Seismic Reservoir Modeling. Dario Grana

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Seismic Reservoir Modeling - Dario Grana


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of the favorable event, generally called the probability of success, is a real number p such that 0 ≤ p ≤ 1. The probability mass function pX(x) is then:

      (1.29)p Subscript upper X Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column p 2nd Column x equals 1 2nd Row 1st Column 1 minus p 2nd Column x equals 0 period EndLayout

      The mean of the Bernoulli distribution is then μX = p and the variance is sigma Subscript upper X Superscript 2 Baseline equals p left-parenthesis 1 minus p right-parenthesis.

      The Bernoulli distribution has several applications in earth sciences. In reservoir modeling, for example, we can use the Bernoulli distribution for the occurrence of a given facies or rock type. For instance, we define a successful event as finding a high‐porosity sand rather than impermeable shale. The probability of success is generally unknown and it depends on the overall proportions of the two facies.

      1.4.2 Uniform Distribution

      A common distribution for discrete and continuous properties is the uniform distribution on a given interval. According to a uniform distribution, a random variable is equally likely to take any value in the assigned interval. Hence, the PDF is constant within the interval, and 0 elsewhere. If a random variable X is distributed according to a uniform distribution U([a, b]) in the interval [a, b], then its PDF fX(x) can be written as:

      (1.30)f Subscript upper X Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than a 2nd Row 1st Column StartFraction 1 Over b minus a EndFraction 2nd Column a less-than-or-equal-to x less-than-or-equal-to b 3rd Row 1st Column 0 2nd Column x greater-than b period EndLayout

      The mean μX of a uniform distribution in the interval [a, b] is:

      (1.31)mu Subscript upper X Baseline equals StartFraction a plus b Over 2 EndFraction

      and it coincides with the median, whereas the variance sigma Subscript upper X Superscript 2 is:

      (1.32)sigma Subscript upper X Superscript 2 Baseline equals StartFraction b minus a Over 12 EndFraction period

      1.4.3 Gaussian Distribution

      Most of the random variables of interest in reservoir modeling are continuous. The most common PDF for continuous variables is the Gaussian distribution, commonly called normal distribution. We say that a random variable X is distributed according to a Gaussian distribution script upper N left-parenthesis upper X semicolon mu Subscript upper X Baseline comma sigma Subscript upper X Superscript 2 Baseline right-parenthesis with mean μX and variance sigma Subscript upper X Superscript 2, if its PDF fX(x) can be written as:

Graph depicts uniform probability density function in the interval [1, 3]. Graph depicts standard Gaussian probability density function with 0 mean and variance equal to 1.

      To compute a probability associated with a non‐standard Gaussian distribution script upper N left-parenthesis upper Y semicolon mu Subscript upper Y Baseline comma sigma Subscript upper Y Superscript 2 Baseline right-parenthesis with mean μY and variance sigma Subscript upper Y Superscript 2, we apply the transformation X = (YμY)/σY and we use the numerical tables of the standard Gaussian distribution. Indeed, the random variable X is a standard Gaussian distribution with 0 mean and variance equal to 1, and Скачать книгу