Advances in the Hydraulic Interpretation of Water Wells Using Flowmeter Logs

. This paper reports on the methodology developed for a new hydraulic interpretation of flowmeter logs, allowing a 10 better characterization of continental hydrological basins. In the course of a flowmeter log, different flow stretches are established mostly corresponding to permeable layers (aquifers), among which there are other stretches mainly corresponding to less permeable layers (aquitards). In such hydrological basins of sufficient thickness, these flow stretches may not have the same hydraulic head. This fact brings about the need for a new hydraulic interpretation that provides the actual distribution of horizontal permeability throughout the aquifer at depth. The modified hydraulic interpretation developed in this study focuses 15 on the differences of the effective pressure gradient (considered as the difference between the hydraulic head in the well and the hydraulic head of each stretch) experienced by the different flow stretches along the well, due to the existence of different hydraulic heads. The methodology has been developed starting from a water well located in a multilayered aquifer within the so-named Madrid Basin (the NW part of the continental basin of the Tajo River), located in the centre of the Iberian Peninsula. In this well, a step-drawdown pumping test was conducted, in which the pumping rate versus drawdown and the specific 20 capacity versus drawdown showed discrepancies with Darcian behaviour and an exponent of the Jacob equation of less than 1. Flowmeter logs were then recorded for different discharge rates and pump depths; the resulting water input from deeper permeable layers did not appear to show the expected relation with respect to drawdown. With the proposed methodology the results comply with the expected linearity and the

