Hydrologic Characterization of
Fractured Rocks for DFN
Models
Useful Definitions and
Concepts
• Transmissivity -- Properties of a
conductor (aquifer, reservoir, single
fracture, fracture zone) (L2/T)
• Permeability, Hydraulic Conductivity -Property of material inside conductor
(L/T)
Definitions, continued
• Storativity -- Storage of a conductor or
conducting feature (dimensionless)
• Specific Storage -- Property of material
in a conductor (1/L)
• Hydraulic Diffusivity -- Ratio of T/S
(L2/T)
– Controls speed of propagation of pressure
effect of a disturbance
– Very (!!!) important for scaling results
Overview
• Useful Concepts
• Steady Flow Methods
– Packer Tests
– Flow Logs
• Transient Flow Methods
– Boundary effects
– Dimension effects
Steady Flow Methods
• Packer Testing
– Falling Head Test
– Constant Pressure/Lugeon Test
• Flow Logging
– Heat pulse
– Spinner
– Hydrophysical
Steady Radial Flow
rw
• Pressure and flow
constant
• Only exists with
constant pressure
boundary
• Generally underestimates due to
skin
R
ln( R rw )Q Q
T

2 π h
h
Packer Test (Fixed Interval Length)
• Used in Civil
Engineering
• Testing at fixed
interval lengths
• Some zones have
no fractures; some
zones have multiple
fractures
• Efficient testing has
some no flows but
Pn - # of no flows/# of tests
L - length of test zone
 ln( Pn )
P10 
L
Oxfilet (Osnes Extraction of Fixed Interval
Length Evaluation of Transmissivity)
• Guess T and P10 of Fractures
• Oxfiet generated fracture along hole
• Oxfilet calculates packer test
transmissivities
• Oxfilet compares measured and
simulated pacer test transmissivities
Fracture
Network
Stats
Packer
Test Stats
Oxfilet Interface
Data and
Simulated
PDF’s
Data and
Simulated
CDF’s
Oxfilet Challenges
• Results non-unique but constrained
(range of combinations of distributions
of T and frequency that will fit a test
• Flow logging preferred method
Flow Log Types
•
•
•
•
Spinner
Heat pulse
Hydrophysical
Induced electromagnetic
Spinner
Hydrophysical Log
(1) Replace fluid with deionized
water
(2) Log fluid resistivity while
pumping
UCM (Electromagnetic Log)
Well Name: KI0025F02
File Name: C:\WELLMAC\WELLDATA\ASPO\TRUE\KI025F02.HDR
Location: ASPO HRL, TRUE Block Scale
Elevation: 0 Reference: Ground Surface
Date: 98-09-01
UCM Probe:9302
Metres
0
Flow
0
(l/min)
Temp
60 16.2
(Deg C)
Fluid_Res
16.8 0.75
(ohmm)
2
-50
Fluid Resistivity
-100
Flow
-150
Temp
-200
Posiva (Finland) Heat
Pulse Flow Log (Äspö)
Heat Pulse Log
FLOW RATE AND SINGLE POINT RESISTANCE LOGS
DEPTHS OF LEAKY FRACTURES
ÄSPÖ, KI0025F03
120
121
122
123
124
124.65
125
125.45
126
127
128.3
Depth (m)
128
129
130
131.1
131
132
133.3
133
134
135
136
137
138
139
140
1E+1
1E+2
1E+3
1E+4
Flow rate (ml/h)
1E+5
1E+6
1E+1
1E+2
1E+3
Single point resistance (ohm)
Thoughts on Flow Logging
• Cumulative logging methods fast and
easy
• Discrete interval logging methods
provide better detail and wide range of
distribution
• Complementary temperature and fluid
resistivity can be useful
Image Logging
Borehole TV (BIPS)
FMI (micro-resistivity)
Hydro-Testing Work Flow
• Steady tests (flow log)
to identify conductors
• Image log or core
analysis to geo-logically
characterize conductors
• Transient tests to
characterize network
away from hole
4.00E+00
3.00E+00
Dimensionless Pressure
2.00E+00
1
1.00E+00
2
0.00E+00
-1.00E+00
3
-2.00E+00
-3.00E+00
-2.00E+00
-1.00E+00
0.00E+00
1.00E+00
2.00E+00
3.00E+00
Dimensioness Time
4.00E+00
5.00E+00
6.00E+00
7.00E+00
Transient Well Tests
Overview of Transient Tests
• Important source (most important?) of
geometric information on fracture
plumbing system
• Cylindrical flow and beyond
• Dimensions, boundaries, and reading
derivative curves
Radial Diffusion Equation
(Radial Cylindrical Flow)
1   h  1 h
r  
r r  r   t
Exponential Integral:
 r2  
q
e x
q 
p( r , t ) 
dx 

 Ei 
4T r2 /( 4t ) x
4T 
 4t  

Semilog Approximation of the Exponential
Integral
u2
u3
u4
 Ei(  u )  05772
.
 ln u  u 


