Percolation theory in research of oil-reservoir rocks ppt

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An Official Publication of The Vietnam National Oil and Gas Group Vol 10 - 2009
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Percolation theory in research
of oil-reservoir rocks
Comprehensive CO2 EOR study - Study on Applicability of
CO
2 EOR to Rang Dong field
Comprehensive CO2 EOR study - Study on Applicability of
CO2 EOR to Rang Dong field
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Contents
01
11
41
Percolation theory in research of oil-reservoir rocks
Distribution rule of lower Miocene
sandstone in Cuu Long basin
Determination of fractured basement
permeability in White Tiger oil field from well
log data by artificial neural network system
using zone permeability as desired output
18

Comprehensive CO2 EOR study - Study on
Applicability of CO
2 EOR to Rang Dong field
24
Novel surfactans for high temperature, high
salinity emhanced oil recovery applications
34
Quasi-dynamic and dynamic random analysis of
mooring system of FPSO installed at White-Tiger field
using hydrostar and ariane-3D softwares
Prediction of aquatic organism impact on rig
submerged structures of oil and gas field
At Cuu Long basin
66
PETROVIETNAM JOURNAL VOL 10/2009
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Introduction
In reality, the reservoir rock space is a very
complex metamerism; however when caculating
according to the common way, in many cases, we
consider the void structure in the rocks as similar
fractal, and use suitable statistical approximate for-
mula to demonstrate the space in form of effective
homogene. When researching the layers, we take
the rock samples from one layer with different col-
lector parameter. To get a parameter value (grain
density, porosity, permeability, saturation etc.) of a
researched object, we calculate the average value
of parameters measured from samples of the same
object. Therefore, the real space is inhomogeneous

(metamerism) when we consider it in a small scale
(core sample), however we consider it homogene-
nous in large scale (formation, layer) with an aver-
age value according to a way of calculation.
For example: In a volume V, with distribution of
parameter values X
i
we can get the average value
−
X
i
of the effective space according to: , or
if the values X
i
are distributed standard.
That way of calculation will not be suitable
when there is a strong inhomogene in the research-
ing space as in the case of fractured basement of
Cuu Long basin.
Percolation Theory will help much in calcula-
tion of permeability characteristic as an accidental
process in complex structures. This theory was
introduced more than half of centuries ago, and has
been applied widely and developed strongly since
middle 1970s in many fields: Matter formation,
material technology, transport and forest-fire protec-
tion, etc. In this series of articles, the author only
would like to introduce the application of percolation
theory in reasearching the percolation process (per-
meation, disffusion) of fluid in void space with com-

plex structure.
Introduction to Percolation Theory
In this writing, the meaning of the term “perco-
lation” is only limited within the permeation or the
penertration of the fluid into the solid matters with
voids. When percolating into solid objects, the fluid
penetrates into sites which has capability of contain-
ing fluid or it flows in bonds, capillary segments con-
necting the sites in the space.
Sites, bonds and types of percolation
Starting from simple cells, for example net of
squares (Figures 2a). Cells with black round spot
are called reservoir sites, white cells are called
empty sites (no reservoir). If we call p the probabili-
Ass. Prof. Dr. Nguyen Van Phon
Hanoi University of Mining and Geology
Percolation theory in research
of oil-reservoir rocks
Abstract
Following the articles about fractal geometry in the research of oil-reservoir rocks [1, 2], in this article,
the author will introduce the application of percolation theory in researching the permeability process of fluid
in void space in general, and fractured rock in particular.
Percolation theory is a mathematical method which has been introduced since the early 1950s, and it
has been applied widely in social and human sciences, and technological sciences since 1970s. Through
this work, the author would like to suggest applying the percolation theory in researching the layers of oil-
reservoir rock, based on the similarity between geometrical forms of percolation process and physical
nature of permeability process of fluid in void space. In the final part of this work, the author proposes the
procedure of calculating the permeability in fractured rocks according to well-log datas, based on applica-
tion of percolation theory.
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ty of a reservoir site in the net, the probability of an
empty site will be (1 - p). Squares with shared bor-
der are called contiguous sites, squares with shared
angular vertex are called adjacent sites. The fluid
can only penertrate from one cell in the net to the
contiguous cell (if that is a reservoir site), it can not
penertrate to a adjacent cell. In a net with 2 or more
contiguous reservoir cells, these cells form a clus-
ter. That way of fluid penertration is called site per-
colation (Figure 2a).
If all the squares cells are reservoir, a channel
allowing a connection between two contigous cells
is called a bond. Bond is a conduit allowing the fluid
to penertrate into the space, it is also a conduit
between 2 contiguous reservoir sites. If we call p the
probability allowing 2 contiguous sites to connect
with each other through a bond, (1 - p) is the prob-
ability ensuring no connection between them
(clogged, disconnected bond). When there are 2 or
more bonds connect contiguous sites continuously,
they form bond group (Figure 2b). That way of fluid
penertration is called bond percolation. In space
with voids such as oil-reservoir rock, these groups
are sites connected with each other through a bond.
Therefore the percolation in oil-reservoir has the
characteristisc of both site percolation and bond

percolation.
Percolation threshold and unlimited group
For low value p, there are only groups with dif-
ferent sizes. When p increases the number of
reservoir sites or the number of bonds also
increases, creating a chance for groups in the net
to increase their size. If p continues to increase,
the groups also grow gradually, and they can inte-
grate to each other through a common bond to
form a bigger group. Until reaching an ultimate
value p = p
c
, the big groups will become unlimited-
size group and the ultimate probability p
c
is called
percolation threshold. The percolation threshold p
c
is an ultimate probability enabling an unlimited
group to form in a large net. With p > p
c
unlimited
group are enlarged more and more, extend form
margin to margin (in 2D) or from face to face (in
3D) of a large net. With p < p
c
there is no unlimit-
ed group in the net.
Percolation threshold p
c

depend on type of
cell (square, triangle, hexagonal, etc.), number of
dimension and type of percolation. Value of perco-
lation threshold p
c
stated in Table 1 is the calcula-
tion result of (2003) with different types of cell net.
Table 1
The Figure 2a shows the layers of reservoir,
the white cells are solid rocks, with no capability of
fluid containing, cells with round black spot are void
space that can contain fluid, then the probability p is
considered the common void ratio of the rock. If
cells (sites) are connected with each other through
a bond, the propotion of void connected are called
open void ratio or connection void ratio P. Then P is
the probability ensuring that any site or bond
belonging to a largest group, P ≤ p. In layers of
reservoir, the value P determines the permeability of
the space.
From above: When p < p
c
, there is only fluid in
connection group with small size. In that situation, if
a well is designed to put at any site, it can easily
penertrate into a small group, the exploiting capaci-
ty of this well will decrease rapidly. To get much
product, and long-term stable exploiting capacity,
the reservoir layer with p > p
c

