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

Abstract

It is a critical concern to properly manage the risks associated with the stability of tailings and dredge sludge dams. This requires the knowledge of the pore water pressure (PWP) within a slurried material during and after the deposition. The evaluation of the PWP can also help to estimate the settlement, which is important for the water management and design of the storage facility. An analytical solution based on the Gibson model has been proposed by the authors to evaluate the excess PWP during the slurry deposition. However, there is no solution available to estimate the excess PWP and volume of draining water after the end of slurry deposition. In this paper, a new solution is proposed to evaluate the PWP, settlement, and volume of draining water after the slurry deposition on a pervious base. The proposed PWP solution is partly validated by numerical modeling with SIGMA/W and laboratory results. The influence of the consolidation coefficient and filling rate on the distribution and evolution of PWP is analyzed. The solution development for estimating the excess PWP, volume of draining water and settlement after slurry deposition on an impervious base has been presented in a companion paper (part I).

Appendices

Appendix A: Derivation process for the proposed solution (Eq. 5)

Equation (3) can be solved by applying the method of separation of variables (Asmar 2004). The function p_w1(z, t₁) can be expressed as follows:

$$ p_{w1} (z,t_{1} ) = Z(z)T(t_{1} ), $$

(13)

where Z(z) and T(t₁) are functions of z and t₁, respectively.

The boundary conditions p_w1 = 0 at z = 0, H can then be met by considering Z(0) = 0 and Z(H) = 0.

Substituting Eq. (13) into Eq. (3) yields

$$ \frac{1}{Z(z)}\frac{{d^{2} Z}}{{dz^{2} }} = - \lambda = \frac{1}{{c_{v} T(t_{1} )}}\frac{dT}{{dt_{1} }}, $$

(14)

where λ is a non-zero constant (λ ≠ 0).

The former equality of Eq. (14) can be rewritten as

$$ Z^{\prime \prime } (z) + \lambda Z(z) = 0 $$

(15)

A general solution of Eq. (15) is given as follows

$$ Z(z) = C_{1} \sin (\sqrt \lambda z) + C_{2} \cos (\sqrt \lambda z) $$

(16)

Taking the boundary conditions Z(0) = 0 and Z(H) = 0 into Eq. (16) leads to

$$ C_{2} = 0 $$

(17)

$$ C_{1} \sin \sqrt \lambda H = 0 $$

(18)

The characteristic value of λ can then be obtained from Eq. (17) as follows:

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

(19)

The characteristic function of Z(z) for Eq. (20) can be expressed as follows:

$$ Z_{\alpha } (z) = A_{\alpha } \sin \left( {\frac{\alpha \pi }{H}z} \right), $$

(20)

where A_α is a series of constants.

The latter equability of Eq. (14) can be rewritten as follows:

$$ T^{^{\prime}} (t_{1} ) + \lambda c_{v} T(t_{1} ) = 0 $$

(21)

The characteristic function of T(t₁) for Eq. (21) can be solved as follows:

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

(22)

where B_α is a series of constants.

Introducing Eqs. (20) and (22) into Eq. (13) leads to

$$ p_{w1} (z,t_{1} ) = \sum\limits_{\alpha = 1}^{\infty } {Z_{\alpha } (t)T_{\alpha } (t_{1} )} = \sum\limits_{\alpha = 1}^{\infty } {C_{\alpha } } \exp \left( { - c_{v} \left( {\frac{\alpha \pi }{H}} \right)^{2} t_{1} } \right) \times \sin \left( {\frac{\alpha \pi }{H}z} \right), $$

(23)

where C_α is a series of constants.

Considering the initial condition (at t₁ = 0) of Eq. (4) for the post-deposition process, the following equation can be obtained:

$$ p_{w0} (z,t_{0} ) = p_{w1} (z,0) = \sum\limits_{\alpha = 1}^{\infty } {C_{\alpha } \sin \left( {\frac{\alpha \pi }{H}z} \right)} $$

(24)

or

$$ \int_{0}^{H} {p_{w0} (z,t_{0} )} \times \sin \left( {\frac{\beta \pi z}{H}} \right)dz = \int_{0}^{H} {\sum\limits_{\alpha = 1}^{\infty } {C_{\alpha } } } \times \sin \left( {\frac{\alpha \pi z}{H}} \right) \times \sin \left( {\frac{\beta \pi z}{H}} \right)dz $$

(25)

The orthogonality of trigonometric function stipulates α = β (otherwise, the right side of Eq. (25) is equal to zero for α ≠ β). One thus has

$$ \int_{0}^{H} {p_{w0} (z,t_{0} )} \times \sin \left( {\frac{\alpha \pi z}{H}} \right)dz = \int_{0}^{H} {\sum\limits_{\alpha = 1}^{\infty } {C_{\alpha } } } \left[ {\sin \left( {\frac{\alpha \pi z}{H}} \right)} \right]^{2} dz = C_{\alpha } \times \frac{H}{2} $$

(26)

or

$$ C_{\alpha } = \frac{2}{H}\int_{0}^{H} {p_{w0} (z,t_{0} ) \times \sin \left( {\frac{\alpha \pi z}{H}} \right)} dz $$

(27)

Substituting Eq. (27) into Eq. (23) leads to

$$ p_{w1} (z,t_{1} ) = \frac{2}{H}\sum\limits_{\alpha = 1}^{\infty } {\exp \left[ { - c_{v} \left( {\frac{\alpha \pi }{H}} \right)^{2} t_{1} } \right]} \times \sin \left( {\frac{\alpha \pi z}{H}} \right) \times \int_{0}^{H} {p_{w0} (z,t_{0} )} \times \sin \left( {\frac{\alpha \pi z}{H}} \right)dz $$

(28)

Appendix B: MATLAB program for solving Eq. (5)

A MATLAB program is given here to solve Eq. (5) and output the (excess) PWP at an elevation z at a given time t1. An example of calculation is conducted by using typical geometry and property parameters with H = 8 m, γ = 20 kN/m³, m = 0.2 m/h, cv = 1 m²/h, t0 = 40 h, and t1 = 5 h.