This document summarizes a paper that quantifies uncertainties in petrophysical properties derived from well logs. It shows that the interpretation model used can be a major source of uncertainty, and that Monte Carlo modeling provides a way to quantify this. Different log analysis models are compared for water, oil, and gas bearing sands. For gas sands, density and sonic porosity estimates without hydrocarbon corrections exceed the actual porosity range. Core data provides the best uncertainty estimates by validating logs, and Monte Carlo can match these when using the proper interpretation model.

1. Copyright 2005, Society of Petroleum Engineers Inc.
This paper was prepared for presentation at the 2005 Asia Pacific Oil & Gas Conference and
Exhibition held in Jakarta, Indonesia, 5 – 7 April 2005.
This paper was selected for presentation by an SPE Program Committee following review of
information contained in a proposal submitted by the author(s). Contents of the paper, as
presented, have not been reviewed by the Society of Petroleum Engineers and are subject to
correction by the author(s). The material, as presented, does not necessarily reflect any
position of the Society of Petroleum Engineers, its officers, or members. Papers presented at
SPE meetings are subject to publication review by Editorial Committees of the Society of
Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper
for commercial purposes without the written consent of the Society of Petroleum Engineers is
prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300
words; illustrations may not be copied. The proposal must contain conspicuous
acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.
Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.
Abstract
Typical petrophysical deliverables for volumetric and
modeling purposes are net reservoir, porosity, permeability,
water saturation and contact locations. These data are usually
provided without quantitative determination of their
uncertainties.
Current computing power renders it now feasible to use
Monte-Carlo simulation to determine the uncertainty in
petrophysical deliverables. Unfortunately, quantitative
uncertainty definition is more than just using Monte-Carlo
simulation to vary the inputs in your interpretation model. The
largest source of uncertainty may be the interpretation model
itself.
This paper will use a variety of porosity interpretation
models to illustrate how the impact of each input on the
uncertainty varies with the combination of input values used in
any given model. It will show that use of the incorrect model
through oil and gas zones may give porosity estimates with
Monte-Carlo derived uncertainty ranges that exclude the
actual porosity.
Core data provides the best means of quantifying actual
uncertainty in the petrophysical deliverables. Methodologies
for deriving uncertainties quantitatively by comparison with
core data will be presented. In the absence of core data,
interpretation models should have been tested against core
data through the same or similar formations nearby. Monte-
Carlo simulation can then be used as an effective means of
quantifying petrophysical uncertainty. Comparisons between
the core comparison and Monte-Carlo techniques will be
made, showing that similar results are achieved with the
appropriate interpretation models.
The methodologies described in this paper are
straightforward to implement and enable petrophysical
deliverables to be treated appropriately in volumetric and
modeling studies. In addition, quantification of petrophysical
uncertainty assists in operational decision-making by letting
users know how reliable the numbers produced actually are,
and what range of properties is physically realistic. Such work
also allows the key contributions to uncertainty to be defined
and targeted if overall volumetric uncertainty must be reduced.
Introduction
Petrophysical evaluations are carried out for a number of
different purposes, including operational decision-making,
volume in place estimation and reservoir modeling. In all
cases, the uncertainty in the deliverables of net reservoir,
porosity, permeability, water saturation and contact locations
are critical. However, these data are usually provided without
quantitative determination of their uncertainties.
This paper will highlight the ease with which uncertainties
can be derived using Monte-Carlo simulation. It will also
illustrate how flexible this technique is when it comes to
working with different interpretation models, which is not
commonly done. The largest source of uncertainty in
petrophysical interpretation may be the interpretation model
itself.
Given the large number of possible interpretation models
for all the different petrophysical deliverables, this paper will
only use the most basic petrophysical deliverable, being
porosity, to illustrate the relationship between uncertainty and
the log interpretation model selected.
It will also be shown that verification of log porosity using
an independent measure such as core porosity can also provide
quantitative uncertainties allowing comparison with the log
derived uncertainties.
The State of Uncertainty in Petrophysics
The requirement for quantification of petrophysical
uncertainty is not a recent development. Many papers are in
the literature describing functions for uncertainty definition
and how to use Monte-Carlo modeling for the same purposes.