........
2  2 ! 3  3! 4  4 !
q
2.246t
p( r , t )  2.3026
log
(MKS units)
2
4T
r
PressureDerivative:
dp
 constant
d (log t )
Exponential Integral Function
Semilog
Log-Log
100
10
14
12
log pD
10
pD
8
1
6
4
0
2
-2.00
0
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
log tD
-2.00
0
-1.00
0.00
1.00
3.00
2.00
log tD
4.00
5.00
6.00
7.00
Derivative Methods
• Plots P/log(t)
• Intent to make semi-line unambiguous
• Effect is a very powerful tool to interpret
geometry from tests
• Derivative is a map of transmissivity
versus distance from the well
• Shape of derivative constrains network
geometry
Exponential Integral and Derivative
100
log pD
10
1
0.1
0.01
0
5
10
15
20
log tD
25
30
35
Calculating Pressure
Derivative in Spreadsheets
5
6
7
8
9
10
11
12
13
14
15
A
B
C
Time
Head or Pressure Change
Derivative
2.33E-02
6.15E+01
2.47E-02
6.37E+01
3.68E+01
3.16E-02
7.38E+01
4.47E+01
3.98E-02
8.52E+01
5.27E+01
4.67E-02
9.39E+01
5.72E+01
5.08E-02
9.86E+01
5.78E+01
6.32E-02
1.13E+02
Formula
in Cell C8:
t   p/
 t,
or approximately6.89E+01
7.96E-02
1.30E+02
7.95E+01
=a8*(b9-b7)/(a9-a7)
9.46E-02
1.44E+02
8.69E+01
If the
derivative is noisy, calculate
derivative over a larger
spread,
for example, at C7 calculate
using rows 10 and 8.23E+01
4
9.73E-02
1.46E+02
1.03E-01
1.51E+02
154.4430288
Note: Averaging deteriorates at beginning and end of data
especially if a larger is used
Dimensionless Variables
(Radial Cylindrical Flow)
1 4t
Dimensionless Time: t D   2
u
r
2T
Dimensionless Pressure: pD 
p
q
Useful Definitions
T  transmissibility = kh /  ( L5 / FT )
T *  transmissivity = K * h ( L2 / T )
S  storativity = ct h ( L2 T 2 / M )
  diffusivity  T / S ( L2 / T )
  viscosity ( FT / L2 )
  porosity (-)
ct  compressibility ( L 2 / F )
S s  specific storage = S / h ( LT 2 / M )
K  conductivity = gk / 
Generalized Radial Flow
p (r , t ) 
4
  1 n / 2
qr
n/ 2
2 n
Kh
3 n
 ( v , u )
Dimension Information from
Well Tests
4.00E+00
3.00E+00
Dimensionless Pressure
2.00E+00
1
1.00E+00
2
0.00E+00
-1.00E+00
3
-2.00E+00
-3.00E+00
-2.00E+00
-1.00E+00
0.00E+00
1.00E+00
2.00E+00
3.00E+00
Dimensioness Time
4.00E+00
5.00E+00
6.00E+00
7.00E+00
Integer Flow Dimensions
Linear Flow:
 e u

p (r , t ) 
  erfc u 

2

2  Kh  u
qr
Cylindrical Flow
q
p (r , t ) 
Ei u
4Kh
Spherical Flow
q
p (r , t ) 
erfc u
4Kr
Spherical (3-D)
x-section area r2
Generalized
Flow, x-section
area  rn-1
Linear (1-D), x-section
area  r0
Cylindrical (2-D)
x-section area r1
Log Slope and Dimension
For Log Plots of Pressure or Inverse Flow Verus Time
Log Slope =  = 1 - n / 2
1< n < 2
For Log Plots of Pressure or Inverse Flow Derivative
Log Slope =  = 1  n / 2
For all n
Boundary and Dimension Effects
1-D
2-D
3-D
Reservoir geometry
Network/Flow geometry
Fracture Intensity (Fracture Area/Rock Mass
Volume) Can Influence Dimension
100
.06
10
Head, meters
0.1
0.1
0.175
1
0.25
Boundary
Effect
0.5
0.1
0.1
1.0
10.0
Time, seconds
100.0
1000.0
Geometric Information From Well Tests
5
Lower Intensity,
Smaller Fractures
= Low Dimension,
Compartments
Log Drawdown (m)
4.5
4
3.5
3
Near Field Domain
2.5
Domain
Boundaries
2
High Intensity, Large
Fractures = High
Dimension, Good
Boundary Connections
1.5
1
0.5
0
-2
-1
0
1
2
3
Log Time (s)
4
5
6
7
Composite Dimension
1.00E+03
Dimensionless Pressure
1.00E+02
Composite
Boundary
1.00E+01
Linear Flow
1.00E+00
Spherical
Flow
1.00E-01
1.00E-02
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
Dimensionless Time
1.00E+05
1.00E+06
1.00E+07
1.00E+08
Comments on Interference
Tests
• Radius of Investigation (very handy !!!)
• Estimate diffusivity from response time
• Independent of dimension
r  2 t
Important Notes on Tests
• Transmissivity can be determined only
from pumping wells in fractured or
heterogeneous rock without assuming
uniform flow over region of influence
• Storativity (diffusivity) can only be
obtained from observation responses
• Observation wells give geometric
information for areas farther from
pumping source than themselves
Composite Dimension
• Dimesional Variation Reflect Local
Scale versus Larger Scale Effects
• May Reflect Borehole Geometry as Well
as Conductive Geometry
Parts of Composite Dimension
Curves
• Early Time Effects (Wellbore Storage,
Finite Borehole)
• Inner Shell (n1)
• Transition (changes in area, property)
• Outer Shell (n2)
• Boundary Effects
Composite Interference
Response
• Response depends on relative
distances of transition radius and
observation well radius
• Inner zone not observed for observation
points near or beyond the transition
radius
Rd=1, n1=1.5, n2=2.5
Dimensionless Pressure
1E+2
1E+1
1E+0
1E-1
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
Dimensionless Time
1.00E+06
1.00E+07
1.00E+08
Rd=85, RD1=100, n1=1.5, n2=2.5
Dimensionless Pressure
1E+2
1E+1
1E+0
1E-1
1.00E+02
1.00E+03
1.00E+04
1.00E+05
Dimensionless Time
1.00E+06
1.00E+07