Notes for ’Hydrogeology’
Anders Damsgaard Christensen - 20062213
[email protected]
Last revision: January 26, 2010 Version 0.4
About this compendium
This document is created on base of the lectures in the course ’Hydrogeology’ held by Keld Rømer Rasmussen at the department of Geology at Århus University, that I followed during the fall semester semester 2009. All figures and tables are recreated recreated and possibly modified without permission, and this document is strictly for personal use only. Redistribution is not allowed. I cannot be held responsible regarding the correctness of factual claims made in this document. This document is created with LATEX
About this compendium
This document is created on base of the lectures in the course ’Hydrogeology’ held by Keld Rømer Rasmussen at the department of Geology at Århus University, that I followed during the fall semester semester 2009. All figures and tables are recreated recreated and possibly modified without permission, and this document is strictly for personal use only. Redistribution is not allowed. I cannot be held responsible regarding the correctness of factual claims made in this document. This document is created with LATEX
Contents
1 Intro duction
7
1.1 Lecture material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2 Main topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2 Geometric elements of open channels 3 Fundamentals
9 10
3.1 Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.1.1 Rectangular channel . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.1. 3.1.22 Trapez rapezoi oida dall-,, Trian riangu gula larr- and and circ circul ular ar channe hannels ls . . . . . . . . . . . .
11
3.1.3 Stationary flow, steady flow . . . . . . . . . . . . . . . . . . . . . .
11
3.2 Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
4 Energy and momentum principles
12
4.1 Hydrostatic pressure distribut bution . . . . . . . . . . . . . . . . . . . . . . .
12
4.2 Ideal fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.3 Real fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.4 Open pen channel flow classification . . . . . . . . . . . . . . . . . . . . . . . .
14
2
Hydrogeology
CONTENTS
4.5 Mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
4.6 Momentum transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
4.7 Energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
4.8 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
5 Critical flow
17
5.1 Head expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
5.2 Froudes number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
5.2.1 Critical depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
5.2.2 Specific energy - E . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
5.2.3 Small threshold at the stream bottom (Q constant) . . . . . . . . .
19
5.2.4 Choking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
6 Friction in fluids
20
6.1 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
6.2 Mannings approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
6.3 Single loss (enkelttab) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
6.4 Loss in pipes with turbulent flow . . . . . . . . . . . . . . . . . . . . . . .
23
6.5 Composite channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
6.6 Compound channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
7 Hydraulic structures
25
7.1 Broad crested weir (Bredt overløb) . . . . . . . . . . . . . . . . . . . . . .
25
7.1.1 Free and ventilated weirs - Generally . . . . . . . . . . . . . . . . .
26
7.2 Sharp-crested weir (Skarpkantet overloeb) . . . . . . . . . . . . . . . . . .
26
3
Hydrogeology
CONTENTS
8 Open channel flow - Summary
28
8.1 Example - Aarhus stream . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
8.2 Losses in hydraulic streams . . . . . . . . . . . . . . . . . . . . . . . . . .
28
8.2.1 Simple (turbulent?) system . . . . . . . . . . . . . . . . . . . . . .
28
8.2.2 Laminar flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
8.3 Drainage of groundwater aquifers . . . . . . . . . . . . . . . . . . . . . . .
29
9 Groundwater flow - basics
30
9.1 Poriosity of a soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
9.2 Hydraulic conductivity - Darcy’s experiment . . . . . . . . . . . . . . . . .
31
9.3 Refraction of streamlines (Brydningsloven) . . . . . . . . . . . . . . . . . .
32
9.4 Aquifer flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
9.5 Confined aquifers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
9.5.1 Derivation of the flow equation . . . . . . . . . . . . . . . . . . . .
33
9.6 Leakage aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
9.7 Unconfined (phreatic / water table) aquifer . . . . . . . . . . . . . . . . . .
37
9.7.1 Linearizing the Boussinesque equation - Homogenous and isotropic aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
9.8 Defining the flow problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
9.8.1 Interaction with the surroundings - Boundary value problems (BVP) 39 4
Hydrogeology
CONTENTS
10 Analytical models
40
10.1 Steady state flow to a well in a leakage aquifer . . . . . . . . . . . . . . . .
40
10.2 Unsteady flow to wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
10.2.1 Recovery test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
10.2.2 Theis eq - Well function - Estimating T and S from type curve plot
43
10.2.3 Storage in confined aquifers . . . . . . . . . . . . . . . . . . . . . .
45
10.2.4 Barometer influences in confined aquifers . . . . . . . . . . . . . . .
46
10.3 Unsteady flow to wells in a confined aquifer . . . . . . . . . . . . . . . . .
46
10.3.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
10.3.2 Transient flow in a confined aquifer . . . . . . . . . . . . . . . . . .
46
10.4 Leaky aquifers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
10.4.1 Leaky aquifer geometry . . . . . . . . . . . . . . . . . . . . . . . . .
47
10.4.2 Drawdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
10.4.3 Type curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
10.5 Unconfined aquifers/Water table aquifers, S&Z ch.11 . . . . . . . . . . . .
49
10.5.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
10.5.2 Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
10.5.3 Drawdown and delayed storage, Neumann type curves . . . . . . . .
51
10.5.4 Early and late effects . . . . . . . . . . . . . . . . . . . . . . . . . .
51
10.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
10.6 Slug testing and stepwise testing, S&Z ch.12 . . . . . . . . . . . . . . . . .
53
10.7 Bounded aquifers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5
Hydrogeology
CONTENTS
10.7.1 Drawdown near positive boundary . . . . . . . . . . . . . . . . . . .
54
10.7.2 Drawdown near negative boundary . . . . . . . . . . . . . . . . . .
56
10.7.3 Pumping situations with multiple positive/negative boundaries . . .
58
10.7.4 Well between two streams in infinite strip . . . . . . . . . . . . . .
58
10.8 Well in uniform flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
10.9 Abstraction - long term influences, S&Z ch.8 . . . . . . . . . . . . . . . . .
60
10.9.1 Release of water - initial phase . . . . . . . . . . . . . . . . . . . . .
62
10.9.2 Release of water - late phase . . . . . . . . . . . . . . . . . . . . . .
63
10.10Salt water intrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
10.11Stream groundwater interaction . . . . . . . . . . . . . . . . . . . . . . . .
64
A Hydrogeology terms
66
B Commonly used formulas
69
B.1 Basic hydraulics in streams . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
B.2 Groundwater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
B.2.1 W(u) - Theis equation/Well function . . . . . . . . . . . . . . . . .
71
B.2.2 Unconfined aquifer: Steady 1D groundwater flow . . . . . . . . . .
71
B.2.3 Confined aquifer: Steady 1D groundwater flow . . . . . . . . . . . .
71
B.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . .
72
6
Chapter 1 Introduction Lectures Monday 9-11 (1525-229), and exercises Tuesday 8-10 (1525-229). Homepage: http://aula.au.dk/courses/HYDROE04/ Mail:
[email protected]
1.1
Lecture material
Osman Akan: Open Channel Hydraulics Chapters 1, 2, 3 and 6. F.W. Schwartz and Hubao Zhang: Fundamentals of Ground Water Main book. Jacob Bear: Hydraulics of Groundwater Two chapters C.T. Jenkins: The influence from pumping near a stream Keld Rasmussen: Exercises in hydrogeology
1.2
Main topics
• Basic hydraulics in streams – Mannings formula – Specific energy (diagrams, Froude’s number) – Linear reservoirs
• Groundwater 7
Hydrogeology
CHAPTER 1. INTRODUCTION
– Confined aquifers – Watertable aquifers (often treated as confined aquifers) – Leakage aquifers (steady state when leakage in cone equals the pumping rate) – Finding hydraulic parameters by simple pumping tests
∗ Cooper-Jacob plot ∗ Theis plot ∗ Delayed storage
– Streams: Assume that there is no loss at the groundwater-stream interface.
Streams are a positive or negative boundary.
