4.1. Single Injection Test in a Homogeneous Aquifer
 This case is designed to provide a straightforward evaluation of the DPP under the simplest conditions. A single injection test is performed with the injection screen placed near the center of the aquifer (5.150 to 5.175 m above the aquifer bottom). The transducers PT1 and PT2 are located 0.15 m and 0.40 m above the screen center, respectively. The input K field is homogeneous and isotropic, with a value set to the depth-averaged Kr of 1.5 × 10−3 m/s. Figure 3(a) shows the contours of injection-induced head changes at steady state for the base scenario. The head changes quickly dissipate with distance from the injection screen. A close examination reveals that, despite an apparent symmetry, the head gradient in the horizontal direction is slightly greater than that in the vertical direction. This is mainly a result of the injection screen being a cylinder. The flow is thus not spherically symmetric in the immediate vicinity of the screen.
Figure 3. Contours of steady-state head changes induced by a single DPP injection test under different scenarios. The head field for the screen clogging scenario is essentially identical to that in the base scenario and is not plotted.
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 Variations are made relative to the base scenario to investigate the importance of screen clogging, a low K skin, and anisotropy. Table 1 summarizes the results of the base scenario and variations of it. In the base scenario, the K estimate calculated with the simple analytical formula (1) is ∼10% larger than the input reference value due to the deviations of test conditions from the assumptions embodied in (1). There are two major differences between the flow during a DPP test and that assumed in (1). First, as discussed above, instead of an ideal point, the injection screen is a cylinder. Second, the actual aquifer domain for the DPP tests is not the assumed infinite domain, particularly in the vertical direction. Because of these conceptual differences, numerical analysis is necessary to obtain more accurate K estimates. Nonetheless, equation (1) provides a good initial estimate that can be used for facilitating the convergence of the numerical inversion. Lowry et al.  reported a similar discrepancy between the K value calculated using (1) and the actual input K, and noted that this discrepancy should decrease with increasing distance between the transducers and the injection screen.
Table 1. Results of the Single-Injection, Homogeneous Aquifer Casea
| ||Δh1 (m)||Δh2 (m)||Kc (m/s)||Kd (m/s)|
|Base||0.0201||0.0081||1.65 × 10−3||1.50 × 10−3|
|Clogging||0.0201||0.0081||1.66 × 10−3||1.51 × 10−3|
|Skinb||0.0194||0.0077||1.71 × 10−3||1.55 × 10−3|
|Anisotropy of K (Kz: Kr)||1:2||0.0206||0.0081||1.59 × 10−3||1.45 × 10−3|
|1:10||0.0212||0.0082||1.53 × 10−3||1.39 × 10−3|
|1:100||0.0214||0.0082||1.50 × 10−3||1.36 × 10−3|
|1:1000||0.0214||0.0082||1.50 × 10−3||1.36 × 10−3|
 In the clogging scenario, a thin zone of low-K material is applied over the entire DPP injection screen. The clogged zone is 0.005 m in thickness with a low K value 1.5 × 10−6 m/s. During the numerical inversion, the clogged zone is not explicitly specified and the estimated K field is assumed to be homogeneous. Table 1 displays the results of the clogging simulation. The head changes at both transducers and the resulting K estimates remain similar to those in the base scenario. This indicates that screen clogging during tool advancement has a negligible effect on DPP performance. The only significant impact of screen clogging is that the injection pressure behind the screen is much higher than that when clogging is not present.
 In the skin scenario, a low-K zone, represented by a cylindrical column of nodes, is placed around the DPP tool. To mimic the anticipated field conditions, the low-K zone extends from the aquifer top to just below the injection screen. The thickness of the skin zone varies from 0.005 to 0.09 m between simulations, and the skin K varies from 1.5 × 10−3 to 1.5 × 10−6 m/s. Other settings remain identical to those in the base scenario. Figure 3(b) shows the induced head changes at steady state for a skin thickness of 0.04 m with a K of 1.5 × 10−4 m/s. Clearly, the presence of the skin has altered the flow field in the immediate vicinity of the screen. However, these effects quickly diminish with distance from the screen, so the difference in the head changes between the two transducers remains close to that in the base scenario (0.012 m).