should be chosen to
put the exploiting well, and the well has to pener-
trate an unlimited group.
A new problem is raised here: With the proba-
bility p in the square net, how we calculate the aver-
age size (average number of sites and bonds) of the
group and the proportion of sites belong to unlimit-
ed group P?
The quantity of groups, average size and space
of group correlation
In net of squares, identify the probability so that
a random cell (site) is a group which has the mini-
mum size s = 1, which means that it is a reservoir
site and independently standing among nonreser-
voir sites. The reservoir site has its own probability,
and around it is 4 adjacent nonreservoir sites with a
probability of (1-p) for each site. These five sites
cells (sites) are independent so they are cooperat-
ed in terms of probability by the product of probabil-
ity: n
1
= p(1 - p)
4
.
For the case of 2 reservoir sites standing
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among 6 adjacent nonreservoir sites, these two
sites can be arranged in vertical or horizontal direc-
tion; therefore, n
2
= 2p
2
(1 - p)
6
. It is easy to conclude
that in a net of squares includes 3 aligned sites,
there will be n
3
= 2p
3
(1 - p)
8
, for around 3 aligned
sites are 8 nonreservoir sites with the probability of
(1 - p) for each site.
We call n
1
, n
2
, n
3
,… the number of groups
which have 1, 2, 3,… aligned sites on the net of
squares. More generally, the number of site includ-
ing S aligned sites, n is the probability so that these

groups can be formed in the net of squares. We
write:
n
S
= 2p
S
(1 - p)
2S+2
(1)
For p < 1, if S  ∞, n
S
 0 is the probability
for a group which has sites S  ∞ aligning in net of
works is very low, nearly reaching 0.
In 3D, on a simple net of squares, each aligned
group including S will have (4S + 2) adjacent non-
reservoir blocks and sites which can be aligned in 3
perpendicular directions , the number of average of
groups (for a net of sites) is calculated as follows:
n
S
= 3p
S
(1 - p)
2S+2
(2)
For the case of hypercubic d-dimensions, each
site has 2d adjacent boxes; for internal sites of a S
group, sites creating lines will have (2d - 2) non-
reservoir sites. If two ends are considered, Group of

S-sites in this case will have (2d - 2)S + 2 adjacent
nonreservoir sites. In this case, the number of
groups are calculated as follows:
n
S
= dp
S
(1 - p)
(2s-2)S+2
(3)
The expression (3) is true for d = 1, 2, 3.
The expression above is only accurate for sim-
ple case; however, natural world is so complicated!
They will not be true for cases in unaliged groups in
the net, for cases of 3 unaligned sites, the alterna-
tives of arrangement is abundant. The Figure below
(Figure 3) shows that group S = 4 sites has 19 dif-
ferent arrangements.
If number of sites S of one group increases, the
number of arrangement (configuration) is increasing
rapidly. For instance, if S = 5, there will be 63 alter-
natives of arrangement; if S = 24, there will be 10
23
different alternatives.
Back to 2D case, for the probability p < p
c
on
the net of squares, there will be only groups of aver-
age size S. The size S of the group is nearly equal
to the correlation length ξ, average distance

between two sites under a correlation group. If p 
p
c
, nearly equal to percolation, the scale (ratio level)
for typical average computation (volume in 3D, area
in 2D) is getting bigger to the scale “mini” around p
c
.
Then, the ratios are equivalent to one another. This
means that adjacent to level p
c
is a fractal which has
the similar structure with scale D~2.5 in 3D [9]. This
explains why at this level, the description of active
space becomes unsuitable for space which has
strong homogene.
Around percolation p
c
, correlation length ξ is
calculated as follows
ξ ~ |p - p
c
|
-x
,(4)
In which ultimate exponent does not depend on
the arrangement of net. In 3D, x ≈ 0.88, 2D, x ≈ 1.33,
[7, 11].
At the level p
c

small groups can connect to
each other, widen the size, increase correlation dis-
tance. In the net, there are sites under different cor-
relation groups and formed unlimited groups.
Point (crack) density in network of limitless group
Assume P (L) is billion parts of point in a net-
work belonging to limitless group, and also average
density of points in limitless group. In square net
having area L
2
, this density is identified:
In which M (L) is number of center points in the
same group in area L
2
(L is positive integral odds 3,
5, 7, 9,…, because it is necessary to have odd num-
ber in length of square to have a square in the mid-
dle of net from which the others is symmantric).
It is clear that M (L) increase gradually in
accordance with area L
2
, P does not depend on L
but only depends to p; p is propositional to P.
Therefor M (L) is L’s function, at ~p
c
, it is proportion-
al to L
2
. P is the probability for any point (crack)
belonging to limitless group, when p is probability

for any point (crack) to contain (connect). If p is con-
sidered as common porosity inaccordance with sur-
veying terms, P is connecting porosity or opening
porosity (P ≤ p).
When logM(L) and logL are represented in loga
couple chart for net having large number of points,
Staufer (2003) found that chart was a line having
angle factor D = 1.9 (Fingure 4). D ≈ 1.9 is fractal
integral number of limitless group in 2D presenta-
tion space. Fractal dimensional numbers of limitless
group do not depend on arranging form of network
(triangle, square…) and only denpend on Euclid
position dimension. In 3D scale D ≈ 2.5.
In Figure 4, line chart shows that:
M (L) ~ L
1.9
,(6)
Meams that M(L) grows with L
1.9
, average den-
sity (5) is not a constant number but decrease L
-0.1
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times in rate grade. The larger scope is, the larger
the difference is. For example, average density P
(amount of workable oil) counted on an area of
reservoir with porosity of approximate p
c

and edge L
= 100km shall be smaller than area counted in sam-
ple with edge L = 10cm, in accordance with space
coefficient (10
6
)
-0.1
~ 0,25. 75% of remained amount
of oil in reservoir is not connected directly with
exploiting wells located in central point. In 3D, corre-
sponding coefficient is much smaller: (10
6
)
-0.5
~ 10
-3
.
In fact, the counting result shall not always bad
because density P that is gradually constant to L
and p is larger than p
c
, at that time there is a corre-
lated length ξ(p), a limitation so that: M(L) L
1.9
to
L < ξ and M(L) L
2
to L > ξ.
Limitation ξ is the limitation of the farest well in
the packing, it shall decrease similar to increasing p

than p
c
. Therefore, explorer shall use a sample with
L that is larger than ξ to calculate amount of oil
which may be exploited more exactly.
Of course, the amount of oil take from reservoir
layer depends on many other factors relating to fluid
flow in pore space and dynamtics characteristics in
osmotic packages such as diffission of fluid in mixed
space and force osmosis which shall be discussed
in another works.
Bethe net
In order to have exact solution for complex
structures, problem above is studied in form of
branch separated tree – Bethe net. Bethe net (or
Cayley tree) is tree shaped net with unlimited
dimensionals. Approximated calculation Bethe is
used to give anwser for tree problems. Therefore
complex structures with unlimited dimensional d are
Bethe net.
In order to understand structures with unlimited
dimensional d, we will start with d = 2: Area of circle
with radius r is πr
2
, its circle is equal to 2πr. Area S
of sphere (3D) radius r is 4πr
2
, and volume V is
propotional with r
3