8
Chapter 2 Geometric elements of open channels
Table 2.1: Relationship between various section elements y d
T P A D R S 0 γ
Flow depth
Vertical distance from the channel bottom to the free surface Depth of flow section Flow depth measured perpendicular to the channel bottom. The relationship between d and y is d = y · cosθ. For most man-made and natural channels cosθ ≈ 1.0, and therefore y ≈ d. Top width Width of the channel section at free surface. Wetted perimeter Length of the interface between the water-channel boundary. Flow area Cross-sectional area of the flow. Hydraulic depth Flow area divided by top width: D = A/T Hydraulic radius Flow area divided by wetted perimeter, R = A/P ă Bottom slope Longitudinal slope of the channel bottom, S 0 = tanθ ≈ sinθ. Specific gravity γ = ρ · g
9
Chapter 3 Fundamentals Textbook: Akan Ch.1 Three types of aquifers: Confined, leaky and phreatic (unconfined or water table aquifer).
3.1
Flow
Vector. Flow through a channel is conversion of potential energy to kinetic energy. The restriction is friction.
3.1.1
Rectangular channel
Width: b, Depth: y The area of the flow: A = by , [m2 ] Wetted perimeter: P = b + 2y , [m] Hydraulic radius: how big the area of the flow is in relation to the wetted perimeter: R(m) =
10
by b + 2y
Hydrogeology
3.1.2
3.2. DISCHARGE
Trapezoidal-, Triangular- and circular channels
Akan, table 1.1. Circular channels: Ex.: partially filled tube. A partially filled tube is better for trans-
porting water than a completely filled tube. Akan, table 1.1.
3.1.3
Stationary flow, steady flow
Flow is said to be steady if the flow conditions do not vary in time. Therefore, the partial derivative terms with respect to time can be dropped from the continuity, momentum and energy equations.
3.2
Discharge
dQ = v dA
·
¯ = For a large cross section with area A: V
Q A
In a channel there is a continuity, that requires inverse behavior of area and velocity. I.e. in a larger channel the velocity of the stream is lower than in a narrower section.
11
Chapter 4 Energy and momentum principles
4.1
Hydrostatic pressure distribution
Piezometric head (Trykniveau): z+
p = h = constant γ
Valid in open water bodies and open (non-confined) groundwater aquifers, where flow is parallel to the surface and bottom. That means the pressure is the same value in a horizontal plane. When we look at pressure, we look at a static fluid. The surface is integrated over the vertical section. Force perpendicular to the vertical surface: F p = γydA Definition of centroid (tyngepunkt): Y c = So that: F p = γY cA
ydA A
The pressure force is always acting slightly below the centroid. The pressure at the bottom of a stream is the overlying weight of the fluid. If the stream/tube is oriented vertical, no pressure is applied at the sides. The pressure decreases with an increasing inclination (hældning). 12
Hydrogeology
4.2
4.2. IDEAL FLUIDS
Ideal fluids
1. Frictionless, i.e. viscosity µ = 0. 2. No loss of energy when flowing. 3. No eddies (hvirvler) will develop because of friction. Bernoulli’s equation applies: p + 12 · ρv2 + γz = constant, which consists of internal energy + kinetic energy + potential energy = constant. 2
Rewrites to: z + γ p + v2g = constant When observing ground water elevation, the height (h) of the groundwater table is a good expression for the energy, because the velocity is relatively very small, and the pressure at the groundwater table equals atmospheric pressure. Se slide "Example: Container with a short, streamlined nozzle", for using Bernoulli’s equation. The potential energy at position A is converted to kinetic energy at position B. The result is Toricellis theorem (when the friction is ignored), that gives the velocity of √ the escaping water at a height: v0 = 2gh
4.3
Real fluids
When there is a loss of√energy due to friction and surface tension, so a velocity coefficient is introduced: v0 = C v 2gh , where C v ≈ 0.95 − 0.99. Real fluids have viscosity µ. x1 is the horizontal axis, x2 is the vertical axis. For laminar flow of Newtonian fluids:
δv 1 δv 2 τ = µ + δx 2 δx 1
For a laminar planar flow - Newtons formula: τ = µ
δv1 δx 2
The velocity of the water molecules is zero (0) at the bottom/wall/side. The velocity rises constantly with the distance from the wall/bottom/side. Groundwater flows with planar flow because of small pipe radius and slow water movement speeds. 13
Hydrogeology
4.4
CHAPTER 4. ENERGY AND MOMENTUM PRINCIPLES
Open channel flow classification
Laminar or turbulent flow? Determined by the ratio of inertial forces to viscous forces. Reynold’s number (Re) is dimensionless. VR ν
Re =
Numerical boundaries depend on the choice of variables. ν : kinematic viscosity, V : average velocity, R: hydraulic radius. 580 ≤ Re ≤ 750. Laminar flow beneath 580, turbulent above 750. A rule of thumb: Groundwater flow laminar, surface flow turbulent. Ratio of inertial forces to gravity (Froude’s number small or large): Fr =
√V gD
Fr < 1 : Subcritical Fr = 1 : Critical Fr > 1 : Supercritical Large rivers with calm surface are in a subcritical state. Mountain streams with waves on top are supercritical. With the same discharge (Q) the water can flow slowly through a large area (subcritical), or fast through a small area (supercritical). This is related to the inclination to the bed.
4.5
Mass transfer
Discharge is Q. Mass (M) transfer for an incompressible fluid in an open channel (pipe) flow is called the mass flux or the mass transfer rate. Rate of mass transfer = Mass flux = ρQ
4.6
Momentum transfer
Momentum is a property of a moving object: p = M · V Momentum is the numerical measure of an objects tendency to keep moving in the same manner. Mass flux at any point: φM = ρdQ = ρvdA. Momentum flux at any point: φ p = v · ρdQ 14
Hydrogeology
4.7
4.7. ENERGY TRANSFER
Energy transfer
The total energy (internal + kinetic + potential) is converted to a size of potential energy, E = [m]. The potential energy is a relative quantity to a reference elevation. The potential energy of an object of mass M is M gzc, where g = gravitational acceleration, zc = elevation of the center of mass above the reference level. In open channel flow: Q = discharge (rate of volume transfer), and ρQ = rate of mass transfer. Therefore, the rate of potential energy transfer through a channel section: Rate of potential energy transfer = E p = ρQgzc Rate of kinetic energy transfer (v: point velocity): 1 ρ E k = ρvdA v 2 = 2 2
·
·
v 3 dA
A
In practice, it is easier to use the average cross-sectional velocity, V. α: energy coefficient or kinetic energy correction coefficient: ρ ρ E k = α V 3 A = α QV 2 2 2 α = 1 (for rectangular channels) α=
v 3dA V 3 A
A
Rate of internal energy transfer: E p = ρeV A = ρevdA = ρeQ
4.8
Conservation of mass
Consider a volume element of an open channel between upstream section U and downstream section D, with length ∆x, and average cross-sectional area A. The mass of water present in the volume: ρA∆x Water enters at upstream section at a mass transfer rate: ρQU , and leaves at downstream section at a rate ρQD . Change in the elements volume over time increment ∆t: ∆(ρA∆x) ∆t 15
Hydrogeology
CHAPTER 4. ENERGY AND MOMENTUM PRINCIPLES
The difference in water volumes that enters and leaves the volume element must equal the change in the elements volume. ρQU
− ρQ
D
=
∆(ρA∆x) ∆t
For a gradually-varied flow A and Q are continuous in space and time, and as ∆x and ∆t approach zero, the equation becomes....: δA δQ + =0 δt δx
....the continuity equation!
16
Chapter 5 Critical flow
Textbook: Akan Ch. 2 cont.
