Week 5

Question 1

Fluid compressibility =

Answer 1

describes how the density of a fluid increases with increasing pressure

Question 2

Incompressible =

Answer 2

volume doesn’t change no matter how much stress is applied

Question 3

Density =

Answer 3

mass/volume

Question 4

Rock compressibility =

Answer 4

describes how the porosity of a rock increases with increasing pressure

• b/c increasing P = squash and reduce grain size = increase pore space

Question 5

Porosity =

Answer 5

pore space/volume

Question 6

/\density =

Answer 6

/\M / volume

Question 7

For infinitesimal /\P, Cf =

Answer 7

1/density x d(density)/dP

Question 8

/\porosity (/\n) =

Answer 8

/\pore space/volume

Question 9

For infinitesimal /\P, Cr =

Answer 9

1/n x dn/dP

Question 10

Concept of solving Cr = … or Cf = … for n and density

Answer 10

1. Integrate both sides
2. Apply constraints
3. Rearrange n = … or density = …

Question 11

Relationship between h and n

Answer 11

h = w + z

where w = P/density x g

SO change h = change P = change n

Question 12

Area x time x density x flow = (unit)

Answer 12

Mass M

• conservation of mass

Question 13

Fluid mass per unit vol =

Mass conservation

Answer 13

/\x/\y/\x/\m = mass in - mass out

Question 14

Finding equations for dm/dt, basic gist:

Answer 14

FIRST EQUATION
1. Mass conservation
/\x/\y/\x/\m = mass in - mass out

2. Reduce
3. Rearrange

SECOND EQUATION
1. m = density x porosity

2. Product rule
   dm/dt = d(np)/dt…
3. Chain rule as n and p are both f(P)

Question 15

Fluid mass per unit vol (m) =

Answer 15

Density x porosity

Question 16

Using equations for dm/dt to form an equation for the specific storage coefficient (and further steps)

Answer 16

1. Ss = n x density x g(Cr+Cf)
2. So both equations for dm/dt = Ss/g x dP/dt
3. Given that the density of water doesn’t vary much, and recalling h = P/(density)g + z:

Ssdh/dt = -d(qx)/dx - d(qy)dy - d(qz)/dz

4. Substitute Darcy’s law
5. In a horizontal confined aquifer of thickness H, qz will be very small
6. Substitute S=HSs
7. Substitute Tx = HKx
8. Substitute Ty = HKy

Sdh/dt = d(Tx(dh/dx))dx + d(Ty(dh/dy))dy

Question 17

Storativity (S) =

Answer 17

in confined aquifers

Volume of water released per unit area over entire aquifer thickness due to fall in potentiometric surface

(drop in P = drop in porosity and density)

S = vol/(A x /\h)

Question 18

Specific yield (Sy) =

Answer 18

in unconfined aquifers

Volume of water released per unit area due to fall in water table elevation

(due to dewatering of pores as water table lowered)

Sy = vol/(A x /\h)

Question 19

Substitutions for an unconfined aquifer

Answer 19

S = Sy

Tx = hKx

Ty = hKy

Question 20

Unconfined aquifer Sy equation

Answer 20

Sydh/dt = d(hKx(dh/dx))dx + d(hKy(dh/dy))dy + W

W = aquifer recharge per unit area

Question 21

Theis solution (1945) boundary and initial conditions:

Answer 21

h = he; r>=0; t=0

h=he; r–>infinity; t>0

2pirHqr=-Qw; r–>0; t>0

Question 22

The Theis solution

Answer 22

s = he-h

= Qw/4piT x E(Sr^2/4Tt)

Question 23

Jacob’s large time approximation

Answer 23

s ~= Qw/4piT [ln(4Tt/Sr^2) - 0.5772]

N.B. remember -ve log flip rule!

Question 24

How does re grow with time? Basic gist

Answer 24

1. Equate Thiem equation and Jacob’s large time approximation
2. re =
3. Gives:
   - with decreasing S, re grows faster
   - with increasing S, re grows slower

Question 25

What is a multi-well pumping test?

Answer 25

Produces water from one well whilst monitoring response in neighbouring (observation) well

Question 26

Using Theis solution to calculate T and S, basic gist:

Answer 26

1. Write Jacob’s approximation in y=mx+c form
   i.e.
   s = mln(t) + c
2. Can calculate T from rearrangement of gradient
3. Can calculate S from rearrangement of C