
Research Article


Effectiveness of Kriging Interpolation Technique for Estimating Permeability Distribution of a Field


O. Anomohanran
and
U. Chapele


ABSTRACT

This study seeks to investigate the effectiveness of using the kriging interpolation technique in estimating the permeability distribution of a field. In carrying out this study, the permeability of three sand layers each of 39 wells in a field were obtained from the record of porosity and water saturation using the Tixier and Timur empirical models. The permeability of the three layers was also determined experimentally from core samples in the laboratory. Result shows strong correlation between core and derived permeability. The derived permeability values were used as input data in the kriging interpolation. Result shows that the kriged permeability for layer one ranges from 0.81 to 3.98 D for Tixier empirical model while the Timur model yielded a permeability range of 0.97 to 3.80 D. For the second layer, the Tixier model yielded a permeability range of 0.40 to 3.69 D while that of Timur gave a range of 0.38 to 3.59 D. For the third layer, the Tixier model gave a range of 0.20 to 0.95 D while the Timur model gave a range of 0.30 to 1.10 D. This study has revealed that there is consistency in the values of the permeability distribution obtained for both the Tixier and Timur models. The study also shows that there is decrease of permeability with increase in depth. The study has shown that the kriging algorithm can be used effectively in generating a two dimensional permeability distribution that is reliable with error ranging from 0.6 to 2.4%. 




Received:
December 04, 2011; Accepted: March 02, 2012;
Published: May 25, 2012 

INTRODUCTION
The permeability of a rock or soil is defined as its ability to transmit fluid
through its pores or voids. Permeability is a property of the soil which is
very useful to researchers interested in investigating the characteristics of
the various soil layers in geophysical exploration of the subsurface (Alipour,
2007; Todd, 2004). Permeability is a parameter which
determines whether a well should be completed and brought on stream or not.
An effective management of reservoirs can be carried out only if the rock properties
such as porosity, water saturation and permeability are well known (Mohaghegh
et al., 1995; Aigbedion, 2007; Merdhah
and Yassin, 2009). Of all these properties, permeability has drawn much
attention because of the complexity involved in predicting it accurately (Balan
et al., 1995; Tang and Cheng, 1996). Nevertheless,
a lot of improvements have been successfully recorded empirically in the determination
of permeability (Bloch, 1991; Yamamoto,
2003; Fernando, 2008).
Once the properties of the rock are known, it is possible to determine the
permeability distribution using records from wells. In doing so, zones or layers
which are found to be similar in properties can be correlated together in a
bid to determine the distribution (Balan et al.,
1995; Mohaghegh et al., 1997; Qobi
et al., 2001).
It is now very possible to draw up relationships between porosity, water saturation
and permeability through the means of empirical models (Kapadia
and Menzie, 1985). The records of well log are used to estimate effective
porosity and irreducible water saturation which becomes the source data in empirically
developed models (Orodu et al., 2009).
Some empirical models based on the relationship between permeability, porosity and irreducible water saturation are the Tixier and the Timur approach. The Tixier model for determining permeability is expressed as: On the other hand, the Timur model for the estimation of permeability is expressed as:
where, φ stands for the rock porosity and S_{wi} is the irreducible
water saturation (Balan et al., 1995; Mohaghegh
et al., 1995).
Kriging is a geostatistical technique used in formulating an unbiased estimates
of regionalized variables at some locations (Bayraktar and
Turalioglu, 2005; Emery, 2005; Balasundram
et al., 2007, 2008). Kriging can be applied
to estimate the value of a variable at a particular point or to estimate the
average value of a block. It is known by the acronym “BLUE” also known
as the “Best Linear Unbiased Estimator” (Negreiros
et al., 2010). Kriging interpolation models have shown success in
its application in many areas of research. These include predicting groundwater
level, soil salinity, soil fertility property and areal storm pattern analysis
(Yasrebi et al., 2008; Sokouti
and Mahdian, 2009; Rabah et al., 2011). Kriging
when applied to research data gives values which best represent the distribution
of the original data (Mohaghegh et al., 1997;
NIH, 1998).
If we consider a situation in which a property is measured at a number of points,
x_{i}, within a region to give values of Z (x_{i}), where i
= 1, 2, 3, …., N, the value of the property at any place x_{o}
can be estimated. The place might be a point or an area of the same size and
shape or a block. Kriging computes the best linear unbiased estimate based on
a stochastic model of the spatial dependence (Tonkin and
Larson, 2002).
In the simplified kriging equation, the kriged estimate is given as:
where, w_{i} is the weight assigned to the sample Z at location x_{i}
(Largueche, 2006; Balasundram et
al., 2007; Akbari et al., 2009).
The kriging weights are unique in the sense that they eradicate the effect
of bias towards input sample values (Vann et al.,
2003).
This study is therefore aimed at investigating the effectiveness of estimating the permeability distribution of a field using the kriging interpolation technique. This is carried out by employing the Tixier and the Timur empirical models of permeability distribution. MATERIALS AND METHODS Core samples of three sand layers each of thirty nine wells from a field were subjected experimentally to permeability test in the laboratory. The porosity φ for the three layers was obtained from the sonic and density logs of the wells while the resistivity log was used to compute the water saturation S_{w}. The porosity and water saturation were used to derive the permeability for the three different sand zones using the Tixier and the Timur empirical relation as stated in Eq. 1 and 2. The values obtained were compared with the core permeability values obtained from laboratory analysis. Since, there was strong correlation between the core permeability and the Tixier and Timur’s derived permeability, the derived permeability were used as input data in the kriging process to derive the permeability distribution of the field.
The study area was divided into a square grid and permeability was estimated
at each of the grid nodes using the Tixier and Timur derived data as the respective
variables. Since we have thirty nine wells in the field, it follows that we
have thirty nine equations corresponding to these points which represent the
locations of the wells. There is also an equation that constrains the sum of
the weighting coefficients to be one. The sum of squares of the differences
between the core permeability and estimated values were obtained. This leads
to the universal kriging system which in matrix form is expressed as Eq.
4 (Negreiros et al., 2010). The result of
the kriging operation is a 40x40 matrix and a 1x40 matrix. The 40x40 matrix
was inverted in this study and its product multiplied with the 1x40 matrix to
obtain the weighing coefficients for estimating the permeability at each grid
location. This technique is applied to the three sand zones in the field of
study:
The values of the coefficients obtained by solving Eq. 4 are substituted into Eq. 5 to yield the permeability at successive grid points:
The values of permeability so obtained are then posted to the grid points and
were contoured to show the permeability distribution. The error associated with
the kriged values was determined by calculating the standard deviation between
the kriged permeability value and the true permeability value.
RESULTS
The record of the porosity and water saturation obtained for this study as
well as the derived permeability using the Tixier and the Timur models are as
shown in Table 1. These were records computed from three sand
zones located at a depth of between 7995 to 8010 feet for zone 1, 8100 to 8130
feet for zone 2 and 8280 to 8300 feet for zone 3. Because of the strong correlation
between the measured and the derived data, the log derived permeability was
used in obtaining the distribution for the reason that they can from the analysis,
accurately and consistently determine permeability.
Table 1: 
Table showing porosity, water saturations, Tixier and Timur’s
derived permeability for the three sand zones 

