We consider two-dimensional steady-state groundwater flow governed by: 1∇⋅Kx∇hx+qsx=0 where x=x1,x2 is a vector of spatial coordinates in domain Ω2; h is hydraulic head; K is (isotropic) hydraulic conductivity; and qs is a source/sink term. We conceptualize K as a spatially heterogeneous random field, associated with a given spatial correlation structure. The source/sink term in Eq. (1) corresponds to a production well associated with an uncertain pumping rate and location in the domain. Propagation of uncertainty related to model parameters and/or forcing terms onto hydraulic heads and fluxes is typically assessed through numerical Monte Carlo (MC) simulations (see, e.g., Ballio and Guadagnini, 2004; Xia et al., 2020, 2024, and references therein).

We consider (non-reactive) solute transport evolving in Ω2 to be described through: 2∇⋅D∇cx,t-∇qxcx,t+qsxθcsx,t=∂cx,t∂t

Here, t denotes time; c is solute concentration; D is the (isotropic) dispersion coefficient; θ is effective porosity; cs is solute concentration corresponding to qs; and qx=-Kx/θ∇hx is an effective velocity associated with solute transport.

Numerical methods (e.g., finite differences or finite elements) are commonly employed to discretize Eqs. (1) and (2) that are then solved within a numerical MC context. The probability distribution of state variables of interest (e.g., heads or concentrations) is then evaluated at N nodes of an aptly designed numerical grid. Consistent with Sect. 1, we refer to the model corresponding to the numerical solution of the above equations as the Full System Model (FSM). When the domain is characterized by a large spatial extent and/or one is interested in exploring the system behavior across long temporal windows, performing numerical MC simulations relying on FSM is associated with a heavy computational burden. To circumvent this issue, we rely on the development and implementation of a Reduced-Order Model (ROM) strategy for solute transport. We note that in this study we employ ROM solely for solute transport because only limited computational costs are associated with the steady-state flow condition we consider, as opposed to simulating transport. Hereafter, we refer to numerical MC analyses grounded on ROM as ROMC.

We denote by m=Y1,Y2,…,YN,ln⁡qs,x1,qs,x2,qsT the vector (of size P=N+3) whose entries correspond to the uncertain model parameters (i.e., the log-conductivity, Y=ln⁡K, field) and flow rate and location of a pumping well. The pumping rate is parameterized in logarithmic form to ensure consistent scaling with the log-conductivity field and to reduce disparities in parameter magnitudes. Such a transformation is consistent with common practice in inverse modeling to improve numerical conditioning, enhance stability of the estimation process, and mitigate potential bias arising from large disparities in parameter magnitudes. In case the pumping rate and location are known, then m=Y1,Y2,…,YNT and P=N. We further denote by d=d1,d2,…,dOT the vector (of size O) of the randomly perturbed observations (i.e., measured head and concentration values). To estimate m, we implement the iES (Luo and Bhakta, 2020; Xia et al., 2024): 7mk+1=mk+KGainkΔdkKGaink=SmkSdkTSdkSdkT+γkCd-1Δdk=d-gmkwithγi=ξitraceSdiSdiT/O.

Here, superscript k is the index of the iteration step; matrices Smk=m1k-m‾k,…,mNMCk-m‾k/NMC-1 (of size P×N, where m‾k=∑j=1Nmjk/NMC) and Sdk=gm1k-gm‾k,…,gmNMCk-gm‾k/NMC-1 (of size O×N, where g⋅ represents model operator being either FSM or ROM) collect the ensemble anomalies of parameters and simulated observations associated with the kth iteration step; Cd is the covariance matrix of observation errors; and ξk is an adaptive coefficient (Luo and Bhakta, 2020) associated with each iteration of the Levenberg-Marquardt (LM; Levenberg, 1944) algorithm. We set ξ0=10 in our showcase application examples (see Sect. 3) and follow Luo and Bhakta (2020) to update its value for the remaining iteration steps.

