Sensitivity analysis of a pumping test on a well with wellbore storage and skin

Abstract

This paper reviews the basic concepts of sensitivity analysis and points out their limitations. A case is then made for logarithmic sensitivity. The magnitude of logarithmic sensitivity alone does not determine the accuracy of an aquifer parameter estimate, especially when the relative measurement errors are not uniform throughout space and time. Deterministic parameter correlations and plausible relative errors in parameter estimates are introduced as imperfect measures of information content in measurements. A plausible relative error in the parameter estimate combines the effect of logarithmic sensitivity with that of relative measurement error. Minimizing the plausible relative errors rather than maximizing the corresponding sensitivities should serve as a guide to identifying the measurements most useful for parameter estimation or as candidate measurements for optimal sampling. Furthermore, avoiding among them measurements with high parameter correlations as much as possible may help ensure that the sensitivity matrix X (or X T X ) is well-conditioned and, thus, that the parameter estimates are accurate. The discussed concepts are then applied to a model of a pumping test conducted on a fully penetrating well situated in a confined aquifer. The model accounts for the wellbore storage and an infinitesimal skin. In contrast to the traditional and normalized sensitivities, the logarithmic sensitivities of the drawdown in the pumping well, the drawdown in an observation well, and the wellface flowrate to transmissivity, T, storativity, S, and the skin factor, η, depend on a small number of parameters. They can thus be represented by a single type curve or a family of a relatively few type curves. The plausible relative errors in T, S, and η estimated from wellbore drawdown rapidly decrease during the wellbore storage phase and reach a plateau or slowly decrease outside the wellbore storage phase. The plausible relative errors from the wellface flowrate rapidly decrease during the wellbore storage phase before reaching a minimum (around the time when the wellface flowrate is equal to about half the pumping rate) and then rapidly increase. This means that transient flowmeter test measurements of drawdown and wellface flowrate should not be made during the early times of the wellbore storage phase.