WATER RESOURCES RESEARCH, VOL. 41, W12420, doi:10.1029/2004WR003877, 2005
Development of a simple lateral preferential flow model with steady state application in hillslope soils Daizo Tsutsumi Research Center for Fluvial and Coastal Disasters, Disaster Prevention Research Institute, Kyoto University, Uji, Japan
Roy C. Sidle Division of Geohazards, Disaster Prevention Research Institute, Kyoto University, Uji, Japan
Ken’ichiro Kosugi Division of Forest Science, Graduate School of Agriculture, Kyoto University, Kyoto, Japan Received 7 December 2004; revised 5 September 2005; accepted 16 September 2005; published 16 December 2005.
[1] A general model describing three-dimensional lateral preferential water flow in a
hillslope with soil pipes was developed. Matrix flow and pipe flow were regarded as separate flow systems and computed using the governing equations (Richards’ and Manning’s equations, respectively), while simultaneously considering the interaction between these two flow systems. The model accommodates both partially filled and full pipe flow, seepage into the pipe, and backflow from the pipe into the surrounding soil matrix. Simulations were conducted for conditions outlined in an earlier bench-scale experiment, including differing internal roughness configurations within the pipe. Both groundwater levels and preferential flow under steady state conditions were simulated for different roughness elements within the soil pipe; previous models have not simulated such conditions. Six different pipe arrangements were also simulated: no pipe (uniform matrix flow), single straight pipe, two sets of discontinuous pipes, branched pipes, and unopened pipe. Branched pipes had the highest discharge, and even the unopened pipe contributed to an enhanced total discharge compared to the no pipe simulation. These simulations demonstrated the versatility of the model under the steady state conditions, although the model can also be applied to transient conditions. Citation: Tsutsumi, D., R. C. Sidle, and K. Kosugi (2005), Development of a simple lateral preferential flow model with steady state application in hillslope soils, Water Resour. Res., 41, W12420, doi:10.1029/2004WR003877.
1. Introduction [2] Several studies have documented the existence of lateral preferential flow networks in hillslopes [Tsuboyama et al., 1994; Sidle et al., 2001; Holden and Burt, 2002], and some have revealed that soil pipes form complex interconnected networks, which may contain many breaks [Noguchi et al., 1999; Terajima et al., 2000; Sidle et al., 2001]. Water flow through preferential flow paths can contribute significantly to the rapid transfer of storm water [e.g., Mosely, 1982; Tsukamoto and Ohta, 1988; Kitahara et al., 1994; Sidle et al., 1995a, 2000], as well as to subsurface soil erosion [e.g., Jones, 1987; Onda, 1994; Bryan and Jones, 1997; Terajima et al., 1997; Uchida et al., 1999]. Consequently, preferential flow is recognized as an important hydrologic and geomorphic control at the hillslope scale, and the development of a theoretical method to analyze preferential flow processes is needed. However, few studies have modeled preferential flow in hillslopes, and there is no general simulation technique for water dynamics (including preferential pathways) at the hillslope scale. [3] Early models that quantified lateral pipe flow focused on the transmission of water through the pipe system using Copyright 2005 by the American Geophysical Union. 0043-1397/05/2004WR003877
hydraulic theory [Gilman and Newson, 1980; McCaig, 1983]. Gilman and Newson [1980] used data obtained from an artificial pump experiment to calculate the parameters of a kinematic subsurface flow model. McCaig [1983] used the Chezy (Manning’s) equation with some simplifying assumptions, such as a linear increase in discharge downslope, constant half-full pipe flow, and a roughness coefficient inversely proportional to the pipe radius. These approaches were analyzed and compared with field data by Jones [1988]. Barcelo and Nieber [1981, 1982] proposed a model that simulated matrix flow using the two-dimensional Richards’ equation, with the presence of a pipe or pipe network. This model also calculated the flux into the pipe using an equation given by Kirkham [1949], assuming a constant pressure head in the pipe. The simulated hydrographs of both matrix and pipe flow discharge demonstrated that the contribution of pipe flow to the storm hydrograph and to the distribution of soil moisture in the hillslope was significant. Nieber and Warner [1991] extended the earlier model by Barcelo and Nieber [1981, 1982] to three dimensions and calculated the flux into the pipe using an equivalent drain technique [Vimoke et al., 1963] that assumes a constant pressure head in the pipe and fully saturated soil. They examined the contribution of the pipe to total discharge by changing factors such as pipe length,
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[6] Therefore this study modeled the three-dimensional lateral preferential water flow in a hillslope with a soil pipe with partially filled and full pipe flow. The model covers both seepage into the pipe from a water table and backflow from the pipe into the unsaturated soil. It was tested using the bench-scale pipe flow experiments of Sidle et al. [1995b]. In addition, simulations were conducted to demonstrate that the new model is applicable to complex, realistic macropore networks, including the discontinuous, branched, and unopened pipes that are common in hillslopes.
