Evaluation of Flow Transmissibility of Rockfill Structures Toshihiro MORII 1 and Takahiko TATEISHI 2 Abstract To predict the hydraulic conditions during and after the construction of such structures as flowthrough dams, cofferdams and gabion weirs, an accurate evaluation of flow transmissibility of the rockfill is needed. The analysis of flow through rockfill is complicated because of the non-linear relationship between the discharge velocity through the rockfill and the hydraulic gradient applied. In the present study, a procedure to evaluate the transmissibility of the rockfill is developed based on the laboratory tests and the non-linear numerical calculation formalized by the finite element method (FEM). One-dimensional permeability tests were conducted to study the hydraulics of the flow through the rockfill. The head loss property of flow through the rockfill was successfully modeled by the Forchheimer equation that takes account of the effects of the size and shape of rock particles as well as the water temperature on the flow hydraulics. The non-linear parameters included in the head loss equation of flow were determined by the one-dimensional permeability tests using the river gravel, and were effectively correlated to the hydraulic mean radius of voids. By the comparison between the measurements in the water flume tests and the FEM calculations, the practical applicability of the proposed head loss equation of flow was confirmed. The stage-discharge rating curves of the prototype flowthrough rockfill dams were estimated by the FEM calculation code incorporating the proposed head loss equation of flow. Almost negligible effect of water temperature on the flow discharge through the rockfill was noticed. The flow transmissibility of the rockfill structure in response to the applied hydraulic gradient can be well evaluated by the proposed head loss equation of flow and the FEM calculation code. A special feature of the developed procedure is that it requires only two design parameters to estimate the hydraulics of flow through the rockfill: The hydraulic mean radius of voids within the rockfill and the water temperature of flow. Keywords: Rockfill, Non-linear flow, Head loss equation, Flow transmissibility, One-dimensional permeability test, Water flume test, Finite element method 1 Introduction There has been a gradual increase in the use of rockfill for hydraulic structures. To predict the hydraulic conditions during and after the construction of such structures as flowthrough dams, cofferdams and gabion weirs, an accurate evaluation of the flow transmissibility of the rockfill is needed. The analysis of flow through rockfill is complicated because of the non-linear relationship between the discharge velocity and the applied hydraulic gradient.
In the present study, a procedure to evaluate the flow transmissibility of the rockfill is developed based on laboratory tests and numerical calculations. Firstly, a head loss equation of flow through rockfill is determined from the one-dimensional permeability tests. A hydraulic mean radius is effectively introduced into the head loss equation of flow to describe the effects of the shape and size of rock particles and the voids within the rockfill on the flow hydraulics. The effect of water temperature is also included in the head loss equation of flow. Secondary, the numerical code based on the finite element method (FEM) is developed to evaluate the flow transmissibility of the rockfill structures. Numerical difficulty due to the non-linearity of the head loss equation of flow is solved by a method of successive approximation. Thirdly, water flume tests are conducted to examine practical applicability of the proposed head loss equation of flow and the FEM calculation code developed. The bulk discharge through the model embankment, and the free surface and the total head within the model embankment are measured, and compared with the calculations by the FEM code incorporating the proposed head loss equation of flow. Lastly, the stage-discharge rating curves of the prototype flowthrough rockfill dams are estimated, and the effect of the water temperature on the flow discharge through the rockfill is investigated. 