Although work such as that of Amaefule & Keelan (1989),
Chen & Fang (1986) and Hook (1983) provides an excellent
foundation on which to calculate uncertainties, the
methodologies are both time consuming to program and
inflexible with regard to interpretation model.
With the computing power available on desktop machines
today, engineers no longer have to use these analytical
techniques to derive uncertainty. Monte-Carlo models are
straightforward to build and no longer time consuming to run.
The literature contains a number of examples of Monte-Carlo
simulation being used to characterize petrophysical
uncertainty, such as the work of Spalburg (2004).
SPE 93125
Quantifying Petrophysical Uncertainties
S.J. Adams, WellEval.com Ltd.

2. 2 SPE 93125
However, none of these examples highlight the uncertainty
owing to the interpretation models being used.
Monte-Carlo Modeling & Assumptions
In order to understand the problems associated with Monte-
Carlo simulation and how best to overcome them, a simple
explanation of the technique is warranted.
Monte Carlo methods use random numbers and probability
descriptors for the input variables to investigate problems
expressed as mathematical formulae. As an example, if a
simple density porosity calculation is considered, as shown in
the equation below:
φd = (ρma – ρ)/( ρma – ρfl)
where ρma is the matrix density, ρ is the density log
measurement and ρfl is the density of the fluid in the pore
space of the zone investigated by the density tool and φd is the
log-derived density porosity.
The input values (ρma, ρ, ρfl) all have uncertainties
associated with them, so the resulting output (φd) will also
have an uncertainty. With Monte-Carlo simulation, the
uncertainty in the output is determined by randomly selecting
input values from their uncertainty distributions and
calculating the output value. The output value is stored then
the input selection and calculation processes are repeated a
large number of times. Finally all the output values are
examined statistically to determine the uncertainty in the
output value.
Monte-Carlo modeling is very flexible, allowing different
interpretation models to be built and the uncertainties tested
quickly. Dependencies between input variables may also be
accounted for in the input value determination.
The downside to Monte-Carlo simulation is that a large
number of cycles (>500) are typically required for meaningful
statistics to be developed. This point is illustrated below where
the equation for density porosity above has been modeled
through a water-bearing sand. In Figure 1 the distribution of
porosities does not begin to approach a reasonable (“normal”)
shape until 500 scenarios or more are run.
0.00
0.05
0.10
0.15
0.20
0.25
0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26
density porosity (v/v)
normalisedfrequency
50
100
500
10000
30000
Figure 1 – Histogram of the density porosities derived using
Monte-Carlo simulation through a water-bearing sand. Only
the number of scenarios (from 50 to 30000) changes between
each curve.
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
10 100 1000 10000 100000
number of scenarios
porosityvalues(v/v)
P90
P50
P10
Figure 2 – The P90, P50 and P10 statistics derived from the
Monte-Carlo density porosity distributions vary with the
number of scenarios modeled.
Figure 2 shows how the statistics derived from the
porosity distributions do not approach the correct values until
at least 500 scenarios have been run.
Note that the required number of scenarios for statistical
accuracy will increase with the number of input variables used
in any particular model.
Porosity Uncertainty Using Monte-Carlo in
Theoretical Cases
To illustrate the significance of the assumptions and models
used for uncertainty quantification, the basic petrophysical
deliverable of total porosity is used.
Monte-Carlo models have been built for density porosity,
sonic porosity and density-neutron porosities. The
uncertainties in these three different methods are compared
through water, oil and gas bearing sand models.
Figure 3 compares the porosities calculated for the same
density log measurement in known water, oil and gas systems.
This Figure serves to illustrate that not correcting for the
presence of hydrocarbons will result in significant errors in
density porosity estimates. Indeed, failure to correct for
hydrocarbons in a gas-bearing zone will result in most likely
porosity estimates that do not include the actual porosity value
in the P90 to P10 uncertainty range.
0
0.1
0.2
0.3
0.4
0.5
0.1 0.15 0.2 0.25 0.3
density porosity (v/v)
normalisedfrequency
water
oil
gas
Figure 3 – Histogram of the density porosities derived using
Monte-Carlo simulation through water, oil and gas-bearing
sands. Note that these porosity estimates are corrected for the
presence of hydrocarbons.