In the case of g⋅ representing the model operator of ROM, we note that the approximation of solute concentration relying on Eq. (4) is compatible with the implementation of Eq. (7). The degree of compatibility of ROM to iES is reduced when considering a typical Karhunen-Loève expansion of ci (i.e., ci≈c+∑j=1nαjipj=c+Pαi). This is related to the observation that c evolves with time and needs to be evaluated at each time step. This, in turn, implies that m numerical solutions of solute concentration through FSM need to be obtained to evaluate c at every outer iteration of iES. Hence, computational advantages of employing ROM are reduced while coding complexity increases. When approximating solute concentration via Eq. (4), we only obtain m numerical solutions of solute concentration through FSM at the first outer iteration of iES. Leading eigenvectors are computed upon relying on these solutions and are then stored. The Fourier coefficients αi associated with time interval t,t+Δt for each MC realization starting from the second outer iteration of iES are obtained solely through solving Eq. (5).

If an outer iteration does not lead to reduction in data misfit, an inner iteration is triggered to progressively decrease the time step until a lower misfit is obtained. When implementing the LM optimization scheme, we set the maximum number of both inner and outer iteration to 10 (see also Luo and Bhakta, 2020). Additionally, a stopping criterion δk-1-δk/δk-1×100%≤10-6 (where δk=1N∑j=1Ndjk-gmjkTCd-1djk-gmjk), is set.

We denote by iES_FSM and iES_ROM the approaches associated with coupling the iES with FSM and ROM, respectively. A workflow for iES_ROM is depicted in Fig. 1. The total number of MC realizations is denoted as NMC. Neglecting the computational cost of the inner iterations and assuming iES comprises Nout outer iterations, the main computational costs of either method can be divided into two components, corresponding to forecast and analysis step (Table 1), respectively. In the forecast step, a number of (Nout+1) MC simulations for groundwater flow and solute transport are required. Otherwise, Eq, (7) is evaluated Nout times in the analysis step. The steady-state groundwater flow is solved through the FSM in both iES_FSM and iES_ROM, with a computational cost of order ON3NMC+1 for each forecast step. The main computational cost for the NMC FSM-based MC realizations of solute transport at a single time step in iES_FSM is ON3NMC+1, while being OsN+N2NMC+1 (where s≈7 or ≈15 in two and three dimensions, respectively) for iES_ROM. These computational efforts correspond to the projection of the full-system stiffness matrix onto the reduced-order space of the system state (i.e., solute concentration). Computational costs associated with solving Eq. (7) coincide for both approaches and are here denoted as C8. We further note that, with reference to iES_ROM, the Nsn solutions of solute concentration obtained through FSM (associated with a computational cost of order ON3Nsn) and the basis functions obtained through SVD (with a computational cost of order OnNsn2) are calculated only once and stored. When the grid mesh employed is large or the simulation time is long, computational savings through iES_ROM compared with iES_FSM become significant.

Figure 3 depicts EY (Fig. 3a), SY (Fig. 3b), and Eobs (Fig. 3c) versus the number of outer iterations for test cases (TCs) 1–6 obtained through iES_ROM and iES_FSM, when well pumping rate and location are uncertain. Note that results obtained through iES_FSM are independent of n (and are identical among TCs 1–6) and are taken as references. Percentage differences (denoted as ΔEY) between the values of EY obtained through iES_ROM and iES_FSM are depicted in Fig. 3d. Corresponding results associated with percentage differences between values of SY (ΔSY) and of Eobs (ΔEobs) are depicted in Fig. 3e and f, respectively.

Values of EY, SY, and Eobs obtained at the end of the iteration procedure through iES_ROM generally decrease with n. When n= 25 or 30, the values of EY and SY based on iES_ROM tend to approach their counterparts obtained through iES_FSM. The latter generally correspond to the lowest values across TCs 1–6. These findings are consistent with the observation (Xia et al., 2020, 2025) that accuracy of ROM for n=30 and FSM are very similar for solute transport. They are also in line with the results of Li and Hu (2013), who documented a high degree of correlation between simulated concentrations provided by their ROM and FSM for non-reactive transport.

Figure 4 depicts EY (Fig. 4a), SY (Fig. 4b), and Eobs (Fig. 4c) for TCs 7–12 obtained through iES_ROM and iES_FSM when the well characteristics are deterministically known. Similar to above, results obtained through iES_FSM are identical among TCs 7–12 and are taken as reference. Values of ΔEY, ΔSY, and ΔEobs are depicted in Fig. 4d, e, and f, respectively.

