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

Abstract

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.

Appendices

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:

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

(14)

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

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

(15)

where λ is a constant (λ ≠ 0).

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

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

(16)

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

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

(17)

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

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

(18)

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

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

(19)

where A_α is a series of constant.

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

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

(20)

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

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

(21)

where B_α is a series of constant.

The excess PWP can then be expressed as follows:

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

(22)

where C_α is a series of constant.

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

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

(23)

where u₀ (x, t₀) is given by Eq. (3).

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

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

(24)

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)

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

(25)

The value of C_α can be calculated as

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

(26)

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

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

(27)

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, γ_sat = 20 kN/m³, γ_w = 9.8 kN/m³, m = 0.5 m/h, c_v = 1 m²/h, t₀ = 16 h, and t₁ = 10 h.