3. SPE 93125 3
Figure 4 compares the porosities calculated from the
density-neutron log combination using the same scenarios as
modeled for the density porosity. In this case, the reduced
neutron response through the gas-bearing sand results in some
correction for the hydrocarbons. However, this “correction”
would not be so apparent were the sands shaley. The porosities
estimated through the oil sands show little correction for the
presence of hydrocarbons.
0
0.1
0.2
0.3
0.4
0.5
0.1 0.15 0.2 0.25 0.3
density-neutron porosity (v/v)
normalisedfrequency
water
oil
gas
Figure 4 – Histogram of the density-neutron porosities
derived using Monte-Carlo simulation through water, oil and
gas-bearing sands. Note that no assumptions have been made
about the presence of hydrocarbons.
Figure 5 compares the porosities calculated from the sonic
log using the same scenarios as modeled for the density
porosity. Again, the porosity estimated for the water-bearing
sands is a match with the density log-based estimate.
However, the porosities for the oil and gas-bearing sands are a
little too high unless corrections for the presence of
hydrocarbons are made. Of course, in the case of this
theoretical model, such corrections are possible using the work
of Batzle and Wang (1992), but in real cases, the porosities
through the hydrocarbon-bearing intervals must be derived
from another source before quantitative corrections for
hydrocarbons can be made.
0
0.1
0.2
0.3
0.4
0.1 0.15 0.2 0.25 0.3
sonic porosity (v/v)
normalisedfrequency
water
oil
gas
Figure 5 – Histogram of the sonic porosities derived using
Monte-Carlo simulation through water, oil and gas-bearing
sands. Note that no assumptions have been made about the
presence of hydrocarbons.
To better illustrate that the various porosity models do
actually give different results and different uncertainty
distributions Figures 6, 7 and 8 compare the water, oil and
gas-bearing sand porosities for the three porosity interpretation
techniques.
In the water-bearing sands (Figure 6), all three techniques
give similar porosity values, but the uncertainty distributions
are slightly different, as should be expected. In the oil-bearing
sands (Figure 7), the best estimate of the porosities differs
significantly between the different interpretation techniques,
yet the range of the uncertainties for each technique remains
similar to that observed for the water-bearing case. And in the
case of gas-bearing sands (Figure 8), the differences in the
best estimate of porosity increase even more between the
interpretation techniques, while the uncertainty ranges remain
virtually unchanged from the water-bearing case.
It is clear from the foregoing that even porosity estimation
from wireline log data can give different answers depending
on the method used i.e. on the logs used and whether or not
hydrocarbon corrections are carried out. Accordingly the
uncertainty estimates also differ depending on the models
used.
0
0.1
0.2
0.3
0.4
0.5
0.1 0.15 0.2 0.25 0.3
porosity (v/v)
normalisedfrequency
density
density-neutron
sonic
Figure 6 – Histogram of the porosities derived using Monte-
Carlo simulation through water-bearing sands.
0
0.1
0.2
0.3
0.4
0.5
0.1 0.15 0.2 0.25 0.3
porosity (v/v)
normalisedfrequency
density
density-neutron
sonic
Figure 7 – Histogram of the porosities derived using Monte-
Carlo simulation through oil-bearing sands.

4. 4 SPE 93125
0
0.1
0.2
0.3
0.4
0.5
0.1 0.15 0.2 0.25 0.3
porosity (v/v)
normalisedfrequency
density
density-neutron
sonic
Figure 8 – Histogram of the porosities derived using Monte-
Carlo simulation through gas-bearing sands.
Porosity Uncertainty Using Monte-Carlo in Real
Cases
To allow the conclusions drawn from the theoretical models
presented in the previous section to be verified, real data has
been selected from cored wells through water, oil and gas
columns. Figures 9, 10 and 11 show the data. Note that only
the “density HC corr” data has been hydrocarbon corrected
(using invaded zone resistivity logs).
In the water-bearing sands (Figure 9), the log-derived
density and sonic porosities have similar average values, close
to those from the core data. While the density-neutron
combination actually overestimates porosities in these sands.