. In d- volume dimensional of ball
shall be propotional r
d
, and surface area S is propo-
tional with r
d-1
. General calculation:
(7)
(Symbol is used to count rate between val-
ues. In several cases, this rate means approximate
limitation; d  ∞)
Expression (7) shows that when dimension up
to unlimited point (d  ∞), then area of the ball outer
will approach to the cubic content. This remark is
true even with grid, cube, multi-cube etc.,
Construction of Bethe network
To construct a simple Bethe network (Figure 5)
to conduct as follow: To point of O origin site, pass-
ing four origin points (Z = 4) adjacent to A. From A
site, four bonds are generated, one connecting to O,
the other three ones connect to B site. From B site,
it connects to (Z - 1) = 3 of new site C, and more
lengthened by this way. Bethe Net is an unclosed
net, which has no branch connecting to O origine
site by any form. Continuation with this process of
branching, we will have an unlimited network, of
which the sites will increase in the distance from the
outside site to the O origin site with a structure d-
its dimension will be incresed: (distance)
d

.
In example in Figure 5: Z = 4, original site is
covered by 4 sites A (the first system), second sys-
tem (or layer) will have 12 B site, the third will be 36
C sites therefore, the point network consisting of
the first system to the last system of 4 x 3
r
-1
site is
the outer site. Then, the network expand to r, the
last system consist of
4 x 3
r-1
/
2 x 3
r
-1
= 2/3 of the
total sites on Bethe net. This is equal to and correct
to
(Z-2)
/
(Z-1)
any Bethe net with Z at random.
From this point of view, we can expand to the
3D case, ratio of area of internal side and volume of
the ball whose radius is of r will reach the approxi-
mation ~1 when Z  ∞. This is fitted with the
expression (7) when 1/d  0. Now we can see that
Bethe net is an abnormal model; thus, when men-

tioning to the percolation of the Bethe net, one very
important thing is to imagine that it only occurs
inside the net but not effect the outerest surface.
The evaluations above imply that probability by
which an infinite unit spreads all over the net is zero,
and the percolation threshold p
c
is given by:
(8)
This calculus is suitable with the site percola-
tion case and bond percolation case. For further
understanding about the percolation threshold p
c
,
take a look back Figure 5. There are four rays at
each site A (Z = 4) among which one connects with
O, (Z - 1), the three remained rays reach the site B,
and the same for the case of site C. etc. Hence, (Z
- 1)
-1
will be ultimate probability for the creation of
infinte unit, and called percolation threshold of the
Bethe net.
Average size (S) of the unit when probability
approximately reach the level p
c
In order to calculate S, we assume T is the
average size of each unit at four branches. T is the
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average number of sites connected with original site
to form each branch A. Each of these separate
branch is continuously divided into three smaller
infinite branches, T will still be average size of the
unit in each branch.
Next to site A, site B may be the containing with
probability p or non-containing with the probability
(1 - p). The non-containing sites will not be very
meaningful while the containing sites contribute (1 +
3T) point for this branch in which one is point B and
3T is three branches extended from this site.
Therefore:
(9)
The size of group originating from the site O is
0 if this site is non-reservoir site or (1+4T) if it is
reservoir site. Therefore:
(10)
Referring to (8), so S can be adjusted
according to (p
c
- p)
-1
for p < p
c
. For p > p
c

, S will
branch off. If p  p
c
with the ultimate exponent x ≈ 1.
S = (p
c
- p)
-1
(11)
Relating to the expression (4) we find that if p
 p
c
, average size S of group and the correlation
length ξ are equal and reach infinity.
By similar inference, it is possible to identify
ratio P of sites under unlimited group in the space
which has p greater than permeability p
c
. P, as ana-
lyzed above, is the probability for the original site O
under unlimited group. It can arrange different abili-
ties so that original site O is connected to 4 neigh-
boring sites (Figure 6). On the drawing, each arrow
is an unlimited daisy chain connected from original
site O Figure 6.
We consider Q as the probability connecting
from O to adjacent site A which is discontinuous
(congestion). According to Figure 6, we find that
probability P is identified referring to the probabili-
ties of three final alternatives c, d or e. If each arrow

is an unlimited daisy chain, O must be of two unlim-
ited chains which consider them as the connection
part of a permeability group.
The probability to exist the arrow between O
and site A is (1 - Q). For case (c), we have probabil-
ity 6Q
2
(1 - Q)
2
(including 6 probabilities of arrange-
ment so that from O there are two arrows and 2
nonarrows which rotate indifferent directions). For
case (d) it will be 4Q(1 - Q)
3
; and for the case (e), it
will be (1 - Q)
4
. Total probability will be:
(12)
In fact, probability has function relation with
probability p. In fact, a chain line connecting from O
to site A is discontinuous if O and A are not connect-
ed (probability1 - p), or if O and A are connected but
fragmented on connecting to A (probability pQ
3
), it
can be computed as follows:
Q = (1-p) + pQ
3
(13)

Equation (12) has the simplest result Q = 1, P
= 0, which means that at that time the system is
under the percolation threshold. Furthermore, this
equation also has two different results, but these
result mentioned below have physical meaning:
(14)
Q reduces from 1 to 0 if p increases from to 1.
In the range p < p
c
, Q = 1, P ≡ O. From two depend-
ent relations between P(Q) and Q(p) it is possible to
find out P(p). Around the threshold p
c
we find that P
change referring to the form (p - p
c
)
2
:
p  p
c
; P ≈ (p - p
c
)
2
(15)
It is possible to express (15) as Figure 7.
Picture 7 shows that the permeation probabili-
ty P or the site density (bond) of the unlimitted group
increases from 0 to 1 when the probability p of the

site increases from p
c
to 1. In the space of p < p
c
the set status is below the permeation threshold P
=0. This means the reservoir rock space has the
critical void ratio (p
c
), the space will have the perme-
ation or the permeability will occurs at that time.
The finner the grain of the clastic rocks is, the
higher the critical void ratio value (p
c
) is; the void
rate of the fractured rock with the kinetic penetration
is usually lower than that of the crumb rock. This
relates to the specific surfaces and the channel
bend of the two mentioned above rocks.
In the basement of Bach Ho oil field and other
fields in Cuu Long basin, the hydrodynamic penetra-
tion occurs at fractures spaces (F
f
) and macrofrac-
tures while the capillary penetration occurs in
microfractures. The result of 270 granitoit fractured
samples analysis (2001) in basement of Bach Ho oil
field by Mr. P. A. Tuan showed that their average gen-
eral void ratio is 3.1% while the average open void
ratio is only 1.88%, it means that the close non-con-
nected void ratio makes up nearly 40% of the gener-

al void ratio. The equivalent numbers in the analysis
of the well-logs in a well of Rang Dong field are
5.455%, 0.617% and 89%
Percolation through fractured rock space
Percolation theory gives us a decription
method of strongly heterogeneous space in reser-
voir rocks. Here meaning of threshold is often
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research carefully. Threshold effect can be seen
clearly in many phenomena occuring in the nature
and human society. The permeability of fractured
rocks space are emphasized here.
Fractured rock space
The process of development and mature of the
rocks result in the appearance of interrupted frac-
tures within rocks due to various reasons: volume
shrinkage (the freezeing process of igneous rock ),
load reduction (weathering and erosion), mechani-
cal stress (tectonic activities), corrosion and elutria-
tion (thermalization activities), etc. Fractures are the
results of the disruption of the initial uninterrupted
structure. To simulate the fractured rock space, we
will look through dished fractures and evaluate them
in terms of dip anglar, strike, aperture, radius, filling
– up level of secondary minerals and density in
rocks.