computer program for flow-log analysis of single holes applicable up to 10 levels, in which the hydraulic head of each zone is determined by minimizing the differences between the flow rates obtained and those of the model, and between borehole's water level and far-field heads.
This communication presents the possibilities of the flowmeter log to provide a hydrogeological explanation of the described anomalous cases. Flowmeter logging is conventionally used to determine variations in the flow velocity along a well casing, 70 allowing water inputs at different depths that contribute to the total discharge rate to be computed. These quantities are used to estimate changes in hydraulic characteristics with depth, thereby improving the management and rational exploitation of aquifers. In addition to this conventional purpose, a method has been developed in this work that uses flowmeter logs to provide information regarding different hydraulic heads in a multilayer basin. Moreover, determining these different hydraulic heads allows hydraulic reinterpretation that explains the abovementioned anomalous behaviours of the pumping test results. 75 To use flowmeter logs, a thorough pre-processing of results is necessary, without which the water inflow values determined in each filter can have very high errors and in turn allow an accurate determination of the head loss inside the well. Although different types of sensors have been used in well logging tools, spinner flowmeters are the most widely used in assessing the productivity of wells. Díaz-Curiel et al. (2020) proposed a complete reformulation for processing spinner flowmeter logs.
Another aspect related to the reliability of the flowmeter log results is the variability caused by differences in the near-wellbore 80 or skin zone in the different layers of the well, for whose solution this work proposes the establishment of 'flow stretches'. In this work, the term 'flow stretch' is primarily used for differentiate sets of screens that corresponding to more permeable units (aquifers) among which there are other stretches (aquitards) mainly corresponding to less permeable units. We have chosen to use the term stretch to avoid controversy with other terms such as "units" which have a different hydrogeological meaning.
Despite its origin, in this study, the term stretch is used both to designate the flow stretches in the well, as well as the sets of 85 layers to which they correspond. These stretches are obtained from a zonation process of the flowmeter log established by Díaz-Curiel et al. (1997), and it starts by generating a flow curve interpolated between water inputs. This curve is transformed into a smooth curve with constant depth increments. To obtain the depth values at which the limits between stretches are located, first, the inflection points of the smooth curve are calculated, and then the average values between those limits are determined. Finally, the upper and lower limits of each stretch of minimum values (the impermeable stretches) are 90 approximated to each other so that the average variance within each permeable stretch is minimal. These stretches show some parallelism with zonation relative to the average grain sizes shown in Díaz-Curiel et al. (1995), which spatial extension is addressed in the discussion section. The use of the flow stretches allows the differences between screens within each stretch to be ignored, and their influence is not evaluated in this work because the average hydraulic conductivity of each flow stretch compensates for them. 95 Regarding the hydraulic interpretation of flowmeter logs, its main advantage lies in the fact that different permeable layers that the well crosses may have different hydraulic properties. These cannot be drawn from the results of a conventional pumping test without using packers. The differences are quantified by water inputs through screens corresponding to each layer and its thickness. In wells with a high technical control budget, the hydraulic characteristics of the different permeable layers can be achieved by using packers. However, despite the high cost of this technique in deep wells, the results do not have 100 to match those obtained during operations with no packers on the pump. The main reason for this difference is that at higher pumping rates, there is significant vertical flow through the gravel pack surrounding the screen (Boman et al., 1997). For example, for a well drilled to 44.5 cm and cased with a 39.2 cm filtering pipe (annulus space ~1400 cm 2 ) with a 2-3 mm gravel pack, the flow through it is larger than the water inflow through an isolated screen of 320 cm (area of ~80 000 cm 2 ) located in front of sands whose permeability is one hundred lower. By isolating each layer, the static and dynamic water levels may be 105 different from those presented in the well when all permeable layers are connected ('dynamic level' refers to the well water level when it reaches a quasi-steady state for a given pumping rate). The influence of pump depth is not analysed in this study, considering that it only affects the flowmeter logs mainly for measurements in front of the screens close to the pump and that the initial study depths are rather below the pump depth.
To achieve hydraulic interpretation from flowmeter logs, most authors (Molz et al., 1989;Rehfeldt et al., 1992;Ruud and 110 Kabala, 1996;Zlotnik and Zurbuchen, 2003a;Barahona-Palomo, et al. 2011;Riva et al., 2012) start from the basis that hydraulic conductivity values for each permeable layer (from each screen) are proportional to the hydraulic conductivity of the entire well up to a multiplying constant. In these studies, the hydraulic conductivity is obtained from measurements by a nearby piezometer during pumping tests using the Theis equation (1935) between the discharge of a well and the water level drawdown a short distance from the well (Theis, 1963). That proportionality is a function of the ratio between the water input 115 at each screen and the pumping rate and the ratio between the thickness of each screen and the saturated thickness of the aquifer. In mathematical form, the hydraulic conductivity value of the permeable layer j is given by Kj= (ΔQ j /Q P )·(Δz j /b)·K P , (Kabala, 1994), where Q j is the water input at layer j, Q P is the extraction rate of the well, b is the aquifer thickness and K P is the hydraulic conductivity of the entire well. Among the different thicknesses in the literature, saturated thickness (Molz et al., 1989;Li et al., 2008), aquifer thickness (Clemo and Barrash, 2003;Riva et al., 2012) and screened casing thickness ( Barahona-120 Palomo et al., 2011;Gueting et al., 2017) used to calculate the hydraulic conductivity of an entire well, the saturated thickness is employed in this work.
Unlike the previous procedure, this study follows the less common methodology established by Rehfeldt et al. (1989) starting from the Thiem equation (1906). Although there are contradictory opinions on the validity of this equation, some more recent studies consider that it is still applicable for determining the hydraulic characteristics of the well (Zlotnik and Zurbuchen, 125 2003b;Schneider and Attinger, 2008;Day-Lewis et al., 2011;Houben, 2015). Rehfeldt et al. (1989) stated that a unique radius of influence R0 value (the distance for which the produced drawdown in the aquifer water table is nil) allows the direct determination of the hydraulic characteristics of different permeable layers. Following the proposal in Rehfeldt et al. (1989), variation in the radius of influence can be neglected because it is included in the logarithm; therefore, its variation affects the hydraulic conductivity computation by less than 10% for all permeable media in a given aquifer. This statement assumes that, 130 for a certain type of aquifer, its radius of influence varies only a few hundred metres around a mean value of approximately one thousand metres (Villanueva and Iglesias 1984).
For these reasons, the goal of this work is to investigate the causes of anomalies in the characteristic curves of pumping tests and to develop a methodology that improves the estimation of the hydraulic parameters in multilayered aquifers. Considering that the hydraulic conductivity (k) of the permeable layers should remain the same at different pumping rates, this advance is 135 based on the fact that the hydraulic head of successive permeable stretches can be different, as already proposed by Bennett and Patten (1960). Although different hydraulic heads are acceptable to determine the hydraulic properties of fractured aquifers (Hess, 1986;Paillet, 2000;Lane, 2002), this is not conventionally taken into account in multilayered aquifers. This methodology has been applied to a 475 m deep borehole drilled in a multilayer detrital aquifer located in the centre of the Iberian Peninsula (Madrid Basin). A step-drawdown pumping test was conducted in this well, showing discrepancies with 140 Darcian behaviour and simultaneously with the non-Darcian coefficients of the Jacob equation. The relation between pumping rates and well drawdown in the step-drawdown pumping test as a whole did not show the expected behaviour for the type of aquifer considered. Moreover, the pump characteristic curves that were obtained do not correspond to any aquifer type. This difference results from the fact that the pumping rate increases with drawdown that has a power greater than 1 and that the specific obtained capacity increases with drawdown. A flowmeter log was collected, and the hydraulic interpretation is 145 presented in this study, showing that the activation of the deepest aquifer stretches is the cause of this hydraulic behaviour, as explained throughout this study.
These results allow the avoidance of the possibly hazardous effects derived from intensive exploitation. As shown in this work, dangerously high arsenic contents occur in the deepest aquifer stretches in the Madrid Basin (López-Vera, 2003). Since the studied well is part of the official network of the Madrid city water supply, it is imperative to limit the spread of this pollutant. 150 As demonstrated by the hydraulic reinterpretation proposed in this paper, this aquifer undergoes strong activation when very high drawdown is applied, producing a sudden increase in its water inputs. This information is key to managing the exploitation network.

Estimating the hydraulic parameters 155
To determine the hydraulic conductivity K of the aquifer obtained through the entire well and each permeable layer, the Thiem solution (1906) is used, which is presented by Eq. (1) as a function of the radius of influence R 0 : where Q is the extraction rate, b is the aquifer thickness, d is the drawdown in the well, and r w is the well radius.
The main drawback to this procedure, which is mentioned by Kruseman and Ridder (1970), is the influence of local well 160 factors on the drawdown values. Excluding friction along the pipe (which depends on depth), the different local well factors that modify the obtained hydraulic conductivity of the permeable layers are 1) the reduction in the cross-sectional area of the well due to the submersible pump; 2) the entrance loss caused by flow through the screen slots; 3) the head loss due to the gravel pack; and 4) the head loss caused by the disturbed zone around the well (referred to as the skin effect) (Hufschmied 1986;Rehfeldt et al. 1989). Some of these factors have been considered in detail regarding flowmeter logs (Ruud and Kabala, 165 1997;Ruud and Kabala, 1999). In this work, these factors are not considered because they do not justify an increase or decrease in the hydraulic conductivity with depth; thus, although any of the four factors may have locally different values, their influence on the hydraulic conductivity obtained at each permeable level is constant for any flow.
As established by Rehfeldt et al. (1989), the hydraulic conductivity of each permeable layer is given by Eq. (2): where q j is the water input produced in each screen and Δz j is the thickness of each screen. Equation (2) has been applied in various studies (Xiang 1995;Oberlander and Russell, 2006), but in this work, it is applied to well flow stretches.

