MATH 1050QCMathematical Modeling in the Environment
Lecture 5. Darcy’s Law.
Dmitriy Leykekhman
Spring 2009
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 1
Experimental setup for pushing water through a geologicalsample
figure 2.7 from C. Hadlock’s book
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 2
Two-dimensional simplification from the previous figure
figure 2.8 from C. Hadlock’s book
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 3
Doubling the pressure
figure 2.9 from C. Hadlock’s book
Question: what do you think would happen to the rate at which waterwould be forced through the sample?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 4
Doubling the pressure
figure 2.9 from C. Hadlock’s book
Question: what do you think would happen to the rate at which waterwould be forced through the sample?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 4
Doubling the length of the sample
figure 2.10 from C. Hadlock’s book
Question: what do you think would happen to the flow rate in suchexperimental setup?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 5
Doubling the length of the sample
figure 2.10 from C. Hadlock’s book
Question: what do you think would happen to the flow rate in suchexperimental setup?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 5
Doubling the pressure and the length of the sample
figure 2.11 from C. Hadlock’s book
Question: what do you expect the net effect of the flow rate to be?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 6
Doubling the pressure and the length of the sample
figure 2.11 from C. Hadlock’s book
Question: what do you expect the net effect of the flow rate to be?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 6
Doubling the cross-section area of the sample
figure 2.12 from C. Hadlock’s book
Question: what do you expect to happen to the total amount of waterforced through the apparatus?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 7
Doubling the cross-section area of the sample
figure 2.12 from C. Hadlock’s book
Question: what do you expect to happen to the total amount of waterforced through the apparatus?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 7
Summary of principles
1. More pressure will increase movement (flow rate) of water.
2. Longer length of aquitard substances will decrease the flow rate.
3. Larger cross section will increase flow rate, but not the speed!
4. Flow rate will depend on the properties of the geologic mediumunder consideration.
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 8
Revision of the experimental apparatus
figure 2.13 from C. Hadlock’s bookD. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 9
Summary of previous principles in mathematical language
1. The flow rate is proportional to the net driving pressure, representedby ∆h
2. The flow rate is inversely proportional to the length L of the sample
3. The flow rate is proportional to the cross-sectional area A of flowpathway
4. The constant of proportionality for the above relationships willnaturally depend on the specific geological medium
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 10
Physical parameters and their relationship
I h1, h2 are the heights of the water above some arbitrary plane.
I ∆h = h1 − h2I L is the length of the medium through which the water is moving.
I A is the cross sectional area.
I Q is the total flow rate, measure in volume per unit time.
I K is the corresponding constant of proportionality (hydraulicconductivity)
Q = K ×∆h× 1L×A
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 11
Typical ranges of values of porosity and hydraulicconductivity
Geological Medium Porosity, η Hydraulic Conductivity K (ft/day)
Gravel 0.25-0.40 100-100,100
Sand 0.25-0.50 0.01-1000
Silt 0.35-0.50 0.001-0.1
Clay 0.40-0.70 0.0000001-0.001
Sandstone 0.05-0.30 0.00001-0.1
Limestone 0.001-0.20 0.0001-0.1
Granite (fractured) 0.0001-0.10 0.0001-10
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 12
Darcy’s Law
Let
i =∆hL
i is called the hydraulic gradient.
Theorem (Darcys Law)
Q = KiA.
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 13
Two monitoring wells
figure 2.14 from C. Hadlock’s bookD. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 14
Two monitoring wells
Question: What information do you need to apply the Darcy’s Law?
i =h1 − h2L
L is given. From the picture we can compute
h1 = 100 ft − 16 ft = 84 ft
andh2 = 96 ft − 18 ft = 78 ft
Hencei = (84− 78)/350 ≈ 0.17
We also need to know K. This can be obtained from the knowledge ofthe types of soil or rock aquifer consists of.
We also need to know A (not given in the picture.)
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 15
Two monitoring wells
Question: What information do you need to apply the Darcy’s Law?
i =h1 − h2L
L is given. From the picture we can compute
h1 = 100 ft − 16 ft = 84 ft
andh2 = 96 ft − 18 ft = 78 ft
Hencei = (84− 78)/350 ≈ 0.17
We also need to know K. This can be obtained from the knowledge ofthe types of soil or rock aquifer consists of.
We also need to know A (not given in the picture.)
