darcys law

Darcys law

Groundwater Hydraulics

Outline

Properties Aquifer Storage

Darcys Law

Hydraulic Conductivity

Heterogeneity and Anisotropy

Refraction of Streamlines

Generalized Darcys Law

Aquifer Storage

Storativity (S) - ability of an aquifer to store water

Change in volume of stored water due to change in piezometric head.

Volume of water released (taken up) from aquifer per unit decline (rise) in piezometric head.

Unit area

Unit decline

in head

Released

water

Aquifer Storage

Fluid Compressibility (b)

Aquifer Compressibility (a)

Confined Aquifer Water produced by 2

mechanisms

1. Aquifer compaction due to increasing effective stress

2. Water expansion due to decreasing pressure

Unconfined aquifer Water produced by draining

pores

gV a

S = rg(a +fb)

S = Sy

Unconfined Aquifer Storage

Storativity of an unconfined aquifer (Sy, specific yield) depends on pore space drainage.

Some water will remain in the pores - specific retention, Sr

Sy = f Sr

Unit area

Unit decline

in head

Released

water

Porosity, Specific Yield, & Specific Retention

yr SS f

Confined Aquifer Storage

Storativity of a confined aquifer (Ss) depends on both the compressibility of the water (b) and the compressibility of the porous medium itself (a).

Unit area

Unit decline

in head

Released

water

Example

Storage in a sandstone aqufier

f = 0.1, a = 4x10-7 ft2/lb, b = 2.8x10-8 ft2/lb, g = 62.4 lb/ft3

ga 2.5x10-5 ft-1 and gbf 1.4x10-7 ft-1

Solid Fluid

2 orders of magnitude more storage in solid

b = 100 ft, A = 10 mi2 = 279,000,000 ft2

S = Ss*b = 2.51x10-3

If head in the aquifer is lowered 3 ft, what volume is released?

V = SAh = 2.1x10-6 ft3

Darcy

http://biosystems.okstate.edu/Darcy/English/index.htm

hL

z1

z2

P1/g

P2/g

Q

Q

L

v

Sand column

Datum plane

Are

a, A

h1 h2

Darcys Experiments

Discharge is Proportional to

Area

Head difference

Inversely proportional to

Length

Coefficient of proportionality is K = hydraulic conductivity

L

hhAQ 21

Q = -KA

h2 - h1

LQ = -KA

Dh

L

Darcys Data

0

5

10

15

20

25

30

35

0 5 10 15 20

Flo

w, Q

(l/

min

)

Gradient (m/m)

Set 1, Series 1

Set 1, Series 2

Set 1, Series 3

Set 1, Series 4

Set 2


Has dimensions of velocity [L/T]

A combined property of the medium and the fluid

Ease with which fluid moves through the medium

k = cd2 intrinsic permeability

= density

= dynamic viscosity

g = specific weight

Porous medium property

Fluid properties

Groundwater Velocity

q - Specific discharge

Discharge from a unit cross-section area of aquifer formation normal to the direction of flow.

v - Average velocity

Average velocity of fluid flowing per unit cross-sectional area where flow is ONLY in pores. A

Qq

ff A

Qqv

dh = (h2 - h1) = (10 m 12 m) = -2 m

J = dh/dx = (-2 m)/100 m = -0.02 m/m

q = -KJ = -(1x10-5 m/s) x (-0.02 m/m) = 2x10-7 m/s

Q = qA = (2x10-7 m/s) x 50 m2 = 1x10-5 m3/s

v = q/f = 2x10-7 m/s / 0.3 = 6.6x10-7 m/s

/

h1 = 12m h2 = 12m

L = 100m

10m

5 m

Flow Porous medium

Example

K = 1x10-5 m/s

f = 0.3

Find q, Q, and v

Hydraulic Gradient

Gradient vector points in the direction of greatest rate of increase of h

Specific discharge vector points in the opposite direction of h

Well Pumping in an Aquifer

Aquifer (plan view)

y

h1 < h2 < h3

x

h1 h2 h3

Well, Q

q

h

Circular hydraulic

head contours

K, conductivity,

Is constant

Hydraulic gradient

Specific discharge

Validity of Darcys Law

We ignored kinetic energy (low velocity)

We assumed laminar flow

We can calculate a Reynolds Number for the flow

q = Specific discharge

d10 = effective grain size diameter

Darcys Law is valid for NR < 1 (maybe up to 10)

NR =rqd10

m

Specific Discharge vs Head Gradient

q

Re = 10

Re = 1

Experiment

shows this

a

tan-1(a)= (1/K)

Darcy Law

predicts this

Estimating Conductivity Kozeny Carman Equation

Kozeny used bundle of capillary tubes model to derive an expression for permeability in terms of a constant (c) and the grain size (d)

So how do we get the parameters we need for this equation?

22

32

)1(180dcdk

f

fKozeny Carman eq.