 During the numerical inversion, the skin is not explicitly specified and the aquifer is assumed homogeneous. Table 1 shows the results for a skin with thickness of 0.04 m and a K of 1.5 × 10−4 m/s. The K estimate from the numerical inversion deviates from the input reference value by less than 10%. Figures 4(a) and 4(b) display numerical K estimates for skins of different K values and thicknesses. When the skin K is smaller than ∼2.0 × 10−5 m/s for a skin thickness of 0.04 m, or the skin thickness is greater than ∼0.06 m for a K of 1.5 × 10−4 m/s, the DPP K estimate becomes smaller than 1.0 × 10−3 m/s (a deviation of 33% from the reference value). In many situations, however, the low-K skin formed by advancement of a direct-push tool is expected to be within one order of magnitude of the aquifer K [e.g., Butler et al., 2002].
Figure 4. The DPP estimated K values for different skin configurations: (a) Thickness 0.04 m, K varied and (b) K 1.5 × 10−4 m/s, thickness varied.
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 The impact of anisotropy is investigated by varying the Kz: Kr ratio from 1:2 to 1:1000, while Kr is maintained constant at 1.5 × 10−3 m/s and other conditions are identical to those in the base scenario. Figure 3(c) depicts the induced head changes at an anisotropy ratio of 1:100. Flow is overwhelmingly horizontal because of the anisotropy. The altered flow field does not greatly affect the difference in the head changes between the two transducers, which increases less than 10% relative to that in the base scenario.
 Different K estimates for the anisotropic configurations are provided in Table 1. Despite the wide range of anisotropy factors, all the equation (1)-calculated K estimates remain close to the reference Kr value, consistent with results reported in Butler et al. . In the highly anisotropic case, the impact of the point injection assumption in (1) becomes minimal so that there is essentially no difference between the calculated and reference Kr values. The numerically inverted K estimates are obtained by assuming that the K field is isotropic and no information on anisotropy is available. The numerical lr2dinv estimates, which are a function of both the vertical and horizontal K, are smaller than the reference Kr value. After the anisotropy factor decreases below 1:100, both the equation (1)-calculated and lr2dinv-inverted K estimates show no further significant changes.
 To further explore how the DPP is influenced by the K field in a single injection test, we compute the sensitivity of the difference in head changes between the two transducers with respect to K,
where ∂Ki is a small perturbation around the base value i at location i; ∂ (Δh2 – Δh1) is the change in the difference (Δh2 – Δh1) caused by ∂Ki. Here the K field is isotropic so that K = Kr = Kz. Scaling the sensitivity by the corresponding parameter value i gives results that are more indicative of the actual influence of K and allows us to compare more appropriately the values computed at different locations in a heterogeneous setting. Note that the difference Δh2 – Δh1 is used instead of Δh1 – Δh2 in (8), because Δh2 – Δh1 is positively related to the K estimate (see equation (1)). Therefore if the sensitivity of Δh2 – Δh1 to the K at location i is a positive value, the DPP K estimate will increase when the K increases at location i, and vice versa.
 In this work Ji is computed using the adjoint state method [Sykes et al., 1985; Sun, 1994]. The adjoint function, ϕ, of the head difference (Δh2 – Δh1) under steady-shape conditions is obtained by solving,
where δ is the Dirac delta function; r1′ and r2′ are the log-transformed radial coordinates for PT1 and PT2. Equation (9) is similar to the equation for h itself (i.e., equation (4)), except with a right-hand side forcing function representing the locations of transducers PT2 and PT1. That is, the adjoint function essentially represents an influence function associated with the observed head difference (Δh2 – Δh1). The boundary conditions for the adjoint equation have the same form as those for the flow equation, with all specified-head boundaries being replaced with ϕ = 0 and all specified-flux boundaries with = 0 or = 0. The normalized sensitivity to the K at location i is then given by,
 Figure 5 shows the adjoint function ϕ for the base scenario (K of 1.5 × 10−3 m/s). The ϕ is a dipole field with the positive pole located at the near transducer and the negative pole at the far transducer. Similar dipole fields, although differing in detail, are obtained for all other scenarios described here. Figure 6 depicts the sensitivity of (Δh2 – Δh1) to K for the base scenario. As shown in (10), the sensitivity is proportional to the dot product of the gradients of the head (Figure 3a) and the adjoint function ϕ (Figure 5). There are two important observations that can be made on the computed sensitivity values in Figure 6. First, very close to the tool between the injection screen and the first transducer, and above the far transducer, the sensitivity is negative indicating that the DPP K estimate will increase when the K decreases in these areas, and vice versa. For most of the area in the vicinity of the tool, the DPP K estimate and aquifer K is positively related. Second, the DPP is most sensitive to the small area surrounding the interval between the injection screen and the transducers. In other words, K features located further than a few tens of centimeters below the screen or above the far transducer have a minimal impact on the DPP estimate. Horizontally, the DPP becomes largely insensitive to K features located more than ∼0.5 m from the tool. Note that the sensitivity results do not change with injection rate due to the use of the steady-shape analysis. The extremely compact domain of high sensitivity enables the DPP to be an effective tool for obtaining detailed K information with minimal interference from the area outside the sampling volume.