2 Head loss equation 2.1 One-dimensional permeability tests and material One-dimensional permeability tests of rockfill column were conducted to study the head loss property of the flow through rockfill. Rock particles were filled and compacted lightly by hand into the cylindrical column 10 cm diameter and about 100 cm length. The mass and the volume of the rockfill specimen were measured to calculate porosity, n, and void ratio, e, of the rockfill specimen. The rockfill specimen being placed vertically, water was supplied into the bottom of the specimen from the water reservoir. A flow rate through the rockfill specimen was measured by mass, and divided by the cross-sectional area of the specimen to determine a discharge velocity of flow through the rockfill specimen, V. The total head along the rockfill specimen was measured by eight piezometers connected 10 cm interval to the wall of the specimen column. A hydraulic gradient, i, applied to the rockfill specimen was determined from the slope of the linear regression line through a plot of the total head along the rockfill specimen. i was increased successively from about 0.05 to 0.8 by seven to eight steps in each test of the rockfill specimen. Temperature of flow through rockfill specimen, T, was also measured at the beginning and the completion of the test. River gravel was used in the one-dimensional permeability tests. Being washed and air-dried, the rock particles were sieved into four classes of size, D1, D2, D3 and D4. Four classes of size, and the representative diameter, rock particle shape and physical properties of the rock particles in each class of size are given in Table 1. The representative diameter of rock particles, d, is defined as an arithmetic mean of particle size. The rock particles were classified according to the Zingg diagram (Garga, et al., 1991) into four shapes of blades, disks, spheroids and rods based on the measurements of three orthogonal lengths of selected rock particles. Their frequencies are given in the rows (4)-(7) of Table 1. The mean shape coefficient of rock particles in each class of size, r, in the row (8) is calculated as a mean of r e weighted by the frequencies measured in the rows (4)-(7), in which r e is the shape coefficient for each shape of rock particles measured by Sabin and Hansen (1994). Sixteen rockfill specimens, that is four classes of rock particles given in Table 1 and four mixed
Class of Particle Particle size, mm Table 1 Size, shape and physical properties of the rock particles used in the one-dimensional permeability tests. Arithmetic mean of particle size d, mm (3) Blades Disks Rock particle shape a) Spheroids Rods Mean shape coefficient r b) (8) Specific gravity of dried particle G (9) Water absorption of particle, % (10) (1) (2) (4) (5) (6) (7) D1 20-25 22.5 0.32 0.22 0.14 0.32 1.9314 2.563 1.4 D2 15-20 17.5 0.22 0.18 0.26 0.34 1.8600 2.550 1.5 D3 10-15 12.5 0.22 0.24 0.20 0.34 1.8774 2.557 1.4 D4 5-10 7.5 0.32 0.16 0.06 0.46 1.8944 2.575 1.6 a) Rock particle shapes such as blades, disks, spheroids and rods are classified according to the Zingg classification (Garga, et al., 1991). Numerals in the cells show frequency of rock particle shapes determined by measuring three orthogonal axes of rock particle. Fifty rock particles are selected from each class of rock particle size to determine the frequency of rock particle shapes. b) r is a mean shape coefficient of rock particles, and is calculated as a mean of r e weighted by the frequency given in the rows (4)-(7). Re is the shape coefficient for each shape of rock particles, and is given by Sabin and Hansen (1994). classes of rock particles repeated twice, were tested. T ranged between 15-23 degrees centigrade. The Reynolds numbers, R e, were calculated to be 9-323. R e was calculated by V m R e = (1) n ν where m is the hydraulic mean radius of voids within the rockfill which will be defined in Section 2.2, and is the kinematic viscosity of water. Typical results of the one-dimensional permeability test of rockfill are shown in Figure 1, where solid lines are quadratic regressions curves. It is shown that the discharge velocity of flow through the rockfill specimen has a non-linear relationship with the hydraulic gradient applied to it, and that much lager rate of water flows through the rockfill as compared with the Darcy flow through soil. 2.2 Head loss equation of flow through rockfill It has been well known that there are two types of equation describing the non-linear head loss property of flow through coarse materials such as rockfill: The Forchheimer equation and the Missbach equation (Basak, 1977; Parkin, 1990; Kazda, 1990). A superior feature of the former compared with the latter, which has been empirically determined, is that it can be theoretically derived from the Navier-Stokes equation. The Forchheimer equation is given by i 2 = A V + B V (2) where A and B are coefficients which depend on the void structures of the rockfill as well as the water viscosity of the flow, and have their dimensions [TL -1 ] and [T 2 L -2 ], respectively. According to the theoretical considerations done by Irmay (1958), Scheidegger (1974), and Bear and Verruijt (1987), the coefficients A and B in Equation (2) can be described as