Measuring Conductivity Permeameter Lab Measurements

Darcys Law is useless unless we can measure the parameters

Set up a flow pattern such that

We can derive a solution

We can produce the flow pattern experimentally

Hydraulic Conductivity is measured in the lab with a permeameter

Steady or unsteady 1-D flow

Small cylindrical sample of medium

Measuring Conductivity Constant Head Permeameter

Flow is steady

Sample: Right circular cylinder Length, L

Area, A

Constant head difference (h) is applied across the sample producing a flow rate Q

Darcys Law

Continuous Flow

Outflow

Q

Overflow

A

Q = KAb

L

Sample

head difference

flow

Measuring Conductivity Falling Head Permeameter

Flow rate in the tube must equal that in the column

Outflow

Q

Qcolumn = prcolumn2 K

h

L

Qtube = prtube2 dh

dt

rtubercolumn

2L

K

dh

h= dt

Sample

flow

Initial head

Final head

Heterogeneity and Anisotropy

Homogeneous

Properties same at every point

Heterogeneous

Properties different at every point

Isotropic

Properties same in every direction

Anisotropic

Properties different in different directions

Often results from stratification during sedimentation

verticalhorizontalKK

www.usgs.gov

Example

a = ???, b = 4.673x10-10 m2/N, g = 9798 N/m3,

S = 6.8x10-4, b = 50 m, f = 0.25,

Saquifer = gabb ???

Swater = gbfb

% storage attributable to water expansion

%storage attributable to aquifer expansion

Layered Porous Media (Flow Parallel to Layers)

3K

2K

1K

W

b

1b

2b

3b

1Q

2Q

3Q

h

h2

h1

Piezometric surface

Q

datum

Layered Porous Media (Flow Perpendicular to Layers)

Q

3K

2K

1K

b Q

L

L3 L2 L1

h1

Piezometric surface

h2

h3

h

Example

Find average K

Flow Q

Anisotrpoic Porous Media

General relationship between specific discharge and hydraulic gradient

K is symmetric, i.e., Kij = Kji.

Principal Directions

Often we can align the coordinate axes in the principal directions of layering

Horizontal conductivity often order of magnitude larger than vertical conductivity

qx = -K xxh

x

qy = -K yyh

y

qz = -Kzzh

z

qx

qy

qz

= -

K xx 0 0

0 K yy 0

0 0 Kzz

h

xh

yh

z

Kxx = Kyy = KHoriz >> Kzz = KVert

Groundwater Flow Direction

Water level measurements from three wells can be used to determine groundwater flow direction

Groundwater

Contours

Groundwater

Flow, Q

x

y

z

Head Gradient, J

hk hj

hi hi > hj > hk

h1(x1,y1) h3(x3,y3)

h2(x2,y2)


Magnitude of head gradient =

Angle of head gradient =

Head gradient =


Set of linear equations can be solved for a,

b and c given (xi, hi, i=1, 2, 3)

3 points can be used to

define a plane

Equation of a plane in 2D Groundwater

Flow, Q

x

y

z

Head Gradient, J

h1(x1,y1) h3(x3,y3)

h2(x2,y2)


Negative of head gradient in x direction

Negative of head gradient in y direction

Magnitude of head gradient

Direction of flow

x

Well 2

(200 m, 340 m)

55.11 m

Well 1

(0 m,0 m)

57.79 m

Well 3

(190 m, -150 m)

52.80 m

Example Find:

Magnitude of head gradient

Direction of flow

y

Contour Map of Groundwater Levels

Contours of groundwater level (equipotential lines) and Flowlines (perpendicular to equipotiential lines) indicate areas of recharge and discharge

Refraction of Streamlines

Vertical component of velocity must be the same on both sides of interface

Head continuity along interface

So

2K

1K

Upper Formation

12KK

y

x

1

2

2q

1q

Lower Formation

qy1= qy2

q1 cosq1 = q2 sinq2

h1 = h2 @ y = 0

K1K2

=tanq1tanq2

Summary

Properties Aquifer Storage Darcys Law

Darcys Experiment Specific Discharge Average Velocity Validity of Darcys Law

Hydraulic Conductivity Permeability Kozeny-Carman Equation Constant Head Permeameter Falling Head Permeameter

Heterogeneity and Anisotropy Layered Porous Media

Refraction of Streamlines Generalized Darcys Law

Darcys Law Examples

Example

a = ???, b = 4.673x10-10 m2/N, g = 9798 N/m3,

S = 6.8x10-4, b = 50 m, f = 0.25,

Saquifer = gabb ???

Swater = gbfb = (9798 N/m3)(4.673x10-10 m2/N)(0.25)(50 m)

= 5.72x10-5

percent of storage coefficient attributable to water expansion

= Swater /S = 5.72x10-5 /6.8x10-4 *100 = 8.4%

percent of storage coefficient attributable to aquifer expansion

= Saquifer /S = 1 (Swater /S ) = 91.6%

Example

Flow Q

Kh,A =K1z1 +K2z2

z1 + z2=

(2.3 m / d)(15 m)+ (12.8 m / d)(15 m)

(15 m)+ (15 m)= 7.55 m / d

Example Flow Q

Kv,A =z1 + z2z1K1

+z2K2

=(15 m)+ (15 m)

15 m

2.3 m / d+

15 m

12.8 m / d

= 3.90 m / d

x = -5.3 deg

Well 2

(200, 340)

55.11 m

Well 1

(0,0)

57.79 m

Well 3

(190, -150)

52.80 m

Example

darcys law

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