We consider space as rocks with different frac-
tures of random distribution. If there are not many
fractures in the space, it is unlikely that such frac-
tures cut each others, low connectivity, zero perme-
ability.The higher the density of fractures is, the
greater the probability for such fractures cut. each
others. If the critical density is to be outnumbered,
there will be a ratio f representing intersecting frac-
tures in the space, which forms the “unlimited
group” (Figure 8) and enables the fractured space
to let the fluid through – that means the non-zero
permeability.
Using the model Bethe net (Figure 5) with Z =
4 to demonstrate the fracture net , we have: f is den-
sity P, and p
c
= 1/3, and p is the probability so that
two random intersecting fractures cut At different
value of p, which is greater than the critical proba-
bility p
c
, then p would be directly proportional to P,
which is the ratio of intersecting fractures and
belong to unlimited group within the scope of stud-
ied volume. As P increases, the probability of leting
fluid through the space also increases. This princi-
ple is also applicable for the conductance of fracture
net if the carrying fluid follows the saturated fluid
(water) in the empty space of fractures. In this case
the Ohm Law and Darcy Law is compatible.

For the purpose of calculation, we demonstrate
the fractured rock space as in Figure 9 and assume
that all fractures have the dished form, radius c,
aperture 2w and density N ≈ 1/ℓ
3
, in which ℓ is the
average distance between fractures (Figure 9).
Estimating pressure p according to c, w and
density N
Assume that p is the pressure to have two ran-
dom intersecting fractures. As the density N and
radius c increase, p also increases. The number of
fractures over a partial volume, the greater the aver-
age dimension c of fractures is, the higher the prob-
ability that these fractures cut each others. The
product Nc
3
is the non-dimensional quantity, so p=0
as one of the two parameters (N or c) is equal to 0,
so it can be assumed that p changes accordingly
with Nc
3
:
(16)
To calculate the pressure p and determine the
percolation zone and permeability, we introduce a
new concept: Peripheral volume V
ex
is the maximal
volume containing a random fracture with center O

to have a second fracture O’ ( with the same radius
c), which is arranged randomly in the volume. The
result is that the two fractures will cut each others.
In the fluid crystal physics (De Gennes, 1976), this
volume is defined as:
V
ex
= π
2
c
3
(17)
At a density of fracture N, the average number
of intersecting instances of each fracture is v = NV
ex
.
The Bethe net in Figure 5 shows that the probability
for an isolated fracture (does not have any fracture)
is p
o
= (1 - p)
4
. This probability can be presented
according to v as follows. Assume V
o
is wide volume,
in which there are disorder distribution of centre O of
fractures with the density N. Probability for a random
point in V
o

falls into a volume V which is smaller (V⊂
V
o
) will be
V
/V
o
. Assume that n is a random in V
o
the
probability p
m
for m points (m<n) falls into V will be
calculated as follows:
(18)
In which:
If we calculate the limit of the expression (18)
as n and V
o
will reach the unlimited pole
n
/V
o
=N we
will have:
(19)
If there is no point falling into V, m=0 then the
limit (19) p
o
= e

-n
will be the probability for a fracture
to be isolated (does not cut any fracture). But then
p
o
= (1 - p)
4
. So we have:
(20)
According to (20) we can see if the density N
 ∞, that means v  ∞ then the probability for p so
that two random intersecting fractures will be nearly
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equal to an unit If N is so small, n << 1 that p ~
v
/4. Because so (20) express
the connection between p, c, ℓ and N.
Take and , the conditions for
percolation threshold p
c
= 1/3 will divide (mặt) (ℓ, c)
into two domains (Figure 10), the percolation
domain in the right side of the dividend line, corre-
sponding to values and .
There will not be percolation if ℓ is too great,
density N is too small or c is too small (small frac-
tures).

This is suitable for (15) as p moves upwards to
p
c
from the greater value (p > p
c
), the pemermabil-
ity will be proportional to (p - p
c
)
2
. This is also cor-
rect for the conductance if the electric current is the
ion current flowing through the saturated fluid in the
fractured voids.
Percolation effect
In fact the intersections between fractures in
the fractured rock space is at random. The fractured
space has the permeability as the intersections
between fractures form an unlimited group and the
number of groups or average dimension of the
group stretchs out, number of springs P belonging
to the unlimited group is greater. To calculate the
permeability of the fractured rock space, let’s get
back to the model (Figure 9), and apply Darcy Law.
In mechanics, call q as the Darcy speed of the fluid,
which is equal to the volume of the fluid through the
cross-section S perpendicular to the speed direction
over an area unit in a time unit: The volume of fluid
is equal to q. If the fluid has the viscosity h and gra-
dien with pressure , then the Darcy Law will be

presented as:
(21)
In which k is the permeability with the perme-
ability factor [m
2
].
Darcy speed is the volume flux (not the actual
speed of the fluid) and can present the connection
between it with the average speed of the fluid in the
porous hole F according to Dupuit-Forcheimer Law.
q = vΦ (22)
In the fractured space, the average speed
―
v of
the fluid between the two parallel sides will apply
Landau-Lifshitz Law (1971):
(23)
From (23) and (21) we can easily conclude that:
(24)
Take the approximate porosity F of the frac-
tured rocks as an replace (24) we can calcu-
late that:
(25)
Here, once again it is proved that in the frac-
tured rock space, permeability k depends on three
micro-structure factors: c, w and ℓ.
Expression (24) and (25) are true for the per-
meability which all fractures will connect with each
others completely, that’ means p ≡ P as the model
of Warren – Root (see instruction documents of

chapter [6], chapter7). But in reality, such cases
occur rarely, because the intersections between the
fractures in the fractured rock space are at random
and we have to apply the permeability theory to
evaluate the intersecting level between the frac-
tures in the net.
From part 3.2 result, peripheral volume V
ex
=
π
2
c
3
and estimated , we can see: If, the
possibility of intersections between any fractures is
low, the space does not have permeability, k = 0; in
contrast, if p > p
c
,
unlimited group is created and the
space is capable of permeability, and increase in
multiplication factor f = (p - p
c
)
2
(see (15) and Figure
10.) This factor is permeability probability P(p)
shown in Figure 7.
In calculation, permeability factor calculated as
(24) and (25) need to be multiplied with f factor

because of permeability effect:
(26)
In which, W is aperture of fracture, p is com-
mon porosity, p
c
is porosity limen for fuildl passing
permeability space, and Φ is leaky porosity or open
porosity, including carven and fracture porosity
which are called secondary one in some docu-
ments.
Define permeability basing on well log datas
According to the porosity result in bore well,
common porosity (p) in fractured rocks is calculate by
average of porosity Φ
D
and Φ
N
at the same depth;
secondary porosity Φ is calculated as following:
(27)
In which Φ
S
is calculated basing on sonic
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method at matrix rock without fracture.
Dimensions w, c and ℓ or fracture density in the