Hydraulic gradient i 1.0 0.8 0.6 0.4 0.2 D4 D 3 D2 0.0 0 2 4 6 8 10 12 D ischarge velocity V, cm/s Figure 1 Typical results of the one-dimensional permeability tests of rockfill. D1 Particle size D1 20-25mm D 2 15-20mm D 3 10-15mm D 4 5-10mm A0 ν B0 A =, B = g g (3) in which g is the acceleration of gravity, and both A 0 and B 0 are the coefficients depending only on the void structures of rockfill. A 0 and B 0 have the dimensions [L -2 ] and [L -1 ], respectively. A reciprocal of A 0 is corresponding to a physical or intrinsic permeability of porous media that has been well established in the theory of Darcy flow through soil. Effects of the void structures on the head loss property of flow through the rockfill should be evaluated by taking the size and shape of the rock particles as well as the size and distribution of voids within the rockfill into account. Martins (1990) shows that all the effects above mentioned could be well described by a hydraulic mean radius of voids, m, which is defined as (Sabin and Hansen, 1994) m e b = (4) 6 r e where b is a representative diameter of rock particles. Thus it may be right to think that the coefficients A 0 and B 0 in Equation (3) are closely related to m. Figure 2 shows such relationships of A 0 and B 0 with m determined from the one-dimensional permeability tests of rockfill. The values of A and B in Equation (2) being firstly determined by the regression analysis as shown in Figure 1, then A 0 and B 0 were calculated by using Equation (3) with the known value of at the water temperature T. In calculating m, b and r e in Equation (4) were effectively replaced by d and r in Table 1, respectively. A close relationship of A 0 and B 0 with m can be seen in Figure 2. According to Scheidegger (1974) and Bear and Verruijt (1987), a power type equation may be accepted to describe the functional relationship of A 0 and B 0 with m shown in Figure 2:
Coefficient A0, cm -2 8000 6000 4000 2000 (a) Nonlinear coefficient A0 0 0.00 0.05 0.10 0.15 0.20 Hydraulic mean radius of voids m, cm Figure 2 Relationships of the nonlinear coefficients A 0 and B 0 with the hydraulic mean radius of voids. Coefficient B0, cm -1 50 40 30 20 10 (b) Nonlinear coefficient B0 0 0.00 0.05 0.10 0.15 0.20 Hydraulic mean radius of voids m, cm 1.5011 1.1547 A 0 = 31.815 m, B0 = 0. 6282 m (5) in which A 0 and B 0 and m are in cm -2, cm -1 and cm, respectively. The hydraulic mean radius of voids being determined from the size and shape of rock particles and the void ratio of the rockfill deposition, and the water temperature of the flow being measured or supposed, the head loss property of the flow through the rockfill will be able to estimated by using Equation (2) with the known values of A and B. 3 Flow transmissibility of rockfill structures 3.1 Water flume tests To examine the practical applicability of the head loss equation of the rockfill proposed in the preceding chapter, a series of the model embankment tests was conducted using the acrylic-walled water flume 20 cm wide and 40 cm deep in the laboratory. Figure 3 shows the setup of the water flume test. The same rock particles as used in the one-dimensional permeability tests were filled into the water flume in layers and compacted lightly with a steel rod to construct the model embankment. The mass and the volume of the model embankment were measured during and after the construction, respectively, to calculate n and e. Both the upstream and downstream slopes of the model embankment were retained by the wire mesh with the squared openings of 2 mm to ensure the stability of the slopes and to prevent any erosion of rock particles from the downstream slope. The total head within the model embankment were measured from the tapping points installed along the base and on the wall of the water flume. The free surface within the model embankment and its exit point on the downstream slope were observed through the acrylic wall of the water flume. The upstream water level, h u, was raised successively from about 10 cm to 30 cm by 4 to 6 steps, and at each step in the water flume test, after the steady state of flow was attained, the bulk discharge through the model embankment was measured by mass at the outlet of the water flume. Then the flow discharge per unit width of the water flume, q, was calculated by dividing the measured bulk discharge by the width of the water flume. No regulation was given on the downstream water level in any water flume tests.
