## 5 Results
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Fit** & **Parameter** & **Estimated Value** & \multicolumn{2}{c}{**95\% Confidence Limits**} \\  & & & **Lower Limit** & **Upper Limit** \\ \hline Sorption & \(k_{\text{a}}\) & \(3.993\times 10^{-2}\) & \(3.576\times 10^{-2}\) & \(4.410\times 10^{-2}\) \\ coefficients only & \(k_{\text{d}}\) & \(8.230\times 10^{-13}\) & \(-6.304\times 10^{-11}\) & \(6.469\times 10^{-11}\) \\ \hline Sorption & \(k_{\text{a}}\) & \(4.459\times 10^{-2}\) & \(4.104\times 10^{-2}\) & \(4.813\times 10^{-2}\) \\ coefficients and & \(k_{\text{d}}\) & \(3.018\times 10^{-12}\) & \(-1.656\times 10^{-10}\) & \(1.717\times 10^{-10}\) \\ injection & \(C_{\text{Ca}^{2+},\text{inj}}\) & 10.64 & 10.08 & 11.19 \\ concentrations & \(C_{\text{urea,inj}}\) & 20.31 & 19.47 & 21.14 \\ \hline Sorption & \(k_{\text{a}}\) & \(4.471\times 10^{-2}\) & \(3.864\times 10^{-2}\) & \(5.079\times 10^{-2}\) \\ coefficients and & \(k_{\text{d}}\) & \(4.534\times 10^{-10}\) & \(-3.247\times 10^{-9}\) & \(4.154\times 10^{-9}\) \\ urease activity & \(c_{\text{u,0}}^{\text{e}}\) & 532.71 & 467.77 & 597.64 \\ \hline Sorption & \(k_{\text{a}}\) & \(4.360\times 10^{-2}\) & \(4.017\times 10^{-2}\) & \(4.702\times 10^{-2}\) \\ coefficients and & \(k_{\text{d}}\) & \(1.874\times 10^{-11}\) & \(-6.849\times 10^{-10}\) & \(7.224\times 10^{-10}\) \\ pKa multiplier & \(f_{\text{PKa}}\) & 1.200 * & 0.8515 & 1.549 \\ \hline \hline \end{tabular}  
* Upper parameter value bound set in the calibration; see Table 1.  
\end{table} 
Table 1: Estimated parameter values and confidence intervals as given by the Model-Independent Parameter Estimation (PEST) protocol [58] for the four investigated sets of calibration parameters calibrating to the concentration measurements of column #1 (0.33 M). Judging from the plots of predicted concentrations (Figures 2 and 3), all four sets of calibrated parameters are able to predict the urea and calcium concentrations over time qualitatively for column #1 (0.33 M). However, only the sets with a multiplier to the pKa or the injection concentrations added to the fitting parameters show the experimentally observed complete consumption of calcium. The residuals between model predictions and experiment measurements are calculated as the sum of the squares of the difference between the model prediction \(x_{i}^{\text{mod}}\) and the experimental observation \(x_{i}^{\text{exp}}\):  
\[r=\sum_{i}\left(x_{i}^{\text{mod}}-x_{i}^{\text{exp}}\right)^{2}. \tag{29}\]  
Table 3 summarizes the residuals for all sets of data (calcium and urea concentrations at 0.1016 m and 0.4064 m, and final calcite distribution along the column) as well as the sum of the residuals for the concentrations. The sum of the residuals can be decreased by including the injection concentrations (\(C_{\text{Ca}^{2+},\text{inj}}\) and \(C_{\text{urea},\text{inj}}\)), the urease activity (\(c_{\text{u},0}^{\text{e}}\)), or the multiplier to the pKa (\(f_{\text{pKa}}\)) to the set of calibration parameters in addition to the ad- and desorption rate coefficients \(k_{\text{a}}\) and \(k_{\text{d}}\). This effect is strongest for the parameters sets adding the injection concentrations or the multiplier to the pKa; see Table 3. The residual reduction for the case including the urease activity is probably related to the relatively high correlation of \(k_{\text{a}}\) and \(c_{\text{u},0}^{\text{e}}\) in this case. In those two cases, the effect is mainly due to the reduction in the residuals for the calcium concentration measurements. The residuals for the urea concentration measurements do not change significantly for any of the investigated sets of fitting parameters. Adding the urease activity does not significantly reduce the sum of the residuals. The lowest total residual can be achieved when adding the concentrations to the fitting parameters, which also results in the lowest residual for the precipitated calcite volume fraction.