Step-drawdown pumping test
In this type of pumping test, the hydraulic behaviour of the well is analysed through the characteristic relationship d =A·Q+B·Q 2 (Jacob 1947) or, in a more general form (Rorabaugh, 1953), as shown in Eq. (3): 175 where Q denotes the consecutive values of the extraction rate in each step, d is the corresponding stabilized drawdown (i.e., when its increase is negligible for an increase in the pumping time), A is a constant that depends on transmissivity, and B and p are fitting constants to the resulting data from the pumping test, where p is greater than 1 (Todd 1980). The second term represents the apparent divergence from the linearity expected by Darcy's law (Darcy 1856), which is addressed in the Sect. 5. 180 This is generally attributed to an increase in head loss due to turbulence as the pumping rate increases. It is also coherent when the dynamic level exceeds the depth of the upper aquifer layers, reducing the specific capacity. Although some authors consider that the Jacob equation (Eq. (3)) can be improved, there are still authors who continue to use it (see Mathias and Todman, 2010).
The conventional interpretation of step-drawdown pumping tests begins with the fact that drawdown for different pumping 185 rates is caused by either the general or extensive characteristics of the aquifer. In this way, confined, semiconfined and unconfined aquifers are distinguished, whose curves, pumping rate versus drawdown and specific capacity versus drawdown, are different in each case ( Figure 1).

Flowmeter data processing method
The need for an exhaustive treatment of the flowmeter logs arose initially to avoid doubts on observed anomalies in the 195 characteristic curves of the step-drawdown test could stem from the reliability of the flowmeter log results. Thus, it had to be shown that such effects were not due to head losses along the well. In addition, considering that the flow velocity used in the Darcy-Weisbach equation is raised to a power of two, the differences between the head losses resulting from considering the actual flow velocity instead of the velocity directly measured by the sonde is greatly amplified.
This exhaustive process of the flowmeter logs will be done according to the laws of pipe hydraulics using the methodology 200 developed by Díaz-Curiel et al. (2020). To obtain the flow velocity at each depth, <V(z)>, a conventional iterative process is used. It begins by taking the measured velocity Vmeas at a given depth as the initial flow velocity and the initial Reynolds number Re ini according to its definition, that is, Re = ρ·<V>·D/μ, where ρ is the water density, D the well diameter and μ the dynamic viscosity. Then, a relationship τ(Re) that provides the flow turbulence exponent τ as a function of the Reynolds number is applied. Knowing the turbulence exponent and the normalized radius r D of the sonde (the ratio of the sonde distance 205 to the well axis with respect to the well radius), a velocity law must be applied; this law is the ratio between the velocity at the normalized distance V(r D ) and the maximum velocity in the well axis Vmax; this allows this maximum value to be obtained.
Then, using the relationship for the velocity factor F vel (τ), defined as the ratio between Vmax and the flow velocity <V>, the first flow velocity is obtained with the corresponding Reynolds number Re ini , which is closer to the actual value. Applying τ(Re), V(r D ), and F vel (τ), a new Re value Re k is obtained (k being the iteration index of the convergence algorithm). This 210 process is repeated until a given convergence criterion c CR is reached, following the flow chart in Fig. 2 (adapted from Díaz-Curiel et al., 2020), to obtain Re(z). (Re/2490) 9.994 + 1 0.2 · (Re/2490) 9.9 + 1 (4) In these equations, the influence of temperature is not considered because viscosity is practically homogeneous along the well due to water circulation during pumping. 220 Once the Reynolds number at each depth is known, the head loss can be obtained by the Darcy-Weisbach equation (Darcy 1857;Weisbach 1845), given by Δh = f·(ℓ/D)·(<V> 2 /2g), where g is the gravity acceleration (m·s −2 ), <V> is the average flow velocity (m·s -1 ), D is inner diameter of the well (m), ℓ is the length of each considered pipe element (m), and f the friction factor (dimensionless) for smooth pipes given by Eq. (7): Re 10.75 + 4850 10 Re 10 + 2850 10 (7) 225 It is important to point out that according to Eq. (7), as in all pipe hydraulics relations, the friction factor decreases with the Reynolds number except for the transition interval between laminar and turbulent regimes.
Applying the rigorous formulation presented to process the flowmeter logs (Eqs. 3 to 7) and considering that the sonde has a significant diameter (r D ), the values of <V>/V(r D ) vary between 0.85 and 0.94. This difference represents a 20% error in the total range of variation of that velocity ratio between 0.5 for laminar flow and 1.0 for fully turbulent flow. However, if the well 230 diameter is smaller (close to the diameter of the sonde), V(r D ) approaches Vmax, resulting that the <V>/V(r D ) ratio presents a greater variation (from 0.50 to 0.83) for the range of Re found in the case studied, than if the diameter of the well analysed is close to 0.2 m as in the case studied in this work.
To estimate the hydraulic parameters from the flowmeter logs, once they have been processed, two specific approaches 235 developed in this work are applied to obtain the actual hydraulic conductivity of the different layers. The first approach is to divide the well into flow stretches with different hydraulic behaviours as a function of the flowmeter results. The second approach is based on the fact that the hydraulic head of the deepest flow stretches of the well do not necessarily match the head of the overall well (Fig. 3). The hydraulic head of a flow stretch is defined as its effective static level, that is, the height of the water level that would be achieved if the well were connected with the aquifer only through this stretch. This work proposes that a flowmeter log allows to know the existence of hydraulic heads that are different for each stretch. This distinction implies changes in the effective 245 drawdown of each stretch, which justifies, as shown in the case study, that the water inputs of the deeper aquifer stretches are not proportional to drawdown.
In most flowmeter logging with several pumping steps, the drawdown used in the Thiem (1906) equation is the same for all of the aquifer stretches in a well, d0(s) = h DL (s)−H SL where h DL (s) is the dynamic level for the 's' pumping step and H SL is the dynamic level of the entire well. However, under the hypothesis presented in this work, the hydraulic head of each stretch, and 250 therefore the corresponding drawdown, can be different. Numerically, the drawdown of each flow stretch т N will be given by the following relation: where h SL (N) is the static level for flow stretch т N . In short, the proposed method consists of replacing the single drawdown d in Eq.
(2) from Rehfeldt by a drawdown for each stretch. 255 The main differences with the method used by Paillet (1998) are that we have chosen to use the Rehfeldt relationship (Eq. 2) for permeability instead of the Davis and DeWeist relationship (1966) relation for transmissivity, given that the thickness of the layers and the productive sections are taken into account. It has also been considered that the different hydraulic heads are below the static water level (the water level in ambient conditions from Paillet, 1998). The procedure developed is based on the linearity of the hydraulic behavior of the aquifer sections and each section is treated separately. 260 The proposed method for obtaining the hydraulic head of each flow stretch is to 1) correct the drawdown values of the total head loss due to flow along the pipeline and 2) modify the height of the hydraulic head for each flow stretch until the straight line fitted to the data, q N (s) versus d N (s), reaches the maximum regression coefficient (where q N (s) is the water input in flow stretch N for the s pumping steps). With a static level value for each flow stretch, the effective drawdown of each flow stretch can be obtained, and although other local well factors may cause differences between screens within each stretch, their 265 influence is not evaluated in this work because the average hydraulic conductivity of each flow stretch compensates for them.

