WELLOG                                 UNCERTAINTY AND PROBABILITY

REVISED 9-03-2007

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

WHY IS IT IMPORTANT?

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;

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

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.

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.

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.

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

HOMOGENOUS RESERVIOR:

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)

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

F =  (rma – rb)/( rma-rf)                           (1)

F = a X F                                                      (2)

Sw = (F x Rw/ Rt)1/n                                        (3)

Where:

F = formation resistivity factor

F = porosity, fraction

rma = matrix density, grams/cc

rb = formation bulk density, grams/cc

rf = average fluid density, grams/cc

a = constant

m = porosity exponent

Sw= water saturation, fraction

Rw = water resistivity, ohm-meters

Rt = true formation resistivity, ohm-meters

n = water saturation exponent

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

Parameter:      Lower Limit:    Upper Limit:

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

rb                    2.360               2.380

rma                 2.580               2.630

rf                     0.900               0.900

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)

NATURALLY FRACTURED RESERVOIR:

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)

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

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

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

HISTOGRAM DISTRIBUTION OF POROSITY:

Matrix Porosity:          Frequency:

.02 to .03                     .10

.03 to .04                     .20

.04 to .05                     .40

.05 to .06                     .18

.06 to .07                     .12

Total                            1.00

Equations used in Monte Carlo simulation example:

F = Fb[1-v(1- Fb)]                                                               (4)

m  =  - log[1 /v F + (1-v) Fb-mb] / log F                              (5)

Sw =  [ Rw/(Fm Rt)] 1/n                                                                                                                   (6)

Ff  = (approx.)  -  Fb                                                             (7)

Swf  = (approx.) krw = (approx.) (mw WOR) /(Bo mo + mw WOR)               (8)

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

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

Where:

F  = total porosity, fraction

Fb = matrix porosity, fraction

Ff =  fracture porosity, fraction

v  =  partitioning coefficient, fraction

m  =  double porosity exponent

mb  =  matrix porosity exponent

Swf  = water saturation in fractures, fraction

Sw  =  total water saturation, fraction

mw = water viscosity, cp

mo = oil viscosity, cp

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

WOR = water oil ratio

Krw = relative permeability to water, fraction

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

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

A = Area, acres

H = net pay, feet

Other nomenclature as defined previously

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.

CONCLUSIONS:

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.

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.

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.

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.

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).

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.

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.

References:

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

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

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

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

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

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

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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