2m m-grid w ire mesh Acrylic wall of wa ter flum e 10 5 40 hu Acrylic floor of w ate r flume Piezometer tap 5 10 Unit: cm Figure 3 Schematic diagram of the water flume test and the typical section of the model embankment constructed using the rock particles. 3.2 Applicability of head loss equation To estimate the flow transmissibility of the rockfill structures, a numerical calculation code was formalized based on FEM. A numerical difficulty due to the non-linearity of the head loss equation of flow, Equation (2), was successfully solved by the method of successive approximation (Kazda, 1990). In each iterative calculation, the flow is assumed to be Darcy flow, and a fictitious hydraulic conductivity, K f, is introduced to calculate the flow velocity (Kazda, 1990; Kells, 1993): K f A + = 2 A + 4B i 2B i (6) Fourteen model embankments with different combination of the rock particle size, the downstream slope and the top width of the model embankment were tested in the water flume. The values of T measured in the tests were 8 to 24 degrees centigrade. R e ranged between 15-145, which was sufficiently included within the range of the one-dimensional permeability tests shown in Section 2.1. In calculating R e by Equation (1), the average velocity through the vertical section within the embankment under the upstream water level was used as V. Figure 4 shows a comparison of q between the water flume tests and the FEM calculations. The first, second and third terms of the test number represent the rock particle size given in Table 1 (D13 is the case of D1 mixed with D3), the specified downstream slope (11=1V: 1H, 12=1V: 2H and V=Vertical), and the specified top width of the embankment in cm, respectively. The model embankment with the rectangular section, the upstream and downstream halves of which were made of the rock particle size D1 and D3, respectively, was also constructed and indicated as D1(u)+D3(d)-V-60 in Figure 4. The height of the model embankments was about 30 cm above the base of the water flume in all tests, and the upstream slope was 1V: 1H except for the vertical D1-V-60 and D3-V-60. In the FEM calculations, the non-linear coefficient A and B were determined by using Equations (3)-(5) with the known values of e, d, r and T. It s can be seen in Figure 4 that the FEM calculations which incorporate the head loss equation proposed in Section 2.2 predict well the measurements in the
Discharge per unit width of water flume q calculated by nonlinear FEM, cm 3 /s/cm 150 100 50 D1-11-10 D2-11-10 D3-11-10 D4-11-10 D13-11-10 D1-12-10 D2-12-10 D3-12-10 D4-12-10 D1-11-30 D1-11-30 D1-V -60 D3-V-60 D1(u)+D3(d)-V-60 0 0 50 100 150 Discha rge per unit w idth of water flume q mea sured in tests, cm 3 /s/cm Figure 4 Comparison of flow discharge between the measurement in the water flume tests and the FEM calculations. Figure 5 Comparison of the free surface and the total head within the embankment between the water flume tests and the FEM calculations. water flume tests. In Figure 5, the free surface and the total head of water within the model embankment are compared between the FEM calculations and the water flume tests at the step of the highest position of the upstream water level in D1-11-10 and D1(u)+D3(d)-V-60. There can be see again good predictions by the FEM in Figure 5.