Consistent with what one can observe in Fig. 3, values of EY, SY, and Eobs obtained at the end of the iteration procedure for TCs 7–12 through iES_ROM generally decrease with n. Except for the cases where n= 5 or 10 (corresponding to low solution accuracy of ROM), values of EY, SY, and Eobs for TCs 9–12 based on either iES_ROM or iES_FSM are lower than their counterparts related to TCs 3–6. These results suggest that the accuracy of conductivity estimates is lower when qs is uncertain compared to the case where qs is deterministic.

Figure 5 depicts the values of Ex1 (Fig. 5a), Ex2 (Fig. 5b), Eqs (Fig. 5c), Sx1 (Fig. 5d), Sx2 (Fig. 5e), and Sqs (Fig. 5f) versus the number of outer iterations for TCs 1–6 obtained through iES_ROM and iES_FSM. Values of Ex1, Ex2, and Eqs obtained through iES_ROM approach their iES_FSM-based counterparts as n increases. This is consistent with the observation that increasing n improves the accuracy of the ROM-based solution (see also Li and Hu, 2013), therefore enhancing the accuracy of the identification of the well attributes.

Figure 6 depicts the estimated (ensemble) Y fields for TCs 1–6 obtained through iES_ROM and iES_FSM, together with their reference Y field. The white circle and cross symbols in Fig. 6 denote the estimated and reference locations of the pumping well, respectively. As n increases, the estimated Y field obtained through iES_ROM (Fig. 6a–f) approaches its iES_FSM-based counterpart and the reference Y field (Fig. 6h). The accuracy of the iES_ROM-based estimate of the location of the pumping well generally increases with n, consistent with the nature of the findings illustrated in Fig. 5. Figure 7 depicts the estimated (ensemble) Y variance fields for TCs 1–6 based on iES_ROM and iES_FSM. The white circle and cross symbols therein denote the identified and reference locations of the pumping well, respectively. These results show that the variance of Y is overestimated when n is small. This is related to the observation that small values of n correspond to large modeling errors (i.e., low solution accuracy) of ROM (as also seen in Li and Hu, 2013, and Pasetto et al., 2017). The latter, in turn, imprint the low accuracy of conductivity estimates (see Fig. 6a in the case of n=5) and yield overestimated values for the variance of Y (see Fig. 7a).

Figure 8 depicts the empirical probability density function (PDF) of x1,qs, x2,qs, and ln⁡qs at the end of the iteration procedure for TCs 1, 2, 4, and 6 as obtained through iES_ROM and iES_FSM, together with their counterparts associated with initial guess (black solid) and reference values (black dashed). One can observe that large values of n yield high accuracy for x1,qs and x2,qs estimates, as visually indicated by the compact supports associated with the empirical PDFs of x1,qs (Fig. 8a) and x2,qs (Fig. 8b). The accuracy of the estimate of qs is already acceptable when n=5.

As an additional element, we explore the way the choice of the value of n impacts the local PDFs of hydraulic head and solute concentration. We do so upon considering the results associated with three reference points (i.e., I, II, and III in Fig. 2d) that are aligned in the direction of the mean groundwater flow. Figure 9 depicts the (sample) PDFs of (hydraulic) head at these three selected locations (Fig. 9a–c) obtained through iES_ROM and iES_FSM at the end of the iteration procedure for TCs 1, 2, 4, and 6. Black solid lines included therein indicate reference head values. Note that the PDFs stemming from iES_FSM peak at values very close to their reference counterparts. Hence, the corresponding empirical PDFs are considered as reference. The logarithm absolute difference (ΔPDF, evaluated as the pointwise log-ratio of the densities and corresponding to a local measure of relative likelihood between two empirical PDFs) between the PDFs of the head at points I–III obtained through iES_ROM based on diverse values of n and their counterpart based on iES_FSM are also shown in Fig. 9d–f, respectively. One can see that a large value of n (e.g., n=30 for TC6) corresponds to high accuracy of the PDF of head, as quantified through a low value of ΔPDF. Although the head solution is obtained by solving FSM, the accuracy of the conductivity estimate is impacted by n. The latter, therefore, impacts the accuracy of heads. Figure 10 depicts results related to solute concentration. As expected, the PDFs stemming from iES_FSM peak at values very close to their reference counterparts also in this case. Consistent with Fig. 9, a large value of n (e.g., 30 for TC6) corresponds to high accuracy in the delineation of the PDF of solute concentration.

