concerning

the numerous ground-water contaminant plumes that seem never to migrate out from the

responsible party’s property to another’s, ending instead somewhere near the boundary.

The averted costs in liabilities add another vast value to this unit.

One accepts as obvious that this geographic coincidence of property boundaries and

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contaminant plumes and their controlling stratigraphy cannot be random coincidence and

that property lines themselves cannot influence the underground. Could there in any way

be some nonapparent control made by the underground environment on the boundaries?

At first glance, of course, this would seem to be impossible. But perhaps it involves

unknowing influence on decisions in platting, somewhat like the suspected subconscious

reading of subtle topographic and other surficial signs by ”water witchers” who then truly

believe that they feel the divining rod move as they pass over an underground ”stream”.

What other reasonable explanation can there be for these fortuitously placed natural

”liners”, and even natural ”slurry walls”, containing the future spread of ground-water

contaminants at the many sites that so desperately need them?

How did the extremely widespread occurrence of this single stratum, or at least an

analogous stratigraphic sequence, escape geologic investigation’s attention prior to the

proliferation of contamination-site hydrologic investigation? Dare one put forth an out-

rageous hypothesis? Could some action of the contaminants themselves ”create” this

layer, say perhaps by somehow chemically reducing permeability. What other physical

or cultural characteristic of ground-water contamination, or of such sites, might cause or

explain this prevailing presence? This remains as grist for future research. In any case, if

this important zone exists almost everywhere – and a host of apparently irrefutable pro-

fessional sources have identified it at scores of sites – it should be recognized and named

formally, as are all significant geologic units no matter how emplaced.

I have sought advice from colleagues regarding a descriptive or distinctive name for the

layer or formation, with a diversity of suggestions received: From the artificial constructs

”Nofurac” or ”Commonly Formations” (no further action or continued monitoring only),

to the Linnean binomial ”Confinus Maximus”, to the Victorian-era-sounding ”The glori-

ous [or ”great” or ”grand”] hydraulic impedance”. My personal favorite, and therefore

the winner, is the ”Nocostus Aquitard”, and thus it is given.

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