Week 5 Flashcards
Fluid compressibility =
describes how the density of a fluid increases with increasing pressure
Incompressible =
volume doesn’t change no matter how much stress is applied
Density =
mass/volume
Rock compressibility =
describes how the porosity of a rock increases with increasing pressure
- b/c increasing P = squash and reduce grain size = increase pore space
Porosity =
pore space/volume
/\density =
/\M / volume
For infinitesimal /\P, Cf =
1/density x d(density)/dP
/\porosity (/\n) =
/\pore space/volume
For infinitesimal /\P, Cr =
1/n x dn/dP
Concept of solving Cr = … or Cf = … for n and density
- Integrate both sides
- Apply constraints
- Rearrange n = … or density = …
Relationship between h and n
h = w + z
where w = P/density x g
SO change h = change P = change n
Area x time x density x flow = (unit)
Mass M
- conservation of mass
Fluid mass per unit vol =
Mass conservation
/\x/\y/\x/\m = mass in - mass out
Finding equations for dm/dt, basic gist:
FIRST EQUATION
1. Mass conservation
/\x/\y/\x/\m = mass in - mass out
- Reduce
- Rearrange
SECOND EQUATION
1. m = density x porosity
- Product rule
dm/dt = d(np)/dt… - Chain rule as n and p are both f(P)
Fluid mass per unit vol (m) =
Density x porosity
Using equations for dm/dt to form an equation for the specific storage coefficient (and further steps)
- Ss = n x density x g(Cr+Cf)
- So both equations for dm/dt = Ss/g x dP/dt
- 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
- Substitute Darcy’s law
- In a horizontal confined aquifer of thickness H, qz will be very small
- Substitute S=HSs
- Substitute Tx = HKx
- Substitute Ty = HKy
Sdh/dt = d(Tx(dh/dx))dx + d(Ty(dh/dy))dy
Storativity (S) =
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)
Specific yield (Sy) =
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)
Substitutions for an unconfined aquifer
S = Sy
Tx = hKx
Ty = hKy
Unconfined aquifer Sy equation
Sydh/dt = d(hKx(dh/dx))dx + d(hKy(dh/dy))dy + W
W = aquifer recharge per unit area
Theis solution (1945) boundary and initial conditions:
h = he; r>=0; t=0
h=he; r–>infinity; t>0
2pirHqr=-Qw; r–>0; t>0
The Theis solution
s = he-h
= Qw/4piT x E(Sr^2/4Tt)
Jacob’s large time approximation
s ~= Qw/4piT [ln(4Tt/Sr^2) - 0.5772]
N.B. remember -ve log flip rule!
How does re grow with time? Basic gist
- Equate Thiem equation and Jacob’s large time approximation
- re =
- Gives:
- with decreasing S, re grows faster
- with increasing S, re grows slower
What is a multi-well pumping test?
Produces water from one well whilst monitoring response in neighbouring (observation) well
Using Theis solution to calculate T and S, basic gist:
- Write Jacob’s approximation in y=mx+c form
i.e.
s = mln(t) + c - Can calculate T from rearrangement of gradient
- Can calculate S from rearrangement of C