5.1
Head expression
From the energy equation: αU V U 2 αD V D2 H = zbU + yU + = zbD + yD + + ∆H 2g 2g ∆H is the head loss of energy between the two boundaries: U upstream and D downstream. zb : Elevation head, [m] (or [ft]) yU : Pressure head, [m] H : Total energy head 2 αU V U 2g
: Velocity head
For groundwater where the velocity is low, the piezometric head or hydraulic head: h = zb + y
17
Hydrogeology
5.2
CHAPTER 5. CRITICAL FLOW
Froudes number
Dimensionless F =
√V gD
Subcritical flow: F < 1 : Surface waves can move up-stream Supercritical flow: F > 1 : Surface waves cannot move up-stream Critical flow: F = 1
5.2.1
Critical depth
The depth where the flow is critical (F = 1). For a rectangular channel (other types in Akan): yc =
3
q2 g
3 q : Specific flow rate (discharge per meter across the stream), [ ms m−1 =
m2 ] s
The critical flow happens at one section in the elevation (not over a distance along the flow). With critical flow, the specific energy has its minimal value ( E min). At constant energy (E) and constant discharge (Q), the flow can exist in two forms (two y-values), i.e. in either supercritical or subcritical form. That is, either subcritical with large depth (y), or in supercritical state with small depth (y). Critical flow is where a maximum in flow can happen at a minimum in energy ( E min).
5.2.2
Specific energy - E
E
≈
αQ2 y+ 2gA2
Velocity coefficient α is usually 1. 18
Hydrogeology
5.2.3
5.2. FROUDES NUMBER
Small threshold at the stream bottom (Q constant)
Energy loss at positive structure. a) Subcritical flow: Stream depth decreases over threshold. b) Supercritical flow: Stream depth increases over threshold. E decreases with ∆z . E decreases, so stream depth y changes corresponding to Figure 2.11, Akan.
5.2.4
Choking
If the threshold is so high, that the energy is too low in the stream to pass it with the constant discharge Q (→ moving left of E min in fig. 2.11, Akan), the water level y will rise, until at least E min is reached over the threshold. The discharge Q will reduce, until the necessary y is developed in front of the threshold.
19
Chapter 6 Friction in fluids Textbook: Akan, ch. 3 A wall: Boundary where the flow velocity equals zero (0), (No slip conditions). This creates a velocity gradient: An area with increasing flow velocities. This zone is called the boundary layer. The atmospheric boundary layer has about 1 km thickness from the surface. Near the boundary: Stress and friction. Flows can be completely laminar, turbulent with a laminar wall layer or fully turbulent. S f : Friction loss τ 0: Friction force per unit area (no matter flow type) τ 0 = γRS 0 where: R = A/P : Hydraulic radius S 0: Slope of the bed γ = ρg : Specific gravity
Insert notes about pipe resistance and Darcy-Weisbach number from Slides + Akan. Fraction of kinetic energy, where f is the Darcy-Weisbach number: τ 0 = f
· 12 ρV
2
The Darcy-Weisbach number is a function of Reynolds number, form and roughness. The Reynolds number is a function of speed. ks = k: The particle diameter of a spheric grain. 20
Hydrogeology
6.1. REYNOLDS NUMBER
• Laminar flow: • Smooth flow (Re < 100.000): • Fully rough turbulent flow:
6.1
f =
0.316 Re0.25
k √1f = −2log 12R s
Reynolds number Re =
VR ν
u∗ = us = ν is the viscosity.
6.2
64 Re
f =
τ 0 ρ
Mannings approximation
Walls in nature are rough.
• For rough cylindrical pipes:
2 V = = 6.62 + 2.45ln f u∗
R k
• For rough rectangular pipes:
2 V = = 6.15 + 2.45ln f u∗
• For rough pipes:
2 V = = 6.40 + 2.45ln f u∗
21
R k
R k
Hydrogeology
CHAPTER 6. FRICTION IN FLUIDS
In nature (Manning):
R < 300 k u∗ = gRS 0
4.7 <
⇒
V u∗
≈ · R 8.1 k
⇒ V ≈ 8.1 M =
25.4 6 k
√
R k
1/6
1/6
gRS 0
(Manning number)
m1/3 s−1
⇒ V = M · R · S = kn · R · S ⇒ Q = AV n
1/2 0
2/3
2/3
1/2 0
The n-value is the Manning factor, used in the imperial system. Mannings number (M) can be found in tables. Often: 10 < M < 40.
6.3
Single loss (enkelttab)
Expansion carnot, where flow is moving from a small pipe into a larger one. Loss of part of the kinetic energy ( αA = 1.1): ∆H E = αA
(V A
2
− V ) B
2g
Flow into large reservoir; loss of the entire kinetic energy: V 2 ∆H E = α 2g Generally: ∆H E = ξ
(V A
− V ) B
2
2g
For a sudden, sharp contraction: ξ = 0.5 For a smoothened contraction (bottleneck type): ξ = 0.1 For a gradual contraction: ξ = 0 Bends:
2
θ ξ = 1.1 90
22
Hydrogeology
6.4
6.4. LOSS IN PIPES WITH TURBULENT FLOW
Loss in pipes with turbulent flow
The losses in the different parts of the system are summed together:
• Inlet loss:
Q2 ∆H I = 0.5 2gA2
• Pipe loss (friction):
∆H F =
• Outflow loss: • Total loss:
L Q M 2 R4/3 A2
·
∆H u =
i
⇒
∆H i = Q2
αQ2 2gA2
L + M 2 R4/3 A2
ξ 2gA2
∆H i = H = k Q2
·
i
⇒Q=
H k
k: Specific resistivity In a pipe system, the section in the brackets in the upper formula is a constant. When plotting Q2 (h), the result will be a straight line. In natural channels, the hydraulic radius (R) may vary. The result is that the Q2(h) plot will curve upwards in the ends.
6.5
Composite channels
Channels that have different roughness in different parts of the cross section (Fig 3.9, Akan). The equivalent roughness of the channel ( ne , Manning koefficient): ne =
N 2 i=1 (P i ni ) N i=1 P i
23
1/2
Hydrogeology
6.6
CHAPTER 6. FRICTION IN FLUIDS
Compound channels
Channels that overflood during high discharges (Fig 3.10, 3.11, Akan).ă The flow is divided into subsections. The conveyance (K) for each subsection: K i =
kn Ai Ri2/3 n
If the energy head is the same for all subsections: Q=
K i S 01/2
24
Chapter 7 Hydraulic structures Textbook: Akan, Akan, ch. ch. 6 Hydraulic structures are often constructed for measuring the discharge via the water level (h) in higher resolution.
7.1 7.1
Broad Broad cre crest sted ed wei weirr (Bre (Bredt dt ov overløb erløb))
Fig 6.7, Akan. On the broad crested weir, the flow transfers from upstream subcritical flow into critical flow on top of the weir, and exits as a jet with surrounding air in supercritical state (nappe). 3 3 E = yc = 2 2 h
3
αq 2 g
⇒q=
≈
4 2 H 2gH = H 2gH 27α 27α 3 3
38h ≈ H ; α ≈ 1 ⇒ q = 0.38h 2 C = √ 3 3 0
2gh
√
1.7h3/2
The above relationship between specific discharge (q) and water level (h) can be calibrated via measurements. Geometric and hydraulic elements: w: Weir height, Q: Discharge, q: Specific discharge, h: Flow height upstream over weir, H: energy upstream ( h < H ), ), yc: Critical depth, b: Weir width. 25
Hydrogeology
7.1.1 7.1.1
CHAPTER CHAPTER 7. HYDRAULIC HYDRAULIC STRU STRUCTURE CTURES S
Free and and ven ventilate tilated d we weirs irs - Genera Generally lly
(As example above)
Q = C 0 bH 2gH
If the upstream velocity is significant:
Q = C bh 2gH
⇒ C = C
0
αq 2 C = C 0 1 + 2g(w + h)2h
7.2
3/2
SharpSharp-cre creste sted d weir weir (Skarp (Skarpk kantet antet ov overloeb) erloeb)
Usage of the energy equation to obtain a relationship between the approach flow characteristics and the discharge over a weir. Rectangular sharp crested weir: Q = kw Lew
3/2
2gh e0
(6. (6.4)
Correction factors. Effective head over crest: he0 = h0 + hk
,
hk = 0.001m 001m
The effective crest length: Lew = Lw + Lk
26
Hydrogeology
7.2. SHARP-CREST SHARP-CRESTED ED WEIR (SKARPKANTE (SKARPKANTET T OVERLOEB) OVERLOEB)
27
Chapter 8 Open channel flow - Summary Different contributions for open channel water:
• Precipitation • Ditches • Groundwater flow • Drains and other man-made structures 8.1
Example - Aarhus stream
Discharge highest in winter. During summer: Generally speaking; Evapotranspiration > precipitation ⇒ Stream water supplied by groundwater contribution. During summer rainfalls, the precipitation will temporary infiltrate the aquifers as groundwater, until reentering the atmosphere into streams.