Geology of the area and well characteristics
The study well (named CNC in this work) is located in the Tajo River Basin on the Iberian Peninsula. More specifically, it is located in the western sub-basin, also 270 known as the Madrid Basin, near the city of Madrid (Spain). The Madrid Basin has a triangular shape and is bound by several mountain systems of igneous-metamorphic nature, which are significant contributing source areas. The structure of the basement corresponds to that of a complex graben and has resulted in a sediment thickness of approximately 1000 m, although in some areas, the thickness can exceed 3000 m. The 275 Tertiary (Miocene) sediments that fill the basin correspond to continental deposits of an arid, endorheic nature that are fed by alluvial fans; these alluvial fans develop edge or detrital facies, intermediate or transitional facies, and central or chemical facies, all of which are characteristic of this depositional system (Navarro et al., 1993).
The aquifer is located in a detrital facies single arkosic unit separated into two 280 lithostratigraphic units, which are differentiated by grain size and, therefore, by hydrogeological characteristics; due to the depositional process of the materials, they are differentiated from one area to another as well as vertically. The lower unit, the Tosco Formation, is composed of arkose that is generally very clayey with clayey sand. The upper unit, the Madrid Formation, consists of arkosic coarse-grained sand, 285 gravel and clay. Although the Madrid Formation is sandier and permeable and overlaps the more clayey Tosco Formation, they are not considered different aquifers (López-Vera 1985) but rather a heterogeneous and anisotropic free aquifer system where more permeable layers are separated by clayey strata with a lower permeability (which qualifies as a multilayer aquifer). 290 A lithological column was compiled from information provided by the detritus from the borehole and conventional well logs; the normal resistivity and natural gamma ray records are presented in Fig. 4. Note that the logs were not corrected for borehole diameter, conductivity or mud density; therefore, there was a notable 295 difference between the larger diameters in the upper part and the base. Three differentiated parts were established: the first (0 to 75 m) was composed of sands alternating with thin layers of clay, the second (75 to 285 m) comprised alternating sandy clay and thin sandy layers, and the third (285 to 485 m) had a predominance of coarse sand and gravel with intercalations of very thick clay layers. As a result of the correlation between lithological stretches 300 in the NW zone of the studied basin (Díaz-Curiel et al., 1995), the permeable stretches can be considered radially homogeneous differentiated aquifers. With the exception of the superficial part, the rest of the permeable stretches can be treated as confined aquifers. This allows the use of the Thiem equation for these stretches and for obtaining hydraulic parameters on a regional scale (as proposed by Rehfeldt) by averaging the stretches as a whole (Sánchez-Vila, 2006).
The borehole was rotary drilled with a diameter of 660 mm down to a depth of 120 m, and then it was drilled with reverse 305 injection of natural mud with a diameter of 445 mm to a final depth of 490 m. The construction details of the well consisted of casing to a depth of 480 m, with a 404 mm inner diameter in the sections of blind pipe and 392 mm in the wire-wrap screen sections; a gravel packing of 2-3 mm grains was added throughout its length. The well was developed by adding previously diluted polyphosphates, and after 12 hours, a series of intermittent pumping was carried out; once the extracted water contained no suspended fines, a 72-hour gauging of increasing pumping was conducted up to a flow rate of 100 l/s. 310