2. Model Description
Figure 1. General concept of the model. Pipe flow is categorized into three types: no (I), partially filled (II), and full (III) pipe flow. Sp [m3/s] indicates source or sink. pipe depth, pipe radius, slope length, slope angle, and pipe spacing. A recent model by Jones and Connelly [2002] divided the sources of pipe flow into several types, with each type assigned an equation derived from monitored hydrological data. In their model, the pipe is considered to be partially full and the water level in the pipe is set according to the monitored phreatic level near the pipe. The model parameters were calibrated using storm events observed in an experimental catchment and were used to predict the pipe flow hydrograph caused by several other storm events. [4] All of the previous models, except that of Jones and Connelly [2002], assumed that the water level and pressure head in the pipe were constant. For actual pipe flow, however, the water level at any point in a pipe depends on how much water flows from upslope and from the soil matrix around the pipe, as well as the water transmission capacity of the pipe. If the pipe flow rate exceeds the transmission capacity, the pipe becomes full and generates a pressure potential gradient. Under such conditions, both pipe flow and seepage into the pipe depend on the pressure potential along the pipe. [5] Actual soil pipes have a wide variety of diameters depending on climate, vegetation, soil type, and the formation factors [Beven and Germann, 1982; Noguchi et al., 1997]. Although the assumption of partially filled flow within the pipe might be appropriate when the pipe is relatively large [e.g., Jones and Connelly, 2002], smaller pipes (less than or equal to several centimeters in diameter) are readily filled. A study using a fiber scope to examine the flow conditions in soil pipes demonstrated that both full and partially filled conditions occurred simultaneously within the same soil pipe [Terajima et al., 2000]. Bench-scale experiments also demonstrated these two pipe flow conditions [Uchida et al., 1995; Sidle et al., 1995b]. Moreover, backflow occurs from the full flowing pipe into the surrounding soil and pore water pressure increases relative to that in conditions without a pipe [Uchida et al., 1995]. Similarly, backflow might be generated in a pipe with a necking point [Beven and Germann, 1982]. Backflow from pipes into the soil is a major trigger mechanism of landslides [Ohta et al., 1981; Pierson, 1983; Sidle et al., 2000].
2.1. General Concept [7] This section presents the equations for the matrix flow and pipe flow together with the boundary conditions. The cases of no pipe flow, partially filled pipe flow, and full pipe flow are discussed separately. Figure 1 shows the general concept of the model regarding matrix flow and pipe flow. The groundwater level is shown in Figure 1 to clarify different pipe flow types; however, the water level in the pipe is not necessarily the same as the groundwater level. As is shown later in section 2.4, the pressure potential at the pipe wall determines pipe flow generation. 2.2. Soil Matrix Flow [8] Richards’ equation was solved numerically using a finite element method [Zienkiewicz, 1971; Istok, 1989] to predict the three-dimensional matrix flow within the soil. Following the model of Barcelo and Nieber [1981], the pipe is assumed to be a line of negligible volume aligned with nodes in the grid (Figure 2); this avoids increasing node density and the resulting computation time and also simplifies the creation of the finite element grid and makes the model more applicable to complicated preferential flow systems. The three-dimensional version of Richards’ equation based on the pressure potential y [m] is C ðyÞ
@y @ @y @ @y ¼ K ðyÞ þ K ðyÞ @t @x @x @y @y @ @y K ðyÞ þ1 þ @z @z
ð1Þ
where z [m] is positive upward, t [s] is time, C(y) [m1] is the soil water capacity, and K(y) [m/s] is the hydraulic conductivity. Some studies have noted mass balance problems in standard y-based numerical solutions of equation (1) [Milly, 1985; Celia et al., 1987]. Since our model will be used for transient flow, it is important to pay attention to this problem, although our application of this model focuses on steady state flow. Our application uses a partially implicit Picard iteration method and time steps that varied depending on the convergence of each iteration (the largest step was 2 sec). To represent C(y) and K(y) for unsaturated conditions (y < 0), the lognormal model proposed by Kosugi [1996] was used:
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( ) dq qs qr ½lnðy=ym Þ2 C ðyÞ ¼ exp ¼ pffiffiffiffiffiffi 2s2 dy 2psðyÞ
ð2Þ
2 lnðy=ym Þ 1=2 lnðy=ym Þ FRND þs K ðyÞ ¼ Ks FRND s s ð3Þ
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Figure 2. Soil domain divided by the finite element grid: (a) front view on the y-z plane, (b) side view on the x-z plane, and (c) 6 tetrahedral finite elements derived from a parallelepiped unit domain. The pipe is indicated by a circle in Figure 2a and a dashed line in Figure 2b. The arrow in Figure 2b indicates the one-dimensional coordinate l [m] along the pipe. The gray shading in the lowermost part in Figure 2b indicates finite elements in which Ks was reduced by a factor of 3.3 to simulate flow resistance at the lower end boundary (simulation 2 in Table 2). where, qs [m3/m3] is the saturated soil water content, qr [m3/ m3] is the residual soil water content, ym [m] is the pressure potential corresponding to the median soil pore radius, s is a dimensionless parameter related to the width of the pore size distribution, and Ks [m/s] is the saturated hydraulic conductivity. The function FRND(x) represents the residual normal distribution and is represented as Z
1
FRND ð xÞ ¼ x
2 1 u pffiffiffiffiffiffi exp du: 2 2p
ð4Þ
For saturated conditions (y 0), K(y) = Ks and C(y) = 0.0 were used. For fully saturated conditions, the left hand side of equation (1) is zero for the entire soil domain, which denotes steady state flow conditions. For variably saturated conditions, the left hand side of equation (1) is also zero in the saturated portion. However, since transient flow in the unsaturated portion affects the saturated domain, changes in pressure potential in the saturated part can be uniquely determined from the partially implicit Picard iteration method used in the solution of equation (1). The stable solution of Richards’ equation under such variably saturated flow conditions was demonstrated by Ohno et al. [1998]. In our applications, also, no instability or oscillation of the solution occurred during variably saturated flow condition.