The data also confirm that the uncertainty ranges are different
for each measurement type. Note too that the log-derived
porosities have not been calibrated to the core data.
What is particularly interesting about Figure 9 is that the
actual average porosity value confirmed by both the core and
hydrocarbon corrected density porosity is less than the P90
estimate from the density-neutron combination.
In the oil-bearing sands (Figure 10), the porosity estimates
are much closer together, with all the uncertainty ranges (P90
to P10) including the actual best estimate porosity from the
core data. What is significant here is that the density porosities
should be corrected for the density difference due to even oil
being lighter than water i.e. the non-hydrocarbon corrected
density porosities are already 0.5 p.u. too high. The density-
neutron porosities are also on average 1.0 p.u. too high.
The largest differences are observed in the gas-bearing
sands (Figure 11), with the sonic and non-hydrocarbon
corrected density porosities being more than 2.0 p.u. too high.
Indeed the P90 to P10 uncertainty ranges for these porosity
models do not include the actual best estimate porosity from
the core data. Even the density-neutron combination
overestimates porosity by 0.5 p.u.
0
0.1
0.2
0.3
0.4
0.5
0.1 0.15 0.2 0.25 0.3
porosity (v/v)
normalisedfrequency
density
density HC corr.
density-neutron
sonic
core
Figure 9 - The log-derived porosity statistics for water-
bearing sands in a medium porosity shaley sand reservoir.
0
0.1
0.2
0.3
0.4
0.5
0.1 0.15 0.2 0.25 0.3
porosity (v/v)
normalisedfrequency
density
density HC corr.
density-neutron
sonic
core
Figure 10 - shows the log-derived porosity statistics for oil-
bearing sands in a similar medium porosity shaley sand
reservoir.
0
0.1
0.2
0.3
0.4
0.1 0.15 0.2 0.25 0.3
porosity (v/v)
normalisedfrequency
density
density HC corr.
density-neutron
sonic
core
Figure 11 - The log-derived porosity statistics for gas-bearing
sands in a similar medium porosity shaley sand reservoir.
A second set of examples from a different region are
presented as Figures 12, 13 and 14.
In the water-bearing reservoir (Figure 12), the log-derived
density and sonic porosities have similar average values, close
to those from the core data. While the density-neutron
combination again overestimates porosities in these sands.
In the oil-bearing reservoir here (Figure 13), the porosity
estimates still show divergent most likely values. In fact the
density-neutron based porosity estimate again does not
encompass the actual best porosity estimate from core within

5. SPE 93125 5
the P90 to P10 range. In this example too, the value of
correcting the density porosities for even oil is apparent.
The largest differences are again observed in the gas-
bearing reservoir (Figure 14), with the sonic and non-
hydrocarbon corrected density porosities being more than 5.0
p.u. too high. Here too, the P90 to P10 uncertainty ranges for
these porosity models do not include the actual best estimate
porosity from the core data. Even the density-neutron
combination does a poor job, underestimating porosity by 2.3
p.u.
0
0.1
0.2
0.3
0.4
0.1 0.15 0.2 0.25 0.3
porosity (v/v)
normalisedfrequency
density
density HC corr.
density-neutron
sonic
core
Figure 12 - The log-derived porosity statistics for water-
bearing sands in a medium porosity limestone reservoir.
0
0.1
0.2
0.3
0.4
0.15 0.2 0.25 0.3 0.35
porosity (v/v)
normalisedfrequency
density
density HC corr.
density-neutron
sonic
core
Figure 13 - The log-derived porosity statistics for oil-bearing
limestone in the same unit as Figure 12.
0
0.1
0.2
0.3
0.4
0.15 0.2 0.25 0.3 0.35
porosity (v/v)
normalisedfrequency
density
density HC corr.
density-neutron
sonic
core
Figure 14 - The log-derived porosity statistics for gas-bearing
limestone in the same unit as Figures 12 and 13.
Porosity Uncertainty By Core Comparison
Since the real cases presented in the previous section also had
core data acquired over the same logged intervals, it is
possible to compare the log-derived total porosities with those
measured on the equivalent piece of core.
