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(1)

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

(2)

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

(3)

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

(4)

Q & A

(5)

Q & A

(6)

Mid-term exam

• From Dr Min’s notebook…

• Closed-book, 75 minutes, English/Korean

(7)

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

(8)

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)

(9)

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 ) /

1

e 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

(10)

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)

(11)

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)

(12)

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)

(13)

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)

(14)

Conduction-convection equation D i ti (2)

Derivation(2)

(15)

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

(16)

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

 

 

   

(17)

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

(18)

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

 

(19)

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

(20)

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

 

solutionp PE

solution

(21)

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)

(22)

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.

(23)

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.

(24)

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

(25)

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

(26)

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)

(27)

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

(28)

Next week

• Reservoir Geomechanics

– Hydraulic fracturingHydraulic fracturing – Borehole stability

C l d

– Coupled process – …

(29)

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

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