Figure 6. The sensitivity of the difference in head changes between the pressure transducers to K for the base scenario.
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 The sensitivity results shown in Figure 6 provide an explanation for the K estimates obtained in some of the single injection test scenarios. First, a skin surrounding the tool occupies both positive and negative sensitivity regions, so the net impact is dependent on both, which in turn are a function of the thickness and the K of the skin. The negative sensitivity region has slightly greater influence on the skin configurations during the initial change of skin K and thickness in Figures 4(a) and 4(b). As a result, the DPP K estimates are slightly larger than the input reference value despite the presence of the low-K skin. As the skin K decreases or the skin thickness increases further, the positive sensitivity region starts to dominate and the resulting DPP K estimates become smaller than the reference value. Second, in the clogging scenario, the DPP K estimate is larger than the reference value after incorporation of the thin low-K zone around the screen. However, because the volume of this clogged zone is extremely small, the increase in the DPP K estimate is minimal.
 An important advantage of the DPP is that the head field needs only be at steady-shape conditions. Figure 7 shows the head changes from the transient simulation of the base scenario for a specific storage of 5.0 × 10−6/m. The individual head changes at the transducers have not reached steady state after 200 s. However, the difference in the head changes between the transducers stabilizes in less than a second. Although more time is obviously required in practice because injection rates do not stabilize instantaneously, the requirement that only steady-shape conditions be attained enables a tremendous reduction in the time required for a DPP test.
Figure 7. Comparison of time required for obtaining steady-shape versus steady state flow in a DPP test. The head changes at the individual transducers (Δh1 and Δh2) do not reach steady state at the end of simulation (200 s), whereas the difference in the head changes (Δh1 – Δh2) becomes constant in less than 0.1 s.
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4.2. Single Injection Test in a Homogeneous Aquifer With a Thin Embedded Layer
 In this case we investigate DPP performance when a thin low- or high-K layer is embedded in an otherwise homogeneous aquifer. A particular objective is to identify how a typical DPP injection test responds to a thin anomalous layer in the vicinity of the tool. Other than the embedded layer, the K field is homogenous and isotropic (1.5 × 10−3 m/s). The embedded layer is 0.1 m in thickness with a K of either 1.5 × 10−5 m/s (a low-K barrier, such as a silty sand) or 1.5 × 10−1 m/s (a high-K channel, such as a gravel). The layer extends horizontally to the edge of the model domain.
 Figure 8 shows the position of the thin layer relative to the injection screen and the near and far transducers. In scenarios 1 through 7, the layer is placed, in turn, below the screen, over an interval encompassing the screen, between the screen and PT1, over an interval encompassing PT1, between PT1 and PT2, over an interval encompassing PT2, and above PT2. In each scenario, only a single anomalous layer is included and the base conditions are present elsewhere. Furthermore, as each configuration is assigned two K values, letter “a” is used to represent a low-K layer and “b” to represent a high-K layer. For example, scenario “1a” indicates that a 0.1 m-thick low-K layer is below the injection screen, while scenario “4b” indicates that a 0.1 m-thick high-K layer encompasses PT1. The induced head changes for scenarios 5a and 5b are plotted on Figures 3(d) and 3(e), respectively. Because the low-K layer impedes flow, the resulting difference in head between the transducers is greater than twice the base case. When the embedded layer is high K, the head difference is slightly below the base case due to the flow converging into that layer.
Figure 8. The position of the thin low- or high-K layer relative to the DPP injection screen and pressure transducers for the different scenarios in section 4.2. The left part of the diagram shows the example of scenario 4 in which the thin layer is represented as the shaded interval encompassing PT1 in an otherwise homogenous aquifer. Scenario numbers are shown on the right.