8.2 8.2.1
Losses in hydraulic streams Simple (turbulent?) system
Quadratic relationship between energy differences and discharge:
√
h=
h(t = 0)
− Kt 28
2
⇒ ∆H ∝ Q
Hydrogeology
8.2.2
8.3. DRAINAGE OF GROUNDWATER AQUIFERS
Laminar flow
Linear relationship between energy differences and discharge: Q = K L ∆H ∆H Q dh = A h(t) = h0 e−Kt dt
⇒
−
∝
⇒
This means, that the reservoir never empties.
8.3
Drainage of groundwater aquifers
Unconfined aquifer: Draining makes air replace water in upper lying pore-spaces, and
the groundwater-table drops. Confined aquifer: Totally filled by water under pressure. The piezometric surface is the
corresponding water-pressure surface (groundwater-table in unconfined aquifer), observed in observation-drillings. Drainage lowers the pressure because of elastic storativity. Lowering of the piezometric surface in a confined by drainage results in much less water than by lowering of the groundwater table in an unconfined aquifer.
29
Chapter 9
Groundwater flow - basics
Textbook: Schwartz & Zhang: Ch. 3, 4 (skip fractured rocks) and 5.1
Chapter 3 can be used as repetition for previously learned content.
9.1
Poriosity of a soil
The total porosity of a rock or soil is defined as the ratio of the void volume to the total volume of material: nT =
V v V T V s = V T V T
−
where nT is the total porosity, V v is the volume of voids, V s is the volume of solids and V T is the total volume. In some cases, porosity i expressed as percentage. 30
Hydrogeology
9.2
9.2. HYDRAULIC CONDUCTIVITY - DARCY’S EXPERIMENT
Hydraulic conductivity - Darcy’s experiment
q=
Q h2 h1 = K = A ∆l Q h2 h1 = K A ∆l kρ w g K = µ
−
−
−K ∆h ∆l
−
K: Hydraulic conductivity (L/T, length/time), used for water flow. k: Permeability ( L2) µ: Dynamic visosity (M/LT) h −h : Hydraulic gradient ∆l q : Darcy velocity / Specific discharge 1
2
The hydraulic gradient is denoted as i. h1 and h2 are the hydraulic heads at points 1 and 2, and ∆l is the distance between the points. i=
h1
−h
=
or
Q = KiA
2
∆l
− dh dl
Darcy’s equation is rewritten as: q = Ki
The Darcy equation is valid for a fully laminar flow in homogenous material. See table 4.7 for example values of parameters. 31
Hydrogeology
CHAPTER 9. GROUNDWATER FLOW - BASICS
In the field, it is possible to install a large number of piezometers in a unit and to contour the resulting head values. The gradient of change in head, and is maximal perpendicular to the lines of equal hydraulic head (equipotential lines).
9.3
Refraction of streamlines (Brydningsloven)
The figure below is a scenario for K 2 > K 1 , fx clay over sand.
Darcy- and continuity equation: dh1 dh2 = K 2c dl2 dl2 (h + ∆h) h
dQ1 = dQ2 = K 1 a dh1 = dh2
≈
−
b 1 = dl1 sinθ1 b 1 c = b cosθ2 : = dl2 sinθ2 dh1 dh2 K 1 b cosθ1 = K 2 b cosθ2 b sinθ1 b sinθ2 K 1 tanθ1 = K 2 tanθ2 That means, if K 1 >> K 2 θ2 >> θ1 . For example, in a confined aquifer, a low permeable a = b cosθ1 :
·
·
⇒
·
·
· ⇒
⇒
·
(medium 1) is overlying a highly permeable (medium 2), the downward flow is bent in a horizontal direction. 32
Hydrogeology
9.4
9.4. AQUIFER FLOW
Aquifer flow
The transmissivity describes the ease with which water can move through an aquifer. Transmissivity has units of [L2 /T ]. T = Kb
K: Hydraulic conductivity, b (or B): Aquifer thickness Aquifer flow can be considered horizontal . The length of a natural aquifer is much
greater than its thickness. This means there is a small height-gradient, and as good as horizontal flow. Vertical flow is only relevant in proximity to well pumping and near springs and streams.
9.5 9.5.1
Confined aquifers Derivation of the flow equation
Following principles are used: Conservation of mass, Darcy equation, Elastic properties and elastic laws. In a compressible medium a volume change will result in a mass change within a control medium (changes in volume of porespaces). Conservation of flow volume (input output = change of storage): Qinflow
−Q
outflow
= Qstorage
Mass conservation [M i M = ρq
,
− M ] = ∆m/∆t o
q : Specific discharge [m /s ]
Left side (M i − M o ). Same equations for x, y and z: (Inflow in x-direction)
M i,x = ρw qx ∆y∆z
M o,x = ρw qx
⇒ ∆M =
−
δ (ρw qx )∆x) ∆y∆z δx
−
(Outflow in x-direction)
δ δ δ (ρw qx ) + (ρw qy ) + (ρw qz ) ∆z∆y∆x δx δy δz
∆M =
q )V T w
− (ρ
33
=
−div(ρ
q)V T w
Hydrogeology
CHAPTER 9. GROUNDWATER FLOW - BASICS
Right side (∆m/∆t). Poriosity is n. Mass of water in control box: m = ρw n ∆x∆y∆z
· ·
δ = (ρ · n)∆x∆y∆z ⇒ δm δt δt w
Combining left and right side, removing ∆x∆y∆z :
−
δ δ δ δ(ρw n) (ρw qx ) + (ρw qy ) + (ρw qz ) = δx δy δz δt
·
Assume that time changes in ρw override spatial changes:
−
δq x δq y δq z 1 δ(ρw n) + + = δx δy δz ρw δt
· ≡ S δh δt s
, [S s ] = m−1
S s : Specific storativity. Delivery from 1m3 in reservoir per unit decline of head (h).
: Time-change of the hydraulic head. S : Storativity of the aquifer, S = S s · B , assuming presence of hydrostatic pressure distribution. B : Aquifer thickness δh δt
qi represents velocities, i.e. flow rate per cross section area. From the Darcy equation: qi =
−K δxδh i
i
δ δx
K x
δh δx
+
δ δy
K y
δh δy
+
δ δz
K z
δh δz
= S s
δh δt
In isotropic aquifers or aquifer layers the hydraulic conductivity (K) is equal for flow in all directions, while in anisotropic conditions it differs, notably in horizontal (Kh) and vertical (Kv) sense. In an isotropic, homogeneous medium: K x = K y = K z = K δ 2 h δ 2h δ 2h δh K + + = S s δx 2 δy 2 δz 2 δt
= 0), fx regional settings): Steady flow (constant speed ( δh δt δh =0 δt
⇒
δ2h δ2h δ2h + + = 0 (Laplace equation) δx 2 δy 2 δz 2
Confined aquifer:
δ δx
K x
δh δx
+
δ δy
K y
δh δy
+
δ δz
K z
δh δz
= S s
δh δt
Now: T = KB and S = BS s , so we multiply with B on both sides and rewrite:
δ δx
T x
δh δx
+
δ δy
34
T y
δh δy
δh = S δt
Hydrogeology
9.6. LEAKAGE AQUIFER
Homogeneous and isotropic aquifer (isotropic: constant hydraulic conductivity): δ2 h δ2 h S δh + = δx 2 δy 2 T δt
1D flow:
9.6
δ2h S δh = δx 2 T δt
Leakage aquifer
Basic equation for 2D flow. A leaky aquifer has an impermeable bed, an aquifier section on top (with parameters B,K), and a semi-permeable section over that (with parameters B’,K’). This material could be a till. There is no significant horizontal flow in the semipermeable layer, but slow vertical downward flow (N), (scale: decimeters/year).
