A quantitative analysis of hydraulic interaction processes in stream-aquifer systems

The hydraulic relationship between the stream and aquifer can be altered from hydraulic connection to disconnection when the pumping rate exceeds the maximum seepage flux of the streambed. This study proposes to quantitatively analyze the physical processes of stream-aquifer systems from connection to disconnection. A free water table equation is adopted to clarify under what conditions a stream starts to separate hydraulically from an aquifer. Both the theoretical analysis and laboratory tests have demonstrated that the hydraulic connectedness of the stream-aquifer system can reach a critical disconnection state when the horizontal hydraulic gradient at the free water surface is equal to zero and the vertical is equal to 1. A boundary-value problem for movement of the critical point of disconnection is established for an analytical solution of the inverted water table movement beneath the stream. The result indicates that the maximum distance or thickness of the inverted water table is equal to the water depth in the stream, and at a steady state of disconnection, the maximum hydraulic gradient at the streambed center is 2. This study helps us to understand the hydraulic phenomena of water flow near streams and accurately assess surface water and groundwater resources.

This paper presents a mathematical description of the hydraulic processes of the stream-aquifer system with a groundwater pumping well near the stream in a symmetrical flow system. Our special focus is on the following issues: 1) under what conditions does the stream start to disconnect from the aquifer? 2) what happens between the stream and aquifer after the stream disconnects from the aquifer? 3) what equations can be used to appropriately describe the movement of the critical point of disconnection below the streambed? 4) what hydraulic gradients at the center of the streambed can be reached at a steady state after disconnection? Thereafter, the application of the numerical method is presented for a free water-table equation and the analytic approach of the boundary-value problem of the inverted water table is discussed to explore the specific hydraulic processes. Finally, a validation of the mathematical analysis is conducted against an experimental result to summarize and conclude this study.

Results
This study examined the physical processes of the hydraulic connectedness of the stream-aquifer systems from connection to disconnection due to groundwater pumping (or drainage) near the stream by using the analytic numerical method combined with the sandbox experiments. The main results from this study are as follows: (1) When a stream disconnects from aquifers, a stream-aquifer system consists of the stream, inverted water table beneath stream, unsaturated zone, and saturated groundwater zone. The water table drawdown due to intensive pumping (or drainage) near the stream can potentially reduce saturations in the aquifer between the well and stream. The relationship between the stream and aquifer would evolve from the hydraulic connection to disconnection if the pumping intensity exceeds the maximum seepage capacity of the stream under a given streambed condition. (2) When hydraulic heads at the free water surface satisfy = are hydraulic gradients in the horizontal (x) and vertical (z) directions, respectively) for a symmetrical stream-aquifer system, the hydraulic connectedness between the stream and aquifer is at a critical disconnection state.
(3) From the simplified boundary-value problem of the inverted water table movement, we deduce that the maximum distance (or vertical length) of the inverted water table beneath the stream is the same as the water depth in the stream. The maximum rate of stream infiltration (per unit time per area) is equal to two times of the hydraulic conductivity of the streambed. This result implies that the maximum hydraulic gradient at the streambed center is 2 when the disconnection occurs. (4) The results of the sandbox experiment, a recent field test and the finite analytic numerical solution are consistent with the mathematical analysis results of the hydraulic disconnection processes. These results will help us to understand the hydraulic phenomena of variably saturated flow near streams and to accurately assess surface water and groundwater resources.

Discussion
The evolution of hydraulic gradients in the center of the streambed from connection to disconnection. To derive our mathematical analysis methods, we develop a lab-scale stream-aquifer system as an example for explaining the evolution of hydraulic gradients under the streambed (Fig. 1), where the hydraulic relationship of the stream-aquifer system evolves from connection to disconnection. The flow domain is 3 m in length, 2 m in height and 1 m in width. The stream is 0.2 m in width and is located along the center line of the flow domain. The water depth in the stream is 0.1 m. There are two ditches (which function equivalently as two pumping wells) located equidistant 1.4 m from the stream bank on each side. The stream stage is initially the same as the water level in both ditches. When the water levels in both ditches were suddenly declined to 0.7 m and then remained constant during the period of simulation, the stream-aquifer system were evaluated from connection to disconnection and finally reached a steady state. The porous medium in the aquifer is silt-fine sand and the corresponding parameters for the unsaturated water flow are summarized in Table 1. The sediments of streambed are the same as the medium in the aquifer. A finite analytic numerical method developed by Dai et al. 27 , Wang et al. 28 , and Zhang et al. 29 for the saturated-unsaturated flow simulation was adopted in the validation. Relative conductivity functions were broadly used by many investigators to simulate water flow in variably saturated zones [30][31][32][33][34] . For simplicity, here, we assume that the soil hydraulic conductivity follows the exponential relative conductivity model 35 .
