introduction to fluid flow in rock (week 6 5 oct)(week 6, 5...

Introduction to fluid flow in rock(Week 6 5 Oct)(Week 6, 5 Oct)

Ki-Bok Min, PhD

Assistant ProfessorDepartment of Energy Resources EngineeringS l N i l U i iSeoul National University

Content of last week and this week’s l tlecture

• Fluid flow in fractured media – last week– Cubic lawCubic law– Permeability defined in fractured rock

Ch t i ti d Di t F t N t k (DFN)– Characterisation and Discrete Fracture Network (DFN)

• Some useful steady state and transient solutions – this weeky– Flow to a well in confined aquifer

Steady state solutiony Transient Theis solution

– Method of measuring hydraulic conductivity and specific storage Curve matching method, Time drawdown method & Distance drawdown method

Fractured rock fluid flowC bi LCubic Law

We take average velocity V =2/3 V

e

We take average velocity. Vav 2/3 Vmax32

22 2

( ) ( )12

e

we

we dQ v wdy p gzdx

d d

x Hydraulic conductivity (K) of

2 2

, / ( )12 12w we d e daverage velocity v Q ew p gz gh

dx dx

2

12wge hq

x

2 2

( 1)12 12w wge geh hQ qA A e

x x

w = 1y y ( )

parallel plate model

d it f fl id12 x 12 12x x

3

12wge hQ

x

3

12e pQ

x

with zero elevation

ρw: density of fluidg: acceleration of gravityμ: viscosity

– Cubic law: for a given gradient in head and unit width (w), flow rate th h f t i ti l t th b f th f t

12 x 12 x

through a fracture is proportional to the cube of the fracture aperture.

Terminology

– Aquifer (대수층): a rock unit that is sufficiently permeable so as to supply water to wells.

– Aquitard (준대수층): beds of lower permeability in the stratigraphic sequence that contain water but do not readily yield water to pumping wellswater to pumping wells.

– Artesian well (자분정): a well that flows at the surface without pumping hydraulic head lies above the ground level*pumping – hydraulic head lies above the ground level*

– Reservoir :1.저수지, 2. A subsurface body of rock having ffi i t it d bilit t t d t it fl id sufficient porosity and permeability to store and transmit fluids

(Schlumberger oilfield glossary, 2009).G th l i i it bl f th l – Geothermal reservoir: reservoir suitable for geothermal energy utilization.

*Oxford Dictionary of earth sciences, 2003

Diffusion equation

sh h h hK K K q S

x x y y z z t

– For a confined aquifer of thickness m. Transmissivity, T is defined as y(hydraulic conductivity defined in a unit width);

2 s sS S mh h S hhK K T

T Km

– Storativity S is the product of Ss and

K t Km t T t

m.

Domenico & Schwartz, 1998

Steady state solutionFl t ll i fi d ifFlow to a well in a confined aquifer

2 2

2 2 2

1 1h h h S hr r r r T t

1 h h 1 0h hrr r r

hr C lnh C r C

– Applying BC (known h1 and h2 at

1r Cr

1 2lnh C r C

Applying BC (known h1 and h2 at r1 and r2, respectively)

1 1 1 2lnh C r C 2 1 2 2lnh C r C

1 21

1 2

( )ln( / )h hC

r r

(2 )h hQ KA K rHr r

1 2( ) 1h hh

1 2( )2 h hQ KH 1 2

1 2

( )ln( / )r r r r

1 2

1 2

( )2ln( / )h hQ KH

r r

Analogy with heat transfer

• Much of solutions for porous media fluid flow can be taken from heat conduction.

Transient Theis solutionFl t ll i fi d ifFlow to a well in a confined aquifer

2 2

2 2 2

1 1h h h S hr r r r T t

r r r r T t

20 /44

z

r S Tt

Q eh h s dzT

/44 r S TtT z

h0: original heat at any distance r from a fully penetrating well at time t equals zeroh: heat at some later time ts: drawdown, difference between h0 and hQ: steady pumping ratey p p gT: transmissivityS: storativity


– Based on the work by Theis (1935)

– Used the analogy with heat transfer

– When a steadying pumping is conducted in a well, the head difference at any given radius is expressed as follows.

