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Solutions to estimate the excess PWP, settlement and volume of draining water after slurry deposition. Part I: impervious base


In coastal engineering, the storage of dredged sludge needs the construction of containment structures. In mining engineering, the surface disposal of tailings slurry requires the construction of tailings dam. In both cases, one needs to estimate the excess pore water pressure (PWP), draining water and settlement of the slurry during and after the deposition. An analytical solution presented by the authors based on a model proposed by Gibson in 1958 can be used to evaluate the excess PWP during slurry deposition. The equation given by Gibson in 1958 for assessing the excess PWP after the end of slurry deposition cannot be used because it contains an error. In addition, there is no existing solution to estimate the settlement and volume of draining water after the end of slurry deposition. In this paper, a new solution based on the Gibson’s governing equation is proposed to evaluate the excess PWP, settlement and volume of draining water after the end of slurry deposition on an impervious base. The proposed solution is partly validated by numerical and laboratory test results. The solution for estimating the excess PWP, settlement and volume of draining water after the end of slurry deposition on a pervious base is presented in a companion paper.

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The authors would like to acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC 402318), Institut de recherche Rovert-Sauvé en santé et en sécurité du travail (IRSST 2013-0029), Fonds de recherche du Québec—Nature et Technologies (FRQNT 2015-MI-191676), Mitacs Elevate Postdoctoral Fellowship (IT12572), and industrial partners of the Research Institute on Mines and the Environment (RIME UQAT-Polytechnique;

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Appendix 1: Derivation process for the proposed solution (Eq. 6)

Equation (4) has the same form as the heat conduction equation. It can be solved by separating the equation variables, a method commonly used to solve heat conduction problems (Asmar 2004). The excess PWP at moment t1 (= t − t0) at an elevation x can then be expressed by two new functions as follows:

$$u_{1} (x,t_{1} ) = X(x)T(t_{1} ),$$

where X(x) is a function of the variable x, T(t1) is a function of the variable t1.

Substituting Eq. (14) into Eq. (4) yields

$$\frac{1}{X(x)}\frac{{{\text{d}}^{2} X}}{{{\text{d}}x^{2} }} = - \lambda = \frac{1}{{c_{\text{v}} T(t_{1} )}}\frac{{{\text{d}}T}}{{{\text{d}}t_{1} }},$$

where λ is a constant (λ ≠ 0).

Considering the former part of Eq. (15) leads to an expression as follows:

$$\frac{{{\text{d}}^{2} X}}{{{\text{d}}x^{2} }} + \lambda X = 0.$$

Solving Eq. (16) leads to a general solution expressed as follows:

$$X(x) = C_{1} \sin \left( {\sqrt \lambda x} \right) + C_{2} \cos \left( {\sqrt \lambda x} \right).$$

The boundary conditions u1 = 0 at x = H and du1/dx = 0 at x = 0 can be met by imposing X = 0 at x = H and dX/dx = 0 at x = 0. Applying these boundary conditions to Eq. (17) leads to the following expression for the eigenvalue λ

$$\lambda = \left[ {\frac{(2\alpha - 1)\pi }{2H}} \right]^{2} \quad \left( {\alpha \, = \, 1,{ 2}, \ldots , \, \infty } \right)$$

The eigenfunction of X(x) to Eq. (17) can be expressed as:

$$X_{\alpha } (x) = A_{\alpha } \cos \left[ {\frac{(2\alpha - 1)\pi }{2H}x} \right],$$

where Aα is a series of constant.

Considering the latter part of Eq. (15) leads to an expression as follows:

$$\frac{{{\text{d}}T(t_{1} )}}{{{\text{d}}t_{1} }} + \lambda c_{\text{v}} T(t_{1} ) = 0.$$

By performing the same processing of X(x), one can obtain the characteristic function of T(t1) to Eq. (20) expressed as follows:

$$T_{\alpha } (t_{1} ) = B_{\alpha } \exp \left\{ { - c_{\text{v}} \left[ {\frac{(2\alpha - 1)\pi }{2H}} \right]^{2} t_{1} } \right\},$$

where Bα is a series of constant.