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 15
Two monitoring wells
Question: What information do you need to apply the Darcy’s Law?
i =h1 − h2L
L is given. From the picture we can compute
h1 = 100 ft − 16 ft = 84 ft
andh2 = 96 ft − 18 ft = 78 ft
Hencei = (84− 78)/350 ≈ 0.17
We also need to know K. This can be obtained from the knowledge ofthe types of soil or rock aquifer consists of.
We also need to know A (not given in the picture.)
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 15
Two monitoring wells
Question: What information do you need to apply the Darcy’s Law?
i =h1 − h2L
L is given. From the picture we can compute
h1 = 100 ft − 16 ft = 84 ft
andh2 = 96 ft − 18 ft = 78 ft
Hencei = (84− 78)/350 ≈ 0.17
We also need to know K. This can be obtained from the knowledge ofthe types of soil or rock aquifer consists of.
We also need to know A (not given in the picture.)
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 15
Block diagram of idealized aquifer
figure 2.15 from C. Hadlock’s book
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 16
Computing the total flow rate
If we assume the value for the hydraulic conductivity K = 100 ft/day,picture gives all the info we need to compute the total flow rate.
i =h1 − h2L
=65 ft − 54 ft
1200 ft
A = 40 ft × 200 ft
Hence the total flow rate
Q = 100 ft/day× 65 ft − 54 ft1200 ft
× (40 ft × 200 ft) ≈ 7, 333 ft3/day
Question: How much in terms of gallons per day?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 17
Computing the total flow rate
If we assume the value for the hydraulic conductivity K = 100 ft/day,picture gives all the info we need to compute the total flow rate.
i =h1 − h2L
=65 ft − 54 ft
1200 ft
A = 40 ft × 200 ft
Hence the total flow rate
Q = 100 ft/day× 65 ft − 54 ft1200 ft
× (40 ft × 200 ft) ≈ 7, 333 ft3/day
Question: How much in terms of gallons per day?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 17
Computing the total flow rate
If we assume the value for the hydraulic conductivity K = 100 ft/day,picture gives all the info we need to compute the total flow rate.
i =h1 − h2L
=65 ft − 54 ft
1200 ft
A = 40 ft × 200 ft
Hence the total flow rate
Q = 100 ft/day× 65 ft − 54 ft1200 ft
× (40 ft × 200 ft) ≈ 7, 333 ft3/day
Question: How much in terms of gallons per day?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 17
Computing the total flow rate
If we assume the value for the hydraulic conductivity K = 100 ft/day,picture gives all the info we need to compute the total flow rate.
i =h1 − h2L
=65 ft − 54 ft
1200 ft
A = 40 ft × 200 ft
Hence the total flow rate
Q = 100 ft/day× 65 ft − 54 ft1200 ft
× (40 ft × 200 ft) ≈ 7, 333 ft3/day
Question: How much in terms of gallons per day?
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 17
Concentration of the contaminant
figure 2.16 from C. Hadlock’s book
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 18
Concentration of the contaminant
Let C denote the concentration of the contaminant ( measured inweight/volume ). Then,
C =contaminant inflow rate
Q, the water flow rate
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 19
Concentration of the contaminant
ExampleSuppose the flow rate in the aquifer Q = 1200 ft3/day and thecontaminant is leaking at the rate of 2 lb3/day.Thus the concentration of contaminant is
C =2 lb/day
1200 ft3/day≈ 0.00167lb/ft3.
Problem: Convert the result to kilogram per cubic meter kg/m3, to gramper cubic centimeter g/cm3, and to parts per million ppm.
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 20
Concentration of the contaminant
ExampleSuppose the flow rate in the aquifer Q = 1200 ft3/day and thecontaminant is leaking at the rate of 2 lb3/day.Thus the concentration of contaminant is
C =2 lb/day
1200 ft3/day≈ 0.00167lb/ft3.
Problem: Convert the result to kilogram per cubic meter kg/m3, to gramper cubic centimeter g/cm3, and to parts per million ppm.
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 20
Summary of physical parameters
I h1, h2 are the heights of the water above some arbitrary plane.
I ∆h = h1 − h2I L is the length of the medium through which the water is moving.
I A is the cross sectional area.
I Q is the total flow rate, measure in volume per unit time.
I K is the hydraulic conductivity.
I i = ∆h/L is the hydraulic gradient.I C is the concentration of the contaminant.
D. Leykekhman - MATH 1050QC Mathematical Modeling in the Environment Course info – 21