Pumping test results
After a process was conducted to eliminate the well storage effect, the static level settled at a depth of 151 m. The pumping test started with a flow of 5 l/s until the hydraulic head stabilized. Two main extraction pumping rates of 30 and 70 l/s and a final rate of 75 l/s were then used, with a total elapsed time of 80.5 hours. All steps were performed for a sufficient time to reach the quasi-steady conditions mentioned in the Sect. 1. The resulting temporary data are shown in Fig. 5  Hydraulic conductivity values were obtained using Eq. (1) (Thiem 1906), taking the saturated thickness to be equal to 329 m (the depth of the well minus the depth of the static level) and considering a radius of influence of 950 m once the quasi-steady 320 state was reached. This last datum is the central value of those shown in Villanueva and Iglesias (1984) for semiconfined and confined aquifers. The resulting hydraulic conductivity values of the well are shown in Table 1, and they increase with drawdown.

335
Let remember that in the characteristic equation d = A·Q + B·Q p for the well for step-drawdown pumping tests, the first term corresponds directly to Darcy's law (1856) for the total volume of the water flow crossing through successive cylindrical layers.
In the second term, the exponent p can be greater than 1; successive drawdown should show a power relationship with an exponent greater than 1 versus the pumping rate. Moreover, the specific capacity should decrease with successive drawdown; otherwise, the well extraction ratio would increase with drawdown. However, in Fig. 6, the opposite pattern is observed, i.e., 340 d(Q) increases with a power that is less than 1, and Q/d increases with drawdown. These anomalies do not seem to arise from errors in the water level measurement, as their values versus time appear to be correct (Fig. 5).

Flowmeter results
The static level H SL was measured at a depth of 157 m before the beginning of flowmeter logging. Flowmeter logs were obtained for pumping rates of 20 l/s (measured dynamic level at 172 m), 30 l/s (dynamic level at 178 m), and 70 l/s (dynamic 345 level at 205 m). The drawdowns of the entire well for each pumping rate, without including the head losses, hence are 15 m, 21 and 58 for 20 l/s, 30, and 70 respectively.
Since the flowmeter logs were collected during pumping operations, measurements could be obtained only below the pump depth. For the pump located at a depth of 191 m, logs were recorded from 200 m to 470 m for pumping rates of 20 and 30 l/s, and for the pump located at 253 m, logs were recorded from 260 m to 470 m for pumping rates of 30 and 70 l/s. 350 Equation 5, that of V(r)/Vmax, used a normalized distance of 0.64, which corresponds to the ratio (r w −r s ) /r w, where r s is the external radius of the spinner frame (the sonde has a device that maintains its hold on the wall) and r w is the inner radius of the well casing. The different diameters (difference < 1%) in the screens were not considered due to the difficulty of executing the iterative process to obtain <V>. The initial velocity values considered in the iterative process were obtained by applying the The accuracy of the measuring equipment was 0.5 l/s, which greatly reduced the reliability of the results between consecutive screens and produced strong variation in the quantified water inputs from each screen.

Head loss results 365
The head loss was calculated using the Darcy-Weisbach equation (Darcy, 1857;Weisbach, 1845), and the friction factor was calculated using Eq. (7). The curve of the friction factor values obtained for pumping rates of 20, 30, and 70 l/s is shown in Fig. 7. Depending on the scale on which the transition interval is analysed, the turbulence of the fluid flow cannot be determined at all points along its path. In this study, we chose to use a fitting expression for smooth pipes given by Eq. (7) because the flowmeter sondes used in well logging reflect the fluid advance on a much larger scale. The friction factor for low 370 pumping rates in the deep screens increased by a maximum of 70% compared to that obtained using conventional equations.
The total head loss below the pump is obtained by integrating the head loss throughout the well based on the flow velocity obtained at each depth (see Fig. 7), that is, the cumulated Δh adding the successive local head loses values relying upon local friction factors and local water velocities. Above the pump depth, the calculation is based on a linear increase in the velocity between the pump depth and the dynamic level. The obtained values of the head loss ∆h(s) for each pumping rate are shown 375 in Table 2, which will be used in the calculation of the effective drawdown produced.  In this case, the friction factor reaches values six times higher at the bottom of the well than at the initially recorded depth, and the value of the head loss is low (0.06 m) because the average velocity in the Darcy-Weisbach equation is raised to a power of two. Therefore, in large-diameter water wells, the influence of the friction factor along the pipeline is negligible. Finally, despite the inclusion of head loss values, the water inputs from some of the stretches still do not maintain the expected proportionality with drawdown, so hydraulic reinterpretation is carried out using the flowmeter results. Figure 8 shows the results from different pumping rates after processing. On the left-hand side of Fig. 8a, the curves for upward flow rates versus depth are shown, and in Fig. 8b, the water inputs deducted in the different screens are shown, while the negative water inputs (outputs) are not shown in Fig. 8b.

