BASE-FLOW ANALYSIS ON HYDRAULIC CONDUCTIVITY using numerical experiments

Department of Civil Engineering, Graduate School of Engineering, Tohoku University

For the efficient use of water resources, it is important to understand the relationships between basin yields and geographical characteristics of a watershed such as basin area, slope gradient, and the permeability of soil and geology layers. One of practical way to understand them is physical consideration on the relationships using distributed runoff model, because they enable us to analyze the effect of a single geographical characteristic on discharges. Similar attempts were made by many researchers [3], [4], but they have not made on the effect of anisotropic hydraulic conductivity of soil and geology layers on base-flow behavior. Hence, this study tried to investigates the relationships between those two using numerical simulations. The model had vertical, two-dimensional, and rectangular cross section and was sloped from the ground surface. The anisotropic soil and geology layers were expressed by employing the hydraulic conductivity in the direction of hill-slope and its normal direction.

Figure 1 shows the structure of the numerical model designed to have vertical, two-dimensional, and rectangular cross section. The model is inclined to

radian from horizontal plane. The segments OA and BC model the upper and the lower end of a watershed, respectively. The segment AB corresponds to the ground surface that has precipitation and evapotranspiration. The segment OC models bottom of the soil and geology layers. The rectangle OABC consists of small cells with the area of D x times D z. The x-axis and z-axis are set as shown in Figure 1 and n_x and n_z are the number of cells in the x-direction and z-direction, respectively. Because this model doesn’t consider surface runoff, precipitation and storage water flow through the model to the segment BC that yields the discharge from this model.

In Figure1, rectangle OABC models soil and geology layers that play important roles for hydrogical processes, and the appropriate governing equation for this model is the Richards’ equation as shown in equation (1).

This equation cannot be solved without the relationships between moisture content

, pressure head

, and hydraulic conductivity K_x (and K_z). Equation (2), Tani’s equation [4], was employed to relate

. The relationship between

and K_x (or K_z) was modeled by equation (3) [1].

According to the model structure, the following equations (4), (5), (6), and (7) were employed as the boundary conditions for the segments OA, CO, BC, and AB, respectively.

Runoff simulations were performed only changing the hydraulic conductivities in x-direction and z-direction, using the numerical model in Figure 1. The other parameters of the model were evaluated as shown in Table 1. The pressure head at every point in the model were set at -0.9 m that is equal to the moisture content of 0.4, as the initial condition of the simulation. Precipitation and evapotranspiration were not provided to the model, and only runoff quantities were calculated for 500 days long.

Name of parameter	n_x	n_z	Δx(m)	Δz(m)	α(radian)	θ_s	θ_r	ψ₀(m)	β
Value of parameter	99	9	1.0	0.5	π/10	0.7	0.3	-0.3	3.5

The simulations can be divided into three groups. The saturated hydraulic conductivity in the x-direction, Ks_x, was only changed while that in the z-direction, Ks_z, was kept as constant in the first group. The variable in the second group simulations was only Ks_z. Effects of both Ks_x and Ks_z on base-flow were investigated by the simulations in these two groups. The simulations of the third group were carried out to observe the combined effects of randomly evaluated hydraulic conductivities in the two directions on the base-flow.

Five cases of experiments were performed by changing the value of Ks_x from 0.002 cm/s to 0.01 cm/s, where the value of Ks_z was fixed at 0.01 cm/s as shown in Table 2. The result of experiments is shown in Figure 4.

Case	Ks_x-1	Ks_x-2	Ks_x-3	Ks_x-4	Ks_x-5
Ks_x(cm/s)	0.002	0.004	0.006	0.008	0.010
Ks_z(cm/s)	0.010	0.010	0.010	0.010	0.010

The peak runoff became greater and the recession curves become steep when Ks_x was great, as shown in Figure 3. Figure 4 shows the relationships between Ks_x and peak runoff and the arrival times. As you see, peak runoff was proportional to Ks_x and the arrival times decreased exponentially as Ks_x become greater.

As shown in Table 3, eight cases of experiments were carried out. Ks_z was varied from 0.00001 cm/s to 0.01 cm/s, while Ks_x was fixed at 0.01 cm/s. Figure 4 shows the results.

Case	Ks_z-1	Ks_z-2	Ks_z-3	Ks_z-4	Ks_z-5	Ks_z-6	Ks_z-7	Ks_z-8
Ks_x(cm/s)	0.010	0.010	0.010	0.010	0.010	0.010	0.010	0.010
Ks_z(cm/s)	0.00001	0.00005	0.00010	0.00020	0.00030	0.00050	0.00100	0.010000

Figure 4 shows that peak runoff decreased and arrival times were delayed when Ks_z became smaller. The relationships between Ks_z, peak runoff and the arrival times are shown in Figure 5. As Ks_z become greater, peak runoff increased logarithmically and the arrival times decreased exponentially.