As a complement to these results, values of the Kullback-Leibler Divergence (KLD) between the (sample) PDFs of head at the three reference points at the last outer iteration obtained through iES_FSM (hFSM) and iES_ROM (hROM) with n=5 (TC1), 10 (TC2), 20 (TC4), and 30 (TC6) are listed in Table S1 (see Supplement). We recall that values of KLD(hROM || hFSM) (or KLD(hFSM || hROM)) quantify (in a global sense) information loss when using hFSM (hROM) to approximate hROM (hFSM). Values of KLD(hROM || hFSM) generally increase with n. This indicates that the difference between PDFs of hROM and hFSM decrease as n increases. While the highest values of KLD(hFSM || hROM) correspond to n=5, no clear decreasing trends with increasing n are observed. Furthermore, the difference between KLD(hROM || hFSM) and KLD(hFSM || hROM) generally decreases as n increases. This is related to the observation that the accuracy of ROM tends to increase as the dimension of the reduced-order model increase. Values of KLD between the empirical PDFs of solute concentrations at the three selected reference points at the last outer iteration obtained through iES_FSM (cFSM) and iES_ROM (cROM) with n=5 (TC1), 10 (TC2), 20 (TC4), and 30 (TC6) are listed in Table S2 (see Supplement).

Figure 11 depicts iES_ROM- and iES_FSM-based values of EY (Fig. 11a), SY (Fig. 11b), and Eobs (Fig. 11c) versus the number of outer iterations for TCs 6 and 13–15. Values of EY and Eobs decrease as the ensemble size NMC increases (while the value of SY increases) regardless of the approach employed. With reference to TC13, we note that when NMC=30 the values of EY decrease during the course of the first outer iterations to then increase during the last outer iterations, values of SY dropping rapidly during the iteration procedure, regardless of the approach employed. This phenomenon is typically linked to the occurrence of filter inbreeding caused by a limited ensemble size (Chen and Zhang, 2006; Xia et al., 2018, 2024). Values of EY and SY for TCs 6 and 13–15 obtained through iES_ROM are overall similar to those associated with iES_FSM. The iES_ROM-based value of Eobs obtained at the end of the iteration procedure for a given TC is typically larger than its iES_FSM-based counterpart. This is linked to the observation that the limited system dimension of ROM induces low accuracy of concentrations and (possibly) heads due to low accuracy of conductivity estimates, pumping rate, and well locations.

Figure 12 depicts the values of Ex1 (Fig. 12a), Ex2 (Fig. 12b), Eqs (Fig. 12c), Sx1 (Fig. 12d), Sx2 (Fig. 12e), and Sqs (Fig. 12f) versus the number of outer iterations for TCs 6 and 13–15 obtained through iES_ROM and iES_FSM. When increasing NMC, values of Ex1, Ex2, and Eqs obtained through either iES_ROM or iES_FSM do not show a clear trend. Values of Sx1, Sx2, and Sqs generally increase with NMC, a result that is consistent with the findings encapsulated in Fig. 11b. Similar findings are also documented by Xu and Gómez-Hernández (2018, their Fig. 17), who show that, when considering joint identification of contaminant sources and hydraulic conductivities, the accuracy of estimates of key attributes characterizing contaminant sources does not necessarily improve after some time and as data assimilation progresses. We further note that jointly estimating conductivity and identifying source/sink term attributes (in terms of flow rate and location) is associated with a highly nonlinear optimization process. Hence, the accuracies of location and pumping rate estimation through iES_FSM are not always higher than those stemming from iES_ROM in terms of the values of the metrics employed (i.e., Ex1, Ex2, and Eqs).

Figures 13 depicts the estimated (ensemble mean) Y fields for TCs 6 and 13–15 obtained through iES_ROM and iES_FSM. Figure 14 depicts the associated Y variance fields for TCs 6 and 13–15 obtained through iES_ROM and iES_FSM. The white (black) circle and cross symbols in Fig. 13 (or Fig. 14) represent the identified and the reference locations of the pumping well, respectively. Visual comparison of Figs. 13 and 6h suggests that the estimated Y fields rendered through an ensemble size NMC=100 (i.e., TC14) obtained through iES_ROM and iES_FSM are the closest ones to the reference Y field. Nevertheless, jointly analyzing Figs. 11a, 13, and 14 reveal that the estimated Y field corresponding to NMC=10000 (TC6) obtained through iES_ROM is the one most closest to the reference Y field in terms of EY (=0.41). Additionally, the identified and reference locations of the pumping well obtained through either iES_ROM or iES_FSM are close to each other, thus supporting the capability of both approaches to identify the well location.

