Bayesian Analysis of the Data Worth of Pumping Tests Using Informative Prior Distributions
Authors: Heße Falk , Isachsen Lars , Müller Sebastian , Sabine Attinger 
Published: March 2020

&lt;p&gt;&lt;span&gt;Characterizing the subsurface of our planet is an important task. Yet compared to many other fields, the characterization of the subsurface is always burdened by large uncertainties. These uncertainties are caused by the general lack of data and the large spatial variability of many subsurface properties. &lt;/span&gt;&lt;span&gt;Due to their &lt;/span&gt;&lt;span&gt;comparably &lt;/span&gt;&lt;span&gt;low costs, pumping test&lt;/span&gt;&lt;span&gt;s&lt;/span&gt;&lt;span&gt; are regularly applied for the characterization of groundwater aquifers. The &lt;/span&gt;&lt;span&gt;classic&lt;/span&gt;&lt;span&gt; approach is to &lt;/span&gt;&lt;span&gt;identify the parameters of some conceptual subsurface model&lt;/span&gt; &lt;span&gt;by means of curve &lt;/span&gt;&lt;span&gt;fit&lt;/span&gt;&lt;span&gt;ting&lt;/span&gt;&lt;span&gt; some analytical expression &lt;/span&gt;&lt;span&gt;to the measured drawdown.&lt;/span&gt; &lt;span&gt;One of the drawbacks of classic analyzation techniques of pumping tests is the assumption of the existence of a single representative parameter value for the whole aquifer. Consequently, they cannot account for spatial heterogeneities. To address this limitation, a number of studies have proposed extensions of both Thiem&amp;#8217;s and Theis&amp;#8217; formula. Using these extensions, it is possible to estimate geostatistical parameters like the mean, variance and correlation length of a heterogeneous conductivity field from pumping tests.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;W&lt;/span&gt;&lt;span&gt;hile these methods have demonstrated their ability to estimate &lt;/span&gt;&lt;span&gt;such&lt;/span&gt;&lt;span&gt; geostatistical parameters, their data worth has &lt;/span&gt;&lt;span&gt;rarely&lt;/span&gt;&lt;span&gt; been investigated within a Bayesian framework. This is particularly relevant since recent developments in the field of Bayesian inference facilitate the derivation of informative prior distributions for these parameters. &lt;/span&gt;&lt;span&gt;Here, informative means that the prior is&lt;/span&gt; &lt;span&gt;based on currently available background data &lt;/span&gt;&lt;span&gt;and therefore may be able to substantially influence the posterior distribution&lt;/span&gt;&lt;span&gt;.&lt;/span&gt; &lt;span&gt;If this is the case,&lt;/span&gt;&lt;span&gt; the actual data worth of pumping tests, as well as other subsurface characterization methods, may be lower than assumed.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;To investigate this possibility, we implemented a series of numerical pumping tests in a synthetic model based on the Herten aquifer. Using informative prior distributions, we derived the posterior distributions over the &lt;/span&gt;&lt;span&gt;mean, variance and correlation length of &lt;/span&gt;&lt;span&gt;the synthetic&lt;/span&gt;&lt;span&gt; heterogeneous conductivity field. &lt;/span&gt;&lt;span&gt;Our results show that for mean and variance, we already get a substantially lowered data worth for pumping tests when using informative prior distributions, whereas the estimation of the correlation length remains mostly unaffected. These results suggest that with an increasing amount of background data, the data worth of pumping tests may fall &lt;/span&gt;&lt;span&gt;even lower, meaning that more informative techniques for subsurface characterization will be needed in the future.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;