space is identified from analysis results of FMI and
FMS datas. For fractured basement at White Tiger
and Dragon oilfield, Institute of Marine Research,
Vietsovpetro considered porosity threshold perme-
able in fractured rocks p
c
= 0,01 in calculation of
hydrocarbon in inplace and oil recovery factor; aver-
age aperture of fractures. From that statistics, we
could calculate permeability of fractured base
based on:
(26)
For example, result of open porosity Φ = 0,018,
common porosity p = 0,031 to be replaced in (26),
we have:
Permeability of fractured rocks depends on
aperture of fractures. Aperture changes twice, per-
meability will changes four times. In addition, per-
meability of fluid in fractured rocks depend on
draught of fractures. The draught of fractures
increase, the permeability decrease. Up to now,
many authors research more this projects.
Suggestion and Conclusion
Reservoir is pore space which its microstruc-
ture is complex and changes along with rock devel-
opment process. Each kind of rock contains pore
microstructure with typical characteristics but asyn-
chronous. Strong asynchronous state is characteris-
tic of fractured stocks: They have two porosities,
two permeability abilities. They are fractured and

internuclear porosity (or block porosity); kinematical
permeability in large fractures and caves, capillary
in internuclear porosity and fractures. Large frac-
tures have small ratio in common porosity but play
an important and decisive role in effective perme-
ability. Minimum fractures and porosity of particles
play an important role in determining the ability of
product area. The penetration of fluid into an space
with 2 porosity is a complex process. In the porosi-
ty space of fractures, they are up to gradient pres-
sure of fluid, the minimum fractures and porosity
among particles are determined by wettability and
capillary force. The physical nature of permeability
in multi-fracture space are suitable with shape of
permeability. The analogy is the base to apply the
theory of permeability in an unsuitable space such
as fracture stones in Bach Ho oil field and other
fields in Cuu Long basin.
The evaluation and use of the permeability
density P(p) as the permeability effect factor is the
specific result of this construction to overcome the
disadvantage of Warren- Root model in order to
determine the k permeability in the fracture rock
space with two void and two permeabilities objects.
The author would like to thank fellows for help-
ing and exchanging experiences and ideas in the
work implementation…
This work is the result of the research project
KHCB 7.1.5206 sponsored by Ministry of Science
and Technology.

References
[1]. Nguyen Van Phon (2007). Fractal geome-
try for researching reservoir (I). Petrovietnam Rev.
Vol. 2 - 2007, pp 23-26.
[2]. Nguyen Van Phon (2007). Fractal geome-
try for researching reservoir (II). Petrovietnam Rev.
Vol. 3 - 2007, pp 14-21.
[3]. Hoang Van Quy, Phung Dac Hai and
Borixov A.V. (1997) Collecting data of geologic
structure, determining oil & gas an condensat con-
tent of Dragon oilfield. Report of Institute of
Scientific Research and Statistics – Vietsovpetro.
Vung Tau 1997.
[4]. Pham Anh Tuan (2001). Physical features,
heterotrophic features and hydrodynamics of oil-
reservoir rocks of complicated structures in the con-
ditions of modeling pressure and temperature of the
formation. Ph.D. thesis – University of Mining and
Geology.
[5]. De Gennos P.G. (1976) The physics of
fluid crystals. Oxford University Press.
[6]. Golf-Racht Van T.D. (1982) Fundamentals
of Fractured Reservoir Engineering. Elsevier
Scientific Publishing Co.
[7]. Guéguen V., and Palciauskas V., (1994).
Introduction to the Physics of Rocks. Princeton
University Press.
[8]. Landau L., and Lifshitz (1971) Fluid
Mechanics. Moscow Edit. “Mir” .
[9]. Mandelbrot B.B. (1982) The Fractal

Geometry of Nature. San Francisco Freeman.
[10]. Snarskii A.A. (2007). Did Maxwell know
about the percolation threshold? Uspekhi
Fizicheskikh Nauk. 177(12). 1341 – 1344.
[11]. Stauffer D., and Akarony A., (2003).
Introduction to Percolation Theory. Taylor and
Francis (2003).
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Fig. 1. Inhomogeneous real space and effective homogeneous space
Fig. 2. Site percolation and bond percolation
Fig. 3. Different configuration of the group
of 4 sites in net of squares
Fig. 4. M(L) is L’s funtion, for network
having area L
2
=10
10
arrange in
rectangle net p
c
= 1/2
(according to Staufer 2003)
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Fig. 5. Construct of Bethe network
Fig. 6. Five difference probabilities in which original site O
can connect to 4 adjacent sites
Fig. 7. The permeation probability P
depends on the probability p
of the site (the bond)
Fig. 8. The development of fractured space as
the density of fractures increases
Fig. 9. Dished fractures, radius c, aperture (w << c)
Fig. 10. Surface percolation zone (ℓ, c)
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Introduction
Lower Miocene sandstone is the first oil-bear-
ing reservoir discovered as the well BH-1 was
drilled and tested in 1975. Over 30 years, oil com-
panies have drilled more than 80 exploration and
appraisal wells with chance of success of about
52%. However, if only exploration wells are consid-
ered, the chance of success remains just about
30%, which is not high while drilling just concen-
trates in the centre of Cuu Long basin. For this fact,
a need of studies to reevaluate the actual potential
of this reservoir is raised in order to increase the
performance of exploration as well as of production.
With this reason, the study of characteristics, origin
and distribution rule of Lower Miocene sandstone
was deployed.

Method
The study was based on collecting and classi-
fying the analyses of BI.1 and BI.2 sandstones:
petrology and sedimentology (grain size, cement
and matrix compositions), reservoir properties
(porosity, water saturation, NTG ratio); and deposi-
tional environments and constructing cross sections
and maps:
- Constructed 6 geophysic-geological cross
sections (3 strike sections through the basin and 3
sections across the basin):
Section 1: NW-SE, at the Northern part of the
basin, through blocks 15-1 and 01-02 (Figure 1).
Section 2 : NW-SE, at the central part of the
basin, through blocks 16-1, 09-1 and 09-3 (Figure 2).
Section 3: E-W, at the Southern part of the
basin, through blocks 16-2, 09-1 and 09-3 (Figure 3).
Section 4: NE-SW, at the Western margin of
the basin, through blocks 15-1, 15-2, 16-1, 16-2 and
17 (Figure 4).
Section 5: NE-SW, at the central part of the
basin, through blocks 01, 15-1, 15-2, 16-1, 09-1 and
09-3 (Figure 5).
Section 6: NE-SW, at the Eastern part of the
basin, through blocks 01, 15-2, 09-2, 09-1 và 09-3
(Figure 6).
- Constructed 12 distribution maps of thick-
ness, grain size, matrix and cement content, poros-
ity, water saturation and net to gross ratio of BI.1
and BI.2 subsequences.