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 Table 2 shows the DPP K estimates for scenarios 1a through 7b. The K field is taken as homogenous in the numerical inversion assuming that the existence of the embedded layer is not recognized during the parameter estimation process. The values shown for each scenario are estimated by lr2dinv, equation (1) using Δh1 – Δh2, equation (2) using Δh1, and equation (2) using Δh2. The differences between the numerical estimates from lr2dinv and the values computed by (1) are relatively small in all scenarios, consistent with the results of Section 4.1. When applying equation (2), K estimates based on the head change at the near transducer, PT1, are closer to those obtained from lr2dinv than are K estimates based on the head change at PT2.
Table 2. The DPP K Estimates for Scenarios When a Thin Low- or High-K Layer is Embedded in an Otherwise Homogeneous Aquifera
| ||Scenario Letter|
|a (Low K)||b (High K)|
|lr2dinv||equation (1)||equation (2)K1||equation (2)K2||lr2dinv||equation (1)||equation (2)K1||equation (2)K2|
 Two observations can be made based on the lr2dinv results in Table 2. First, the K estimate inverted from the DPP is most sensitive to conditions in the interval between the screen and PT1. Second, the DPP K estimate is not a simple geometric or any other mean of the aquifer K and the K of the embedded layer. Instead, the DPP K estimate is dependent on both the position and the K value of the layer. In some scenarios (e.g., 2b, 3a, 3b, and 4b), the DPP K estimates are heavily affected by the embedded layer, clearly demonstrating the insufficiency of a single test for determining the K of thin layers. As shown in the following sections, a joint inversion of multiple DPP tests is necessary for quantifying such small-scale K variations.
 The sensitivity results in Figure 6 provide some explanation for the K estimates obtained in Table 2. First, the sensitivity is negative above the far transducer PT2. When a low-K layer is placed above PT2 (scenario 7a), the DPP K estimate determined from lr2dinv or equation (1) is larger than the reference value. Similarly, the K estimate is smaller than the reference value when the inclusion is a high-K layer (scenario 7b). Second, the sensitivity is positive between transducers PT2 and PT1 and below the injection screen. The DPP K estimate is smaller than the reference value when a low-K layer is included in these areas (scenarios 5a and 1a), and is larger when there is a high-K layer (scenarios 5b and 1b). Third, the sensitivity contains both negative and positive values between PT1 and the injection screen. The impact of an anomalous K layer in this interval is complicated, as the DPP K estimate depends on the thickness and value of the layer.
 When an anomalous layer occurs in the vicinity of the DPP, additional data and analyses are necessary for identifying its position and properties accurately. As mentioned in Section 2, the use of equation (2) with the individual steady state head changes at PT1 and PT2 can provide some insight into the location of the layer in certain circumstances. Table 2 shows the K values computed using (2) with the head changes at PT1 and PT2 (i.e., K1 and K2), respectively, for scenarios 1a through 7b. The most striking differences between K1 and K2 occur when 1) the low-K layer encompasses PT1 (scenario 4a) or is between PT1 and PT2 (scenario 5a), or 2) the high-K layer is located between PT1 and PT2 (scenario 5b) or encompasses PT2 (scenario 6b). It is interesting to note that in these scenarios, K2 is always larger than K1, regardless of the specific low- or high-K value assigned to the embedded layer. Therefore a simple calculation using equation (2) can serve as a qualitative indicator of an anomalous high- or low-K layer between PT1 and PT2. To determine the specific K of that layer, however, more data and analyses are needed. In the following sections, we investigate whether simultaneous analysis of multiple injection tests at different depths can be used to quantify spatial variations in K more accurately.
4.3. Multiple Injection Tests in a Heterogeneous Aquifer
 To characterize vertical K variations in a heterogeneous setting, multiple DPP injection tests need to be conducted at different depths. The 12 K zones measured at the GEMS research site (Figure 2) are used as the input K during the forward simulations. Five different scenarios are considered here. In scenario I, the same K zonation as in the forward simulations is used in the inverse analysis, assuming that prior information on the K structure is available. In the field such information can be obtained by monitoring the injection rate and pressure during DPP advancement [Dietrich et al., 2008]. A total of 18 injection tests are simulated with the first injection interval 9.150 to 9.175 m above the bottom of aquifer, followed by a sequence of tests at 0.5-m intervals. The last injection interval is 0.650 to 0.675 m above the base of the aquifer. The head changes from all tests are analyzed simultaneously for inverting the K values. Figure 9 shows the joint K estimates from lr2dinv at different depths in scenario I along with the 12 reference values. The numerical DPP K estimates are in very close agreement with the reference values when the K zonation is assumed known during the inverse estimation process.
