One of the most interesting hydrogeological aspects of well pumping tests is that their results not only allow us to estimate the permeability and transmissivity obtained in the well, but that they can also be used to infer the behaviour of the aquifer when the lithological distribution of the basin in its location is known. This estimate of basin behaviour will be less accurate when knowledge of the lithology is local. In the case of step-drawdown pumping tests, this inference is generally known when the characteristic curves of the test show a conventional evolution, i.e. when the drawdown versus the extraction rate curve shows an increasing slope and the specific capacity decreases with drawdown. This is the case when, inside the well, in the near-wellbore zone or in the aquifer, head losses occur, whether linear or polynomial, whose effects are well recognized in step-drawdown pumping test curves (Helweg, 1994; Kawecki, 1995; Mathias and Todman, 2010).

These models provide an accurate representation of the aquifer behaviour for any pumping time. Among other results, Mathias and Todman (2010) found that the best fit was achieved by using a non-linear coefficient, called the “well loss coefficient”, which can be different for each step. The values of this coefficient were obtained using an analytical formula derived to relate this coefficient to the Forchheimer parameter. When the drawdown versus the extraction rate curve presents an increasing slope, as in the case of the step-drawdown test from Clark (1977) (a similar behaviour can be seen in the generic curves 2 and 3 shown in Fig. 1), there are different hydrogeological explanations. However, the slope decreases and the specific capacity versus the drawdown increases, i.e. when the hydric behaviour improves with increasing flow rate, as in the last two stages of the step-drawdown test from Van Tonder et al. (2001) (a similar behaviour can be seen in the generic curve 4 shown in Fig. 1). The only explanation known to date is that the well is not properly developed. In the following text, when such results occur, they are referred to as anomalous cases.

In step-drawdown pumping tests, there is no unified criterion for the duration that each step should have. Thus, in contrast to the values used for the characteristic curves of these tests in some studies (Shapiro et al., 1998; Karami and Younger, 2002), in this study, it is considered that the steps must be performed for sufficiently long periods to have reached quasi-steady states. These states have been reached when the temporal variation of the drawdown is less than 1 % of the total drawdown for each step. This criterion has been adopted by considering its equivalence with the criterion established by Zha et al. (2017) on the temporal derivative of drawdown for quasi-steady state conditions and by considering the validity of assimilating the drawdown in the well to the average behaviour of the different levels in a multilayer aquifer.

A situation that is not often considered in studies on great continental detrital basins that are hundreds of metres deep is that the diverse permeable layers crossed by water wells can have different hydraulic heads. If this difference exists, then the permeability value determined for each permeable layer is incorrect, leading to an error in the estimation of the flow rate provided by each layer and causing a very important deviation in aquifer modelling. Although this possibility has been cited in several publications (Molz et al., 1994; Crowder, 2002; Le Borgne, 2006), no methodology has been published to quantify its effects in water wells in large continental detrital basins.

Flowmeter logging is conventionally used to determine variations in the flow velocity along a well casing, 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 process to provide information regarding different hydraulic heads in fractured rock media from flowmeter logs was proposed in several works by Paillet. Paillet (1998) showed the results of two flowmeter logs obtained with a heat-pulse flowmeter (lower limit of ∼0.1 l min-1 and upper limit of ∼20.0 l min-1) in Waupun (Wisconsin, USA). These flowmeter logs were measured under ambient and injection conditions at about 4 l min-1 and analysed for pumping or injection rates typically 1–5 l min-1. We think that the relationship used to estimate the transmissivity Tk of each fracture k, starting from the flow into the borehole qk, is qkb-qka=2πTkwa-wb⋅ln⁡R0/rw, where a and b address the ambient and stressed conditions, respectively, wa,b are the water levels in the borehole for these conditions, R0 is the distance to the “outer edges” of the fracture, and rW is the borehole radius. This relationship does not depend on the unknown value of the far-field head in the aquifer Hk. Later, in Paillet (2000), ∑Tk⋅Hk=wa⋅∑Tk is used to determine Tk. In this work “the results of high capacity tests, where the effects of ambient hydraulic-head differences would not be significant” were discussed. The hydraulic head values (4.54, 4.91, 4.91, 4.91, and 4.91 m below ground level) obtained for the four productive intervals found in one of the analysed boreholes are also presented in that work, although the process followed is not reflected in this paper. In Paillet (2000) the hydraulic head estimates (centimetres above open hole water level) in the same borehole (+28, -11, -11, and -11 cm above open hole water level) are shown. Based on this methodology, Day-Lewis et al. (2011) presented a 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 the borehole's water level and far-field heads.

This communication presents the possibilities of the flowmeter log providing a hydrogeological explanation for the described anomalous cases. With this aim, in this work a method has been developed that uses flowmeter logs to provide information regarding different hydraulic heads in a multilayer basin. Determining these different hydraulic heads allows hydraulic reinterpretation that explains the abovementioned anomalous behaviours of the pumping test results.

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 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 to differentiate sets of consecutive screens located at depths of the more permeable units (aquifers), among which there are other stretches mainly corresponding to less permeable units (aquitards). 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 to designate both the flow stretches in the well as well as the sets of 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 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), whose 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.

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 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 cm2) 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 cm2) located in front of sands whose permeability is 100 lower. By isolating each layer, the static and dynamic water levels may be 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 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 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=(ΔQj/QP)⋅(Δzj/b)⋅KP (Kabala, 1994), where Qj is the water input at layer j, QP is the extraction rate of the well, b is the aquifer thickness, and KP 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-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, 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, for a certain type of aquifer, its radius of influence varies by only a few hundred metres around a mean value of approximately 1000 m (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 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 for determining 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 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 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. 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.