Thickness
The distribution maps of thickness are shown
in Figures 7 and 8. In Figure 7, BI.2 subsequence is
thinnest (80-100m) to the Eastern and Western
basin margins. Along the central NE-SW axis of the
basin, the subsequence thickness varies in range of
100-500m. The thickness tends to increase towards
the centre of the basin. Maximum value reaches
Dr. Ngo Thuong San, Dr. Cu Minh Hoang
Petrovietnam Exploration and Production Corporation
MSc. Pham Vu Chuong
Salamander Energy Limited
Distribution rule of lower Miocene
sandstone in Cuu Long basin
Abstract
This article presents the study results of distribution rule of thickness, lithology and sedimentology,
reservoir characteristics and depositional environments of Lower Miocene sandstones (subsequences BI.1
and BI.2) in Cuu Long basin. The study was performed by constructing cross sections and maps for the
entire basin. The results show that good reservoirs can be found in the Northern part at BI.2, while they can
be found in the Southern part at BI.1.
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nearly 800m in block 09-1, a part of blocks 16 and
09-2.
Figure 8 shows that the thickness of BI.1 tends
to increase towards basin centre. However, the
thickest (approximately 900m) locates on block 16,

a part of block 09-1 and 09-3. Generally, BI.1 is thin-
ner than BI.2, except in block 16 and 09-3.
Grain size
Maps showing grain size distribution of BI.2
and BI.1 are presented in Figures 9 and 10. Figure
9 show that the grain size of BI.2 sands varies from
very fine-fine (<0.25mm) in block 16, the Southern
of block 09-1, the Eastern of block 09-2 and the
Northwestern of block 15-1, to medium (0.25-
0.5mm) in the remaining area, except in the centre
of block 01-02 where grain size becomes very
coarse (0.5-1mm).
Figure 10 shows that the grain size of BI.1
sands varies from very fine-fine (<0.25mm) in the
Nothern of block 16, 09-1, the Southern of block 09-
2, to medium (0.25-0.5mm) in the Southern of block
16, the Northern of block 15-1 and 01, the Eastern of
block 15-2, to coarse (0.5-1mm) in the Southern of
block 01, 15-1 and the Northeastern of block 15-2.
Commonly, the grain size tends to larger in
the Northern part of the basin than in the Southern.
Especially in BI.1, the boundary between the two
area can be seen clearly (black bold line in Figure
10).
Matrix and Cement
The average contents of matrix and cement
were collected and mapped over the whole basin for
BI.2 and BI.1 subsequences individually (Figures 11
and 12). Figure 11 shows that the matrix and
cement content of BI.2 varies from 4 to over 30%.

The contents less than 15% locate in the Northern,
a part of block 16 and 09-1.
Figure 12 shows that the matrix and cement
content of BI.1 also varies from 4 to over 30%.
However, the contents less than 15% gather into
bands from R, BH, TGT, HST, HSD to TL, DD, P to
SD.
Porosity
Porosity distributions of BI.2 and BI.1 are pre-
sented in Figures 13 and 14. Figure 13 shows that
porosity of BI.2 sands varies from 12% (TGC) to
27% (SD). Very good porosity locates in area from
SD to the Eastern of block 15-2 (22-27%). In block
01-02 and the Northeastern margin of block 16
(TGT), porosity is slightly lower with value of 18-
20%. There possibly is a boundary separating the
map into two areas (the black bold line in Figure 13).
Figure 14 shows that porosity of BI.1 varies
from 13% (in block 09-2: COD) to 21% (block 15-1:
SD). Good porosity locates in a trend from BH to
TGT, HST, HSN, RD to SD (19-21%), and the
Northeastern (block 01-02 and Eastern of block 15-
2) and the Southwestern basin margin (R-DM and
the Southern of block 16) with values of 18-20%.
It is clear that good porosity mostly gathers in
block 01-02, 15-1, centre and the Eastern of block
15-2, the Eastern of block 16, 09-1 and the Western
of block 09-3. Porosity tends to decrease down-
wards from BI.2 to BI.1 in block 01-02, 15-1 and the
Eastern of block 15-2. In block 01-02, porosity of

BI.1 is lower than those of BI.2 probably due to
strong extrusive activities in this period. Meanwhile,
in the centre of block 15-2, the Eastern of block 16
and block 09-1, porosity of BI.1 is higher than those
of BI.2. The reason probably is BI.2 sediments were
deposited far from sedimentary source in deltaic
environment to shallow marine with more abun-
dance of clays.
Water saturation
Water saturation distributions of BI.1 and BI.2
are presented in Figures 15 and 16. Figure 15
shows that in BI.2 low water saturation areas are in
the Northern to the Eastern and the centre of block
15-2. In blocks in the Southern, water saturation is
high (over 80%), except block 09-3, a part of block
09-1 and 16. The boundary between the two areas
is presented by the black bold line in the map.
In reverse to BI.2, water saturation in BI.1
varies in range of 40% in a trend from block 09-1 to
the Eastern of block 16 and the centre of block 15-
2, to more than 80% in block 15-1, the Eastern of
block 09-2, and a part of block 16. In Northern area,
sands are mostly water filled (Figure 16).
Similar to the distribution of porosity, low water
saturations gather in block 01-02, 15-1, the centre
and Eastern of block 15-2, the Eastern of block 16,
09-1 and the Western of block 09-3. However,
water saturation increases from BI.2 to BI.1 in block
01-02, 15-1 and the Eastern of block 15-2.
Reversely, in the centre of block 15-2, the Eastern

of block 16 and block 09-1 water saturation
decrease from BI.2 to BI.1. The reason is the rela-
tionship between water saturation and porosity. In
the Northern area (SD), in spite of high porosity in
BI.1, water saturation is still high. The reason prob-
ably is that there is no top seal to against oil to
move upwards to BI.2.
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Net to Gross ratio
The maps showing the distribution of this
parameter of BI.1 and BI.2 are presented in Figures
17 and 18. Figure 17 shows that the Net to Gross
ratio of BI.2 varies from 0% to more than 70%. The
area with high Net to Gross ratio (>30%) is the
Northern of block 01-02, 15-1, the centre and east-
ern of block 15-2, except SV area.
Figure 18 shows that in BI.1 the area with high
Net to Gross ration is just R, BH, TGT, HST and PD,
TL, DD to P. In the Northern area, in spite of high
porosity, Net to Gross ratio is still very low due to
water-filled sands.
Generally, there exists a boundary with high
Net to Gross of BI.2 in the Northern of the basin and
high Net to Gross of BI.1 in the Southern of the
basin as shown in Figure 17.
Conclusion
From the analysis of reservoir distribution rule,

it can be concluded that good reservoirs can be
found in the Northern part at BI.2, while they can be
found in the Southern part at BI.1. The possible rea-
sons are:
- Different sedimentary sources of the two
areas. Sediments of the Northern area might be
supplied by the paleo Dong Nai river or other paleo-
rivers in the central Vietnam with short transporta-
tion pathway. Sediments of the Southern area might
be supplied by paleo-rivers in the Southern Vietnam
with rather far transportation pathway.
- Either structures of BI.1 in the Northern area
were destroyed by extrusive activities or there is no
top seal due to lack of claystone.
- In BI.2, although structures exist, reservoir
capacity is still low due to abundant claystone.
Reference
1. La Thi Chich (2001), Petrology, Publishing
house of Ho Chi Minh City National University, p.
265-322.
2. Nguyen Ngoc Cu et al. (1998), “Oil-bearing
reservoir formations in Vietnam”, Vietnam
Petroleum Institute science conference, Hà Nội.
3. Pham Tuan Dung and Pham Van Hung
(2001), “Geological structure of Lower Miocene No
23 reservoir, Bach Ho field”, Petroleum conference,
Hà Nội.
4. Nguyen Van Dung (2004), Petrological char-
acteristics, postdepositional deformations and their
impacts on porosity and permeability of Oligocene-

