多伦多大学课件CIV 549 - Saturated Flow color student

U i compressibility Using ibilit of f water t and d porous medium di our continuity equation becomes:
q x q y q z h Ss x y z t

Where Ss is the specific storativity of the aquifer

If you divide by total volume you get the mass conservation equation per unit volume
w q x w n x t
Saturated Flow – Governing E Equations i
Saturated Flow – 2D Flow for C fi d A Confined Aquifers if
Assuming
no vertical variation of K’s K s and no horizontal gradients and that h is constant vertically vertically, we get:
The
right hand side can be simplified using compressibility p y of water and p porous media
Saturated Flow – Governing E Equations i
q x q y q z 1 w n x y z w t
Saturated Flow – Governing E Equations i
If
the material is isotropic the equation becomes:
2h 2h 2h 2 2 2 0 x y z
This
is referred to as the Laplace p equation equation, q , and it is one of the most useful equations used in hydrogeology y g gy
q x q y q z 1 w n x y z w t

Using Darcy’s and assuming that the fluid mass is not changing with time (steady state flow):
h h h Kx Ky Kz 0 x x y z y z

How water is released from an aquifer q depends p on the type of aquifer (confined or unconfined). Storativity (S) [-]: volume of water that an aquifer will store or expel from storage per unit surface area per unit change in head Specific Storativity (Ss) [1/m]: Volume of water released per unit volume of aquifer due to a unit decline in head Specific Yield (Sy) [ [-]: volume of water that is drained by gravity drainage / total volume


dVT / VT d e
The compressibility p y of a fluid is defined as:
d w / w dP
Saturated Flow – Storage ( (section i 3 3.3 3&3 3.9 9i in F Fetter) )
CIV 549 Saturated Flow
Governing Equations and g Flow Regional
Saturated Flow
In

the next section we will discuss
How different aquifers store water Derive the governing equation of groundwater flow Look at flow nets Discuss regional groundwater flow

If we assume that density doesn’t doesn t change spatially, and include all three dimensions we get an equation that deals with volumes of fluid per unit volume (instead of mass):
b b h h h Ky dz dz S s K x x y y 0 t 0 x
h h h Ky b Kx b S s x y y t x bK T , bS s S h h h Ty Tx S t x x y y


Saturated Flow - Storage

Unconfined Aquifer

Water is released by draining pores S = Sy + bSs Since Sy >> Ss, typically S=Sy Water is released via two mechanisms: aquifer compaction (increase of effective stress) and water expansion (due to decreasing pressure)
z x
y

What is the conservation of mass in the x x-direction? Mass inflow rate = w q x yz
Mass outflow rate = x q x x w q x x yz w qx xyz Net outflow rate = x

This will be the end of Unit 1
Saturated Flow – Compressibility (section ( i 3 3.10 10 & 4 4.7 i in F Fetter) )


How much water an aquifer q can store and/or release can be calculated by knowing the compressibility of the subsurface R Recall ll th that t compressibility ibilit of f a porous medium di i is:
Saturated Flow – Governing E Equations i
What
happens to the equation under transient conditions? (i.e. if there is storage)
q x q y q z 1 w n x y z w t
And
b
integrating it in thபைடு நூலகம் vertical direction
b h h h h Ky K z dz d Ss dz d Kx x x y y z z t 0 0
Saturated Flow – Governing E Equations i

Conservation of fluid mass states: mass inflow rate – mass outflow rate = change in mass storage with time
M Mass i in z

Confined Aquifer

S s w g n S bS s
Saturated Flow – Storage Example (F (Fetter, pg. 102)

An unconfined aquifer with a storativity of 0.13 has an area of 123 mi2. The water table drops 5.23 ft during a drought. How much water was lost from storage?
Saturated Flow – 2D Flow for C fi d A Confined Aquifers if
Ground surface Head in confined aquifer Confining Layer
h h b hh h h K K K dz S dz z K y b y x Kx ss b S y y y y tt 0 x x x x 0 x y Bedrock bK T , bS s S
Mass accumulated
M Mass out t y y
z x