Pressure derivative with respect to natural logarithm of time aka Bourdet derivative is defined as follows:

The calculation algorithm uses the weighted slopes  $//<![CDATA[ \begin{array}{l}\Delta P/\Delta X\end{array} //]]>$ of the pressure change versus the change in time:

where  $//<![CDATA[ \begin{array}{l}\Delta P_1 = P_j - P_{j-1}\end{array} //]]>$,  $//<![CDATA[ \begin{array}{l}\Delta P_2 = P_{j+1} - P_j\end{array} //]]>$ are the functions of the pressure change with respect to the point of interest and  $//<![CDATA[ \begin{array}{l}\Delta X_1 = X_j - X_{j-1}\end{array} //]]>$,  $//<![CDATA[ \begin{array}{l}\Delta X_2 = X_{j+1} - X_{j}\end{array} //]]>$ are the functions of the change in time.

Modified Time Function

For buildup analysis, the Horner time modified by Agarwal is used:

where  $//<![CDATA[ \begin{array}{l}t_p\end{array} //]]>$ is duration of time the well produced at constant rate. So, the time function is

Superposition Time Function

In case the well produced at variable rate before shut-in, the superposition time function is used:

where  $//<![CDATA[ \begin{array}{l}q_{n}\end{array} //]]>$ is the rate before shut-in,  $//<![CDATA[ \begin{array}{l}\displaystyle \^q_{n} = \begin{cases} q_{n+1}, & \text n=1\\ q_{n+1} - q_{n}, & \text n>1 \end{cases}\end{array} //]]>$ is the modified rate.

Points per Cycle

Suppose  $//<![CDATA[ \begin{array}{l}N\end{array} //]]>$ is the number of data points,  $//<![CDATA[ \begin{array}{l}n\end{array} //]]>$ is the number of points per logarithmic cycle, and  $//<![CDATA[ \begin{array}{l}X_{j}\end{array} //]]>$ is the  $//<![CDATA[ \begin{array}{l}j\end{array} //]]>$-th data point, $//<![CDATA[ \begin{array}{l}\mathrm{j=1,2,...,N}\end{array} //]]>$. If inequality  $//<![CDATA[ \begin{array}{l}X_{j+k}-X_j< \displaystyle \frac{1}{10^{n+1}}\end{array} //]]>$,  where  $//<![CDATA[ \begin{array}{l}k=1,2,3,...\end{array} //]]>$, is valid, then points  $//<![CDATA[ \begin{array}{l}X_{j+k}\end{array} //]]>$ are removed.

L-spacing

In order to smooth the derivative, the so-called 'L-spacing' method is applied. Suppose the smoothing parameter  $//<![CDATA[ \begin{array}{l}L\end{array} //]]>$ is given. For every  $//<![CDATA[ \begin{array}{l}j=2\dots N-1\end{array} //]]>$ there exist points  $//<![CDATA[ \begin{array}{l}m=j-k_1\end{array} //]]>$ and  $//<![CDATA[ \begin{array}{l}n=j+k_2\end{array} //]]>$ such that  $//<![CDATA[ \begin{array}{l}X_n-X_j\gt L\end{array} //]]>$ and  $//<![CDATA[ \begin{array}{l}X_j-X_m \gt L\end{array} //]]>$.  Thus, the logarithmic derivative formula changes to:

where  $//<![CDATA[ \begin{array}{l}\Delta \^{P_1} = P_j - P_{m}\end{array} //]]>$,  $//<![CDATA[ \begin{array}{l}\Delta \^{P_2} = P_{n} - P_j\end{array} //]]>$,   $//<![CDATA[ \begin{array}{l}\Delta \^{X_1} = X_j - X_{m}\end{array} //]]>$,  $//<![CDATA[ \begin{array}{l}\Delta \^{X_2} = X_{n} - X_{j}\end{array} //]]>$. In practice, it is recommended to choose L between 0 and 0.5.

See also

Formal science / Mathematics / Calculus

Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing / Pressure Transient Analysis (PTA) /  PTA Diagnostic Plot / Logarithmic Derivative

References

1. Bourdet, D. "Well Test Analysis: The Use of Advanced Interpretation Models", Elsevier, 2002.
2. Bourdet, D., Ayoub, J.A., and Pirard, Y.M. "Use of Pressure Derivative in Well-Test Interpretation", SPE Formation Evaluation, 1989, June, pp. 293-302.