[9] A finite element grid with 462 nodes and 1800 elements was used in the calculations, for the purpose of simulating the bench-scale experiment described in section 3. The soil domain divided by the finite element mesh and the method for establishing the tetrahedral finite elements are shown in Figure 2. In Figure 2, x, y, z axes for the spatial rectangular coordinate, x0 axis along the slope length, and z0 axis along the sloping box height at the upslope end are defined. Several types of boundary conditions were imposed within the soil domain to calculate matrix flow. Along the sides, bottom, and surface boundaries, zero flux boundary conditions were imposed. Since the water level zw0 [m] was set on the upslope boundary in the bench-scale experiment described in section 3, zero flux boundary conditions were applied to the portion above the water level (z0 > zw0 ), and a Dirichlet boundary condition imposing static water pressure was assumed y ¼ z0w z0 cos as
ð5Þ
below the water level (z0 z0w), where as [degrees] is the slope angle. At the downstream boundary, seepage face boundary conditions of zero flux above the variable water
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table and a Dirichlet boundary condition setting y = 0 below the variable water table, were imposed. [10] Along the pipe, different boundary conditions were imposed according to the categorization of the pipe flow as no, partially filled, or full pipe flow. When the soil is completely unsaturated around the pipe, which was categorized as no pipe flow, the soil matrix flow was computed as if no pipe boundary existed. For the soil around a partially filled pipe, the Dirichlet boundary condition was imposed (y = 0, assuming the ‘‘open channel flow’’ condition), and the seepage from the surrounding soil matrix into the pipe was computed. This approach is similar to that of Nieber and Warner’s [1991] model, based on the work of Fipps et al. [1986]. For the soil around the full flowing pipe, the Neumann boundary condition giving the flux Sp [m3/s] was imposed; by imposing the Neumann boundary condition along the pipe, the pressure potential was obtained. The flux Sp is specified by balancing the pipe flow rate shown in the next subsection. 2.3. Pipe Flow [11] For both partially filled and full pipe flow, the relationship between the pipe flow rate and flux into the pipe from the surrounding soil matrix is expressed as, Z
l
Q p ðl Þ ¼ 0
Sp0 ðuÞdu
ð6Þ
where Qp [m3/s] is the pipe flow rate, Sp0 [m3/s/m] is the flux per unit pipe length, and l [m] is the length along the pipe (positive from upstream to downstream). To compute the pipe flow numerically, the pipe was divided into np segments. The ends of the segments correspond to the nodes used in the finite element calculation of the soil matrix flow (Figure 2). Modifying equation (6), the flow rate within the kth segment (Qp,k) is expressed as
Qp;k ¼
k X
Sp;i
ð7Þ
i¼1
where, Sp,i [m3/s] is the seepage (or backflow) at the node corresponding to the upper end of the ith segment. As with the soil matrix flow, the pipe flow was calculated for three different types of flow within the pipe: (1) no, (2) partially filled, and (3) full pipe flow. 2.3.1. No Pipe Flow [12] Since no water flows within the pipe, it is not necessary to consider the interaction between pipe flow and soil matrix flow. 2.3.2. Partially Filled Pipe Flow [13] Seepage from the soil matrix into the pipe was obtained by imposing the Dirichlet boundary condition to the pipe boundary. The pipe flow rate was computed by summing the seepage from the upper point where pipe flow appears, using equation (7). This method of calculating the flow rate within the pipe is similar to that used by Nieber and Warner [1991] and Jones and Connelly [2002], although our method of calculating seepage from the soil matrix into the pipe is different. The concept of partially filled pipe flow within the divided segments is summarized in Figure 3a.
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2.3.3. Full Pipe Flow [14] In one of the very few studies of the hydraulic properties of pipe flow, Kitahara [1989] collected soil blocks from the field containing intact soil pipes, and estimated the hydraulic properties of full flowing pipes in a laboratory experiment. Kitahara [1989] concluded that pipe flow could be computed using Manning’s equation using a range of roughness coefficients (0.036 to 1.364). In our model, full pipe flow was calculated using Manning’s equation following the experimental results of Kitahara [1989]. The flow rate within the pipe, Qp, is expressed as Qp ¼
1 1 2=3 df =2 R A nm dl
ð8Þ
where nm [m1/3s] is the roughness coefficient, R [m] is the hydraulic radius, f (= y + z) [m] is the hydraulic potential, and A [m2] is the cross-sectional area of the pipe. In the actual model calculations, the flow rate within individual pipe segments, Qp,k, is computed as Qp;k
1 1 2=3 fp;kþ1 fp;k =2 ¼ R A lp;k nm
ð9Þ
Although we assumed a cylindrical pipe, it is possible to simulate various field pipe configurations by simply modifying the cross-sectional area A and the hydraulic radius R. [15] By modifying equation (7), the flux Sp,k at a node, which is the junction of pipe segment k, is specified as Sp;k ¼ Qp;k Qp;k1
ð10Þ
Sp,k is applied to the node as a point source or sink in the soil matrix flow calculation shown in the previous subsection. According to equation (10), for increasing flows (Qp,k > Qp,k1), seepage occurs from the soil matrix into the pipe as a point sink (Sp,k > 0). For decreasing flows (Qp,k < Qp,k1), backflow occurs from the pipe to the soil matrix as a point source (Sp,k < 0). The concept of full pipe flow within the divided pipe segments is summarized in Figure 3b. 2.4. Calculation Procedure [16] The calculation procedure is summarized as a flowchart in Figure 4. An integer, Tk, was defined to classify the pipe flow type into no (Tk = 1), partially filled, (Tk = 2), and full (Tk = 3) flow. After data input, the distribution of the pressure potential y(1) was calculated with the initial and S(0) assumptions of T(0) k p,k for each segment k (=1, 2, . . ., np). The pipe flow rate Q(1) p,k was calculated using the result of the soil matrix flow calculation. The subroutine ‘‘pipe flow categorization’’ (Figure 5) determined the pipe flow type T(1) k . Note that the pressure potential used to determine pipe flow generation in this subroutine is not that of the surrounding matrix, but is that of the pipe wall (the nodes along the pipe line). If the newly categorized pipe (0) flow type T(1) k differed from the former Tk , the procedure described in Figure 4 was repeated using T(1) k . When segment k is categorized as ‘‘flowing full’’, the values of S(1) p,k obtained from equation (10) must theoretically be (0) identical to the assumed values of S(0) p,k. If the assumed Sp,k
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Figure 3. Concept of the pipe flow within a divided segment. (a) Partially filled flow. Sp,n was determined in the soil matrix flow calculation specifying the pressure potential (y = 0) and then integrated to obtain Qp,n. (b) Full pipe flow. Qp,n was calculated using Manning’s equation, which is a function of the potential gradient; then the flow differences between adjacent pipe segments were used to determine the flux from the soil matrix to the pipe or from the pipe to the soil matrix (if Qp,k > Qp,k1, then the flow difference Sp,k is a sink term, and if Qp,k < Qp,k+1, then Sp,k+1 is a source term in the soil matrix flow calculation). (0) is inappropriate, S(1) p,k is not identical to Sp,k. To establish the correct solution for Sp as well as for Qp and y, Sp,k was then assumed to be ðnÞ
ðn1Þ
Sp;k ¼ ð1 lÞSp;k
ðnÞ
þ lSp;k
ð11Þ
where l is a ratio that determines the contribution of S(n) p,k and S(n1) p,k , n is the iteration number, and the procedure was (n1) 3 repeated until jS(n) p,k Sp,k j < e1; where e1 [m /s] is an acceptable error. The calculation was continued until the distribution of the soil pressure potential, y, reached a steady state (jy(n) y(n1)j < e2; where e2 [m] is an acceptable error. For these calculations, values of l = 0.001, e1 = 108 m3/s, and e2 = 105 m were used.