The sum of flow in the aquifer: Horizontal flow (incl. elastic component) + vertical supply from semipermeable top layer (N). Assumption: Time-constant watertable on top of semi-permeable layer. Vertical (not horizontal!) head gradient in semipermeable layer (from Darcy eq): φ0 φ B
−
35
Hydrogeology
CHAPTER 9. GROUNDWATER FLOW - BASICS
φ0 is the head of the upper watertable, and φ is the head at the bottom of the semipermeable layer. B’ is the thickness. Instead of φ, h is used as the hydraulic head in the main aquifer.
Resulting downward supply to main aquifer from semipermeable layer: φ0 h φ0 h N = K = B σ
−
−
,
B σ= K
N: Downward flow into aquifer, B’: thickness of semipermeable layer, K’: Hydraulic conductivity of semipermeable layer, σ: Leakage coefficient.
Aquifer with 2D flow:
δ δx
δ δx
T x
δh δx
δh T x δx
+
δ δy
δ + δy
T y
δh δy
δh T y δy
δh + N (x, y) = S δt
K + (φ0 B
− h) = S δh δt
√
Leakage factor: λ = σT Homogenous, isotropic leaky aquifer (isotropic: constant hydraulic conductivity): δ2 h δ 2 h (φ0 h) S δh + + = δx 2 δy 2 λ2 T δt
−
1D flow: δ 2 h (φ0 h) S δh + = δx 2 λ2 T δt
−
36
Hydrogeology
9.7
9.7. UNCONFINED (PHREATIC / WATER TABLE) AQUIFER
Unconfined (phreatic / water table) aquifer
In the unconfined aquifer, the water table is the upper limit of the saturated zone. This surface is slightly sloping, and the flow is therefore not horizontal, see fig (a) below.
The flow has a vertical component, and therefore ds > dx. However, if θ cos > 0.99; so: q=
o
≤8
then
dh Kh −Kh dh ≈ − ds dx
This means, that there is a non-linear ( h2 ) relationship. These circumstances reflect figure (b) above, with vertical equipotential lines. This is called Dupuit’s approximation. In the unconfined case S y >> S s , so both sides are multiplied with h.
δ δx
δh K x h δx
δ + δy
δh K y h δy
= [h S s + S y ]
·
δh δt
≈ S δh δt y
In a homogenous and isotropic aquifer (Boussinesque equation): δ δx
9.7.1
h
δh δx
+
δ δy
h
δh δy
=
S y δh K δt
Linearizing the Boussinesque equation - Homogenous and isotropic aquifer
When H is large and variations in h are small, so that an average thickness of the aquifer can be introduced, i.e. h ∼ B : δ δx
B
δh δx
+
δ δy
B
δh δy
=
S y δh K δt
37
2
2
S δh S δh = ⇒ δδxh + δδyh = KB δt T δt 2
2
Hydrogeology
CHAPTER 9. GROUNDWATER FLOW - BASICS
Rewriting both sides using: δh 1 δ(h2 ) h = δt 2 δt δ δx
δh h δx
δ + δy
δh h δy
S y δh = K δt
⇔
1D flow:
9.8
and
S δh S 1 δ(h2 ) = K δt K 2h δt
δ(h2 ) δ(h2) S y δ(h2 ) + = δx 2 δy 2 Kh δt
≈
S y δ(h2 ) S y δ(h2 ) = KB δt T δt
δ 2 (h2 ) S δ(h2 ) = δx 2 T δt
Defining the flow problem
• The governing flow equation (differential equation) • Initial conditions • Boundary conditions The initial- and boundary conditions need to be known, in addition to the gorverning flow equation, to calculate the flow. The flow is described by a partial differential equation (PDE) and the solution requires determining:
• The geometry (x,y,z) of the flow domain must be known beforehand. has a bounding surface to a confining layer, or a curve F(x,y,z).
• The hydraulic parameters (K, S , B) • The initial conditions ( φ = φ (x,y,z, 0)) • The interaction with the surroundings (boundary conditions) s
1
Seek φ = φ(x,y,z,t) as the dependent variable. 38
The domain
Hydrogeology
9.8.1
9.8. DEFINING THE FLOW PROBLEM
Interaction with the surroundings - Boundary value problems (BVP)
1. Known hydraulic head - Dirichlet BVP: φ = φ2 (x,y,z,t) , example: a large lake, the sea. 2. Known flux - Neuman BVP: Example: Impermeable material textbook for formulas.
→ no flow at boundary:
q = 0 and
δh δx
= 0. See
3. Semipermeably boundary (Mixed BVP) - Cauchy type See textbook for formulas When there is a water table with recharge, the position and shape of the boundary condition (water table) is not known, but is part of the problem. ’Vicious circle argument’. Can lead to application of iterative methods in order to obtain a solution.
39
Chapter 10 Analytical models
Textbook: S&Z ch. 9 Insert missing notes f07.pdf
10.1
Steady state flow to a well in a leakage aquifer
Steady flow in an infinite aquifer is not possible. Flow into well: Q(r) = qv + Q(r + ∆r) qv is the vertical downward flow from the semi-permeable upper layer into the aquifer. Q(r + ∆r) is the horizontal flow from the surrounding aquifer. r is the distance from the well, rw is well radius. Axis-symmentrical flow is assumed. Solution: φ0
− φ(r) = αI
0
r r + βK 0 λ λ
I 0 : Modified Bessel function of first type and zero order. K 0 : Modified Bessel function of second type and zero order. K 1 : Modified Bessel function of second type and first order.
40
Hydrogeology 10.1. STEADY STATE FLOW TO A WELL IN A LEAKAGE AQUIFER
If x << 1 ⇒ K 0 (x) ≈ ln( 1,123 ) x If the leakage aquifer is unlimited: x
→ ∞ ⇒ I (x) → ∞, α = 0 0
⇒ φ − φ = βK ( λr 0
0
Q ⇒ s = φ − φ = 2πT (r
K 0 [r/λ] w /λ)K 1 [rw /λ]
0
Near the pumping well: s(r) =
41
Qw 2πT
Hydrogeology
CHAPTER 10. ANALYTICAL MODELS
With increasing distance to the well, the lateral input decreases ( Q(r)), and the vertical downflow increases (qv ). Well function: ζ : Error ’Distance with influence’ formula is a good approximation, no matter what u is.
10.2
Unsteady flow to wells
Notes from Bear-cont. (Hydraulics of Groundwater) S&Z: Ch 10 Notes on Barometer curves Investigating T and S of reservoirs by pumping tests.
10.2.1
Recovery test
Investigation methods: 1) Pumping for a limited time period, 2) Pumping (almost) continuously in a water works. Notes on above figure: y-axis: φ: Drawdown. The pumping takes place until t = t p = 500 (Pumping rate: Qw ). At t=500, t’ begins counting from zero. The recovery begins. 42
Hydrogeology
10.2. UNSTEADY FLOW TO WELLS
Mathematically this can be used. An equal size opposite discharge into aquifer is begun (−Qw ) at pump shutdown. t = t p + t
Q t Q t p + t s (r, t) = ln = ln 4πT t 4πT t This is valid for large times; u < 0.01. The data is plotted in a Cooper-Jacob recovery plot
(log-lin plot).
Plotting (t, s)-points should be linear in the log-lin coordinate system. When u is small, (fx drawdown in pumping well), choose two (t, s)-points. They should be one decade apart ((t1 , s1 ) and (t2 = t1 · 10, s2 )). ∆s = s2 s1 2.3Q T = 4π∆s
−
10.2.2
Theis eq - Well function - Estimating T and S from type curve plot 1 4T t = 2 u r S
43
Hydrogeology
CHAPTER 10. ANALYTICAL MODELS
S: Storativity, (not drawdown, s).