Water inputs
Note that the accuracy provided by the equipment is 0.05 l/s, which greatly reduces the reliability of the results between 390 consecutive screens and produces strong variations in quantifying the water inputs from each screen. For this reason, following the criteria described in the Sect. 3, the flowmeter log is divided into different flow stretches based on the average productivity of each flow stretch. Table 3 shows the water inputs from the different flow stretches for each measured depth interval. A "top" stretch has been added to the top of the well above the pumping depth, where the different water inputs are unknown. The water input in the 395 upper part (which includes flow stretch т1 for the case of a pumping rate of 70 l/s) is obtained by the difference between the pumping rate and the deduced flow rate at that depth.
At a pumping rate of 70 l/s, the water input from flow stretch т 1 is not known, and the input of this flow stretch may increase in proportion to the pumping rate, which should imply that the upper part of the well would remain constant (e.g., due to the dynamic level dropping below some of the upper layers). 400 Both Fig. 8 and Table 3 show that the water input from flow stretch т 2 is very low, even for high pumping rates, and it is close to the flowmeter accuracy; hence, that flow stretch is omitted from the analysis. Table 3 shows that for pumping rates of 20 and 30 l/s, the water inputs from the upper part of the well and from flow stretches т 1 , т 3 , and т 6 increase in a way that is practically proportional to the flow (with a ratio ≈ 1.5) and fits a confined aquifer. Flow stretches т 4 and т 5 have negligible water inputs, reaching negative inputs for a pumping rate of 20 l/s. However, for the 70 l/s 405 pumping rate, there is an abrupt change in the hydraulic behaviour of the well. On the one hand, the whole water input from the upper part of the well and from flow stretch т 1 do not increase proportionally to the pumping rate (ratio=2.33). On the other hand, flow stretch т 6 shows a sharp increase in the water inputs, and flow stretches т 4 and т 5 present an apparent activation.
Given the possibility that an increase in the head loss could justify such behaviour, values for each pumping rate were calculated and added. 410  Regarding the existence of different hydraulic heads, note that the negative water input in flow stretches т4 and т5 for the pumping rate of 20 l/s corroborates the validity of the hypothesis in this work. These negative inputs reflect the fact that when the drawdown is located above the hydraulic head of these flow stretches, the water flow does not occur inwards towards the 420 well but rather outwards, reducing the upward vertical flow. In any case, the fact that the deeper flow stretches have a hydraulic head below the static level of the well explains that the pumping rate versus drawdown curve adjusts to a power function with an exponent greater than 1, and the specific capacity versus drawdown curve is ascending. There are several studies in the literature that mention negative water inputs as those obtained in this case, but they do not present the hydraulic interpretation thereof, most of them correspond to flow-logs measured in ambient conditions (Paillet et al., 2000;Butler et al., 2009;Day-425 Lewis et al., 2011).

Hydraulic reinterpretation
The permeability of each stretch has been calculated using Eq. (2). Instead of the contribution of each layer qj, the sum total of the contributions of each stretch q N (s) is considered (see table 3). The unique initial drawdown d considered in Eq. (2) has been modified by the drawdown of the entire well d 0 (s) = h DL (s)−H SL − Δh(s) for each pumping rate (s) (Δh(s) being the 430 head losses showed in table 2). The static level H SL is 157 m (as determined before the flowmeter logging was conducted) and the dynamic levels h DL (s) are 172 m for pumping rate of 20 l/s, 178 m for pumping rate of 30 l/s y 205 m for pumping rate of 70 l/s. The thickness of each layer Δz j has been replaced by the thickness of each stretch Δz (т N ) (depth intervals in Table 3).
The radius of influence (R 0 ) considered is 950 m (as in the previous calculations), and the well radius (r w ) is 0.404/2=0.202 m.
The characteristic curves of each stretch are shown in Fig. 9.a. 435 Analysing the specific capacities of different flow stretches, т 1 and т 3 show the expected proportionality for a confined aquifer. However, this is not the case for flow stretches т 4 , т 5 and т 6 , whose d N (s) versus q N (s) data fit to a power function with exponents of 0.22, 0.37 and 0.67, respectively (see Fig. 9.a). Not only does this not reflect Darcian behavior, but it also indicates an exponent p in the Jacob equation of less than 1, as is the case with the well as a whole (see Fig. 6).
However, if it is considered that flow stretches т 4 , т 5 and т 6 have different hydraulic heads, the results vary. Through an iterative 440 process, the value of the static level (hydraulic head) of each flow stretch for which the total water input of the flow stretch versus the drawdown acquires greater alignment can be determined. This means that when the data are fitted to a straight line, the regression coefficient is maximum. In other words, the resulting exponent in the Jacob equation when the data are fitted to a power function is p=1. Thus, for flow stretch т 6 , the static level for which inputs versus drawdown acquire greater alignment occurs at a depth of 165 m. Similarly, the resulting static level for flow stretch т 5 is located at a depth of 175 m. For a pumping 445 rate of 70 l/s, flow stretch т 4 undergoes an "activation" effect (even higher than flow stretch т 5 ) when the dynamic level exceeds the true static level of т 4 , which is computed at a depth of 177.5 m. Summarizing, the hydraulic heads h SL (N) obtained with this criterion are 157 m for т 1 and т 3 ; 177.5 m for т 4 ; 175 m for т 5 ; and 165 m for т 6 . Figure 9.b shows the regression lines of water inputs versus drawdown for each stretch, with the corresponding relationships and R 2 coefficients. 450