3. Previous Experimental Study [17] Sidle et al. [1995b] conducted a bench-scale experiment in a sloping box (1.0 m long, 0.6 m wide, 0.4 m deep, with slope angle as = 12.8, Figure 6) filled with uniform
sand to evaluate the effect of pipe flow on the overall hydrologic regime. Five artificial pipes, 0.2 m in length, 13 mm ID, and 18 mm OD, were constructed. The inner perimeter of the pipes was coated with different amounts of glass beads (average diameter 2 mm) to adjust the coefficient of roughness (Manning’s nm) of the individual pipes. The roughness nm of each pipe was determined in hydraulic experiments (for segments 1 to 5, nm,1 = 0.158, nm,2 = 0.049, nm,3 = 0.042, nm,4 = 0.044, nm,5 = 0.325). Each pipe was wrapped with three layers of polyester felt. In each of the pipe flow tests, five 0.2-m sections of pipe, each with a different nm value, were arranged in various spatial combinations for a total pipe length of 1.0 m. The composite pipe was placed 0.05 m above the bottom and in the middle of the box, and was oriented perpendicular to the slope contours, and parallel to the sidewalls of the box. The box was filled with sand (Ks = 0.000314 m/s) to a depth of 0.4 m. The parameters in equations (2) and (3) were determined from an observed water retention curve for the Toyoura standard sandy soil (qs = 0.368 m3/m3, qr =
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Figure 4. Flowchart of the overall calculation procedure. Tf indicates the pipe flow type. The suffix (n) indicates the number of repetitions for convergence of the pipe flow Qp, and e1 and e2 are the acceptable errors of Qp and y, respectively. 0.044 m3/m3, ym = 0.525 m, s = 0.363), which is similar to the soil used in the experiment. Polyester felt was secured to the upper and lower faces of the soil block with wire mesh, and the upstream end of the composite experimental pipe was sealed. Six piezometers were installed at the following positions relative to the downslope end of the box: x0 = 0.0, 0.1, 0.2, 0.4, 0.6, and 0.8 m (Figure 6). Access holes for the piezometers were placed at the bottom and adjacent to the side of the box, about 0.29 m from the pipe. During the experiment, water was supplied to the upper end of the box at several constant water level conditions (zw0 ). Steady state piezometric levels along the length of the box,
pipe flow rate, matrix flow rate, and total discharge were measured for six to eight hydraulic conditions determined by the fixed water level at the upper end of the box (Figure 6). [18] Two experiments (runs 1 and 2) were used to test the model. In run 1, the pipe segments were arranged in the order 1, 2, 3, 4, 5 from upstream to downstream; the pipe segment with the greatest roughness (segment 5, nm,5 = 0.325) was located at the lowest slope position (x0 = 0.0– 0.2 m from the downslope end). In run 2, the pipe segments were arranged in the order 1, 5, 2, 3, 4 from upstream to downstream; the segment with the greatest roughness (segment 5) was positioned next to the uppermost portion of
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Figure 5. Flowchart of the subroutine ‘‘pipe flow categorization.’’ The pipe flow types were represented by the indicator Tf, where (1) Tf = 1, no pipe flow; (2) Tf = 2, partially filled flow; and (3) Tf = 3, full flow. M(f) is the threshold value of the flow rate computed from Manning’s equation assuming a full pipe; Qp,k is as in equation (11). the composite pipe (x0 = 0.6 – 0.8 m from the downslope end). Table 1 shows the water levels, zw0 , imposed to the upslope end boundary; seven levels in run 1, and six in run 2. In the model simulation, the composite pipe was divided into 10 portions (np = 10) using finite element grids. The observed experimental results (runs 1 and 2 [Sidle et al., 1995b]) are compared with model simulations in section 6.
4. Method for Assessing the Model Performance [19] The criterion used to assess the performance of the model with regard to both piezometric profiles and flow rates followed that of Perrin et al. [2001]: n X
Vobs;i Vcal;i
i¼1 CR ¼ 1 X n i¼1
Vobs;i Vobs
2
2
ð12Þ
where, Vobs,i and Vcal,i are the observed and calculated values of the pressure potential at the measured point i for the piezometric profiles, respectively, or the flow rates for each water level condition i at the upstream end, and Vobs is the averaged value of Vobs,i. This statistical measure varies from 1 to 1.0, where 1.0 represents a perfect agreement. If Vcal is replaced by a linear regression, this measure represents the coefficient of determination r2. Perrin et al. [2001] tested the performance of 19 rainfall-runoff models simulating 429 catchments using the assessment criterion (equation (12)), and reported that the mean assessment criteria averaged over these catchments ranged from 0.393 to 0.512 with a minimum of 0.003 and a maximum of 0.845. Although these values were obtained from catchment-scale model simulations, they give a basis for assessing the performance of our model simulating benchscale experiment.
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Figure 6. Schematic of the bench-scale experiment: (a) side and (b) front views [after Sidle et al., 1995b].