Early Miocene sandstone reservoirs in Su Tu Den
field, block 15-1, Cuu Long basin, Master thesis,
University of Natural Sciences of Ho Chi Minh City.
5. Pham Xuan Kim (1988), Characteristics of
petrology, petrofacies, formation environment and
distribution of Early Oligocene-Early Miocene reser-
voirs in Cuu Long basin, Vietnam Petroleum
Institute.
6. Chu Duc Quang (2004), Depositional envi-
ronment and organic facies of Oligocene-Early
Miocene block 15-1 Cuu Long basin, Master thesis,
University of Natural Sciences of Ho Chi Minh City.
7. Pham Tuan Dung, Phung Dac Hai, Tran
Xuan Nhuan (2006), “Formation Characteristics Of
Early Miocene Deposits In The Bach Ho and Rong
Fields”, Technical Forum Clastic_Carbonate
Reservoir, Ho Chi Minh City.
8. F.K.North (1985), Petroleum Geology, Unwin
Hyman, USA.
9. G.M.Friedman & J,E.Sanders (1978),
Principle of Sedimentology, John Wiley & Sons,
NewYork, Chichester, Brisbane Toronto.
10. Tran Van Hoi, Phung Dac Hai, Tran Xuan
Nhuan, Pham Tuan Dung, Bui Nu Diem Loan
(2003), “The Geological Characteristics Of Clastic
Reservoir In The White Tiger And Dragon Fields”,
Technical Forum: Cuu Long basin Production-
Chalenges and Opportunities, Ho Chi Minh City.
11. Howel Williams, Francis J. Turner, Charles
M. Gilbert (1982), Petrography: An Introduction To

The Study Of Rocks In Thin Sections, W. H.
Freeman and Company, San francisco, USA.
12. J.H.Barwis, J.G.McPherson & J.R.J
Studlick (1990). Sandstone Petroleum Reservoirs,
Springer-Verlag.
13. Maurice Tucker (1989), Techniques In
Sedimentology, Blackwell Scientific Pub
14. O. Serra (1989), Sedimentary
Environments From Wireline Logs, Schlumberger.
15. Reineck-Singh (1980), Depositional
Sedimentary Environments, Springer-Verlag Berlin-
Heidelberg-Newyork.
16. Ngo Thuong San và nnc (1993),
“Stratigraphy and Lithology of The Mekong Basin”,
Proceedings of the international seminar on the
stratigraphy of Southern shelf of Vietnam. Ha Noi.
17. Supakorn Krisadasima, Nguyen Tien Long,
Hoang Thanh Bang, Ngo Quang Hien, Chanwichai
Suksawat and Nguyen Thanh Long (2006),
“Overview of Clastic Reservoir Potential in Block 9-
2 Cuu Long Basin, Vietnam”, Technical Forum
Clastic_Carbonate Reservoir, Ho Chi Minh City.
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Fig. 1. Geophysic-geological cross section along NW-SE direction in the Northern area
Fig. 2. Geophysic-geological cross section along NW-SE direction in the central area
Fig. 3. Geophysic-geological cross section along NW-SE direction in the Southern area

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Fig. 4. Geophysic-geological cross section along NE-SW direction in the Western margin
Fig. 5. Geophysic-geological cross section along NE-SW direction in the cetral area
Fig. 6. Geophysic-geological cross section along NE-SW direction in the Eastern margin
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Fig. 7. Distribution map of BI.2 thickness
Fig. 9. Distribution map of BI.2 grain size
Fig. 11. Distribution map of BI.2 matrix
and cement content
Fig. 12. Distribution map of BI.1 matrix
and cement content
Fig. 10. Distribution map of BI.1 grain size
Fig. 8. Distribution map of BI.1 thickness
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Fig. 13. Distribution map of BI.2 porosity
Fig. 15. Distribution map of BI.2 Sw
Fig. 16. Distribution map of BI.1 Sw
Fig. 17. Distribution map of BI.2 NTG Fig. 18. Distribution map of BI.2 NTG
Fig. 14. Distribution map of BI.1 porosity
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Determination of fractured basement per-
meability in White Tiger oil field from well log
data by artificial neural network system
using zone permeability as desired output
MSc. Tran Duc Lan
R&E Institute - Vietsovpetro
Abstract
Recently, some authors have suggested using the Artificial Neural Network (ANN) method
to determine permeability from log data. An ANN is built from the permeability of cores. This
method is highly applicable in sedimentary rocks[1], [4]. However, due to the limitation in coring
methods and laboratory measurements, core permeability is not truly representative for the per-
meability in the fractured basement rock in well bores.
In our study about fractured basement in White Tiger Field, offshore Vietnam, instead of
using core permeability, we have used zone permeability as desired outputs of an ANN to cal-
culate permeability profile in wells. Zone permeability, which was estimated from built-up pres-
sures and production log test data has high reliability and directly represents for the permeabil-
ity of the well bore rock.
We have determined the permeability for 16 wells in the field where both zone permeabili-
ty and well log data are available. For quality control, the calculated permeability is compared
reversely to zone permeability. The correlation coefficients are very high, commonly greater than
0.98.
Having calculated permeability and input well log data i.e. samples (more than 55,000 sam-
ples) in these 16 wells, we have been building a system of dozens of ANNs (called ANN sys-
tem) to determine permeability for wells which have only well log data.
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Preface
In White Tiger and Dragon fields, offshore
Vietnam, the main reservoirs are in fractured and
vuggy basement. Permeability is one of the most
important reservoir properties. This property has a
significant impact on petroleum fields operations
and reservoir management. Joint Venture
Vietsovpetro Company has been conducting sever-
al researches on determination of permeability
using the rate of drilling penetration (ROP), mud
loss data, core data, drilling stem testing data
(DST), built-up pressures analysis (BUP) and pro-
duction logging data (PLT). The results are promis-
ing and applicable in sedimentary rocks. However,
these traditional methods encounter limitations
when applied to fractured basement rocks. Due to
the difficulty in taking core in fractured and faulted
zones, we normally core in fresh rock intervals, the
core permeability hence are not representing for the
permeability in reservoir targets. Using BUP and
PLT data, permeability are determined for wide
intervals from 4m to several hundred meters. These
spare zone permeability values are not enough for
oil reserve calculation and for planning of produc-
tion and development strategy. Following, we will
present a more detailed and reliable method to
determine permeability (in the sense of permeabili-

