Darcys Law

Darcy’s lawGroundwater Hydraulics

Daene C. McKinney

Outline• Properties – Aquifer Storage• Darcy’s Law• Hydraulic Conductivity• Heterogeneity and Anisotropy• Refraction of Streamlines• Generalized Darcy’s 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

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 Retentionyr SS

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?DV = SADh = 2.1x10-6 ft3

Darcy

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

Darcy’s Experiments

• Discharge is Proportional to – Area– Head differenceInversely proportional to – Length

• Coefficient of proportionality is K = hydraulic conductivity

L

hhAQ 21

Darcy’s Data

Hydraulic Conductivity• 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 ρ = densityµ = dynamic viscosityg = specific weight

Porous medium property

Fluid properties

Hydraulic Conductivity

Groundwater Velocity

• q - Specific dischargeDischarge from a unit cross-section area of aquifer formation normal to the direction of flow.

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

Qq

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

FlowPorous medium

Example

K = 1x10-5 m/sf = 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

Dh

Circular hydraulic head contours

K, conductivity, Is constant

Hydraulic gradient

Specific discharge

Validity of Darcy’s Law

• We ignored kinetic energy (low velocity)• We assumed laminar flow• We can calculate a Reynolds Number for the flow

q = Specific discharged10 = effective grain size diameter

• Darcy’s Law is valid for NR < 1 (maybe up to 10)

Specific Discharge vs Head Gradient

q

Re = 10

Re = 1

Experiment shows this

a

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

Darcy Law predicts this

Estimating ConductivityKozeny – 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

Kozeny – Carman eq.

Measuring ConductivityPermeameter Lab Measurements

• Darcy’s 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 ConductivityConstant 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

• Darcy’s Law

Continuous Flow

OutflowQ

Overflow

A

Sample

head difference

flow

Measuring ConductivityFalling Head Permeameter

• Flow rate in the tube must equal that in the column

OutflowQ

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

verticalhorizontal KK

www.usgs.gov

Layered Porous Media(Flow Parallel to Layers)

3K

2K

1K

W

b

1b

2b

3b

1Q

2Q

3Q

Dh

h2

h1

Piezometric surface

Q

datum

Layered Porous Media(Flow Perpendicular to Layers)

Q

3K2K1K

bQ

L

L3L2L1

Dh1

Piezometric surface

Dh2

Dh3

Dh

Anisotrpoic Porous Media• General relationship between specific

discharge and hydraulic gradient

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

Groundwater Flow Direction

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

Groundwater Flow Direction

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 Direction

Negative of head gradient in x direction

Negative of head gradient in y direction

Magnitude of head gradient

Direction of flow

xq = -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

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

1KUpper Formation

12 KK

y

x

1

2

2q

1q

Lower Formation

Summary• Properties – Aquifer Storage• Darcy’s Law

– Darcy’s Experiment– Specific Discharge– Average Velocity– Validity of Darcy’s Law

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

• Heterogeneity and Anisotropy– Layered Porous Media

• Refraction of Streamlines• Generalized Darcy’s Law