Table 2 lists values of EY, SY, Eobs, Ex1, Ex2, Eqs, Sx1, Sx2, and Sqs at the end of iteration procedure for TC16 (characterized by σobs=0.001), TC6 (σobs=0.01), and TC17 (σobs=0.1) obtained through iES_ROM and iES_FSM. Values of EY, SY, Eobs, Ex1, and Eqs generally increase as the quality of observations deteriorates, i.e., σobs increasing from 0.001 to 0.1. These results are also consistent with prior findings by Xia et al. (2018) according to which accuracy of conductivity estimates increases as the quality of observations improves. Values of Ex2 obtained through iES_ROM and iES_FSM do not monotonically decrease as σobs decreases. This is typically related to the strong nonlinear nature associated with the optimization process (see also Xu and Gómez-Hernández, 2018).

Figure 15 depicts iES_ROM- and iES_FSM-based values of EY (Fig. 15a), SY (Fig. 15b), and Eobs (Fig. 15c) versus the number of outer iterations for TCs 6 and 18–19. Values of EY (or SY) for TCs 18 (where the number of monitoring wells is Nm=9), 19 (Nm=18), and 6 (Nm=55) obtained through iES_ROM are similar to their iES_FSM-based counterparts and decrease as Nm increases. Values of Eobs obtained through iES_FSM decrease as Nm increases, while iES_ROM-based results do not display a clear trend with Nm. This result may be attributed to the fact that, while increasing the number of monitoring wells enhances the amount of information available for estimating hydraulic conductivity, errors introduced through model reduction influence the evolution of the solute concentration mismatch between observations and simulations during the iterative calibration process.

Figure 16 depicts the values of Ex1 (Fig. 16a), Ex2 (Fig. 16b), Eqs (Fig. 16c), Sx1 (Fig. 16d), Sx2 (Fig. 16e), and Sqs (Fig. 16f) versus the number of outer iterations for TCs 6 and 18–19 obtained through iES_ROM and iES_FSM. Values of Ex1 (Ex2, Eqs, Sx1, Sx2, Sqs, or SY) for TCs 18 (for a number Nm=9 of monitoring wells), 19 (Nm=18), and 6 (Nm=55) obtained through either iES_ROM or iES_FSM decrease as Nm increases. Values of the same metric (i.e., Ex1, Ex2, Eqs, Sx1, Sx2, or Sqs) obtained through iES_ROM and iES_FSM are overall close to each other.

Figure 17 depicts the empirical PDF of x1,qs, x2,qs, and ln⁡qs at the end of the iteration procedure for TCs 18, 19, and 6 obtained through iES_ROM and iES_FSM, together with their reference counterparts (black dashed lines). One can observe that increasing Nm leads to improved accuracy of the identification of pumping well attributes, as suggested by the reduced support and location of the peaks of the PDFs of x1,qs (Fig. 17a), x2,qs (Fig. 17b), and ln⁡qs (Fig. 17c) obtained through either iES_ROM or iES_FSM and observed as Nm varies from 9 to 55. On the basis of these results, it is hard to tell which approach provides higher accuracy of pumping well identification, solely in terms of Fig. 17. To complement these findings, Table S3 (see Supplement) lists the values of KLD between the empirical PDFs of x1,qs (x2,qs, or ln⁡qs) obtained through iES_FSM (denoted as pFSM) and iES_ROM (denoted as pROM) with n=30, considering Nm=9 (TC18), 18 (TC19), and 55 (TC6), respectively. Values of KLD(pjROM || pjFSM) (with j=x1,qs, x2,qs, and ln⁡qs show an overall decreasing trend as Nm increase, while KLD(pjFSM || pjROM) consistently decreases with Nm. These results are consistent with the observation that increasing the number of monitoring wells improves the accuracy of conductivity estimates (as also seen by Tong et al., 2010, and Xia et al., 2018) as well as pumping rate and well location through both approaches, thus, in turn, reducing discrepancies between the corresponding PDFs.