Twenty-three cases of simulations were performed in order to investigate the relationships between Ks_x, Ks_z, and base-flow. Ks_x and Ks_z were set as shown in Table 4 and the simulation results will be discussed in the next section.

Case No	1	2	3	4	5	6	7	8
Ks_x(cm/s)	0.002	0.004	0.006	0.008	0.010	0.010	0.010	0.010
Ks_z(cm/s)	0.01000	0.01000	0.01000	0.01000	0.01000	0.00001	0.00010	0.00100
Case No	9	10	11	12	13	14	15	16
Ks_x(cm/s)	0.008	0.006	0.004	0.002	0.008	0.006	0.004	0.002
Case No	1	2	3	4	5	6	7	8
Ks_z(cm/s)	0.00100	0.00100	0.00100	0.00100	0.00010	0.00010	0.00010	0.00010
Case No	17	18	19	20	21	22	23
Ks_x(cm/s)	0.008	0.006	0.004	0.012	0.012	0.012	0.012
Ks_z(cm/s)	0.00001	0.00001	0.00001	0.01000	0.00100	0.00010	0.00001

From Figure 2, 3, 4 and 5, it was obvious that peak runoff quantities of base-flow were related to both Ks_x and Ks_z. However, Ks_x was more effective to the peak runoff quantities for this model, because they varied widely when Ks_x was ranged between 0.002 cm/s to 0.01 cm/s. On the other hand, it was difficult to choose effective parameter to the arrival time of the peak runoff from Ks_x and Ks_z, because hydrograph became too flat to find out the peak runoff when Ks_x was small though arrival time varied widely with small change of Ks_x comparing to the results of the experiments 2.

The gradients of the recession curves were also changed with Ks_x and Ks_z. It was also obvious that Ks_x would be effective to the gradients of the recession curves. Detailed discussions on the relationship will be made in the next section.

The target of this section is the effect of the hydraulic conductivities in the both directions on the characteristics of the long-term base-flow recession. First, each hydrograph was drawn for every case. Then exponential function model, as shown in equation (8), was applied to the runoff data that were 400 days after the beginning of each simulation and later.

The parameter b in equation (8) characterizes the recession curves of the base-flow. Figure 6 shows the relationship between the parameter b, Ks_x, and Ks_z. The horizontal axis corresponds the case number of the Table 4. It is estimated that Ks_x is more effective to the base-flow recession curve than Ks_z because parameter b and Ks_x were synchronized each other.

Figure 8 was illustrated to visualize the relationships from other viewpoint. The locations of the bar graphs correspond to the black dots in the Figure 7. Two results of case 17 and case 18 were interesting in Figure 10. It is easy to imagine that parameter b become small and recession curve become mild when both Ks_x and Ks_z were small, but parameter b in case 17 and 18 were bigger that those in case 13 and 14, respectively. This means that recession curves become steep when Ks_z is small as well as when it is great. These results can be obtained because the model becomes impermeable in z-direction when Ks_z is less than 10^-5 cm/s and delayed time of runoff can be smaller than that calculated with bigger Ks_z.

Investigating the effect of anisotropic hydraulic conductivities on the base-flow runoff, following concluding remarks were obtained.

All the findings were based on the results of the numerical simulations under restricted conditions, however they can be useful information for the estimations of the “basin-averaged hydraulic conductivity” in the hill-slope direction and its normal directions. The “basin-averaged hydraulic conductivity” can be one of the important input parameters for the constructions of distributed runoff models. The integration of the subsurface runoff model and surface runoff model will allow to evaluate the effects of land-use in future.

This study was supported by JSPS Research Fellowship for Young Scientists and Grant-Aided for Scientific Research (Grant-in-Aid for JSPS Fellows) in part. We wish to thank the Grant in Aid for Scientific Research (Representative: Professor S. Ikebuchi). Information on watersheds were given from branch offices of Ministry of Construction, Japan, branch offices of The Water Resources Development Public Corporation (WARDEC), Japan and prefecture offices in Japan.

[1] Brutsaert, W., (1968). The permeability of a porous medium determined from certain probability laws for pore size distribution, Water Resources Research, Vol.4, No.2: 425-434.

[2] Hashimoto, T and Morita, T (1982). A Study on the Land-use Hydrological System Model, J. of Japan Society of Civil Engineering, Vol.325, pp.45-50 (in Japanese).

[3] Kobatake, S and Ishihara, Y (1983). Synthetic Runoff Model for Flood Forecasting, J. of Japan Society of Civil Engineering, Vol. 337, pp.129-135 (in Japanese).

[4] Tani, M., (1982). The properties of a water-table rise produced by a one-dimensional, vertical, unsaturated flow, J. of Japan Forest Society, 64: 409-418 (in Japanese).

BASE-FLOW ANALYSIS ON HYDRAULIC CONDUCTIVITY USING NUMERICAL EXPERIMENTS