5. Method for Simulating Various Pipe Configurations [20] Further simulations were conducted for six different pipe arrangements using the same setup and hydraulic properties as used in the bench-scale experiments of Sidle et al. [1995b]: (1) no pipe (uniform matrix flow), (2) straight pipe (one 1.0-m long pipe in center of the box; y = 0.3 m), (3) discontinuous pipe 1 (two 0.4-m pipes situated in the center of the box; y = 0.3 m), (4) discontinuous pipe 2 (two 0.6-m pipes shifted in the y direction; y = 0.2 and 0.4 m), (5) branched pipe (one 0.4-m pipe and two 0.68-m angled pipes connected at a point; x = 0.4 m, y = 0.3 m), and (6) a clogged pipe (one 0.9-m pipe whose outlet was located at a point inside the soil domain; x = 0.1 m, y = 0.3 m); these arrangements are illustrated in section 6.2. All the pipes were located 0.05 m above the impermeable layer. The pipe diameter and roughness coefficient were fixed at D = 0.013 m and nm = 0.05 m1/3s, respectively. The constant water level at the upslope end of the box was set at zw0 = 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, and 0.35 m; the water level zw0 was changed only after steady state conditions were obtained. For the branched pipe calculation, if the flow at the junction was categorized as partially filled, the pipe flow rate just downstream from the junction was obtained by adding the pipe flow rates in the branches just upstream from the junction and the seepage rate at the junction. [21] As noted, soil pipes form complex interconnected networks and containing many breaks. Pipe arrangements 3,
4, and 5 assumed such irregular connections of macropore networks. In addition, pipe arrangement 6 represented a clogged (or unopened) pipe. The simulated results for these irregular pipe conditions are compared to the simple infiltration (pipe arrangement 1), and a straight pipe (pipe arrangement 2).
6. Results and Discussion 6.1. Model Testing [22] Table 2 summarizes the different conditions in two bench-scale experiments (runs 1 and 2), and the two simulations (simulations 1 and 2) discussed later in this subsection. As shown in Table 2, homogeneous soil was assumed in simulation 1, whereas the flow resistance at the Table 1. Upslope Water Levels zw0 in the Previous Experimental Study by Sidle et al. [1995b] Upslope Water Levels zw0 , m Run 1
Run 2
a
0.095a 0.140a 0.188 0.230a 0.281 0.334a
0.050 0.088 0.135a 0.217 0.256a 0.308 0.355a
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a
Selected in Figures 7, 8, and 11.
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Table 2. Summary of the Conditions for the Model Simulationsa Run 1
Run 2
Simulation 1
Pipe segment with the greatest roughness coefficient was located at the lowest slope position. Observed Ks is homogeneously applied to the soil.
Pipe segment with the greatest roughness coefficient was located next to the uppermost portion of the composite pipe. Observed Ks is homogeneously applied to the soil.
Simulation 2
Pipe segment with the greatest roughness coefficient was located at the lowest slope position. Flow resistance is assumed at the downslope end, where Ks is reduced by a factor of 3.3. Ks of the remainder is increased by a factor of 1.2.
Pipe segment with the greatest roughness coefficient was located next to the uppermost portion of the composite pipe. Flow resistance is assumed at the downslope end, where Ks is reduced by a factor of 3.3.
a
Two sets of soil conditions (simulations 1 and 2) were assumed to simulate two different bench-scale experiments (runs 1 and 2).
downslope end is assumed in simulation 2. Note that the soil condition is assumed to be homogeneous (simulation 1) before the flow resistance at the downslope end is assumed later. [23] Observed and simulated pressure potentials of runs 1 and 2 for the piezometers shown in Figure 6 are depicted in Figure 7, under the four conditions of upslope water level zw0 . The composite pipe is described using gray lines, and the segment with the greatest roughness coefficient (segment 5, nm,5 = 0.325) is highlighted with darker gray in Figure 7. Comparing the observed values (symbols) between runs 1 and 2, the piezometric levels show different tendencies,
especially the higher piezometric levels in the lower half of the soil domain in run 1 versus those in run 2. For example, when the upslope water level, z0w, was set to 0.355 m in run 1, the piezometric levels in the lower half are around 0.2 m (Figure 7a); whereas they are about 0.1 m when the water level, z0w, was 0.334 m in run 2 (Figure 7b). These different tendencies of the piezometric levels are significant when the piezometric levels were above the pipe positions (z0w > 0.10). Since the only difference between runs 1 and 2 in the experiments is the location of the roughest pipe segment in the composite pipe, pipe roughness clearly affects the piezometric level within the soil domain.
Figure 7. Profile of the piezometric level in the sloping box with a pipe. The axis along the slope length is indicated by x0. The pipe is indicated by a gray line; the dark gray segment indicates the portion of the pipe with the greatest roughness coefficient n5 = 0.325. The simulated results (lines) are compared to the observed results (symbols). The different symbols and lines can be identified by the water levels at the upslope boundary. 9 of 15
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Table 3. Mean Assessment Criteria Used to Evaluate the Simulated Versus Observed Values of the Piezometric Profiles and Flow Rates (Pipe Flow, Matrix Flow, and Total Discharge) for the Seven Different Upslope End Water Levelsa Simulation 1
Piezometric level Flow rates Pipe flow Matrix flow Total discharge
Simulation 2
Run 1
Run 2
Run 1
Run 2
0.02
0.75
0.40
0.48
0.04 0.88 0.93
0.54 20.5 0.94
0.34 0.94 0.97
0.95 0.60 0.98
a Simulations were conducted for two different downslope boundary conditions (simulation 1, homogeneous soil domain; simulation 2, with flow resistance at the downslope boundary).