When making such a comparison, there are two factors to
bear in mind. Firstly, the depth match between log and core
data must be excellent so that the same intervals are in fact
being compared. Secondly, the porosity resolution at the log
scale is not the same as that derived from the core plug scale.
If the core porosity data is not “filtered” back to a similar
resolution to the log-derived data, then the variability (or
uncertainty) implied by the comparison will be larger than it
should be.
Figures 9, 10, 11, 12, 13 and 14 all show very similar most
likely porosity estimates and uncertainty distributions for the
hydrocarbon-corrected density and the core porosities. This
observation implies that the best match to the core porosities is
using these hydrocarbon-corrected density porosities.
Although the other porosity estimation techniques can provide
reliable porosities in some circumstances, provided the
hydrocarbon influence on the log measurements being used
are taken into account.
Overall Porosity Uncertainty Interpretation
From the foregoing, it is apparent that the uncertain ranges
estimated using Monte-Carlo simulation are interpretation
model dependent. It is still possible for calculated uncertainty
ranges not to include the actual reservoir porosities, if an
inappropriate porosity interpretation model is used.
The best way to ensure that the appropriate interpretation
model is selected is by comparison with core data. If no core
data is available, then the work presented herein suggests that
hydrocarbon-corrected density porosities should be used. If it
is not possible to carry out these calculations, then whatever
model is selected should either include hydrocarbon correction
or model the likely range of hydrocarbon densities in the
uncertainty analyses.
Uncertainty in Other Petrophysical Deliverables
Of course the techniques discussed and conclusions drawn
from the work presented in this paper are equally valid for
other petrophysical properties such as water saturation,
permeability, net reservoir and contact locations.
Although not detailed in this paper, since the impact of the
porosity uncertainties illustrated is sufficient to illustrate the
value of model uncertainty quantification, it is good practice to
derive uncertainties in all petrophysical deliverables so that
users are aware of any limitations in the data presented.
Conclusions
Monte-Carlo simulation is well suited to uncertainty
quantification in the current petrophysical environment.
However, simply calculating uncertainty is insufficient unless
it can be shown that the interpretation model applied is
appropriate. This conclusion is true for all petrophysical
deliverables, not just porosity as presented in this paper. Good
quality core data provides an excellent basis on which to
determine the appropriate interpretation model.

6. 6 SPE 93125
With Monte-Carlo modeling, care should also be taken to
ensure that sufficient scenarios are run to determine valid
statistics on the output values. Generally a few tens of
scenarios are insufficient. Typically greater than 500 runs are
required.
Petrophysical evaluation should attempt to determine
uncertainties in at least the critical items of porosity and water
saturation. Knowledge of the possible range of values enables
Operators to make better data gathering and completion
decisions. Reservoir modeling studies are also more likely to
include scenarios approaching the real reservoir.
Acknowledgements
The author would like to acknowledge the feedback received
from many clients over the years that have seen the value of
uncertainty quantification in their petrophysical deliverables
once the data was made available to them.
References
Amaefule, J.O. & Keelan, D.K.: “Stochastic Approach to
Computation of Uncertainties in Petrophysical Parameters,”
Society of Core Analysts, Paper No. SCA-8907, 1989.
Batzle, M. & Wang, Z.: “Seismic Properties of Pore Fluids,”
Geophysics, Vol. 57, No. 11 (November 1992); P. 1396-1408.
Chen, H.C. & Fang, J.H.: “Sensitivity Analysis of the
Parameters in Archie’s Water Saturation Equation,” The Log
Analyst, pp39-44, September-October 1986.
Hook, J.R.: “The Precision of Core Analysis Data and Some
Implications for Reservoir Evaluation,” SPWLA 28th
Annual
Logging Symposium, June 27-30, 1983.
Liu, N. & Oliver, D.S.: “Evaluation of Monte Carlo Methods
for Assessing Uncertainty," SPE Journal (2003) 8.
Spalburg, M.R.: “Bayesian Uncertainty Reduction for Log
Evaluation,” SPE 88685, 11th Abu Dhabi International
Petroleum Exhibition and Conference held in Abu Dhabi,
U.A.E., 10-13 October 2004.