The Theis eq is plotted in a log-log plot (Well function), with x-axis: 1/u, y-axis: W(u)
Some drawdown (s,t) data is recorded in another well than the pumping well (observational well). The (s,t)-dataplot is put on top of the Well-function plot (typecurve). The two curves are the same except for a displacement. The point (1,1) is used to calculate the displacement (figure below and the book uses another point). 44
Hydrogeology
10.2. UNSTEADY FLOW TO WELLS
T and S are found from the y- and x displacement. From y: T =
Q 4πs1
Where s1 is the drawdown for W (u) = 1. From x: S =
| rt | 4T = 1 · 4T 2 1
S =
10.2.3
4T tu r2
Storage in confined aquifers
Once pumping is begun from a confined aquifer, a small influx (downward and upward) is flowing from the confining layers. This distorts the Theis curve. In this situation, a different type curve must be used, that accounts for the small storage in the confining layers. 45
Hydrogeology
10.2.4
CHAPTER 10. ANALYTICAL MODELS
Barometer influences in confined aquifers
Normally: Pressure 0 (atmospheric) at watertable in an unconfined aquifer. In a confined aquifer the piezometric surface is dependent on atmospheric pressure. High atmospheric pressure ⇐⇒ Low piezometric level Air pressure must be monitored when observation data is measured from a confined aquifer. As time goes on, the confined aquifer re compensates for the changed atmospheric pressure.
10.3 10.3.1
Unsteady flow to wells in a confined aquifer Elasticity
The porous skeleton and the water is incompressible. In a confined aquifer the confining upper layer is carried by granular stress (particle contact) and porewater pressure. Elasticity modulus of water: β =
δρ w /δp ρw
Elasticity modulus of skeleton (grains): α =
δdz/δp dz
The expansion/contraction takes place along the vertical direction (up/down):
⇒ [α + nβ ]gρ
w
δh δt
≡ S δh δt s
S s : Storativity
10.3.2
Transient flow in a confined aquifer
Transient: Midlertidlig, kortvarig Impermeable top + bottom. When pumping starts, a cone of depression is created in the piezometric surface. With time, the cone expands out laterally. Water is supplied by elasticity. u=
r2 S 4T t
46
Hydrogeology
10.4. LEAKY AQUIFERS
s = φ0
Q W (u) − φ = 4πT w
W (u): Well function; exponential integral
s: Drawdown at position r at time t. The Well function: W (u) =
where ξ → 0 for u → 0 i.e. t → ∞. If u < 0.01: s=
−0.5772 − ln(u) + ξ
Q 2.25T t 2.303Q 2.25T t ln 2 = log10 2 4πT r S 4πT r S
The minimum distance for no influence from well at time t: s=
Q 2.25T t ln 2 =0 4πT r S
2.25T t =1 r 2 S
10.4 10.4.1
⇒ r = 1.5
Leaky aquifers Leaky aquifer geometry
47
Tt S
Hydrogeology
10.4.2
CHAPTER 10. ANALYTICAL MODELS
Drawdown
A: No leakage, Theis eq. B: Leakage, no storage in semipermeable layer ( S s = 0) C: Leakage, and storage in semipermeable layer ( S s = 0) Confined aquifer with leakage. With existence of a constant head above semipermeable layer: δ2 φ 1 δφ φ0 φ S δφ + + = δr 2 r δr λ2 T δt
−
Where
φ0 −φ λ2
is the leakage component.
The drawdown: B=
Q s(r, t) = W (u,r/B) , 4πT
b T /K : Leakage factor
Q: Amount of pumping r: Distance from well t: Time b’: Thickness of the confining layer K’: Hydraulic conductivity of the confining layer 48
r2 S u= 4T t
Hydrogeology 10.5. UNCONFINED AQUIFERS/WATER TABLE AQUIFERS, S&Z CH.11
10.4.3
Type curves
r (Distance from well) is helt constant, and 1/u becomes another value for time ( 1/u ∞ ⇒ t → ∞).
→
The type curve for a leaky, confined aquifer will stabilize after a certain time-period, dependent on the thickness and hydraulic conductivity of the semipermeable layer.
Three types of curves:
1. Confined aquifer: Theis’ eq 2. Leaky, confined aquifer with no storage in semipermeable layer 3. Leaky, confined aquifer with storage in semipermeable layer ( S s )
Data from observational wells (t,s)=(time,drawdown) is fitted with a typecurve.
10.5
Unconfined aquifers/Water table aquifers, S&Z ch.11
= Water table aquifers = Phreatic aquifers 49
Hydrogeology
10.5.1
CHAPTER 10. ANALYTICAL MODELS
Geometry
Note on the geometry of a water table aquifer: The section of the well that contains of screen does often in reality not completely cover the top to bottom of the aquifer. If the screen section is only partially covering the aquifer thickness, there is a loss due to convergence and compression of the flow lines in the aquifer near the well.
10.5.2
Pumping
s : Correted drawdown s < 0.02: Drawdoen less than 2% of aquifier height
⇒ Approximation similar to Theis’ eq.
Drawdown in a watertable aquifer is much less than in a confined aquifer with the same amount of pumping. δ2 s 1 δs S y δs + = δr 2 r δr KH 0 δt
50
Hydrogeology 10.5. UNCONFINED AQUIFERS/WATER TABLE AQUIFERS, S&Z CH.11
10.5.3
Drawdown and delayed storage, Neumann type curves
Initially rapid drawdown, after a while drawdown will almost stop. After a longer period it starts again = Delayed storage type curve (note double log).
10.5.4
Early and late effects
The vertical conductivity is often lower because of lower grain size layers (drapings), which means anisotropic conductivity. This means that for some time, drawdown from well does not reach the upper laying water table. The response will in the beginning lay on the elastic properties of the aquifer. This means two u-values (early and late). Two Theis curves: s=
Q W (uA , uB , β ) 4πT
For early data (elasticity): 1 Tt = 2 uA sr
Late data (specific yield): 1 Tt = uB S y r 2
K z r2 β = K r b2 K z : Vertical hydraulic conductivity, K r : Horizontal hydraulic conductivity.
To determine the long-time effects a lot of testing and modeling is needed. 51
Hydrogeology
10.5.5
CHAPTER 10. ANALYTICAL MODELS
Summary
The first part of the curve i.e. the early data is similar to the response in a confined aquifer: Q s= W (uA, β ) ; 4πK b
r2 S uA = 4Kbt
The second part is somewhat similar to leakage, i.e.: K z r2 β = K r b2
The third part is a typical water table response with storativity S y , but where the start of the uB -curve is determined by β : Q s= W (uB , β ) ; 4πK b
r2 S y uB = 4Kbt
For a piezometer or an observation well which have short screens, the response depends on the vertical position of the screen as well as on the anisotropy, i.e. the ratio between K z and K r . 52
Hydrogeology
10.6
10.6. SLUG TESTING AND STEPWISE TESTING, S&Z CH.12
Slug testing and stepwise testing, S&Z ch.12
A solid bar of iron is dropped into the well, that rises the water level in the well. This additional head makes the water flow from the well into the aquifer. After equilibrium is reattained, the steel bar is rapidly lifted up. The lower hydraulic head in the well makes the water flow into the well from the aquifer. This gives local, quick and cheap information about the aquifer. The general solution for the hydraulic conductivity for a Hvorslev slug test is: K =
A 1 H 1 ln F t2 t1 H 2
−
A: Cross sectional area of the well H 0 : Initial water level (used for normalizing) H 1 and H 2 : Water levels just after slug is placed/removed. F : Shape factor, see table 12.1 in textbook.
A common Hvorslev test where length L > 8R, where R is the radius of the piezometer: F =
⇒
2πL ln(L/R)
R2 ln(L/R) K = 2LT 0
·
Where T 0 is the time for which the rise of the water table is 37% of the initial column of water i.e. before removal of the slug. With a confined aquifer a more advanced method is used (Cooper-Bredehoeft-Padadopulos).
