Introduction to fluid flow in rock (Week 7 12 Oct)
(Week 7, 12 Oct)
Ki-Bok Min, PhD
Assistant Professor
Department of Energy Resources Engineering S l N i l U i i
Seoul National University
Last week’s lecture
• Some useful steady state and transient solutions
– Terminology
– Flow to a well in confined aquifer
Steady state solution
Transient Theis solution
– Method of measuring hydraulic conductivity and specific storage
Curve matching method, Time drawdown method & Distance drawdown method
Modification of transient equation Ti d d M th d ( l ) Time-drawdown Method (example)
• 305 m from a well pumping at a rate of 5.43 x 103 m3/day
D d f 10 t 100
• Drawdown from 10 to 100 sec is 0.24 m.
3 3
2
1
3 2
2.3 2.3 5.43 10 /
4 log 4 3.14 0.24
4 1 10 / t
Q m day
T s t m
m day
• We can select any point in the graph. When s=0 is selected for
4.1 10 m /day
2 3Q 2 25Tt
convenience,
3 2
0
2 2
2.25 2.25 4.1 10 / 5 min 93025 1440 i /
Tt m day
S d
2
2.3 2.25 4 log
Q Tt
s T r S 1
2 2
3 2
93025 1440 min/
4.1 10 /
r m day
m day
Domenico & Schwartz, 1998
1
Q & A
Q & A
Mid-term exam
• From Dr Min’s notebook…
• Closed-book, 75 minutes, English/Korean
Today
• Mixing laws - Effective thermal conductivity, heat capacity
• Forced convection and free convection
• Forced convection and free convection
• Conductive-convection equation
– Derivation
– Peclet NumberPeclet Number – 1D equation
• Dimensional analysis
• Free convection
• Free convection
Effective thermal conductivity Mi i L
Mixing Laws
• Model can predict an effective (equivalent) thermal
conductivity of fluid-saturated rock with known properties (porosity, kf, ks).
• Laboratory measurement of thermal conductivity on all rock Laboratory measurement of thermal conductivity on all rock types of possible porosity can be time consuming.
W i ht d ith ti (l t l )
• Weighted arithmetic mean (largest value)
(1 )
e f s
k k n k n
– ke: effective (equivalent) thermal conductivity of saturated rock – kf: thermal conductivity of fluid (e.g., water)
f
fluid (water) – ks: thermal conductivity of solid (e.g., rock)
– n: porosity
Solid (rock) fluid (water)
Effective thermal conductivity Mi i L
Mixing Laws
– Weighted arithmetic mean (largest value) – by considering volume fraction
(1 )
e f s
k k n k n
ke: effective (equivalent) thermal conductivity of saturated rock
( )
e f s
e ( q ) y
kf: thermal conductivity of fluid (e.g., water)
ks: thermal conductivity of solid (e.g., rock)
ks: thermal conductivity of solid (e.g., rock)
n: porosity
Weighted harmonic mean (lowest value) – Weighted harmonic mean (lowest value)
/ (1 ) /
1e f s
k n k n k
– Maxwell model (somewhere between)
2 (3 2 ) / (3 )
e f f s f s
k k nk n k n k nk
Effective thermal conductivity Mi i L
Mixing Laws
l t
largest
experiment
smallest
Mixing law models predictions of thermal conductivity as a function of porosity for water- saturated quartz sand compared with experimental measurement (Somerton, 1992)
Effective volumetric heat capacity Mi i L
Mixing Laws
• Heat capacity of fluid-saturated rock
(1 )
c n c n c
c nw wc (1 n)s sc
– ρw: density of fluid
– ρ : density of solid (e g rock)ρs: density of solid (e.g., rock)
– cw: heat capacity of fluid (e.g., water) – cs: heat capacity of solid (e.g., rock) – n: porosity
Solid (rock) fluid (water) Solid (rock)
Convection
F d ti d f ti
Forced convection and free convection
• Forced convection
– Flow is caused by external forcesFlow is caused by external forces
• Free (natural) convection
– Flow driven by density variation ( thermal expansion of water)
• Forced and free convections are two limiting cases and they can exist together (mixed convection)
can exist together (mixed convection)
Conduction-convection equation D i ti (1)
Derivation(1)
– Fourier’s law
, ,
x y z
T T T
q k q k q k
– Fourier’s law (conduction) + fluid motion (convection)
x y z
x y z
T
,
T x T w w x
q k T n c Tv
x
q k T n c Tv
Contribution from convection!
,
,
T y T w w y
T z w w z
q k n c Tv
y
q k T n c Tv
– qT,x’’: heat flux in x-direction (W/m2)
– v: velocity (m/sec)
, z
v: velocity (m/sec)
– ρwcw: volumetric heat capacity (J/m3K)
Conduction-convection equation D i ti (2)
Derivation(2)
Conduction-convection equation D i ti (3)
Derivation(3)
div c T
q
t q
T w w
k T n c T
q v Contribution from convection!