K: Permeability 
The values obtained were interpolated at the nodes of square grid. The contour
maps of the permeability distribution so obtained are presented as Fig.
1 to 6. The result of this interpolation for the Tixier
method for zones 1, 2 and 3 is as shown in Fig. 1, 3
and 5, respectively while the result for the Timur interpolation
for the three zones are as shown in Fig. 2, 4
and 6.

Fig. 1: 
Permeability contours in mD using Tixier’s equation
for zone 1 

Fig. 2: 
Permeability contours in mD using Timur’s equation for
zone 1 

Fig. 3: 
Permeability contours in mD using Tixier’s equation
for zone 2 

Fig. 4: 
Permeability contours in mD using Timur’s equation for
zone 2 

Fig. 5: 
Permeability contours in mD using Tixier’s equation
for zone 3 

Fig. 6: 
Permeability contours in mD using Timur’s equation for
zone 3 
DISCUSSION The result in Fig. 1 shows that the permeability for zone 1 increases from well B towards well H and increased as we move northward. This trend changes with a decrease from that point as we move downward. This stress leads to a reduction in porosity and thus a decrease in permeability. This situation also plays out in Fig. 2 for the Timur’s permeability distribution. The kriged permeability ranged from 0.813.98 D for the Tixier model and 0.973.80 D for the Timur model. The permeability distribution as shown in both Fig. 3 and 4 for zone 2 lies side by side in a northwest trend close to the fault. Large spacing occurs between the contours as we move southwards. The permeability on the average for this sand zone is slightly lower than what was obtained for zone 1. It ranged from 0.403.64 D for the Tixier model and 0.383.59 for the Timur model.
For zone 3, the permeability ranged from 0.20.95 D for the Tixier model and
0.31.1 D for the Timur model as is shown in Fig. 5 and 6,
respectively. These values are lower than what was obtained for zone 2 indicating
a decrease in permeability with increase in depth. This finding is in support
of the study of Saar and Manga (2004) that permeability
is depth dependence. It is also in agreement with the findings of Morrow
et al. (1994) that the values of permeability are based on rock type
and depth. Figure 5 and 6 shows the permeability
zone running from west to north east. Both contours show an increase in permeability
northwards. The error obtained from each grid point for the various zones ranges
from 0.62.4% which confirm reliability of the method used (Anomohanran,
2004). Result has also shown that the two models produced a similar trend
for the three layers mapped in this study showing that the kriging method used
in the permeability distribution is suitable for estimating the permeability
at any point in the field.
This study has shown that there is strong correlation between the core permeability
and the derived permeability using the Tixier and the Timur models. This view
supports the findings by Shang et al. (2003)
that there is a better correlation between measured and estimated permeability.
CONCLUSION In this study, geostatistical techniques using the Tixier and Timur equations were applied to the field data to determine the permeability. The permeability distributions derived from the Tixier and Timur models produced a similar pattern in contours for each of the three sand zones investigated. The contours show areas of equal permeability and can be used in predicting the best direction and position to drill a well. This study has also shown that there is an observed decrease in permeability with increase depth. The results of this study emphasize that permeability have spatial dependence and that understanding such structure may provide insight into the permeability of the field.

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