ty sections – permeability profile).
Artificial neural network introduction
The first artificial neuron was produced in
1943 by the neurophysiologist Warren McCulloch
and the logician Walter Pits. But the technology
available at that time did not allow them to do too
much. The ANN theory then was developed further
by Minsky and Papert. In 1969, they published a
book to summarize criticized issues in ANN theory
and presented valuable study for later develop-
ment of ANN. Since then, ANN was restored and
applied widely [3].
In fact, ANN is a computer program. It is pro-
grammed based on mathematical models using
ANN theory. An ANN has abilities to learn and to
run. In another word, an ANN has two main
processes. These are training and running
processes [2].
A model of an ANN is shown in Figure 1. It is
used to determine permeability from well log data.
There are three layers in the ANN, which are input
layer, hidden layer and output layer. The input layer
contains 6 nodes corresponding to logging curves
of GR, DT, NPHI, RHOB, LLD and LLS (or MSFL).
The hidden layer contains 7 nodes and the output
layer contains 1 node representing permeability.
Between layers, there are connections and weights.
In the model, an ANN divides complex prob-
lems into simpler tasks. Each task is solved at a rel-
evant node by a so-called processing element (PE).

A PE receives value with sum weighted inputs and
returns outputs by solving the corresponding activa-
tion function. We used sigmoid functions as activa-
tion functions in hidden and output layers especial-
ly for discrete input data. Sigmoid function is nonlin-
ear and derivative in the whole determination field.
Its output varies from 0 to 1. We have also tried tan-
gent hyperbolic functions for calculation. This type
of activation function has the output varies from -1
to 1. However, the calculated permeability from
these two functions appears to be similar.
An example formula of a sigmoid function is as
equation (1) and Figure 2 below:
The permeability in Figure 1 is calculated as
equation (2).
Where ƒ is simplified sigmoid function without
parameter b. In this case b is replaced by w
b[in-o]
and w
b[h-out]
which are weighting factors deter-
mined from the input-hidden layer connection and
hidden-output layer connection, respectively.
After the training process, weighting factors are
set up for each connection using least mean square
error method and back propagation algorithm.
Having weighting factors, we can easily calculate
permeability (Perm) using equation (2).
Using zone permeability as desired output of an
ANN to determine fractured basement perme-

ability from well log data
Determination of permeability profile for the wells
having well log data and zone permeability
One of the most important characteristics in
training process of an ANN is the statistical analysis
based on the majority of samples. It ensures that
the output values are closest to the desired output
values of predominant samples.
An example is in the Table 1. The table con-
tains 2 groups of samples. Group A contains sam-
ples from 1 to 5. Group B contains samples from the
6 to 10. Sample 11 has the input value similar to that
(1)
(2)
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in group A but the desired output is similar to the
desired output in group B. After training and running
processes, the output value of sample 11 will be
similar to the out put values in group A.
By using zone permeability values as desired
outputs for the statistical analysis in an ANN, we
suggest a procedure to determine permeability from
well log data as in Figure 3.
We calculated permeability for 16 wells, which
have enough both zone permeability and well log
data, belong White Tiger oil field. Figure 4 and

Figure 5 shows the relation of calculated permeabil-
ity and data.
Figure 6 shows the zone permeability from
ANN in Y axis and zone permeability form BUP-PLT
in X axis. The correlation coefficients, which indi-
cates the similarity between X and Y data, are as
high as 0.98.
Determination of permeability for the wells
having only well log data
Having calculated permeability and input well
log data i.e. samples (more than 55,000 samples) in
these 16 wells, we can calculate permeability for
other wells in White Tiger field which have only well
log data. However, this amount of samples is too
big for the training process of an ANN. According to
us, this number should be less than 5,000. In order
to optimize the training and re-training processes,
we divide these samples into smaller groups. Each
group has a specific value range and each sample
is assigned to only one group. For samples, which
do not belong to any existing groups, they will be
omitted during calculation. After classification step,
training process is done for each group separately.
In together, we will have a system of ANNs.
The ultimate test for any technique that bears
the claim of permeability prediction from well log
data, is accurate and verifiable prediction of perme-
ability for wells from which only the well log data is
available. We built the ANN system including 44
ANNs. For developing this ANN system, we used 68

zones for training and 16 zones for cross-validation
testing. Figure 7 is the cross plot of zone permeabil-
ity which predicted by ANN system (kz-ANN) on
cross-validation testing data set against actual zone
permeability (kz-PLT). The correlation coefficient is
greater than 0.89.
We used this ANN system to calculate the per-
meability profile for 85 wells in the fractured base-
ment reservoir of White Tiger oil field. Figure 8
shows the permeability profiles, which were predict-
ed by the ANN system, of some wells having only
well log data.
Conclusions
The success in using zone permeability as
desired output for statistical analysis in an artificial
neural network to determine permeability from well
log data has opened a new trend in application of
ANN. Desired output data is not set separately for a
single sample, instead we use averaged number to
for a group of samples. This averaging method is
highly practical, especially when we can not choose
the desired outputs for each particular input data.
By dividing input data i.e. samples into smaller
groups, we can manage a huge amount of input
data for training processes. An ANN is designed for
one specific group of input samples. Depends on
the number of groups, we will have an ANN system.
This ANN system is flexible. It is easy to add new
input data to save running time in re-training
processes.

References
[1]. Lê Hải An, 2000, Phương pháp tính độ
thấm từ tài liệu địa vật lý giếng khoan bằng mạng
nơron, Hội nghị Khoa học lần thứ 14, quyển 4 Dầu
khí, Hà Nội, tr 5-7.
[2]. Trần Đức Lân, 2005, Giải pháp nơron nhân
tạo và phương pháp làm giàu các tham số trong
nghiên cứu Địa chất, 3, Tạp chí DK, Tr.23-31.
[3] Christos Stergiou, 2004, NeuroSolution,
NeuroDimension, Inc. 1800 N. Main Street, Suite
D4 Gainesville, FL 32609..
[4] Mohaghegh, S., Balan, B., Ameri, S.(1995),
State-Of-The-Art in Permeability Determination
From Well Log Data, SPE 30979.
Table 1
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Fig. 1. The model of an ANN [6-7-1] is used to determine permeability from well log data
Fig. 2. Graphic of a sigmoid function given a = 0.5 and b = - 8
Fig. 3. Procedure to determine permeability from well log data
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Fig. 4. Prediction models permeability values (K-AN) vs. zone permeability (K-TV)
for wells belong White Tiger oil field Fractured Section
Fig. 5. Comparison of determined permeability by ANN (PerAnn) and other well log data

Fig. B is a zoom in of fig. A at zone 3600-3660m
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Fig. 6. Zone permeability from BUP-PLT (K-TV)
vs. zone permeability from ANN (K-AN1z)
Fig. 7. Comparison between actual zone permeability
kz-PLT and prediction zone permeability kz-ANN
with 16 zones of testing data set
Fig. 8. Permeability profile prediction by ANN system from the wells having only well log data
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