To estimate the hydraulic parameters from the flowmeter logs, once they have been processed, two specific approaches 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 us to know the existence of hydraulic heads that are different for each stretch. This distinction implies changes in the effective drawdown of each stretch, which justifies, as shown in the case study, the water inputs of the deeper aquifer stretches not being proportional to drawdown.

In most works on hydraulic interpretation of flowmeter logs, a unique hydraulic head given by the static level HSL of the entire well is considered (Molz et al., 1989; Rehfeldt et al., 1992; Ruud and Kabala, 1996; Zlotnik and Zurbuchen, 2003a; Barahona-Palomo, et al., 2011; Riva et al., 2012). Thus, the drawdown used in the Thiem (1906) equation is the same for all of the aquifer stretches in a well, d0(s)=hDL(s)-HSL, where hDL(s) is the dynamic level for the “s” pumping step and HSL is the dynamic level of the entire well. However, under the hypothesis presented in this work, the hydraulic head of each stretch, and therefore the corresponding drawdown, can be different. Numerically, the drawdown of each flow stretch TN will be given by the following relation: 8dN(s)=hDL(s)-hSL(N), where hSL(n) is the static level for flow stretch TN. In short, the proposed method consists of replacing the single drawdown d in Eq. (2) from Rehfeldt with a drawdown for each stretch.

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 DeWiest relationship (1966) relation for transmissivity, given that the thicknesses of the layers and the productive sections are taken into account. The advantage of this option is that it is not necessary to know the storage coefficient of each contribution interval studied. 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 behaviour of the aquifer sections, and each section is treated separately.

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, qN(s) versus dN(s), reaches the maximum regression coefficient (where qN(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 influence is not evaluated in this work because the average hydraulic conductivity of each flow stretch compensates for them.

Regarding the linearity predicted by Darcy's law, in this work, the variation corresponding to non-linear flow is a different process than the change in the flow from a laminar to turbulent regime. Takhanov (2011) determined that the onset of non-linear 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 non-linear in the gravel pack and only becomes turbulent on the screen and inside the pipe. In this work, the regime change is less gradual than that predicted by the Forchheimer equation (1901) and does not increase with the same power after the transition. It should be taken into account that both the values of the friction factor and the particle Reynolds number established for the Forchheimer flow decrease as Re increases. To analyse the flow linearity in water wells, the hydraulic characteristics of the flow in the aquifer levels must be quantifiable by the velocity near the well. Moreover, a value of 0.01 m s-1 can be considered the maximum velocity in groundwater near wells. For example, if a water well with a very high extraction rate of 100 l s-1, radius of 0.2 m, total length of 400 m, and screened length of 80 m (20 %) is considered, then the average water input velocity would be 0.005 m s-1. Considering the results in van Lopik et al. (2017), non-linear behaviour starts at velocities greater than 0.01 m s-1, so the maximum water input still exhibits Darcian behaviour.

Among the possible explanations for the difference in hydraulic head values for the deeper flow stretches are hydrogeological 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 (T2 and T4) contradict this hypothesis, since flow stretch T2 is less permeable than flow stretch T4, but flow stretch T3 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 T4 and T5, thereby producing a drop in the static level of these flow stretches.

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 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.

Under the consideration that vertically there is a high hydraulic connection (similar to the 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 the 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 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, 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 Q2 is the turbulent flow 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.

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; Brill and Arirachakaran, 1992; Kabir and Hasan, 2006; 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.

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 north-western 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, as 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 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 us to carry out the hydrological and hydraulic division of the studied basin that had not been done before, and such division involves more precise obtaining of the permeability values in each stretch (and hence in its corresponding aquifer), which was not been done before. Certainly, the new procedure developed to obtain the hydraulic head 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, with 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 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 well logs; see Fig. 5) that are hydraulically separated from the rest of the aquifers.

The described results can be added to the conceptual models developed both from Toth's (1962) scheme (López Vera et al., 1977; Navarro et al., 1993; Heredia et al., 2001; Martínez-Santos et al., 2010) and from the more recent models on the flow model considering the land subsidence obtained by the A-DInSAR technique (Ezquerro et al., 2014; Béjar-Pizarro et al., 2017; Boni et al., 2020). It should be emphasized that the hydrogeological hypotheses that can be made as the scheme included in Fig. 11 must be contrasted with results in more wells within the siliciclastic basin of Madrid. The usual sub-division of this basin (which forms part of the Tagus basin) is into the lower and upper formations, whose contact according to the hydrogeological map (scale 1:20000, IGME, 1991) of the Spanish Geological Survey is “gradual and arbitrary” and therefore does not provide information on their depth. However, according to the correlation sections of the well logs shown in Caparrini (2006), the bottom of the coarser-grained upper formation is located above the depths analysed in the studied well. It should be noted that the same fact occurs with the hydrogeological models of the Madrid basin, in which the variations in the water table are far above the upper depth analysed in the studied well. In this sense, the hypothesis that can be put forward on the basis of the data from the analysed well is that, within a radius of 10 km around the well, a hydraulic differentiation must be considered from a depth of ∼ 200 m onwards.

The division of the studied well also allows us to propose a strategy regarding the arsenic propagation in the Madrid basin. The obtained results indicate that the stretch of the studied well that is “activated” when the dynamic level exceeds the “hydraulic head” of the aquifer to which it corresponds is 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 not be 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 stretches should have the same hydraulic head.

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 us to reinterpret the hydraulic behaviour of any well in which the 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.

The verification of the existence of different hydraulic heads for the different stretches with depth entails a substantial change 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 characterize parts of the basin.

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 the stretches with different hydraulic heads and to determine the spatial extension of this behaviour.