10.7
Bounded aquifers
= Aquifer without infinite horizontal extent and isotropy. Negative boundary: Impermeable limit zone Positive boundary: Boundary which can supply water. For example a stream, that is
not affected by loss of water into aquifer, because the discharge in a stream is much greater. 53
Hydrogeology
CHAPTER 10. ANALYTICAL MODELS
1) Semi infinite aquifer 2) Buried valley: When the drawdown reaches the valley sides, water must be supplied by valley up- or downstream, because lateral extent is not infinite. 3) Strip aquifer 4) Strip aquifer 5) Infinite aquifer with positive boundary (stream). The hydraulic head can never be changed at stream. 6) When left boundary (with lower K) is reached by drawdown, discharge drops.
10.7.1
Drawdown near positive boundary
The hydraulic head can never be changed at the positive boundary (fx stream). A fictional recharge well is placed symmetrically on the opposite side of the stream. The recharge well will have the same recharge ( | − Qw |) as the real well has discharge (| + Qw |). This will create constant head at stream location. No matter how big the discharge at a well is, the system will be unaffected on the other side of a stream. 54
Hydrogeology
10.7. BOUNDED AQUIFERS
Pumping near stream:
Qw r Qw (x + x0 )2 + y 2 s(r) = ln = ln 2πT r 4πT (x x0)2 + y 2
−
55
Hydrogeology
CHAPTER 10. ANALYTICAL MODELS
Discharge from stream: The influence of the pumping well on the stream, i.e. how
much water the stream loses by passing the well field. The ecosystem can be damaged by well pumping near upstream areas. The inflow from the stream at x=0 over a unit of length on y is integrated, and the specific discharge is found. Q/Qq : Amount of water extracted from section of stream/y-axis in relation to the total discharge from well. Q 2 = tan−1 Qw π
d x0
x0 : (Shortest) distance between stream and well d: Length of stream (starting at midpoint) [tan−1 ]: rad
10.7.2
Drawdown near negative boundary
There can be no velocity components from boundary, since it acts as an impermeable boundary. A fictional well is placed symmetrically opposite of the limit zone with respect to the real well. The imaginary well will have the same discharge ( +Qw ) as the real well. The symmetrical nature creates a no-flow condition at the boundary exactly between the two = 0). The drawdown at the boundary will be twice the drawdown in an unlimited wells ( δφ δx aquifer. 56
Hydrogeology
10.7. BOUNDED AQUIFERS
r: Distance to observational well from real well r’: Distance to observational well from imaginary wellă rw : Radius of the well The drawdown at the impermeable boundary r = r : Qw r s(r = r ) = ln πT rw
In real situations the radius of the well ( rw ), the value can be uncertain because of disturbance of the formation while drilling (especially if the drilling happens with a wider bit than the screen). 57
Hydrogeology
10.7.3
CHAPTER 10. ANALYTICAL MODELS
Pumping situations with multiple positive/negative boundaries
The discharge/recharge has the same numerical value in each situation below. All situations must balance out in relation to the symmetry lines.
10.7.4
Well between two streams in infinite strip
When trying to balance this, an infinite amount of imaginary wells are placed at a longer and longer distance from the real well. There is a solution, see litterature (will not be required at exam).
10.8
Well in uniform flow
Combination of natural flow and drawdown. The result is obtained by adding (superposition). Constant flow: Linear hydraulic head. All water within the groundwater divide will be flowing into the well. 58
Hydrogeology
10.8. WELL IN UNIFORM FLOW
Influx from regional flow q0, [m3/s per m2 = m/s]. Aquifer thickness: B, regional flow has head 0. Stagnation point is examined (singular point). Steady flow to well: φQ =
Qw ln(x2 + y 2 ) 4πT
Steady and isotropic flow. When two gradients are perpendicular: δφ δψ = δx δy
⇒
δφ q0 B Qw 2x δψ = + = δx T 4πT x2 + y2 δy q0 B Qw 2x 1 ψ= dy + dy T 4πT x2 + y 2 q0 B Qw y Ψ= y+ tan−1 T 2πT x
⇒
At the divide: Ψ = 0 q0 B y= T
−
Qw y tan−1 2πT x
⇒ 59
tan−1
y = x
⇒
− 2πqQ By 0
w
Hydrogeology
CHAPTER 10. ANALYTICAL MODELS y = x
±tan 2πqQ By 0
with + for y > 0, - for y < 0
w
Stagnation for: Ψ = 0 for y = 0
(xs , 0) =
10.9
−Q
w
2πq 0B
Abstraction - long term influences, S&Z ch.8
See additional material: ’Noget om følgevirkninger af grundvandsindvinding’ by Henrik Kjærgaard.
With groundwater extraction, the regional hydraulic pattern is changed. The groundwatercatchment can be altered. 60
Hydrogeology
10.9. ABSTRACTION - LONG TERM INFLUENCES, S&Z CH.8
In the figure: The water table is higher in the till (solid line) than the hydraulic head in the aquifer (dashed). This creates a gradient, so water will move down into the aquifer. Near the stream the aquifer has higher hydraulic head, which creates an upward flux of water. The specific yield of the till is about 3 − 5%. The zone with upward movement is often smaller than the downflow area. It is common to find lenses of (glacial) coarser grained material in the stream zone. With pumping, the groundwater divide is moved away from the well. In the last figure, the time is increased. The stream now acts as a water supplier, and not a zone of upflow. When the watertable in the till is lowered, the upper zone is now oxigenized. This changes the solubility of a lot of metals (Sulfur, cadmium, nickel, arsen, etc), which can be a problem 61
Hydrogeology
CHAPTER 10. ANALYTICAL MODELS
in dissolved state in the groundwater. The lowered hydraulic head in the aquifer increases the intergranular stress. This may lead to consolidation and lowering of the topographic surface.
10.9.1
Release of water - initial phase
Release from aquifer, and additional release from aquitards. 62
Hydrogeology
10.9.2
10.10. SALT WATER INTRUSION
Release of water - late phase
The watertable starts to decline. The aquifer can over the long term not deliver unlimited amounts of water. Drawdown uses the entire storage capacity.
10.10
Salt water intrusion
Saltwater has higher density than fresh water. Hydraulic head: φ=z+
p ρg
,
ρf < ρs
For obtaining equal pressure, a freshwater column is 4% higher than a saltwater column: ρs = 1.04, ρf = 1.00
63
⇒
hf = 1.04 hs
·
Hydrogeology
CHAPTER 10. ANALYTICAL MODELS
If an aquifer contains saline water, the apparent hydraulic head (point water pressure head) looks lower. A salinity test will show, that the hydraulic head must be multiplied with a factor, to find the equivalent fresh-water head.
More: Ghyben - Herzberg equilibrium on interface; See slides. These formulas are simplified though, as there is no seepage. Glover (1964) is a better approximation (note: x-direction opposite).
10.11
Stream groundwater interaction
Article by C.T. Jenkins
What happens to a stream on short term basis, not long term (steady state) as calculated before.
Example: Full hydraulic contact through watertable aquifer. Stream is linear (line-source). The influence is integrated over stream length.
Parameters: r2 S sdf = u= 4T t 4t l2 S sdf = T sdf : Stream depletion factor, dimensionless and constant q/Q : Dimensionless pumping rate, the amount of stream water pumped up by well.
(1=100%) V /(Qt) : Dimensionless volume 64
Hydrogeology
10.11. STREAM GROUNDWATER INTERACTION
In the graph, 1 − q/Q is added to make it possible to read more precise values. First t/sdf is calculated, and then q/Q is found on curve A.
65
Appendix A Hydrogeology terms Aquifer: A geologic unit capable of supplying useable amounts of groundwater to a well
or spring. Classification of a water-bearing unit as an aquifer may depend upon local conditions and context of local water demands. Aquitard: A bed or unit of lower permeability, that can store water but does not readily
yield water to pumping wells. Cappilary fringe: The lowest part of the unsaturated zone in which the water in pores is
under pressure less than atmospheric, but the pores are fully saturated. This water rises against the pull of gravity due to surface tension at the air-water interface and attraction between the liquid and solid phases. Confined aquifer: An aquifer that is completely saturated and overlain by a confining
unit. Darcy: A unit of permeability equal to 9.87 10−13 m2, or for water of normal density and viscosity, a hydraulic condctivity of 10−5 ms
∼
·
Drawdown (s): The difference between static water level (water table in unconfined
aquifers or potentiometric surface in confined aquifers), and pumping water level in a well. Effective porosity (ne ): The percent of total volume of rock or soil that consists of
interconnected porespaces, as used in describing groundwater flow and contaminant transport.