Figure 9. Drawdown versus water inputs for different flow stretches in the case study. a) dN(s) # qN(s) with a unique hydraulic head for all the stretches. b) dN(s) # qN(s) with the modified hydraulic heads for each stretch obtained considering that p is at least equal to one in the Rorabough equation).
With these differentiated static levels, the hydraulic conductivities of each flow stretch were obtained using a next change of 455 Eq. (2) (Rehfeldt et al. 1989) replacing d 0 (s) by d N (s) = h DL (s)−h SL (N)−Δh(s), which values are presented in Table 4.
The successive relationships used to arrive to the actual permeability with depth have been: Thiem (1906) →

Rehfeldt et al. (1989) →
Adapting to stretches → With actual тN hydraulic heads It must point out that the k N is the same for the different (s) because the ratio q N (s)/d N (s) is the same for any pumping rate (d N (s) versus q N (s) are fitted to a straight line) 460  As mentioned at the beginning of Sect. 4, the precision of flowmeter logs does not allow to obtain reliable hydraulic conductivity values of each permeable layer to make a more detailed characterization of each stretch. However, that analysis could be undertaken by considering the average water input and average thickness.

Discussion 470
Regarding the linearity predicted by Darcy's law, in this work, the variation corresponding to nonlinear flow is a different process than the change in the flow from a laminar to a turbulent regime. Takhanov (2011) determined that the onset of nonlinear flow occurs prior to the change to turbulent flow; in fact, some authors have considered that turbulent flow does not occur in porous fine-grained media in their natural state (Green and Duwez, 1951;Bakhmeteff and Feodoroff, 1937). In this sense, Houben (2015) established a linear laminar regime in the aquifer that becomes nonlinear in the gravel pack and only 475 becomes turbulent on the screen and inside the pipe. In this work, the regime change is less gradual than that predicted by the reasons, such as the presence of flow stretches with different vertical transmissivities. However, the approximated values for the hydraulic conductivity in the less permeable stretches (т 2 and т 4 ) contradict this hypothesis, since flow stretch т 2 is less permeable than flow stretch т 4 , but flow stretch т 3 is not affected by a similar effect. Another possible explanation is that the change in the effective drawdown is due to the existence of nearby extraction wells, which overexploited the aquifers corresponding to flow stretches т 4 and т 5 , thereby producing a drop in the static level of these flow stretches. 490 Concerning the reliability of the final permeability values, one aspect that must be considered in estimating hydraulic parameters from flowmeter results is that the analysis was conventionally performed through the screens assembled in the casing. However, the distribution of these screens only approximately matches the permeable layers that the well crosses.
Hence, differences between the thicknesses of the permeable layers and the assembled screens may exist, as well as permeable layers that are not faced with a screen, whose effects are minimized by gravel packing. This effect adds to that produced by 495 the local factors of the aforementioned well, which is an additional reason for differences in the water inputs of the different screens within each stretch.
Therefore, although the results of the pumping tests and the flowmeter results yield a similar hydraulic conductivity value for the entire well, after considering the possible hydraulic head difference that justifies and relates the anomalies reported over the pumping test data, this value moves away from the actual hydraulic conductivity of the aquifer. 500 Under the consideration that vertically there is a high hydraulic connection (similar to horizontal one), it is common practice in hydrogeology to model large aquifers as an equivalent porous medium. In addition to obtaining water balance results, such models have a wide application in many basins (De Filippis et al., 2016). However, this study focuses on a case where the vertical hydraulic connection is much lower than the horizontal one, as can be deduced from the existence of different hydraulic heads found. In this study, it is considered that the low hydraulic connection due to the existence of one or several wells in a 505 basin of the size studied does not significantly affect the lateral variations of the hydraulic head along the basin. In contrast with several works taking into account the hydraulic head field (Yeh et al., 1996;Axness and Carrera, 1999), in this study it is considered a single hydraulic head for distances smaller than the radius of influence. When wells are continuously screened, on a small scale it can be taken into account that the hydraulic head does not show as abrupt a change as is considered in great continental hydrological basins. In these basins this effect, which causes the conventional hydraulic head field over distance, 510 is included in the hydraulic parameter relationships from the hydraulic head gradient. It is also considered that in the interior of a large diameter well, such as water wells in large continental basins, there is no change with depth of the effective hydraulic head. In oil wells, this possibility is considered because of the strong variations in vertical flow velocity and the use of smaller diameters, leading to higher head losses.
Regarding Jacob's well equation (Eq. (3)), some authors say that the coefficient that multiplies Q 2 is the turbulent flow 515 coefficient, but others say that when the characteristic curve is not linear, it is because turbulent flow occurs. However, it is not clear what this "turbulent flow" refers to. It does not seem to refer to the change in flow in the pipeline but to the water in the aquifers acquiring turbulent flow. Regarding the friction factor, water flow in granular aquifers is not turbulent, although the obtained Re value would correspond to turbulent flow if the thickness of the aquifer is used to calculate the Reynolds number instead of using a mean pore diameter through which the water circulates. 520 This contrasts with the complex flow regime in oil wells where gas and liquids of different characteristics are combined, the behaviour of which has been analysed in many publications (Nind, 1965;Hasan and Kabir, 1988;Bri and Arirachakaran, 1992;Kabir and Hasan, 2004;Wu et al., 2017). This may lead one to believe that such behaviour is also common for water. However, if we consider, for example, a pressure gradient reaching 5 atm and an average pore diameter reaching 2 mm, the Re number obtained for water flow is <100, which does not correspond to turbulent flow. 525 Related to the spatial extension of the different hydraulic heads obtained, there are two facts that should be considered. On the one hand, hydrogeologists who have studied the Madrid Basin are already aware of the increase in arsenic that occurred at other points in the NW part of the basin for high drawdowns (López-Vera 1985). This would confirm that the hydraulic head of the arsenic-contaminated stretches is lower. On the other hand, although has already mentioned, this part of the basin is classified as a heterogeneous and anisotropic free aquifer system (Samper, 1999;Yélamos and Villarroya, 2007), other studies 530 on borehole correlation in this area show that the stretches established from logs reach distances of more than 10 km (Caparrini, 2006). This study has allowed to carry out the hydrological and hydraulic division of the studied basin that had not been done before, and such division involve a more precise obtaining of the permeability values in each stretch (and hence in its corresponding aquifer) which was neither been before. Certainly, the new procedure developed to obtain the hydraulic head 535 differences in heterogeneous granular basins and the results obtained for the first time in the Madrid basin may allow hydrogeological hypotheses to understand the large-scale structure of aquifers concerning recharge. According to the results obtained, the fact that the Madrid Basin is considered a single aquifer should be replaced, at least from a depth of 200 m, by a sequence of stretches -aquifers-differentiated by their different permeability values. From 345 m depth (the one of stretch 4), it was also found that the aquifers corresponding to stretches 4, 5 and 6 have different "hydraulic heads" than the upper 540 aquifers. One hypothesis would be that this means different "recharge pathways". So that it could be deduced that above 345 m the Madrid Basin can be considered a single heterogeneous aquifer (with different sub-aquifers of different permeability), and below 345 m, the Madrid Basin consists of a sequence of confined aquifers (the last three coarse-grained ones shown in the welllogs, see Fig. 4) that are hydraulically separated from the rest of the aquifers. It should be emphasized that the hydrogeological hypotheses that can be made as the previous scheme must be contrasted with results in more wells within the NW arc of the Madrid Basin.
The division of the studied well also allows proposing a strategy regarding the arsenic propagation in the Madrid basin. The obtained results indicate the stretch of the studied well that is "activated" when the dynamic level exceeds the "hydraulic head" 550 of the aquifer to which it corresponds, is the rather connected to a point -or zone-where the arsenic focus is. As the exploitation of that stretch in different points of the basin will cause the contaminant to move towards those points, that critical dynamic level should be not allowed Finally, regarding the application of this methodology to other aquifers of the same type, there is no hydrogeological hypothesis that implies that in other great continental basins in which large impermeable-type stretches are found, all the permeable 555 stretches should have the same hydraulic head.