2
T w w
k T n c T T c T
t
v v
For Steady state flow
conduction
2
T w w
k T n c T c T
t
v v 0
y
convection
Conservation expression that describe the manner in which energy is moved from one point to another by means of bulk fluid motion and by conduction
diffusion equation Di i l A l i Dimensional Analysis
• Dimensionless group dictate the nature of diffusion process or demonstrate the competition between two rate process
L l h f h d l (1D) l h
/ / x x L y y L
+ indicates a dimensionless quantity L h t i ti l th
L: length of the model (1D) or some length associated with fluid movement (2D)
/ / e z z L t t t
L : some characteristic length te : some characteristic time he : some characteristic head / e
h h h L
2 S h S L2 h
2 2 2
L
2h SKs ht 2 s
e
S L h
h Kt t
diffusion equation Di i l A l i Dimensional Analysis
• Fourier Number, NFO;
2 2
/ /
s FO
e e
S L K L c
N t t
2 /
te L c Transient behavior will NOT be observable
2 /
te L c Transient behavior is observable
• For heat diffusion,
2 /
FO
e
N L
t
Conductive-convection equation Di i l A l i
Dimensional Analysis
• Dimensionless group dictate the nature of diffusion process or demonstrate the competition between two rate process
L l h f h d l (1D) l h
/ / x x L
y y L
+ indicates a dimensionless quantity L h t i ti l th
L: length of the model (1D) or some length associated with fluid movement (2D)
/ / e z z L t t t
L : some characteristic length te : some characteristic time
Te : some characteristic temperature / e
T T T
2
2 w w ( / T)
T e
n c vL L c k T
T T
k t t
Conductive-convection equation P l t N b (N )
Peclet Number (N
PE)
2
2 w w ( / e)
T e
n c vL L c k T
T T
k t t
w w w w
PE
n c vL c qL
N k k
q : specific discharge
L : some characteristic length Transport by bulk fluid motion
• Peclet number (NPE): expresses the transport of energy by
T T
k k L : some characteristic length
Transport by conduction
Peclet number (NPE): expresses the transport of energy by bulk fluid motion to the energy transport by conduction.
Reflects a competition between forced convection and
• Reflects a competition between forced convection and conduction.
• Large NPE convective transport dominates
Conductive-convection equation 1D f
1D form
2 2 T w w
x
n c
k T T T
c x c v x t
• vx: mean groundwater velocity in the x-direction
• It is assumed that the temperature of the fluid and the solids are equal.
2
2 w w z 0
T
n c
T T
z k v z
0 0
exp 1
exp 1
PE L
PE
N z T T T T L
N
solution p PE
solution
Conductive-convection equation 1D form
1D form
Ground surface Water table
T0 T0 Tz
vz,
z Semi confining layer
L
TL
Artesian aquifer
T T T d t t
Artesian aquifer
T0,Tz, TL : measured temeprature vz, : leakage rate (pore linear velocity)
(Domenico and Schwartz, 1998)
Convective heat transfer
• What will be the factors that make these two cases different?
Saturated Porous media - Fluid NOT flowing
T0=100°C TL=15°C
Saturated Porous media Fluid Flowing
T0=100°C TL=15°C
- Fluid Flowing
• In reality, the velocity cannot be taken as constant. diffusion equation of porous media flow and conduction-convection equation have to be solved simultaneously.
Conductive-convection equation 1D f
1D form
T0at z=0 v=vz TLat z=L
• In reality, the velocity cannot be taken as constant. diffusion equation of porous media flow and conduction-q p convection equation have to be solved simultaneously.
Thermal expansion
– Linear thermal expansion
0
l T T
l
α: coefficient of linear thermal expansion (unit: /K) To: reference temperature
l
o p
T: new temperature
– Volumetric thermal expansion – Volumetric thermal expansion
0
V 3
T T
V
– Thermal expansion coefficient of Rock
Berea Sandstone: 1.5×10-5,
Boom clay: 3.3×10-6
Water: 6.6×10-5
Convective heat transfer
F ti
Free convection
• Linear pore velocity for free convection (under hydrostatic pressure)
03 ( 0)
kg T T z
v n x
q
v n
n
Convective heat transfer
F ti
Free convection
– In a buoyancy-driven fluid system (steady-state),
( )
g c Lk T T
2 0 ( 0)
w w 0
T
g c Lk T T
T T
k
q : specific discharge
L : some characteristic length
0 w w ( 0)
RA
T
g c Lk T T
N k
L : some characteristic length
– Rayleigh number (NRA): expresses the transport of energy by free convection to the energy transport by conduction.
– Used to establish the conditions for the onset of free convection.
– Onset of free convection: ~ 40 based on horizontal layers with Onset of free convection: 40 based on horizontal layers with impermeable boundaries (from Domenico and Schwartz, 1998)
(Domenico and Schwartz, 1998)
Today
• Mixing laws - Effective thermal conductivity, heat capacity
• Forced convection and free convection
• Forced convection and free convection
• Conductive-convection equation
– Derivation
– Peclet NumberPeclet Number – 1D equation
• Dimensional analysis
• Free convection
• Free convection
Next week
• Reservoir Geomechanics
– Hydraulic fracturingHydraulic fracturing – Borehole stability
C l d
– Coupled process – …
References
• Domenico PA, Schwartz FW, 1998, Physical and Chemical Hydrogeology, John Wiley & Sons, Inc.
• Gueguen Y and Palciauskas V, 1994, Introduction to the physics of rocks, Princeton University Press
physics of rocks, Princeton University Press
• Somerton WH, 1992, Thermal properties and temperature- l t d b h i f k/fl id t El i
related behavior of rock/fluid systems, Elsevier