**WELLOG****
UNCERTAINTY AND PROBABILITY**

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**REVISED ****9-03-2007**

**© ****WELLOG**** 2007**

**All Rights Reserved**

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**How certain or uncertain am I that my calculation is correct? What
is the probability of being correct?**

**WHY IS IT IMPORTANT?**

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**If we know the uncertainties and are able to quantify the
probability of the answers we give, then we can make better decisions.**

**THE SOURCE:**

**The following information was obtained from an article in the Oil
and Gas Journal and is presented in summary;**

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** A summary of ****“Uncertainty in Log Calculations can be Measured”**** written by Dr. Roberto Aguilera for the Oil
& Gas Journal; (see references)**

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**Because log analysis is based upon empirical relationships, the
results of log analysis involve considerable uncertainty. The amount of
uncertainty can be measured quantitatively using Monte-Carlo simulation.**

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**A Monte-Carlo simulation takes the probability distributions of
the input data and generates random values within the pre-established minimum
and maximum values of the data.**

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**By repeating a calculation many times, the analyst obtains enough
information to plot various answers, for example water saturation vs.
probability of occurrence. Understanding the probability permits better
decisions.**

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**Several distributions
have been presented in the literature. Some of the most useful are the
histogram, triangular, and rectangular. (Reference 1-3)**

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**HOMOGENOUS RESERVIOR:**

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**This example illustrates the use of the ****Monte Carlo**** simulation on a log
calculation when most of the input data have a rectangle distribution. (TABLE
1)**

** (from Reference 2)**

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**Given three basic equations: (from References 4, 6)**

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**
****F**** =
(****r****ma – ****r****b)/(**** r****ma-****r****f)
(1)**

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**
F = a X ****F**^{m }**
(2)**

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**
Sw = (F x Rw/ Rt) ^{1}^{/n }
(3)**

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**Where:**

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**
F = formation resistivity factor**

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****F**** = porosity, fraction**

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****r****ma = matrix density, grams/cc**

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****r****b = formation bulk
density, grams/cc**

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****r****f = average fluid
density, grams/cc**

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a = constant**

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m = porosity exponent**

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Sw= water saturation, fraction**

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Rw = water resistivity, ohm-meters**

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Rt = true formation resistivity, ohm-meters**

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n = water saturation exponent**

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**Range**** of ****Parameters**** using rectangle
distributions: TABLE 1 (from Reference 2)**

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**Parameter: Lower
Limit: Upper Limit: **

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**Rt****
19.000
21.000**

**Rw****
0.055
0.075**

**n****
1.800
2.200**

**a****
0.620
0.620**

**m****
2.000
2.300**

**r****b
2.360
2.380**

**r****ma
2.580
2.630**

**r****f
0.900
0.900**

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**Calculated water saturation spread and probability of occurrence for each water saturation is shown in this figure. The
figure indicates that Sw ranges between 27 and 54 percent with the most likely
value between 36 percent and 39 percent. It further indicates there is a 90
percent chance of having a water saturation of 45 percent or less. The zone
under evaluation was perforated and produced water-free oil. (Ref. 2)**

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**NATURALLY FRACTURED RESERVOIR:**

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**The problem of uncertainty is even more pronounced in the
evaluation of a naturally fractured reservoir. There are many qualitative and
quantitative techniques which are used with varied degrees of success. (from Reference 7, 8) For this example, histogram,
triangular, and rectangle distributions of the input data are used. (TABLE 2) Equations 4 thru 10. (from Reference
3)**

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**Range of parameters using rectangle, triangular and histogram
distributions: TABLE 2 (from Reference 3)**

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**Parameter:
Distribution:
Lower:
Most Likely:
Higher:
Source of data:**

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**v****
Triangular
0.10
0.20
0.40
Core and pressure analysis**

**mb****
Rectangular
1.90
2.10
Analysis of inter-granular cores**

**Rt****
Triangular
30.00
35.00
50.00
Logs**

**Rw****
Rectangular
0.05
0.07
SP Log, water analysis, Rw tables, ****F****-R cross plots.**

**Bo
Rectangular 1.30
1.45
PVT analysis and empirical charts.**

**h****
Triangular
90.00
150.00
Logs, core analysis**

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**HISTOGRAM DISTRIBUTION OF POROSITY:**

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**Matrix
Porosity: Frequency:**

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**.02 to
.03
.10**

**.03 to
.04
.20**

**.04 to
.05
.40**

**.05 to
.06
.18**

**.06 to
.07
.12**

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**Total
1.00**

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**Equations used in ****Monte Carlo**** simulation example:**

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****F**** = ****F****b[****1-v(1-**** F****b)]
(4)**

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**
m = - log[1 /v**** F**** + (1-v)**** F****b ^{-mb}] / log **