Hydraulic conductivity (K ): The proportionality constant in Darcy’s law – a measure of a porous medium’s ability to transmit water. K incorporates properties of both
the medium and fluid. 66
Hydrogeology Hydraulic head (h): A measure of the potential energy of groundwater, it is the level
to which water in a well or piezometer will rise if unhindered. Total hydraulic head is the sum of two primary components: elevation head and pressure head. The third component, velocity head, is generally negligible in groundwater. Hydrostratigraphic unit: A formation, part of a formation, or group of formations with
sufficiently similar hydrologic characteristics to allow grouping for descriptive purposes. Permeability (k): A proportionality constant that measures a porous medium’s ability to
transmit a fluid. It is a function of the medium’s physical properties. Permeability is dependent solely on properties of the porous medium, and is related to the hydraulic conductivity (K ), the dynamic viscosity ( µ) and the density of the fluid ( ρ). Perched groundwater: Unconfined groundwater separated from an underlying zone of
groundwater by an unsaturated zone. This usually occurs atop lenses of clay or low-permeability material above the groundwater table. Porosity (n): The ratio of void space to total volume of a soil or rock. Potentiometric surface: A surface constructed from measurements of head at individual
wells of piezometers that defines the level to which water will rise from a single aquifer. Saturated zone: The zone in which 100% of the porosity is filled with water. Specific capacity: The yield of a well per unit of drawdown ( s). Specific retention (S r ): The volume of water that remains in a porous material after complete drainage under the influence of gravity. S r + S y = n (total porosity) Specific storage (S s ): The amount of water per unit volume of a saturated formation
that is stored or expelled per unit change of head due to the compressibility of the water and aquifer skeleton.
Specific yield (S y ): The volume of water that drains under the influence of gravity from
a porous material.
Static water level: The level to which water rises in a well or unconfined aquifer when
the level is not influenced by pumping. Storativity (S ): The volume of water that a permeable unit releases from or takes into
storage per unit surface area per unit change in head. In unconfined aquifers it is equal to specific yield ( S y ). The storativity in a confined aquifer is the product of specific storage ( S s) and the aquifer thickness ( b). Transmissivity (T ): A measure of the amount of water that can be transmitted horizontally through a unit width by the full saturated thickness of an aquifer. T is equal to the product of hydraulic conductivity ( K ) and saturated aquifer thickness ( b).
67
Hydrogeology
APPENDIX A. HYDROGEOLOGY TERMS
Unconfined aquifer: (alternate terms: phreatic- or watertable aquifer). An aquifer that
is only partially filled with water and in which the upper surface of the saturated zone is free to rise and fall. Unsaturated zone: The zone in which soil/sediment/rock porosity is filled partly with
air and partly with water. Water table: The top of the zone of saturation – the level at which the atmospheric
pressure is equal to the hydraulic pressure. In unconfined aquifers the water table is represented by the measured water level in observation wells. Well efficiency: The ratio of theoretical drawdawn (drawdown in the aquifer at the radius
of the well) to the observed drawdown inside a pumping well. Well yield: The volume of water per unit of time discharged from a well by pumping or 3 free flow. It is commonly reported as a pumping rate ( Q) in ms .
68
Appendix B Commonly used formulas
B.1
Basic hydraulics in streams
Rectangular channel
Area of the flow Wetted perimeter Hydraulic radius Froudes number F r < 1: Subcritical F r = 1: Critical F r > 1: Supercritical Gravitational acceleration Critical depth (F r = 1) Specific energy Velocity coefficient Manning number
A=b y P = b + 2y by R = b+2y F r = √V g·y
[m2] [m] [m] [ ]
g
m2 s
·
−
≈ 9.81
yc =
3
q2 g
[m] 2
αQ E y + 2gA 2 α 1 √6 M = 25.4 k M = by( by Q)2/3√S
≈ ≈
Velocity (M) Discharge(M) Discharge Hydrostatic pressure distribution Reynolds no. (< 580: Laminar, > 750: Turbulent) Kinematic viscosity (water, T = 20o C ) Specific flow rate 69
−
√ V = M · R · S √ Q = A · M · R · S Q = A · V 0
b+2y
2/3
0
2/3
z + γ p = h = constant Re = Vν R ν = µρ = 1 10−6 q = Q/b
·
[J ] [ ] [m1/3 s−1 ] [m1/3 s−1 ]
0
[ ms ] 3 [ ms ] 3 [ ms ] [m] [ ] 2 [ ms ] 2 [ ms ]
−
· ·
Hydrogeology
APPENDIX B. COMMONLY USED FORMULAS
Conservation laws
Rate of change of mass Continuity equation* Time Displacement in flow direction Momentum of a moving point Conservation of momentum* Longitudinal channel bottom slope Friction slope
∆(ρA∆x) ∆t δA + δQ δt δx
[ kgs ]
=0
t x p = mV 1 δV + V g δV + g δt δx S 0 = sinθ F f S f = γA∆x
[s] [m] [kg δy δx
+ S f
− S = 0 0
·
m ] s
[ ]
−
*) If the flow is steady (the flow conditions not varying with time), the partial derivative terms with respect to time can be dropped from the continuity, momentum and energy equations. See page 18, Akan. Loss in pipes
Inlet loss Pipe loss (friction) Outflow loss Total loss
2
Q ∆H I = 0.5 2gA [m] 2 L ∆H F = M 2 R4/3 A2 Q [m] αQ2 ∆H u = 2gA [m] 2 See section 6.4 [m]
·
Hydraulic structures: See chapter 7.
B.2
Groundwater
The Darcy equation is valid for fully laminar flow in homogenous material. Darcy’s equation
Hydraulic conductivity Permeability Dynamic viscosity (water, T = 20o C ) Hydraulic gradient Darcy’s equation Transmissivity Storativity (assuming HPD) Specific storativity Porosity
K = kρµw g k µ = 1.002 10−3 −h1 i = δh = h2∆l δl q = Ki or Q = KiA T = K b S = S s B S s nT
·
−
·
70
[ ms ] [m2 ] [P a s] [ ] 2 3 [ ms ] or [ ms ] 2 [ ms ] 2 [ mm ] 3 [ mm ] [ ]
−
−
·
Hydrogeology
B.2.1
B.2. GROUNDWATER
W(u) - Theis equation/Well function
W(u) is the Well function , that is called the exponential integral, E 1 , in non-hydrogeology literature. The value can be calculated with Matlab’s expint(u). A series of W (u) values can be looked up in table 9.2 (p. 225) in Schwarz & Zhang.
B.2.2
Unconfined aquifer: Steady 1D groundwater flow
When observing ground water elevation, the height (h) of the groundwater table is a good expression for the energy, because the velocity is relatively very small, and the pressure at the groundwater table equals atmospheric pressure. Condition: For expressing the flow as 1D flow (Dupuits approximation), the height variations of the saturated zone must be less or equal than 10% the mean height of the saturated zone: ∆hsat ¯ h
≤ 0.1
The problem: The amount of increase in specific discharge (dq = qout − qin ) over a flow distance (dx) is the supplied by seepage (W): δq =W δx
,
q=
−T δh δx
The differential equation for 1D flow in a homogeneous and isotropic aquifer:
⇒
δ (q) = δx
δ2 h T 2 = W δx
−
2
δh W =− x+c ⇒ δδxh = − W ⇒ T δx T
1
2
W x ⇒ h = − 2T
2
+ c1 x + c2
By determining the boundary conditions, the constants can be found, thus solving the differential equation.
B.2.3
Confined aquifer: Steady 1D groundwater flow
A confined aquifer is totally filled by water under pressure. The piezometric surface ( φ) is the corresponding water-pressure surface (groundwater-table in unconfined aquifer), observed in observation-drillings. Drainage lowers the pressure because of elastic storativity. 71