20 /4( )

4 4

z

r S Tt

Q e Qh h s dz W uT T

22

4 4sr Sr Su

T K

/44 4r S TtT z T

h0: original heat at any distance r from a fully penetrating well at time t equals zero

4 4Tt Kt

h: heat at some later time ts: drawdown, difference between h0 and hQ: steady pumping rate (m3/sec)y p p g ( )T: transmissivity (hydraulic conductivity x thickness), m2/secS: storativity (specific storage x thickness), dimensionless


– Expressed in terms of exponential integral,2 3 4

0.577216 ln ( )2 2! 3 3! 4 4!

z

u

e u u udz u u Ei xz

Initial and boundary conditions are expressed as follows

22

4 4sr Sr Swhere u

Tt Kt

– Initial and boundary conditions are expressed as follows0 0( ,0) , ( , )

li 0

h r h h t hh Q f t

(2 ) 2h hK Q T

– Assumption: Homogeneous and isotropic medium

0lim 0

2r

Qr for tr T

(2 ) 2K rm Q Tr

r r

g p

The aquifer is infinite in areal extent and infinite amount of water is stored in the aquifer.

Well is of intinitesimal diameter and fully penetrates the aquifer.2D approximation hydraulic head does not vary in the third dimension


– Well function W(u)2 3 4

0.577216 ln ( )2 2! 3 3! 4 4!

z

u

e u u udz u u W uz

( )4Qs W u

T

2 2! 3 3! 4 4!z

2

4r SuTt

Th h f d d 4Tt

Type curve: curves showing the

The shape of drawdown curve as a function of time at a fixed distance r.

Type curve: curves showing the relation between W(u) and u (or 1/u more precisely)

If r2S/4T is too big with respect to observation time t, expected drawdown at the point of observation r becomes smallthe point of observation r becomes small

Transient Theis equationC t hi dCurve matching procedure

– Data collected at a distance of 150 m from a well pumped at a rate of 5 43x103 m3/day5.43x103 m3/day.

– W(u) =1, s = 0.2 m, 1/u = 103, t=990 min (0 7 day)(0.7 day)

– Transmissivity:3 3

3 25.43 10 / 1( ) 2.2 10 /4 4 3.14 0.1Q m dayT W u m day

s m

– Storativity;

3 3 3 14

2 2 2

4 4 1 10 2.2 10 / 7 10 2.7 10225 10

uTt m day daySr m


Modification of transient equation

0 for small u2 3 4

0.577216 ln ( )2 2! 3 3! 4 4!

z

u

e u u udz u u W uz

2Q r S

0 0.5772 ln4 4Q r Sh h s

T Tt

04 4ln 0 5772 ln ln1 78Q Tt Q Tth h s 0 2 2ln 0.5772 ln ln1.78

4 4h h s

T r S T r S

0 2 2

2.25 2.3 2.25ln log4 4Q Tt Q Tth h s

T S T S ln 2.3logx x

2 24 4T r S T r S

Modification of transient equationTi d d M th dTime-drawdown Method

• One observation well for various times

• If a drawdown observation is made in a single well for various times a plot a single well for various times, a plot of drawdown versus the logarithm of time will yield a straight line.

10 1 1 2

2.252.3 log4

TtQh h sT r S

20 2 2 2

2.252.3 log4

TtQh h sT r S

2 21 2

1 1

2.3 log ln4 4

t tQ Qs sT t T t


Modification of transient equationTi 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 32

13 2

2.3 2.3 5.43 10 /log4 4 3.14 0.24

4 1 10 /

tQ m dayTs t m

m day

• We can select any point in the graph. When s=0 is selected for

4.1 10 /m day

2 3 2 25Q Ttconvenience,3 2

02 2

2.25 2.25 4.1 10 / 5min93025 1440 i /

Tt m daySd

2

2.3 2.25log4

Q TtsT r S

12 2

3 2

93025 1440min/4.1 10 /

r m daym day


1

Analogy with heat transferTime drawdown Method analogy with thermal transferTime-drawdown Method – analogy with thermal transfer

Modification of transient equationDi t d d M th dDistance-drawdown Method

• Observation at two or more points at one instant of time2.3 2.25lQ Tt

1 21

log4

QsT r S

2 2

2.3 2.25logQ Tts 2 22

g4 T r S

2 21 1

1 2 2 2

2.3 log lnr rQ Qs s 1 2 2 21 1

og2 2

s sT r T r


Content of today’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

• WednesdayWednesday– Exploration techniques

By Tae Jong Lee from KIGAM (Korea Institute of Geosciences and Mineral Resources)

– One question gives 2 points. (attendance of one lecture = 2 points)

References

• Domenico PA, Schwartz FW, 1998, Physical and Chemical Hydrogeology, John Wiley & Sons, Inc.

• Schlumberger oilfield dictionary, http://www.glossary.oilfield.slb.com/default.cfmhttp://www.glossary.oilfield.slb.com/default.cfm

• de Marsily G, 1986, Quantitative Hydrogeology, Academic Ipress, Inc.

• Oxford Dictionary of Earth Sciences, 2003, Oxford University y , , yPress

introduction to fluid flow in rock (week 6 5 oct)(week 6, 5...

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