Weekly Report:
07/11/07
This report is divided into two parts. In first part I will describe two experiments on
Diffusion editing. First experiment was done on DI water sample while second
experiment was done on North Burbank core saturated with Brine solution (1% NaCl). In
second part of the report, I will describe the modeling efforts for including the effects of
internal field gradients on diffusional coupling between micro-macro pore.
1) D-T2 map for DI Water:
The parameters used are:
Δ = 60 ms
δ = 3 ms
τ = 300 μs
The g values were taken in the range of 500 to 13,000. These values were logarithmically
spaced. Figure 1, 2 and 3 show the results. We observe that the value of diffusivity
calculated in the experiment match very well with reported value in the literature for pure
water (represented by horizontal dashed line in Figure 3).
Fig 1: Raw data from the experiment
Figure: T2 distribution for DI water.
Figure 3: D-T2 map for DI water
2) D-T2 map for North-Burbank core saturated with 1% Brine solution:
Since North-Burbank sandstone contains significant internal field gradients, in our
experiments we expect to get higher estimate for the diffusivity of water. For NB core,
the magnetization decays very rapidly in the presence of applied gradients. Hence,
several trial experiments were performed to select the parameters in order to get some
signal data.
The parameters used are:
Δ = 10 ms
δ = 2 ms
τ = 300 μs
Number of scans = 200
The g values were taken in the range of 500 to 13,000. These values were logarithmically
spaced. Even after 200 scans, the signal strength is not considerable and the noise levels
are too high. In Figure 4, we show the raw data for three values of the gradients used
(g=0, g= 500 and g = 13,000). We observe that all three curves are on the top of each
other and there is not enough resolution between different curves.
Figure 4: Raw data from experiment for NB core
Figure 7, shows the D-T2 map for NB core saturated with brine solution. We observe that
the observe value of the diffusivity is higher than that reported in the literature for pure
water. This apparent higher values of diffusivity can be attributed to high internal field
gradients (sometimes as high as 300 Gauss/cm) found in NB sandstone.
Issues to be resolved:
1) Reduce the noise from the raw data by carefully selected the operating
parameters.
2) Obtaining enough resolution among the raw data for different values of g.
Figure 5: Raw data from the experiment for NB core
Figure 6: T2 distribution for NB core saturated with brine solution (τ = 300 μs)
Figure 7: D-T2 map for NB core saturated with brine solution
3) Modeling Internal Field gradients in NB sandstone:
Internal Field gradients: Internal field gradients are observed in porous media because of
the presence of minerals of different magnetic susceptibility. This susceptibility contrast
results in inhomogeneous magnetic field in the porous media even though the applied
external field is homogeneous. North Burbank sandstone contains clay flakes which can
cause internal gradients as high as ~300 Gauss/cm.
2L2
2L1
βL2
2yo
Figure 8: A schematic showing macropore lined with clay flakes. Fluid molecule relax at
micropore while diffusing between micro and macropore.
Above figure shows the schematic of a macropore lined with clay flakes. Clay flakes may
be assumed to have a fixed spacing. By taking advantage of the symmetry of the
schematic, we can simply the physical model as described by Figure 9.
yo: half width of the clay flake
zo : length of the clay flake
β : microporosity fraction
L2 : half length of the pore
L1 : half width of the pore
Δχ : magnetic susceptibility contrast
Bo : applied magnetic field in z direction
L1
Symmetry Planes
L2
z
βL2
y
yo
Bo
Figure 9: Simplified schematic of pore structure lined with clay flakes.
The first step is to calculate the change in the magnetic field due to the presence of clay
flake. Gigi. Q. Zhang (2001) showed that the magnetic field can be expressed as:
B   A
(1)
Where A is a vector potential. We can also define Aδ: as the vector potential deviating
from that of the homogeneous, applied magnetic field, Bo. We obtain the following
expression for the Aδ corresponding to our geometry.
ASheet  
Bo 
[( zo  z ) log[( y  yo ) 2  ( z  zo ) 2 ]
4
 z z 
2
2
2 zo  2 | y  yo | tan 1  o
  z log ( y  yo )  z 
|
y

y
|
o 

 z 
2 | y  yo | tan 1 
]
 | y  yo | 
(2)
Where yo is the half width and zo is the length of the clay flake. Also note that, zo = βL2.
The z component of magnetic field is:
B z  
Asheet
y
(3)
B z  
Bo 
2( zo  z )( y  yo )
2 | y  yo | ( zo  z )
[

2
2
2
4 {( y  yo )  ( zo  z ) } 
  zo  z  

2
1  
  ( y  yo )
y

y
o  

 

 z  z  2 z | y  yo |
2 tan 1  o

2
2
 | y  yo |  ( y  yo )  z

 z 
2 | y  yo | z
 2 tan 1 
]
2
| y  yo | 
  z  


( y  yo )2 1  
  y  yo  


(4)
Governing equations for decay of transverse magnetization:
Assuming, M = Mx+ iMy, the Bloch-Torrey equation can be written as:
M
M
 i B z M 
 D2 M
t
T2, B
if,
M  m.e

t
  io t 
T
2, B




(5)
(where ωo = γBo), equation 5 can be expressed as:
m
 i B z m  D 2 m
t
(6)
The expression for Bδz in equation 5 is given by equation 4.
In the next report, I will discuss the approach to make the governing equations/boundary
conditions dimensionless and also the strategy to solve the equations for m.