The excess PWP can then be expressed as follows:

$$u_{1} (x,t_{1} ) = \sum\limits_{n = 1}^{\infty } {X_{\alpha } (x)T_{\alpha } (t_{1} ) = } \sum\limits_{\alpha = 1}^{\infty } {C_{\alpha } \exp \left\{ { - c_{\text{v}} \left[ {\frac{(2\alpha - 1)\pi }{2H}} \right]^{2} t_{1} } \right\} \times \cos \left[ {\frac{(2\alpha - 1)\pi }{2H}x} \right]} ,$$

where Cα is a series of constant.

Considering the initial condition (i.e. at t1 = 0) of the post-deposition process leads to the following expression:

$$u_{0} (x,t_{0} ) = u_{1} (x,0) = \sum\limits_{\alpha = 1}^{\infty } {\left\{ {C_{\alpha } \cos \left[ {\frac{(2\alpha - 1)\pi }{2H}x} \right]} \right\}} ,$$

where u0 (x, t0) is given by Eq. (3).

Equation (23) can be re-written as follows:

$$\int\limits_{0}^{H} {u_{0} (x,t_{0} ) \times \cos \frac{(2\beta - 1)\pi x}{2H}{\text{d}}x} = \int\limits_{0}^{H} {\sum\limits_{\alpha = 1}^{\infty } {\left\{ {C_{\alpha } \left[ {\cos \frac{(2\alpha - 1)\pi x}{2H}} \right]} \right\}} } \times \cos \frac{(2\beta - 1)\pi x}{2H}{\text{d}}x,$$

where β = 1, 2, …, ∞.

The orthogonality of trigonometric function stipulates that the right side of Eq. (24) becomes zero for α ≠ β and can be rewritten as follows for α = β (Asmar 2004)

$$\int\limits_{0}^{H} {u_{0} (x,t_{0} ) \times \cos \frac{(2\beta - 1)\pi x}{2H}{\text{d}}x} = \sum\limits_{\alpha = 1}^{\infty } {C_{\alpha } } \int\limits_{0}^{H} {\left( {\cos \frac{(2\beta - 1)\pi x}{2H}} \right)^{2} {\text{d}}x = C_{\alpha } \times \frac{H}{2}} .$$

The value of Cα can be calculated as

$$C_{\alpha } = \frac{2}{H}\int\limits_{0}^{H} {u_{0} (x,t_{0} ) \times \cos \frac{(2\alpha - 1)\pi x}{2H}{\text{d}}x.}$$

Substituting Eq. (26) into Eq. (22) yields

$$u_{1} (x,t_{1} ) = \frac{2}{H}\sum\limits_{\alpha = 1}^{\infty } {\exp \left\{ { - c_{\text{v}} t_{1} \left[ {\frac{(2\alpha - 1)\pi }{2H}} \right]^{2} } \right\}} \times \left[ {\cos \frac{(2\alpha - 1)\pi x}{2H}} \right]\int\limits_{0}^{H} {u_{0} (x,t_{0} ) \times \left[ {\cos \frac{(2\alpha - 1)\pi x}{2H}} \right]} {\text{d}}x.$$

Appendix 2: MATLAB program for solving Eq. (6)

A MATLAB program is given to solve Eq. (6) and output the excess PWP at an elevation x at a given time t1 in the slurried material. A sample calculation is performed by considering H = 8 m, γsat = 20 kN/m3, γw = 9.8 kN/m3, m = 0.5 m/h, cv = 1 m2/h, t0 = 16 h, and t1 = 10 h.

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Zheng, J., Li, L. & Li, YC. Solutions to estimate the excess PWP, settlement and volume of draining water after slurry deposition. Part I: impervious base. Environ Earth Sci 79, 124 (2020).

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