Probability Distributions for Monte Carlo

Overview

The choice of probability distribution for each uncertain input parameter is one of the most important decisions in Monte Carlo analysis. The distribution should reflect both the physical nature of the parameter and the available data. Using the wrong distribution can bias results significantly.

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Distribution Types

Normal Distribution

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$

Petroleum applications: Porosity from log analysis, formation temperature, well spacing.

Caution: Normal distributions allow negative values. For parameters that must be positive (permeability, thickness), use lognormal or truncated normal instead.

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Lognormal Distribution

If $X$ is lognormally distributed, then $\ln(X)$ is normally distributed:

$$f(x) = \frac{1}{x\sigma_{ln}\sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu_{ln})^2}{2\sigma_{ln}^2}\right) \quad \text{for } x > 0$$

Petroleum applications: Permeability (well-established log-normal character), OOIP, EUR, well costs, production rates.

Key Relationships

$$\text{Mean} = \exp(\mu_{ln} + \sigma_{ln}^2/2)$$

$$\text{Median} = \exp(\mu_{ln})$$

$$\text{P10/P90 ratio} = \exp(2 \times 1.282 \times \sigma_{ln})$$

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Triangular Distribution

$$f(x) = \begin{cases} \frac{2(x-a)}{(b-a)(c-a)} & a \le x \le c \\ \frac{2(b-x)}{(b-a)(b-c)} & c < x \le b \end{cases}$$

Petroleum applications: Net pay thickness, recovery factor, drilling days, water saturation.

When to Use Triangular

• Limited data (< 10 data points)
• Expert-based estimates available
• Clear physical bounds exist
• Quick screening studies

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Uniform Distribution

$$f(x) = \frac{1}{b - a} \quad \text{for } a \le x \le b$$

Petroleum applications: Contact depths (OWC, GOC) when only bounding wells are available, exploration parameters with high uncertainty.

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PERT Distribution

The PERT (Program Evaluation and Review Technique) distribution is a modified beta distribution using three-point estimates:

$$\text{Mean} = \frac{a + 4c + b}{6}$$

Compared to triangular: PERT gives a smoother, more bell-shaped curve with less probability at the extreme values. It is generally preferred when the min and max represent true physical limits rather than observed extremes.

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Truncated Normal Distribution

A normal distribution bounded by minimum and maximum values:

$$f(x) = \frac{\phi((x-\mu)/\sigma)}{\sigma[\Phi((b-\mu)/\sigma) - \Phi((a-\mu)/\sigma)]} \quad \text{for } a \le x \le b$$

Petroleum applications: Porosity (bounded by 0 and ~0.40), water saturation (bounded by $S_{wi}$ and 1.0).

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Constant (Deterministic)

$$f(x) = \delta(x - c)$$

Used for parameters that are known with certainty or treated as fixed in a particular analysis. Useful for fixing some parameters while varying others in sensitivity studies.

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Distribution Selection Guide

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Parameter Estimation

From Data (Statistical)

When sufficient data exists:

From Expert Judgment

Converting P10/P90 to Distribution Parameters

For lognormal:

$$\sigma_{ln} = \frac{\ln(P10) - \ln(P90)}{2 \times 1.282}$$

$$\mu_{ln} = \frac{\ln(P10) + \ln(P90)}{2}$$

For normal:

$$\sigma = \frac{P10 - P90}{2 \times 1.282}$$

$$\mu = \frac{P10 + P90}{2}$$

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Related Topics

• MC Overview — Monte Carlo methodology and workflow
• MBE Overview — Reserves estimation as Monte Carlo application

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References

1. Rose, P.R. (2001). Risk Analysis and Management of Petroleum Exploration Ventures. AAPG Methods in Exploration No. 12.

2. Murtha, J.A. (1997). "Monte Carlo Simulation: Its Status and Future." Journal of Petroleum Technology, 49(4), 361-373. SPE-37932-JPT.

3. Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd Edition. Wiley.

4. Law, A.M. (2015). Simulation Modeling and Analysis, 5th Edition. McGraw-Hill.

5. SPE/WPC/AAPG/SPEE/SEG (2018). Petroleum Resources Management System (PRMS).