Conclusions
The improvements developed in this work are represented by the following advances in the hydraulic interpretation of flowmeter logs: a) The method developed from the flowmeter allows to reinterpret the hydraulic behaviour of any well in which the 560 characteristic curve d(Q) increases with a power less than 1 and the characteristic curve Q/d increases with drawdown, which until now was considered anomalous due to poorly measured data or due to changing aquifer characteristics with pumping time.
b) The processing of flowmeter logs provides an increase in the quantified values of water inputs in the deepest permeable media for low pumping rates. This increase modifies the obtained values for hydraulic conductivities in the studied well data 565 that approach Darcian behaviour but do not reach it.
c) The division of the wells into flow stretches with different hydraulic heads provides hydraulic reinterpretation that explains the possible anomalies produced in the step-drawdown pumping tests. As occurs in the well in this study, both the characteristic curve of the pumping test and the specific capacity versus the drawdown curve show unexpected slopes, the anomalous nature of which is not justified by non-Darcian behaviour. 570 d) In particular, the resulting values of the different hydraulic heads make it advisable, in any well located in the Madrid Basin, not to use pumping rates for which the dynamic level goes beyond the depth corresponding to the drawdown of 165 m in the studied well. Once it is determined if flow stretches т4 and т 5 have a greater arsenic content than flow stretch т 6 , the mentioned depth can be changed to that corresponding to a drawdown of 175 m in the studied well.
The verification of the existence of different hydraulic heads for the different stretches with depth entails a substantial change 575 in the hydrogeological knowledge of a basin such as the one studied. It can also be concluded that the corresponding determination of the actual hydraulic properties of the different stretches is essential for modelling the hydraulic behaviour of the basin. Likewise, although it does not have a spatial extension corresponding to the entire basin (as there are characteristics that do not necessarily have to be maintained, depending on the position with respect to the different source areas and the distance to them), the extension of up to 10 km is sufficiently interesting to characterise parts of the basin. 580 As a future line of action, this study proposes the execution of step-drawdown pumping test and flowmeter logs with various flow rates in wells progressively distant from the studied one to verify that the stretches with different hydraulic heads maintain and to determine the spatial extension of this behaviour.
Authors contributions. JDC conceptualized the paper. JDC develop the methodology. JDC, MJM and contributed to the writing 585 of the paper, with BBV and LAL reviewing and editing the paper. JDC and NC curated the data and led the investigation. BBV and LAL does the formal analysis and prepared visualization data. JDC supervised the project.
Data Availability Statement. The data used in this manuscript are available for download in the following link https://data.mendeley.com/datasets/gx8dwgvygn/1. 590 Competing interests. The authors declare that they have no conflict of interest.