[24] The simulated piezometric levels (lines) also showed different tendencies between runs 1 and 2, in a similar way to the observed bench-scale results. Since our model considers pipe roughness and consistent water exchange between the pipe and soil matrix, this confirms that the roughness of the pipe wall and the flow conditions within the pipe strongly influence the surrounding soil matrix flow. However, the simulations overestimated the piezometric response in the middle to upper portion of the slope, and underestimated it in the lowest part of the slope in both runs. The mean assessment criteria CR averaged over all upslope water levels shown in Table 1 (seven levels for run 1, and six for run 2) were listed in Table 3 (simulation 1). These values (CR = 0.02 and 0.75 for runs 1 and 2, respectively) also indicate that the simulation do not agree well with the observed piezometric levels, especially for run 1. [ 25 ] The simulated three-dimensional groundwater regimes within the sloping soil domain, which were not measured in the bench-scale experiment, are shown in Figure 8. In run 1, as the fixed water level at the upslope end rose, the groundwater table rose accordingly (Figures 8a – 8d). Except for the lowest fixed water levels (Figure 8a), the groundwater table was much higher than the pipe level, and the pipe flow was full flow (Tp = 3). Backflow from the pipe into the surrounding soil matrix can be seen in Figure 8b, where the groundwater just above the lower part of the pipe has a convex shape. In run 2, when the fixed water level at the upper end of the hillslope rose, the upslope portion of the groundwater table also rose, while the groundwater table at the lower end of the slope remained relatively constant (Figures 8e – 8h). The concave shape of the groundwater table indicates active seepage into the pipe (Figures 8f – 8h). Simulations showed that the pipe flow downstream from the roughest segment was partially filled flow (Tp = 2), and no backflow occurred from the pipe into the surrounding soil matrix. The comparison of the simulated groundwater tables in runs 1 and 2 is in agreement with the previous experimental findings [Sidle et al., 1995b] that both pipe flow and matrix flow during run 1 were regulated by full pipe flow in the segment with a large roughness coefficient located downslope. Because of the low transmission capacity of this segment the upper portion of the pipe could not function as an effective drain. In addition, it is clear that seepage into the lower part of the pipe in run 2 controlled pipe flow, and the position of the high-roughness element in the pipe did not strongly affect the shape of the groundwater table.
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[26] Before the main experiment, Sidle et al. [1995b] made preliminary measurements of the difference in the pressure potential between the side and center of the box. They concluded that the difference was negligible, and they did not measure the pressure potential at the center of the box in the main experiment. The inlets for the piezometers in both the preliminary investigation and the main experiments of Sidle et al. [1995b] are indicated in Figure 6b. Although Sidle et al. [1995b] did not detect the difference in pressure potential between the side and center of the box, our model simulations showed significantly distributed water pressure along the y axis. One possible explanation for this contradiction is that the measured pressure potentials were limited within the range of 0.10 to 0.30 m in the preliminary investigation of Sidle et al. [1995b]. This is confirmed by Figure 9, which plots the relationship between the pressure head at the center and side of the box simulated in our study (solid circles) and observed in the preliminary investigation (open circles). In general, good agreement is
Figure 8. Simulated three-dimensional groundwater table within the sloping soil domain for runs (a –d) 1 and (e – h) 2 for the four water levels fixed at the upslope end (the values are indicated in each plot).
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Figure 9. Plots of the pressure head in the center versus the side of the box for the simulations in our study (solid circles) and observed in the preliminary investigation by Sidle et al. [1995b] (open circles). observed where the pressure heads at the center exceed 0.10 m. However, there are no observed data for pressure heads smaller than 0.10 m where the simulated data show a significant difference between the side and center of the box. Therefore, unlike the observations, the simulated results are able to show the concave and convex shapes of the groundwater table (Figures 8b, 8g, and 8h).
[27] Figure 10 compares the observed and simulated results with respect to the relationships between the water levels at the upslope end and the flow rates (observed data, solid circles; simulated data for simulation 1, open circles). In both runs 1 and 2, as the upslope end water level rose, the observed flow rates increased in a linear fashion. In run 1, the contribution of the observed pipe flow to the total
Figure 10. Comparison of the flow rates between the observed (solid circles) and simulated results without (open circles, simulation 1) and with (triangles, simulation 2) resistance at the downslope end of the hillslope. 11 of 15
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Figure 11. Profile of the piezometric level in the sloping box with a pipe with resistance at the downslope end of the soil domain (simulation 2). The simulated results (lines) are compared to the observed results (symbols). The differences between the simulated and observed piezometric levels were slightly improved for run 1 and became worse for run 2. discharge was very small relative to the matrix flow. In run 2, the contribution of the observed pipe flow to the total discharge was larger than that of the matrix flow, unlike run 1. These contributions of pipe flow and matrix flow to the total discharge and the linear increase of the total discharge were simulated well (simulation 1 indicated by 6). Good agreement of the simulated and observed total discharge were confirmed by the assessment criteria; CR = 0.93 for runs 1, and CR = 0.94 for run 2 (simulation 1 in Table 3). Meanwhile, the simulated pipe flow and matrix flow fit the observed data in both runs 1 and 2 poorly. In particular, the CR for the pipe flow in run 1 and matrix flow in run 2 is quite low, 0.04 and 20.5, respectively (simulation 1 in Table 3). The three most probable reasons for these discrepancies are (1) resistance to the matrix flow discharge caused by the polyester felt attached to the lower end of the experimental soil domain, (2) inadequate assumptions related to the soil-water retention curve, and (3) errors due to the seepage calculation method (the singlenode approach using the specified potential). [28] The first reason was suggested by the fact that the observed piezometric levels in the lowest part of the slope were higher than the simulated values (Figure 7), particularly because the pressure potentials at the downslope end boundary should be zero theoretically. This higher seepage face observed at the downslope end boundary might be due to the flow resistance. To study whether a higher seepage plane and a better fit to the observed flow rates can be obtained with flow resistance at the downslope end of the flume, extra simulations (simulation 2) were carried out in which the hydraulic conductivity K(y) of the soil was reduced by a factor of 3.3 for the lowermost finite elements (x0 = 0.0– 0.1 m: shown as gray elements in Figure 2) in
