Application of Co-Kriging and Ordinary Kriging for Selecting Additional Well Locations
Rong Lu*, Colorado School of Mines
SUMMARY
Well performance index (WPI), which is an indicator on how
much producing potential a well has, is proposed for Cana
Field using the available information from the completion
database. I used ordinary kriging and co-kriging to create in-
terpolation maps for WPI across the region. The interpolation
results can be used to predict WPI values for locations that
have not gone through drilling programs, thus guiding opera-
tor to ﬁnd the next drilling locations. Different kriging models’
performance are compared using cross-validation. It is shown
co-kriging with clean ﬂuid volume has the best performance.
Recommendations are given regarding new well locations.
INTRODUCTION
Unconventional oil and gas resources development has gained
much more attention since the last decade, due to the advance-
ment in hydraulic fracturing (HF, or “frac”) technology. In or-
der to develop shale gas reservoirs, which have extremely low
permeability, HF has to be applied. In the process ﬂuids and
solids under high pressure are pumped into the formation to
break the rock. As fractures are created, more reservoir contact
are obtained and the shale gas would ﬂow through the fractures
to the wellbore. One question the industry are interested in is,
where to drill/frac the wells in unconventional shale plays.
In this work, a well performance index (WPI) is used to eval-
uate the production potential for existing wells (“samples”),
which leverages the initial pressure and production rates infor-
mation. Then geostatistical methods, including ordinary krig-
ing and co-kriging, are used to predict producing potentials
for un-drilled regions (“unsampled” locations). My goal is to
compare the performance of different estimators:
1. ordinary kriging (OK) on variable of interests
2. co-kriging (CK) using secondary variables with differ-
ent levels of correlation
using cross-validation, and then propose new drilling location
candidates for Cana Field from the found best estimator.
The Cana Field dataset (Lu, 2014) is a well completion database.
It contains more than 400 shale gas wells drilled in Cana Wood-
ford Shale in Oklahoma; for each well, there are completion
and HF job parameters, and initial production data up to the
ﬁrst 90 days.
THEORY & METHOD
In this section how to ﬁnd new locations for HF wells is dis-
cussed. From the literature (Krivoruchko and Wood, 2014), a
well performance index (WPI) is used to estimate how much
production potential a well has, using the initial 90 days’ pro-
duction and pressure data:
WPI =
90
X
i=1
dailyProdRatei ×dailyPressurei
(1)
It is assumed in this work that pressure is constant over the ﬁrst
90 days and that fracture pressure can be used for estimation
purposes, then the equation becomes (the hat indicates it is an
estimator for the true WPI):
d
WPI = avgProdRateFirst90Days(MCFE/day)×90×
fracGradient×TVD
(2)
Now all the information needed is available from the dataset.
A visualization of the spatial distributions and relative values
of d
WPI is shown in Figure 1. The approach for prediction,
known as “kriging”, is essentially a linear estimator using the
known information. The core idea is to assign weights wi to
each known data point z(xi) located at xi, and by applying a
linear summation the property’s value at unsampled location
x0 is obtained:
bZ(x0) =

w1
w2
...
wn

·


z1
z2
...
zn

=
n
X
i=1
wi(x0)×Z(xi)
(3)
Weights, wi, are determined by minimizing variance of estima-
tion (Pebesma, 2004). The performance measure of this lin-
ear estimator will be discussed later. In OK, only the variable
of interests (primary variable or primary data) is studied and
the weights will add up to unity. CK also takes advantage of
secondary-data, which have spatial correlations with primary
data. Then the covariance between two or more regionalized
variables is leveraged.
Assumptions and data pre-processing
Besides using the approximated WPI (Equation 2), only co-
located primary/secondary data is included in this study, which
ensures a linear model of coregionalization (will be addressed
later). Since the original values of variables of interests are
huge, both primary and secondary variables are log10 trans-
formed.
Secondary variables
Clean ﬂuid volume and proppant volume are chosen to be the
secondary data respectively. Their corresponding correlations
with primary data is shown in Figure 2. It can be seen ﬂuid
volume has higher correlation with WPI than proppant volume.
Toolset
Stanford Geostatistical Modeling Software (SGeMS) is used
for the purpose of exploratory data analysis (EDA). For the
ﬁnal results generation and presentation, they are completed
arXiv:2203.12164v1  [stat.AP]  23 Mar 2022

Using co-kriging and ordinary kriging for ﬁnding new well locations
x
y
3920000
3940000
3960000
3980000
520000
540000
560000
580000
Figure 1: This shows a bubble plot for WPI samples. Larger
points indicate larger WPI values. X and Y axes are for easting
and northing (units in meters), respectively.
by two R packages: gstat (Pebesma, 2004) and sp (Pebesma
and Bivand, 2005). These packages provides:
• OK and CK algorithms
• API for cross validation
• API for ﬁtting linear models of coregionalization
Strategy for veriﬁcation
In this work k-fold cross validation is conducted for compar-
ing the performance of different estimators. Figure 3 shows
an illustration when the number of folds, k, equals 5. Since
the number of WPI samples is 190 (not huge), leave-one-out
(LOO) is performed: each run I use 189 data points to “krige”
the value at the location of the 190th point, and then calculate
the residual by comparing kriging result with the true value.
Two performance measures are adopted:
1. Mean error:
ME = 1
n
n
X
i=1
(yi −byi)
(4)
2. Root-mean-square error:
RMSE =
v
u
u
t1
n
n
X
i=1
(yi −byi)2
(5)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Clean Fluid Volume, gallon
1e7
0
1
2
3
4
5
6
7
8
WPI, MCFE*psi
1e9
(a)
1000000
2000000
3000000
4000000
5000000
Proppant Volume, lbs
0
1
2
3
4
5
6
7
8
WPI, MCFE*psi
1e9
(b)
Figure 2:
Scatter plots between primary and secondary vari-
ables. (a) WPI and ﬂuid volume have correlation coefﬁcient
of 0.58; (b) WPI and proppant volume have correlation coefﬁ-
cient of 0.37.
Figure 3:
This illustrates the case of 5-fold cross-validation.
The original sample is randomly partitioned into 5 equal sized
subsamples. 4 of the 5 subsamples will be used to train the
model, being tested against the held-out 1 subsample. The pro-
cess repeats 5 times.