3.3 Stage-discharge rating curve of flowthrough rockfill dam The practical applicability of the head loss equation of flow proposed in Section 2.2 as well as the FEM calculation code developed in the preceding section is now made clear. A special feature of the head loss equation of flow is that it can predict the non-linear head loss property of flow through the rockfill by knowing only two parameters: the hydraulic mean radius of voids and the water temperature of flow. In this section, the stage-discharge rating curve of flowthrough rockfill dams will be predicted by using the FEM calculation code together with the proposed head loss equation of flow. The flowthrough rockfill dam has been constructed to reduce the peak flow of flood or to recharge the surface water to groundwater (Parkin, 1990; Hansen, et al., 1995). The effect of water temperature on the flow discharge is also examined in this section. Two prototype flowthrough dams 5 m and 10 m high above the foundation are prepared in the numerical calculations as shown in Figure 6, in which the dam height above the foundation is H. The upstream and downstream slopes are 1H: 1V, and the top width of the dam is H/3 in both prototype dams. The foundation is assumed to be rigid and impervious. The hydraulic mean radius of voids m is estimated to be 2.0 cm in the dam 5 m high and 4.0 cm in the dam 10 m high according to the assumption that the representative diameter of rock particles d directly proportional to H is used to construct the dam and m increases proportionally with d, and based on the test result that m was 0.12 cm in D1-11-10 with H=0.3 m and d=2.25 cm. In the FEM calculations, h u was raised successively from 0.2H to H by six steps. No tail water was supposed. Figure 7 shows the stage-discharge rating curves of the flowthrough rockfill dams calculated by the FEM code incorporating the proposed head loss equation of flow. The stage-discharge rating curves are compared between T=10 and 30 degrees centigrade in each dam. It is noticed from Figure 7 that the flow discharge through the rockfill increases exponentially with the upstream water level. Another important fact that can be seen in Figure 7 is that the effect of water temperature on the flow discharge through the rockfill is very small and may be negligible. This means that the flow through the rockfill depends dominantly on the size and shape of the rock particles and the voids distributed within the rockfill, and leads to two practically important results: The first is that no consideration of the water temperature is needed in estimating the flow discharge through the rockfill, and the second is that no standardization of the water temperature condition is needed in any laboratory permeability test to determine the head loss equation of flow. hu 1 : 1 d 1 : 1 H 2H+H/3 Figure 6 Representative section of the flowthrough rockfill dam employed in the FEM calculations to determine the stage-discharge rating curves.
4 Conclusions Figure 7 Stage-discharge rating curves of the prototype flowthrough rockfill dams. In this study, the head loss property of flow through the rockfill was successfully modeled by the Forchheimer equation that takes account of the effects of the size and shape of rock particles as well as the water temperature on the flow hydraulics. The non-linear parameters included in the head loss equation of flow were determined by the one-dimensional permeability tests using the river gravel, and were effectively correlated to the hydraulic mean radius of voids. By the comparison between the measurements in the water flume tests and the FEM calculations, the practical applicability of the proposed head loss equation of flow was confirmed. The stage-discharge rating curves of the prototype flowthrough rockfill dams were estimated by the FEM calculation code with the proposed head loss equation of flow. Almost negligible effect of water temperature on the flow discharge through the rockfill was noticed. The flow transmissibility of the rockfill structure in response to the applied hydraulic gradient can be well evaluated by the proposed head loss equation of flow and the FEM calculation code. A special feature of the developed procedure is that it requires only two design parameters to estimate the hydraulics of flow through the rockfill: The hydraulic mean radius of voids within the rockfill and the water temperature of flow. Acknowledgements The present study was supported by the Grant-in-Aid for Scientific Research (C), No. 12556036, made by the Ministry of Education, Science, Sports and Culture of Japan. The authors are grateful to Messrs. T. Sada, T. Kobayashi and S. Soyama for their help to conduct the laboratory tests. References Basak, P. (1977): Non-Darcy flow and its implications to seepage problems, Journal of the Irrigation and Drainage Division, Proceedings of the American Society of Civil Engineers, 103(4), pp. 459-473. Bear, J. and A. Verruijt (1987): Modeling Groundwater Flow and Pollution, D. Reidel Publishing, pp. 41-43. Garga, V. K., R. Townsend and D. Hansen (1991): A method for determining the surface area of quarried rocks, Geotechnical Testing Journal, 14(1), pp. 35-45.
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