The calculation algorithm of Logarithmic Derivative uses the weighted slopes of the pressure change versus change in time:
\begin {equation}
\left( \frac{dP}{dX} \right)_j=\frac{\displaystyle\frac{\Delta P_1}{\Delta X_1} \Delta X_2 + \displaystyle\frac{\Delta P_2}{\Delta X_2}\Delta X_1}{\Delta X_1 + \Delta X_2},
\end {equation} |
where ,
are the functions of the pressure change with respect to the point of interest and
,
are the functions of the change in time.
The straightforward application of the formula results in the noisy data.
The most popular practical solution is to apply Sparsing and Smoothing.
Given the approved point , the next approved point
is going to be when
holds true, where:
| total number of source data points | |
number of points per logarithmic cycle |
Suppose is the number of data points,
is the number of points per logarithmic cycle, and
is the
-th data point,
. If inequality
, where
, is valid, then points
are removed.
In order to smooth the derivative, the so-called 'L-spacing' method is applied. Suppose the smoothing parameter is given. For every
there exist points
and
such that
and
. Thus, the logarithmic derivative formula changes to:
\begin {equation}
\^{\left( \frac{dP}{dX} \right)_j}=\frac{\displaystyle\frac{\Delta \^{P_1}}{\Delta \^{X_1}} \Delta \^{X_2} + \displaystyle\frac{\Delta \^{P_2}}{\Delta \^{X_2}}\Delta \^{X_1}}{\Delta \^{X_1} + \Delta \^{X_2}},
\end {equation} |
where ,
,
,
. In practice, it is recommended to choose L between 0 and 0.5.
|
| Fig. 1 – Calculation scheme for logarithmic derivative subject to L-spacing. |
Formal science / Mathematics / Calculus
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing / Pressure Transient Analysis (PTA) / PTA Diagnostic Plot / Logarithmic Derivative