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**
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Sw = [ Rw/****(F**^{m}** Rt)] ^{1/n
}(6)**

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****F****f
=**** (approx.) - ****F****b
(7)**

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Swf = (approx.) krw = (approx.) (****m****w WOR) /(Bo ****m****o +**** m****w
WOR)
(8)**

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OOIPf =
7758.4 AH**** F**** (1-sw)/ Boi
(9)**

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**
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OOIPt =
7758.4 AH**** F**** (1-swf)/ Boi
(10)**

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**Where:**

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**F**** =**** total porosity,
fraction**

**F****b = matrix porosity, fraction**

**F****f = fracture
porosity, fraction **

**v =**** partitioning
coefficient, fraction**

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m = double porosity exponent**

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mb = matrix
porosity exponent**

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Swf = water saturation
in fractures, fraction**

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Sw = total water saturation, fraction**

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****m****w = water viscosity, cp**

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****m****o = oil viscosity, cp**

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Boi = initial oil formation volume factor, bbl/stock-tank
bbl**

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WOR = water oil ratio**

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Krw = relative permeability to water, fraction**

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OOIPf = original oil in place in the fractures, stock-tank
bbl**

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OOIPt = total original oil in place, stock-tank bbl**

**
A = Area, acres**

**
H = net pay, feet**

**
Other nomenclature as defined previously**

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**These equations are not claimed to be perfect but represent an
additional tool for evaluation of fractured media. The main task here, the
quantification of uncertainty, can be carried out with these or any other set
of equations based on the preference and experience of the analyst.**

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**CONCLUSIONS:**

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**Values of porosity were calculated using equation 4 and the ****Monte Carlo**** simulation approach.
The results are shown in the figure (3-A).
The calculated total porosities range between a minimum of 2.6 percent and a
maximum of 9.9 percent. The figure also indicates that there is a 50 percent
probability that the total porosity will be 5.6 percent or greater and an 80
percent probability that it will be larger than 4.3 percent.**

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**The calculated values of the double-porosity exponent, m are also
presented in figure (3-B).
Note that they range between 1.321 and 1.678 compared with the input values of
matrix porosity exponent (mb) which ranged between
1.9 and 2.1.**

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**Figure (3-C)
shows that the total water saturation, Sw, varies between 9.7 and 57 percent.
It also shows that there is a 50 percent chance of having a water saturation of
22 percent or less and an 80 percent probability that Sw will be 27 percent or
less.**

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**Fracture porosity is found to range between 0.47 percent and 3.7
percent as shown in figure (3-D).
There is a 50 percent probability of having a fracture
porosity equal to or greater than 1.8 percent.**

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**Values of water saturations in the fractures are calculated to
range between 21 and 77 percent with the most likely value of 41 percent as
shown in figure
(3-E).**

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**The variation of oil-in-place within the fracture system is very
marked. In fact, it ranges between 1.85 and 13.46 million stock-tank barrels,
with a most likely value of 5.1 million stock-tank barrels. The figure shows
that there is an 80 percent probability that the oil-in-place in the fractures
will be 3.5 million stock-tank barrels or larger. **

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**As in the previous example, this case shows that small variations
in input data can result in astronomic differences in the results.
Consequently, it is recommended to quantify the uncertainty of log
calculations, whenever possible, to reach better decisions.**

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**References:**

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**1. McGray, A. W., Petroleum Evaluations
and Economic Decisions, Prentice Hall Inc. Englewood Cliffs, NJ, 1975, 201.**

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**2. Walstron, J. E. Mueller, T. D., and
McFarlane, R. C. “Evaluating Uncertainty in Engineering Calculations”, J. of
Pet. Tech., Dec. 1967, 1595**

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**3. Aguilera, Roberto, “The Uncertainty of Evaluating Original
Oil-in-Place in Naturally Fractured Reservoirs”, SPWLA Nineteenth Annual
Logging Symposium, June 13-16, 1978.**

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**4. Archie, G. E. “The electrical Resistivity Log as an aid in
Determining Some Reservoir Characteristics,” Trans. AIME, 1952, 146, 54-67.**

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**5. Log Review I, Dresser Atlas, Houston, 1974.**

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**6. Log Interpretation Principles, Schlumberger, 1969.**

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**7. Aquilera, Roberto, and Van Poolen, H. K. How to Evaluate Naturally Fractured
Reservoirs for Various Well Logs” OGJ, Dec. 25, 1978, 202-208.**

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**8. Aquilera, Roberto, and Van Poolen, H. K. , “Porosity and
Water Saturation can be Estimated from Well Logs,” OGJ, Jan. 8, 1979, 101-108.**

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**Contact ****WELLOG**** if you have questions about uncertainty at info@wellog.com .**

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