Using co-kriging and ordinary kriging for ﬁnding new well locations
RESULTS & VERIFICATION
Ordinary kriging
The prediction map generated by OK is shown in Figure 4
distance
semivariance
0.05
0.10
0.15
10000
20000
30000
(a)
log10(WPI), OK predictions
8.8
9.0
9.2
9.4
9.6
(b)
Figure 4:
(a) Variogram modeling of the primary variable
WPI; (b) WPI map generate by OK.
Co-kriging with ﬂuid volume
In CK, the spatial covariance between primary and secondary
data is represented by a cross variogram, as presented in Fig-
ure 5 along with the output WPI map.
Co-kriging with proppant volume
CK results using proppant volume as secondary data is shown
in Figure 6. Note the proppant volumes’ experimental vari-
ogram has a ﬂuctuation behavior naturally but the chosen var-
iogram model smooths it out. This is also to ensure a linear
model of coregionalization (will be addressed later).
distance
semivariance
0.00
0.01
0.02
0.03
10000
20000
30000
lt_wpi.lt_fl
0.000
0.005
0.010
0.015
lt_fl
0.00
0.05
0.10
0.15
lt_wpi
(a)
log10(WPI), CK predictions, Fluid covariable
8.8
9.0
9.2
9.4
9.6
(b)
Figure 5:
(a) Fitted direct and cross variograms between
log10(WPI) and log10(Fluid). “lt” and “ﬂ” stands for log10
and ﬂuid volume respectively; (b) WPI map generated by CK
with ﬂuid volume.

Using co-kriging and ordinary kriging for ﬁnding new well locations
distance
semivariance
0.000 0.005 0.010 0.015 0.020
10000
20000
30000
lt_wpi.lt_prop
0.000 0.005 0.010 0.015 0.020 0.025
lt_prop
0.00
0.05
0.10
0.15
lt_wpi
(a)
log10(WPI), CK predictions, Proppant covariable
8.8
9.0
9.2
9.4
9.6
(b)
Figure 6:
(a) Fitted direct and cross variograms between
log10(WPI) and log10(Proppant). “lt” and “prop” stands for
log10 and proppant volume respectively; (b) WPI map gener-
ated by CK with proppant volume.
Estimator comparisons
Using the performance measures discussed before (Equation 4
and Equation 5), the comparisons are summarized in Table 1.
It can be seen CK results are more accurate than OK result,
though CK takes much longer time to run. CK with ﬂuid is
chosen as the best model since its modeled variogram honors
the experimental variogram better than CK with proppant did
(refer to Figure 5(a) and Figure 6(a)).
Mean Error
RMSE
Running Time
OK
-0.0014
0.249
1.86 sec
CK with ﬂuid
-0.0007
0.232
6.74 sec
CK with proppant
-0.0009
0.229
6.67 sec
Table 1: Comparison of the Three Interpolations
DISCUSSION - CONCLUSION
One of the beneﬁts CK provides is that it might give better pre-
dictions when the main attribute of interest (primary variable;
WPI in this case) is sparse, but related secondary information
is abundant. The raw data from Cana dataset contains more
ﬂuid volume (also proppant volume) than WPI, motivating me
to take the CK approach. For modeling, one needs to ﬁt both
the direct and cross-variograms simultaneously, the results of
which must produce a positive deﬁnite system. Easiest way to
ensure this is to ﬁt a linear model of coregionalization: all the
variograms have same shape and range, with different partial
sills and nuggets. As presented in Figure 7, when including all
the secondary data (ﬂuid volume), ﬂuid volume has naturally a
shorter range than the primary variable. Thus ﬁnally only co-
located data points are included for modeling which happen to
make secondary variables have approximately the same range
as the primary variable.
distance
semivariance
0.00 0.02 0.04 0.06 0.08 0.10
10000
20000
30000
lt_wpi.lt_fl
0.00
0.01
0.02
0.03
lt_fl
0.00
0.05
0.10
0.15
lt_wpi
Figure 7:
Experimental variograms when including all the
available secondary data. “lt” and “ﬂ” stands for log10 and
ﬂuid volume respectively.
In summary, it has been shown that co-kriging performs better
than ordinary kriging on the Cana dataset. The CK with ﬂuid
volume model provides the best predictive power regarding

Using co-kriging and ordinary kriging for ﬁnding new well locations
which regions have higher producing potentials, from where
new drilling locations are proposed (Figure 8).
log10(WPI), CK predictions, Fluid covariable
8.8
9.0
9.2
9.4
9.6
Figure 8:
New wells are recommended to be located at the
“dollar sign” area, whereas the zones showing warnings have
lower producing potentials.

Using co-kriging and ordinary kriging for ﬁnding new well locations
REFERENCES
Krivoruchko, K., and N. Wood, 2014, Using Multivariate In-
terpolation for Estimating Well Performance Understand-
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Lu, R., 2014, Investigation of post hydraulic fracturing well
cleanup physics in the cana woodford shale: Colorado
School of Mines.
Pebesma, E. J., 2004, Multivariable geostatistics in s: the gstat
package: Computers & Geosciences, 30, 683–691.
Pebesma, E. J., and R. S. Bivand, 2005, Classes and methods
for spatial data in R: R News, 5, 9–13.