where, h is the soil-water pressure head (m); K(h) is the unsaturated hydraulic conductivity (md −1 ); K s is the saturated hydraulic conductivity (md −1 ); β is an arbitrary constant. The value of β can be estimated through curve fitting of experimental data, and it is 6.6 in this study. The ground surface boundary condition of two sides of the stream was assumed to be the boundary of the prescribed negative pressure head with four scenarios. The first scenario assigns a pressure of − 2 m while the second and third scenarios are − 1 m and − 0.5 m, respectively. In the fourth scenario, the pressure of the ground surface for two sides of stream changes with a uncertain interval between 0 m at stream bank and − 2 m at the ground surface edge of two sides of the stream.
The boundary condition of the stream is a prescribed pressure head, which is the same as the water depth in the stream. The boundary conditions of the saturated zones on both sides of the ditches are also a prescribed pressure head, which is equal to the actual water head in the ditches. Accordingly, the boundary condition of the unsaturated zones on both sides of the ditches from the ground to the water surface in the ditches is a prescribed negative pressure head, which is given by linear interpolation between negative pressure of the ground and zero pressure. Figure 2a illustrates the evolution of the pressure water head and total water head of seepage field with time for the ground surface boundary condition of 2 m negative pressure simulated by the finite analytic numerical method of the saturated-unsaturated flow. Figure 2b shows the simulated patterns of the pressure water head and total water head of seepage field at steady-state seepage in scenarios 1, 2, 3, and 4 of the ground surface boundary conditions.
The numerical simulation results computed from the saturated-unsaturated flow model [27][28][29] clearly show that, as long as the rate of drainage in the ditches exceeds the seepage capacity of the stream under a given streambed condition, the relationship between the stream and aquifer can be evolved from the hydraulic connection to disconnection (Fig. 3). The results of the numerical simulations are consistent with those of the laboratory experiments 18 . This demonstrates that the numerical approach proposed in this study can be used to describe the stream-aquifer relationship, which includes three hydrologic steps from connection to disconnection: connection ( Fig. 2a,b), critical disconnection (Fig. 2c), and entire disconnection (Fig. 2e,f). Note that Lamontagne et al. 15 developed a nomogram to estimate the height of the groundwater mound without the limiting assumption of horizontal flow. With their approach, the steep gradients right in the center of the mound can be simulated and their results are the same as those shown in Fig. 2c,d,f in this study. By inspecting the flow patterns of the stream-aquifer system for the four ground surface boundary conditions in Fig. 3a,d, we can see that the boundary conditions have a great impact not only on the flow patterns or hydraulic gradients of the stream-aquifer system, but also on the river recharge to aquifer. The hydraulic gradients at the streambed can be used to assess its influence on the stream recharge to the underlying aquifer. Figure 4a describes the variations of the hydraulic gradients at the center of the streambed with stream water depth of 0.1 m, ditch discharge level of 0.7 m, and four different ground surface boundary conditions. By assuming a ground surface pressure of − 2 m, we analyze the change in the hydraulic gradients from the hydraulic connection to disconnection. Figure 4a shows that the hydraulic gradient quickly increases during the period of 0 to 0.0198 d. During that time the stream keeps connected with the aquifer. When time reaches 0.0198 d, the relationship between stream and aquifer becomes the critical disconnection state. At that time the curve of the hydraulic gradient shows an obvious turning point. From 0.0198 to 0.05 d the stream disconnects from the aquifer, and the hydraulic gradient increases slightly. From 0.05 to 0.125 d the hydraulic gradient shows very little change. After 0.125 d the flow of the stream-aquifer system reaches to a steady state, and the hydraulic gradient reaches to the maximum value (approximately 2 at the center of the streambed). Similar trend in the hydraulic gradient variation for the ground surface pressure of − 1 m can be seen in Fig. 4a. When the ground surface pressure is assumed to be − 0.5 m, the hydraulic gradient at the center of streambed is slightly less than 2 after disconnection. The reason is that there is some unsaturated water flowing into the stream-aquifer system from the ground surface boundary (see Fig. 3c). This reduces the river recharge to aquifer compared with the cases which have higher negative pressures. Therefore, the less of the negative pressure on the ground surface boundary, the more water flows into the aquifer from the ground surface boundary and much less hydraulic gradient at the center of streambed after disconnection (see Figs 3c,d and 4a). More significantly, the maximum capacity of stream recharge to aquifer per unit area per time is no more than K 2 (K is the hydraulic conductivity of the streambed) when the stream disconnects from aquifer at the steady state. This result is consistent with that obtained by the analytic method.