Table 3 lists the values of EY, SY, Eobs, Ex1, Ex2, Eqs, Sx1, Sx2, and Sqs at the end of the iteration procedure for TCs 20 (characterized by a mean μ=-0.5 of the initial ensemble of Y), 6 (μ=1.2), and 21 (μ=2.0) obtained through iES_ROM and iES_FSM. We recall that the mean value employed to generate the reference Y field is equal to 0.8. When the discrepancy between μ and the mean value of the reference Y field increases, the error metrics employed display an overall increase, Ex1 and Eqs constituting notable exceptions. This finding is consistent with the behavior documented by Xia et al. (2024) who considered two correlation-based localization approaches to assess conductivity estimation accuracy with respect to the mean of the initial ensemble of Y.

Table 4 lists the values of EY, SY, Eobs, Ex1, Ex2, Eqs, Sx1, Sx2, and Sqs at the end of the iteration procedure for TCs 22 (characterized by a variance σY2=0.01 of the initial ensemble of Y), 6 (σY2=1.0), and 23 (σY2=2.0) obtained through iES_ROM and iES_FSM. We recall that the reference Y field is characterized by a unit variance. The values of EY and Ex2 obtained through both approaches increase as the discrepancy between σY2 and the variance of the reference Y field increases. The values of Eobs, Ex1, and Eqs obtained through both approaches generally increase with σY2. Similarly, values of metrics employed to quantify variability of the final ensemble of realizations (i.e., SY, Sx1, Sx2, and Sqs) consistently increase with σY2.

A joint analysis of the results illustrated in Sect. 4.1, 4.2, and 4.3 suggests that EY and SY provided by both approaches show a consistent behavior as a function of the key feature of interest. Otherwise, the response of the metrics associated with the pumping well attributes provided by both approaches reflects the enhanced nonlinearity of the associated optimization process. Additionally, the accuracy of the conductivity estimate possibly contributes more to the minimization of the objective function than that of pumping well identification. Additionally, the values of the metrics in Sect. 4.1, 4.2, and 4.3 provided by the two approaches are generally consistent with each other, thus supporting the representativeness of the iES_ROM-based results.

Table 5 lists percentage differences of the values of the performance metrics considered (i.e., EY, SY, Eobs, Ex1, Ex2, Eqs, Sx1, Sx2, and Sqs) at the end of the iteration procedure for TCs 24–28 obtained through iES_ROM, considering their counterparts through TC6 as references. These results show that the values of EY and SY systematically decrease as Nsn increases from 30 to 1000, while the other metrics display an overall decreasing pattern. This is related to the observation that a larger snapshot size corresponds to a higher accuracy of basis functions (Pasetto et al., 2014). Otherwise, it is worth noting that snapshots are evaluated only once throughout the entire data assimilation processes, thus resulting in a limited computational cost.

A single forward simulation for TC28 requires approximately 13 min of CPU time on the hardware platform used in this study (13th Gen Intel^® CoreTM i7-13700K 3.40 GHz with 32 GB RAM). The total CPU time required to complete TC6 upon relying on iES_FSM is 122 minutes, whereas the corresponding CPU time required to complete TC28 through iES_ROM is 28 min, thus representing a speedup of approximately a factor of 9. Percentage differences associated with EY and SY are equal to 0.50% and 0.21%, respectively, suggesting that the computational gain is achieved with negligible loss of accuracy. To further quantify the approximation error introduced by the ROM, we evaluate the residual mean square errors between the concentration field obtained from the first realization of the initial ensemble using the full-scale model and the corresponding ROM solutions with reduced dimensions n= 5, 10, 15, 20, 25, and 30. The ensuing error values are 0.8630, 0.4156, 0.2699, 0.1909, 0.1312, and 0.1336, respectively, demonstrating systematic error reduction as the reduced dimension increases. We further note that all of the associated coefficients of determination are higher than 0.99. CPU time savings can become more pronounced during data assimilation for a groundwater system of large size, due to the higher memory requirements of iES_FSM for storing and computing large-dimensional vectors and matrices as compared to iES_ROM.

Additionally, we emphasize that relying on realizations of Y associated with (spatial) statistics different from their theoretical counterparts linked to the initial ensemble of Y fields can contribute to deteriorate the quality of the selected snapshots. Low quality snapshots yield low quality basis functions and low accuracy of ROM outcomes (see our results in Sect. 4.1; Pasetto et al., 2014; Xia et al., 2020). These elements, in turn, contribute to deteriorate the accuracy of conductivity estimates and pumping well attributes. Additional studies should be devoted to assess the potential of techniques (including, e.g., greedy algorithms) that might contribute to increase the quality of snapshots.