Figure 9. DPP K estimates for different scenarios in Section 4.3 when multiple injection tests are conducted in a heterogeneous K field. Also shown for comparison are the reference 12 zone K values from the GEMS research site and the 18 K values calculated from the individual tests with the analytical formula (1). The K estimates of scenario I are essentially the same as the reference values. In scenario II a low-K skin is included. In scenario III the reference K zonation is assumed unknown during the parameter estimation process.
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 Figure 10 displays the calculated sensitivity for the 18 injection tests of scenario I. Results are for the root mean square of sensitivity over all 18 injection tests,
where Jik is the sensitivity of the head change to the K value in grid cell i for the k-th injection test. Figure 10 indicates that the distribution of high sensitivity zones is consistent with the position of each injection test. Conceivably, when the depth interval of successive DPP advancement is sufficiently small, the high sensitivity domains would form a continuous vertical column surrounding the DPP borehole. Furthermore, the sensitivity results are strongly dependent on the base K values. There are several low-K layers between elevations 2.5 and 4.5 m. Correspondingly, the sensitivity values computed within this interval are much larger than those at other depths. Horizontally, the sensitivity values become relatively small when the distance is greater than ∼0.5 m away from the DPP.
Figure 10. The sensitivity of the difference in the head changes between the two transducers to K for the 18 injection tests. The 12 reference K values in scenario I in section 4.3 are used as the base K. Results are the root mean square of the sensitivity values over all 18 injection tests.
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 In scenario II, we include a low-K skin (K = 1.0 × 10−5 m/s, thickness 0.02 m) in the forward simulation. For simplicity, the skin is extended through the entire aquifer thickness. Again, the low-K skin is not explicitly specified during numerical inversion, assuming that no information on the skin is available. All other settings remain identical to those in scenario I. Figure 9 displays the DPP K estimates after incorporating the low-K skin in the forward simulation. The differences between the DPP estimates and the reference K values are insignificant, indicating that this low-K skin does not have a large impact on DPP K estimates in a heterogeneous field.
 In many cases, prior information on the K structure may not be available. In scenario III, we investigate how the DPP performs under such conditions by disregarding the reference K zonation in the inverse analysis. Instead of using the reference zonation, a total of 18 K zones are defined during the parameter estimation process. Each zone corresponds to the injection test at a certain depth and is bounded by the zones for the two adjacent tests. As a result, the thickness of most zones is 0.5 m, except for the top and bottom zones, which are bounded, respectively, by the upper and lower boundaries of the aquifer. The top zone is 1.15 m thick and the bottom zone is 1.10 m thick. All other settings are identical to those in scenario I. The DPP K estimates in scenario III are presented in Figure 9. Despite the local mismatch at some depths due to the different zonations, the K estimates from the DPP are in good agreement with the reference values overall. While prior knowledge of the K structure is essential for estimating the K values to a high degree of accuracy, the DPP can be used, with some degradation in accuracy, to determine both the structure and magnitude of K.
 To assess the applicability of the steady-shape analytical formula in heterogeneous settings, equation (1) is applied to each of the 18 tests in scenario III. Figure 9 displays the 18 analytical K estimates. Similar to the numerical lr2dinv estimates, the analytical results are in good agreement with the reference values overall, suggesting that equation (1) can be applied to heterogeneous settings when the distance over which K is varying is larger than that between the screen and the far transducer.
 In scenario IV, we investigate the ability of the DPP to quantify K features that are thin relative to the DPP tool dimensions. Previous simulations in Sections 4.1 and 4.2 have shown that a single DPP injection test provides an effective K for the area immediately surrounding the 0.4-m interval between the injection screen and the far transducer. Simultaneous analysis of multiple injection tests at different vertical positions allows the DPP to quantify K features that are thinner than 0.4 m, given that the vertical interval between successive DPP tests is sufficiently small. To demonstrate the fine resolution the DPP can achieve, a 0.1 m-thick high- or low-K layer is added to the K profile from GEMS (Figure 11). The K value is set to 5.0 × 10−2 m/s and 5.0 × 10−6 m/s for the added high- and low-K layers, respectively. Three different cases, referred to as A, B and C below, are investigated with this configuration.