Medium type Residual water content (θr) Saturated water content (θs) Saturated hydraulic conductivity k s (cm/h) The hydraulic gradient variation with time on the symmetrical line. Figure 4b  As can be seen from Fig. 4b, no matter the stream is disconnected with aquifer or not, the time-variation of the hydraulic gradient on the symmetrical line from streambed center to impermeable base can be divided into three zones. The first zone (termed as the inverted water table zone) is located at the certain depth below streambed (see Fig. 4b section AB). The essential features of the first zone are: 1) The depth of the zone is nearly equal to stream water depth; 2) The time-variation of the hydraulic gradient shows a slightly growing trend from connection to disconnection; 3) The vertical hydraulic gradient is linear with the vertical coordinate distance z; 4) The vertical hydraulic gradient at the streambed center reaches 2 at the steady state after disconnection.
The second zone (termed as the saturated zone) is located at the certain height above the impermeable base (see Fig. 4b section C′ D). The time-variation of vertical hydraulic gradient shows a slightly decreasing trend from connection to disconnection. The vertical hydraulic gradient is also linear with the vertical coordinate distance z, but with moderated slopes compared with that of the first zone. The third zone (termed as the unsaturated zone) is located between the first and the second zones (see Fig. 4b section BC and BC′ ). There are several essential features in the seepage flow: 1) The vertical gradient has bigger amplitude of variation from connection to disconnection, especially under disconnection condition; 2) When stream disconnects from aquifer, there exists an unsaturated zone between the inverted zone and the saturation zone. Unsaturated flow in this zone dominates in the flow and transport processes of the stream-aquifer system. The nonlinear flow in the unsaturated area depends on the relative conductivity functions which are variable with moisture contents and hydraulic pressures. This may be the reason to cause a larger variation amplitude of the vertical hydraulic gradient in the unsaturated zone than that in the saturated zone; 3) The vertical hydraulic gradient in this zone is nonlinear with the vertical coordinate distance z.
The variation of vertical hydraulic gradient at the free water surface of the regional water table. Figure 5a  for a symmetrical stream-aquifer system, the hydraulic connectedness of the stream-aquifer reaches the critical disconnection state. Figure 5b,c show that the maximum thickness of the inverted water table zone beneath the stream varies with the water depth in the stream at the steady state for entire disconnection by using the laboratory sandbox experiment and finite analytic numerical method, respectively. It can be seen that the higher the stream stage, the thicker the inverted water table zone. In addition, the slopes of the curves in Fig. 5b,c are nearly equal to 1. This implies that the maximum thickness of the inverted water table zone is the same as the water depth in the stream at the steady state for entire disconnection, which agrees with the above conclusion obtained by the analytic method.

The maximum thickness of the inverted water table zone beneath the stream.