both runs 1 and 2. In addition, to improve the fit between the simulated and observed results, the hydraulic conductivity K(y) was increased by a factor of 1.2 for the remaining finite elements in run 1 only. This small modification of the hydraulic conductivity is justified since Sidle et al. [1995b] reported that the hydraulic conductivity might have been altered by the method of packing the soil in each experiment. These modifications are summarized as simulation 2 in Table 2. Given these modified K(y) values, the overall discrepancies between the simulated and observed pipe flow and matrix flow rates were reduced (simulation 2, indicated by the triangles in Figure 10). For run 1, the assessment criterion (CR) for the matrix flow and total flow rates increased from simulation 1 to simulation 2; and for run 2, CR for all flow rate components increased substantially from simulation 1 to simulation 2 (Table 3). The deterioration in the fit of the simulation to the observed pipe flow rates in run 1 (Figure 10) was confirmed by the decrease in CR for the pipe flow rates (Table 3). The differences between the simulated and observed piezometric levels improved only slightly for run 1, and were actually worse for run 2 (Figure 11). These small or negative changes in the piezometric levels from simulation 1 to simulation 2 were evaluated statistically using the assessment criterion, CR, which compares the simulated and observed piezometric levels (Table 3). Therefore applying a smaller hydraulic conductivity value at the downslope end was not sufficient in itself to account for all the discrepancies between the simulated and observed results. The soil-water retention curve for the actual soil used in the previous bench-scale experiment was not used because data for the actual retention curve was not available and the effect of the retention curve on the steady state flow
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mechanism triggering landslides. In the cases of both discontinuous pipes 1 (Figure 12c) and 2 (Figure 12d), groundwater levels rose at midslope, and then decreased in the lower part of the slope. In the case of the straight (Figure 12b) and branched pipes (Figure 12e) the overall water level was lowered. [30] The relationships between the water level at the upslope end (z0w) and flow rate are shown for different the pipe arrangements (Figure 13). All the flow rates increased as the water level at the upslope end z0w rose, and the relationship for the flow rates among pipe arrangements did not change for each water level z0w (i.e., the order of the pipe flow rates is branched > straight > discontinuous 2 > discontinuous 1 > no pipe = unopened for each zw0 ). Corresponding to the groundwater response, the pipe flow and total discharge of the branched pipe (Figure 12e) were the largest among all the pipe arrangements for all water levels zw0 , indicating that a highly branched macropore network within hillslope soils would yield a greater discharge than a single soil pipe. Although the flow rate from the clogged pipe (Figure 12f) was zero, the matrix flow rate was the largest and the total discharge was larger than for the discontinuous pipe 1 (Figure 12c). Therefore, although
Figure 12. Three-dimensional groundwater regimes within the sloping soil domain of the six numerical simulations (thick lines indicate the locations of the pipes; D = 0.013 m, nm = 0.05 m1/3s). The water level at the upslope end was fixed at 0.2 m. problem seemed to be small. However, according to the modeling study of Clement et al. [1996], vadose zone flow may have a significant influence on seepage face conditions in laboratory-scale experiments. Therefore the lack of an actual soil-water retention curve, which influences the unsaturated hydraulic conductivity, may have contributed to errors in the seepage face conditions at the downslope end of our simulations. Moreover, a single-node approach using a specified potential may underestimate the flux into soil pipes from the surrounding matrix [Fipps et al., 1986]. Consequently, use of this method may contribute to some of the discrepancies between the simulated and observed flow rates. Therefore an equivalent drain technique [Vimoke et al., 1963] may be a better method for the simulation if very high accuracy is required. 6.2. Simulation of Various Pipe Configurations [29] The steady state groundwater table and flow rates for six different pipe arrangements (indicated by thick lines in Figure 12) were obtained. Figure 12 shows the simulated three-dimensional groundwater regimes when the water level at the upslope end, z0w, was set 0.2 m. Figure 12 shows that the groundwater regimes were strongly influenced by individual pipe arrangements. Compared to the ‘‘no pipe condition’’ (Figure 12a), groundwater levels were lowered in all cases with pipes, except for the clogged pipe condition (Figure 12f). The rise in groundwater near the lower end of the pipe in Figure 12f suggests that pipe clogging causes local increases in pore water pressure, which might be a
Figure 13. Comparison of the flow rates between the six numerical experiments for various pipe configurations. The pipe flow, matrix flow, and the total discharge are plotted with the seven different water levels at the upslope end.
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the pipe outlet was located within the soil matrix, the rapid flow within the pipe contributed to the discharge. [31] These numerical experiments demonstrated that our model could simulate preferential flow within a hillslope, not only with a single soil pipe, but also with discontinuous pipes, branched pipes, or pipes whose outlets were located within the soil matrix. The combination of such irregular pipe segments can produce a more complex preferential flow system, without any additional modeling assumptions.
7. Conclusion [32] In the three-dimensional preferential water flow model described in this study, the soil matrix flow and the pipe flow were computed separately using their governing equations (Richards’ and Manning’s equations, respectively) and pipe flow was categorized into three types: (1) no, (2) partially filled, and (3) full pipe flow. The model can simulate various conditions of preferential flow (both partially filled and full pipe flow, and the seepage into the pipe and backflow from the pipe to the surrounding soil matrix) and pipe hydraulic conditions (pipe roughness and diameter). Model simulations of both piezometric levels and flow rates compared well with published bench-scale experiments [Sidle et al., 1995b]. The influence of the position of high-roughness pipe segments on the piezometric levels and flow rates was simulated especially well. Earlier models did not obtain such results, which are important for predicting pore water accretion in hillslopes. Furthermore, numerical experiments conducted for six different pipe arrangements showed that the model could also be applied to irregular pipe networks (discontinuous pipes, branched pipes, and pipes whose outlets were located within the soil matrix) without any additional model assumptions. [33] The model was designed to avoid excess computation time and expense by using the normal FEM grids for soil matrix flow calculations; no special grids were used for various pipe configurations. This simple approach makes it easier to develop and apply this model to more complex preferential flow systems. By combining this model with a hydrogeomorphic model of storm flow generation that includes complex preferential flow networks [Sidle et al., 2000, 2001], practical application is possible at the hillslope and small catchment scales. Experimental studies measuring the roughness coefficient of actual soil pipes in forested hillslopes [e.g., Kitahara, 1989] would be useful. As possible next steps for this research, this preferential flow model could be used to elucidate the mechanism of rapid discharge emanating from a hillslope during a storm event or analyzing the stability of hillslopes in which soil pipes influence the distribution of pore water pressures.