Figure 11. DPP K estimates after adding a 0.1 m-thick (a) high-K channel and (b) low-K barrier to the GEMS K profile. The extra layer is placed at elevations 6.65–6.75 m. The added low-K layer in Figure 11(b) is poorly estimated in case A. By adding a few closely-spaced injection tests below the layer, the low-K barrier is characterized much more accurately in cases B and C.
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 In case A, the model settings are identical to those of scenario I in Figure 9 except that the extra layer is added to the forward simulation. As a result, 13 reference K zones are involved in the forward simulations. In the inverse parameter estimation, the reference 13-zone structure (including the new thin layer) is assumed known and specified explicitly. A total of 18 tests at 0.5-m intervals are conducted at the same depths as those in scenario I. Figure 11(a) shows the results when the extra layer is specified as a high-K preferential flow channel. The DPP K estimates from case A are essentially the same as the reference values at all depths. Figure 11(b) shows the results when the extra layer is specified as a low-K flow barrier. In this case, the DPP K estimate from case A does not match with the K value for the thin layer. Instead of the input value of 5.0 × 10−6 m/s, the DPP K estimate is 1.4 × 10−2 m/s.
 In case B, the vertical interval between successive DPP tests is reduced from 0.5 to 0.1 m between 6.150 and 6.775 m, resulting in a total of 23 tests across the aquifer. All other settings remain identical to those in case A. During the numerical parameter estimation, the K structure is assumed known and specified explicitly. Figure 11(a) indicates that, as the true K distribution (including the high-K channel) has already been identified when 18 tests are used, the addition of more tests does not yield any change in the DPP estimates. Figure 11(b), on the other hand, shows that the accuracy of the DPP K estimates is improved dramatically after five additional tests are conducted in the interval immediately below the thin low-K layer. It should be emphasized that such an adjustment to the vertical interval between tests is practically feasible under field conditions, as qualitative K information can be obtained by monitoring the injection pressure continuously during DPP advancement [Dietrich et al., 2008]. Therefore whenever there is a significant reduction in K as indicated by a sharp increase in injection pressure, the advancement interval can be reduced to characterize that low-K feature.
 Case C is similar to B except that the reference K zonation is assumed unknown during the parameter estimation process. A total of 23 K zones are used in the inverse simulation. Each zone is specified in accordance with the injection test at a certain depth. Figure 11(a) shows that, despite the local mismatch produced by the different zonations for the forward and inverse simulations, the DPP K estimates are in good agreement with the reference K values, including at the thin high-K channel. Figure 11(b), however, shows that without using the reference K zonation, the accuracy of the DPP K estimates decreases, particularly in the area immediately below the thin low-K layer where spurious oscillation in the K estimates is observed. Thus prior knowledge of the K structure is more critical in the case of a thin low-K layer.
 To demonstrate the importance of simultaneous analysis of DPP tests in the presence of small-scale K variations, the analytical formula (1) is applied to each of the 23 tests in case C. Figure 11 indicates that the analytical K estimates do not match the reference K values for either the high-K channel or the low-K barrier. While equation (1) is still effective for quantifying K variations at a scale greater than the interval between the injection screen and the far transducer (>0.4 m), simultaneous analysis of all DPP tests is necessary for accurately obtaining the K information at smaller scales (<0.4 m).
 In scenario V, we explore how the DPP responds to lateral variations in K by adding a low- or high-K zone, which has an inner boundary at a radial distance from the DPP, to the test configuration of scenario III. Results show that when that inner boundary is at lateral distances greater than 0.51 m from the DPP, the K estimates are essentially the same as those in the reference case (i.e., the DPP K estimates are unaffected by that zone, consistent with the sensitivity results in Figures 6 and 10). Because of the assumed symmetry in the angular direction, the added zone is in the form of a hollow cylinder with an outer boundary that extends to the edge of the model domain. For a planar or block feature, the influence on the K estimates would be smaller for the same distance from the DPP to the inner boundary of the zone.