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GIROUD et al. D Leachate Flow in Leakage Collection Layers Due to Geomembrane Defects (a) Case I μ ≤ 2, 0 ≤ x ≤ L(1 − μ 2 ) (b) Case II μ ≤ 2, L(1 − μ 2 ) < x < L L L x (c) Case III μ ≥ 2 (i.e. arrow greater than L ) 0
GIROUD et al. D Leachate Flow in Leakage Collection Layers Due to Geomembrane Defects S the case where the defect in the primary liner is located at the high end of the leakage collection layer (x = L), regardless of μ (Case IV, Figure 13d). Furthermore, an interpolation method will be provided for the Case II (Figure 13b), which is defined by Equation 145 (μ ≤ 2) and the following equation derived from Equations 58 and 109: F m L 1 - x L (147) 2 HG I K J £ £ In summary: (i) solutions are provided for μ ≤ 2 (analytical solution for Cases I and IV, and interpolation method for Case II); and (ii) no solution will be provided for μ > 2 (Case III, Figure 13c), with the exception of the solution for Case IV, which does not depend on μ. The reason no analytical solution is proposed for two cases (Cases II and III in Figure 13) is that volume calculations are extremely complex. The considered volume is the cone formed by the leachate phreatic surface and truncated by three planes: the plane of the secondary liner, and two vertical planes located at the high end and the low end of the leakage collection layer slope. (However, in Case I, the phreatic surface does not meet the vertical plane located at the high end of the leakage collection layer slope; therefore, in this case, the phreatic surface is truncated only by two planes, the plane of the secondary liner and the vertical plane located at the low end of the leakage collection layer slope.) The intersection of the leachate phreatic surface with the secondary liner plane is a parabola, as extensively discussed in Section 4, whereas the intersections of the phreatic surface with vertical planes are hyperbolas. The volume to be calculated can be decomposed into two volumes: the volume of the leakage collection layer containing leachate downstream of the defect (x >0);andthe volume of the leakage collection layer containing leachate upstream of the defect (x < 0). A simple expression (in spite of the hyperbolic truncation at the low end of the slope) can be obtained for the downstream volume using Darcy’s equation as shown in Section 5.1.2. However, the hyperbolic truncation makes the calculation of the upstream volume quasi-inextricable. (The senior author has obtained the equation of the hyperbolic intersection at the high end of the slope and has found that the integration of the hyperbola to obtain the volume of the cone could be done analytically but would require lengthy calculations that would be beyond the scope of this paper.) The only two cases where the upstream volume is simple are Cases I and IV (Figure 13): in Case I, the volume is that of a cone whose base and height are known; and, in Case IV, the volume is zero. In Case II, an interpolation method will be proposed, and, in Case III, no solution will be proposed; however, Case III is rare since μ is rarely greater than 2 as discussed in Section 4.5. 5.1.2 Leachate Thickness in Case I In Case I (Figure 13a), the parabola is not truncated. The volume of the leakage collection layer containing leachate (above the parabolic wetted zone) can be decomposed into two volumes, as explained in Section 5.1.1: the volume located downstream of the defect, and the volume located upstream of the defect. GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NOS. 3-4 257
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