In addition, a recent field test in the Ordos Basin, China, was designed to see if there is an inverted water table beneath the stream, and if its maximum thickness is equal to the water depth in the stream (at the steady state under the condition of streambed sediments being the same as the underlying aquifer materials). A site was selected in an area with uniform fine sand in the unsaturated zone, where the depth of groundwater is about 8 m according to observation wells near the testing site. A quadrate test tank (1 × 1 × 1 m 3 ) held a bed of uniform fine sand. Three boreholes with a diameter of 40 mm located at the center line of the tank, 0.3 m and 0.5 m from another side of the tank, respectively, were drilled from the ground surface to 2.7 m deep. These boreholes were used to measure moisture contents by TDR during the test period. Several overflow holes are designed on one side of the tank, which are used to keep the artificial stream stage at a constant level with the tank. The height from the tank bottom (similar to streambed surface) to the overflow hole represents the simulated water depth in the stream. The moisture contents were firstly measured as the initial condition. Then water is added to the tank at a sufficient large rate so that the water in the tank is kept at a constant level (0.4 m in this test). The saturated water content of fine sand is 0.322 cm 3 /cm 3 . The moisture contents were measured from three holes at 10-20 minutes intervals over the entire test period. The test was stopped until the steady state was reached. A sketch of the field test is shown in Fig. 6(a,b). Figure 6c shows the distributions of the moisture contents in the three boreholes at different depths. It can be seen that there exists a saturated zone below the tank bottom in the centre borehole at the steady state. The thickness of the saturated zone below the tank bottom in the centre borehole is almost equal to the water depth in the tank, which validates the result of Eq. (19) in the method section. This is equivalent to the maximum hydraulic gradient with a value of 2 at the streambed under steady-state condition for the disconnected stream. The measured moisture contents in another two boreholes remain no change during the test.

Methods
The prerequisite of the total water head at the water table for the critical disconnection condition.  and Wang et al. 18 have discussed the hydrogeological controls of disconnection and transient effect during the transition from connection to disconnection in great detail. The point when the water-table curve starts to divide into two parts can be termed as the critical disconnection point. At this point, the water-table curve converges at a spot P 0 (see Fig. 7a) on the central axis below the streambed for a symmetrical flow system. There are two water-table curves at the critical disconnection point (P 0 ) in the vertical profile. One is located above P 0 spot which is called the inverted water table 18 , and the other is called the regional water table which is located below P 0 spot. Actually, the differences of the different water table curves are mainly the variations of the hydraulic gradients along the water table curves, which cause changes in the geometric shape of water table curves.
From the theory of the differential geometry, P 0 spot at the critical disconnection state is a singular point on the vertical profile of the water table curve. As long as the first and second-order derivatives of the curve exist, the prerequisite can be obtained while the horizontal and vertical hydraulic gradients of the water table curve in the vertical profile reaches certain values at the critical disconnection state.
The free water surface equation can be used to estimate the prerequisite when P 0 is reached. The equation for an isotropic, homogeneous, unconfined aquifer in the vertical profile at the steady state can be expressed as 36  at the free water surface of regional water table on the symmetrical line from the critical disconnection to entire disconnection (a), the relationship between the maximum thickness of inverted water table zone below streambed and the water depth in stream at steady state for entire disconnection observed by the laboratory sandbox experiment (b), and by the finite analytic numerical method (c). varies with the discharge level in the ditch, the shape of water table curve changes accordingly. In addition, Eq. (3) can also be used to determine a vertical water exchange, i.e. ( , ) W x z of the free water surface at steady state in the ditch. Actually, ( , ) W x z is the water balance at the water table and Eq. (4) can be used to analyze variations of the water table curves at different discharge levels in the ditch. Therefore, a new function of ( , ) f X Z is defined, which is the shape function of the water table curve in the − x z profile. It describes the shape of the water table curve at steady state for given boundary conditions in a flow domain. ( , ) f X Z can be written as groundwater table are tangent at P 0 point, the exchange flux between the two free water tables is equal to ∂ ∂ K H z . This is consistent with the Darcy's law. Above mathematical derivations demonstrate that the hydraulic connectedness of the stream-aquifer system is at the critical disconnection point when the hydraulic head at the water for a symmetrical stream-aquifer system, in which the aquifer is an isotropic and homogeneous system with two ditches (or pumping wells) located equidistantly (L) on each side of the stream with an equal drainage (or pumping) rate.