Notation A C(y) CR e1 e2 FRND(x) K(y) Ks
cross-sectional area of pipe [m2]. specific water capacity [m1]. assessment criterion. acceptable error for Sp convergence [m3/s]. acceptable error for y convergence [m]. residual normal distribution. hydraulic conductivity [m/s]. saturated hydraulic conductivity [m/s].
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l length along the pipe, positive from upstream to downstream [m]. n iteration number for convergence of calculation. nm roughness coefficient [m1/3s]. np number of pipe segments divided for model simulation. Qp pipe flow rate [m3/s]. Qp,k flow rate within the kth pipe segment [m3/s]. R hydraulic radius [m]. Sp water flux on boundary between soil matrix and pipe [m3/s]. 0 Sp water flux per unit pipe length [m3/s/m]. Sp,k water flux corresponding to the upper end of the kth segment [m3/s]. t time [s]. Tk integer defined to classify the pipe flow (no pipe flow, Tk = 1; partially filled flow, Tk = 2; full flow, Tk = 3). Vobs observed values for assessment criterion CR. Vcal calculated values for assessment criterion CR. Vobs mean value of Vobs. x, y, z axes indicating the spatial rectangular coordinate [m]. x0 axis along the slope length [m]. z0 axis along the sloping box height [m]. zw0 water level at the upstream boundary [m] as slope angle [degrees]. f hydraulic potential [m]. l ratio determining the contribution of S(n) p,k and . S(n1) p,k q soil water content [m3/m3]. qr residual soil water content [m3/m3]. qs saturated soil water content [m3/m3]. s dimensionless parameter related to the width of the pore size distribution. y soil pressure potential [m]. ym soil pressure potential corresponding to the median soil pore radius [m]. [34] Acknowledgments. This study was partly supported by a grant from the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT). We thank T. Mizuyama and M. Fujita of Kyoto University, H. Kitahara of Shinsyu University, and T. Uchida of National Institute for Land and Infrastructure Management of Japan for their helpful comments and suggestions and anonymous reviewers for their constructive comments and suggestions.
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Pierson, T. C. (1983), Soil pipes and slope stability, Q. J. Eng. Geol., 16, 1 – 11. Sidle, R. C., Y. Tsuboyama, S. Noguchi, I. Hosoda, M. Fujieda, and T. Shimizu (1995a), Seasonal hydrologic response at various spatial scales in a small forested catchment, Hitachi Ohta, Japan, J. Hydrol., 168, 227 – 250. Sidle, R. C., H. Kitahara, T. Terajima, and Y. Nakai (1995b), Experimental studies on the effects of pipeflow on throughflow partitioning, J. Hydrol., 165, 207 – 219. Sidle, R. C., Y. Tsuboyama, S. Noguchi, I. Hosoda, M. Fujieda, and T. Shimizu (2000), Stormflow generation in steep forested headwaters: A linked hydrogeomorphic paradigm, Hydrol. Processes, 14, 369 – 385. Sidle, R. C., S. Noguchi, Y. Tsuboyama, and K. Laursen (2001), A conceptual model of preferential flow system in forested hillslopes: Evidence of self-organization, Hydrol. Processes, 15, 1675 – 1692. Terajima, T., T. Sakamoto, Y. Nakai, and K. Kitamura (1997), Suspended sediment discharge in subsurface flow from the head hollow of a small forested watershed, northern Japan, Earth Surf. Processes Landforms, 22, 987 – 1000. Terajima, T., T. Sakamoto, and T. Shirai (2000), Morphology, structure and flow phases in soil pipes developing in forested hillslopes underlain by a Quaternary sand-gravel formation, Hokkaido, northern main island in Japan, Hydrol. Processes, 14, 713 – 726. Tsuboyama, Y., R. C. Sidle, S. Noguchi, and I. Hosoda (1994), Flow and solute transport through the soil matrix and macropores of a hillslope segment, Water Resour. Res., 30, 879 – 890. Tsukamoto, Y., and T. Ohta (1988), Runoff processes on a steep forested slope, J. Hydrol., 102, 165 – 178. Uchida, T., K. Kosugi, N. Ohte, and T. Mizuyama (1995), An experimental study on influence of soil pipe for groundwater table (in Japanese), Trans. Jpn. For. Soc., 106, 505 – 508. Uchida, T., K. Kosugi, and T. Mizuyama (1999), Runoff characteristics of pipeflow and effects of pipeflow on rainfall-runoff phenomena in a mountainous watershed, J. Hydrol., 222, 18 – 36. Vimoke, B. S., T. D. Yura, T. J. Thiel, and G. S. Taylor (1963), Improvements in construction and use of resistance networks for studying drainage problems, Soil Sci. Soc. Am. J., 26, 203 – 207. Zienkiewicz, O. C. (1971), The Finite Element Method in Engineering Science, McGraw-Hill, New York.
K. Kosugi, Division of Forest Science, Graduate School of Agriculture, Kyoto University, Kyoto, Kyoto 606-8501, Japan. R. C. Sidle, Division of Geohazards, Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto 611-0011, Japan. D. Tsutsumi, Research Center for Fluvial and Coastal Disasters, Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto 611-0011, Japan. (
[email protected])
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