step tests - aquifer analysis software - Aquiferwin32 Derivative Analysis
Analysis of pumping test data and step tests can be greatly improved using a plot known as a derivative plot. Although rather simple in concept, such a plot can be very instrumental in accurately reducing the data from a pump test. It is beyond the scope of this document to go into the theory of analyzing a derivative plot and the user is referred to texts like Horne, 1995, and journal articles like Spane & Wurstner, 1993, for the details. Suffice it to say that the characteristics of curves representing the first-order pressure derivative versus time can be more distinctive than the traditional type curves. The difference is the result of the sensitivity of the derivative to small variations in the pressure change that occur during a pump test. This sensitivity can be used to identify wellbore storage effects, boundary effects and the establishment of radial flow conditions. The following graph demonstrates the marked difference of the derivative plot from the traditional leaky-confined type curves that flatten out with time.
In the above equation, the differentiation interval or L-Spacing is used to be consistent with the derivatives of the data. In the special case where the L-Spacing has been set to 0 to cause the adjacent data points to be used, the type curves will be generated using an L-Spacing of .1.
The implementation of the derivative analysis within Aquiferwin32 was adapted from two sources. The first-order pressure derivative of the data is performed as per Spane & Wurstner, 1993, and the algorithm from their DERIV program was adapted. As indicated in their paper, the differential algorithm is based on the preferred algorithm listed by Bourdet et al., 1989; the algorithm calculates the first derivative of the pressure change with respect to the natural logarithm of the change of time. Two options are available for calculating the data slopes before and after a given point, LEAST SQUARES and FIXED ENDPOINT. The fixed endpoint uses the points preceding and following the point of interest by the specified distance along the x axis. The least squares regression option uses all the points preceding and following the point of interest within the specified distance in the calculation.
Spane & Wurstner, 1993, recommend the fixed end point options for data from published type curves or data devoid of significant noise. For noisy test data the least squares option is preferred. In Aquiferwin32 the least squares option is the default. The distance along the x-axis to use in the aforementioned calculations is referred to as the L-Spacing. The L-Spacing ranges from 0 to .5 in which 0 uses the points immediately adjacent to the point of interest. Values greater than .2 smooth out noisy data but can also cause a loss of resolution. Since Aquiferwin32 directly calculates the values for type curves, the pressure derivatives are directly calculated using the equation presented by Horne, 1995.
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