Pape, H.

Earth-prints Repository

Simulation of NMR relaxation time 
distribution for a fractal cluster of pores
with surface relaxation, dephasing and 
diffusion
H. Pape
Applied Geophysics and Geothermal Energy 
E.ON Research Center of Energy
RWTH Aachen University, D-52056 Aachen, 
Germany
simulation of NMR-
relaxation
using a corresponding
electrical circuit
Simulation of NMR-relaxation
Start with a pore radius distribution
and a model of diffusion coupled pores
End with a relaxation time distribution
cluster of pores
Simulation of surface relaxation
for one pore class
and longitudinal magnetization
corresponding
electrical circuit
relaxation time
Simulation of surface relaxation
for one pore class
and transverse magnetization
corresponding
electrical circuit
relaxation time
Simulation of surface relaxation and 
diffusion
for two connected pores
corresponding
electrical circuit
relaxation curve
of pore cluster
( )
1
n
i
i
P t P
=
=∑
0 1tP= =normed to
for isolated pores
expi i
i
tP V
T
⎛ ⎞= −⎜ ⎟⎝ ⎠
relative volumes Vi with
1
1
n
i
i
V
=
=∑
Ti is a function of pore
radius ri
1
exp
n
i ij
j j
tP b
T=
⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠∑
1
n
ij i
j
b V
=
=∑
for connected pores
The relaxation times Tj are 
common for all connected 
pores, however the 
coefficients bij are different.
Some bij may be negative.
NMR relaxation
( )
1 1
exp
n n
ij
j i j
tP t b
T= =
⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑
relaxation curve of pore cluster
from coefficients of connected pores
Y AY′ =
1
2
3
P
Y P
P
′
′ ′=
′
1 2 3Y P P P=
( ) ( )( ) ( )
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( )( )
11 1
,1 ,1_ 2 ,1_ 2
1 1
1 111 1
,2 ,1_ 2 ,2_ 3,1_ 2 ,2 _ 3
2 2 2
111
,3 ,2_ 3,2_ 3
3 3
11 11
0
16 16
0
NMR diff diff
NMR diff diffdiff diff
NMR diffdiff
R R R
C C
R R RR R
C C C
R RR
C C
−− −
− −−− −
−−−
+ −
+ +− −
+−
A =
1
,1_ 2 1_ 2 ,1_ 2diff crossR D l A
−=
1
,2 _ 3 2 _ 3 ,2 _ 3diff crossR D l A
−=
( ) 1,NMR i iR Sρ −=
analytical calculation of NMR relaxation
by solving a system of linear differential equations 
example for a cluster of 
three pore classes 
relaxation time distribution
for sub-clusters 
,
1
exp
l n l
k l i ij
i k j i k j
tP P b
T= = =
⎛ ⎞⎛ ⎞= = −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑ ∑
sub-clusters 
clusters
relaxation function
of sub-cluster 
100 1000 T1 (ms)
100 1000 T1 (ms)
0.001
0.010
0.100
1.000
M
100 1000  T1 (ms)
0.001
0.010
0.100
1.000
M
100 1000 T1 (ms)
100 1000 T1 (ms)
100 1000 T1 (ms)
0.001
0.010
0.100
1.000
M
relaxation time distribution
for longitudinal magnetization 
positive
coefficient
negative
coefficient
isolated pores connected pores
coefficient
indication of 
diffusion process
sharing of 
relaxation time
P
P
P
100 1000 T1
0.0001
0.0010
0.0100
0.1000
1.0000
P
coefficients of isolated
pores
positive
negative
coefficients
of cluster{
approximate
coefficients
of cluster
T1 (ms)
100 1000 T1 (ms)
0.0001
0.0010
0.0100
0.1000
1.0000
P 
100 1000 T1 (ms)
0.0001
0.0010
0.0100
0.1000
1.0000
P
relaxation time distribution
for longitudinal magnetization
for the complete cluster
lines are shifted to 
smaller relaxation times
the resulting spectrum is
compressed
similar results by Arns et al.,2005
on the base of digitized core images 
and NMR response simulation by random walkers
0.0001
0.0010
0.0100
0.1000
1.0000
P
10 100 1000 T2 (ms)
0.0001
0.0010
0.0100
0.1000
1.0000
P
10 100 1000  T2 (ms)
0.0001
0.0010
0.0100
0.1000
1.0000
P
10 100 1000 T2 (ms)
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
10 100 1000 T2 (ms)
0.0001
0.0010
0.0100
0.1000
1.0000
P
10 100 1000 T2 (ms)
relaxation time distribution
for transverse magnetization 
negative
coefficient
positive
coefficient
isolated pores connected pores
coefficient
indication of 
diffusion process
sharing of 
relaxation time
0.0001
0.0010
0.0100
0.1000
1.0000
P
10 100 1000 T2 (ms)
coefficients
approximate
coefficients
of cluster
0.0010
0.0100
0.1000
1.0000
P
10 100 1000 T1,2 (ms)
10 100 1000 T1,2 (ms)
0.0001
0.0010
0.0100
0.1000
1.0000
P
relaxation time distribution
transverse magnetization compared to 
longitudinal magnetization
isolated pores 
pores coupled by diffusion 
longitudinal
transverse
longitudinal
transverse


Simulation of NMR relaxation time distribution for a fractal cluster of pores with surface relaxation, dephasing and diffusion

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Simulation of NMR relaxation time distribution for a fractal cluster of pores with surface relaxation, dephasing and diffusion

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