A method predicting pumping-induced ground settlement using back-analysis and its application in the Karla region of Greece

Abstract

In many arid planar regions of the world, ground subsidence induced by the lowering of the water table line due to pumping has recently caused damage to houses and other overlying structures. The depth of the water table lowering is usually tens of meters, the depth of the underlying soil layers may be hundreds of meters, and the region where the lowering is applied may extend tens of square kilometers. In this aspect, the problem under consideration differs drastically from other geotechnical engineering problems and the application of the physical models may have the serious deficiency that required geotechnical information may be incomplete and very costly to obtain: The change in water table variation and the depth of rock are usually known from results of pumping borings and geophysical investigations, but the location, width, compressibility and consolidation characteristics of the clay layers, are usually not known. New space technologies, such as the phase shifting interferometry radar method, provide cost-effective measurements of past displacement data. Based on past displacement measurements, an alternative approach is proposed to predict ground subsidence induced by the lowering of the water table. In particular, the work derives a simplified equation and corresponding methodology which predicts ground subsidence in terms of water table history, based primarily on data of past ground subsidence. This equation was derived and validated based on a state-of-the-art proposed model predicting one-dimensional ground subsidence induced by water level lowering in planar regions. Based on the derived simplified expression, a method predicting the risk at the built environment due to future ground subsidence induced by water level lowering was proposed and applied successfully in a well-documented case study of ground subsidence: the Niki village at Thessaly, Greece.

Appendices

Appendix 1: Derivation of Eq. (11d)

$$s_{t} U\left( t \right) \, = \sum\limits_{k = 1}^{n} {s_{i} U_{i} \left( t \right)} \Rightarrow$$

(20)

$$\begin{aligned}& \left( {s_{1} + s_{2} + \ldots . + s_{n} } \right) \, U\left( t \right) \, = \, \left( {s_{1} + s_{2} + \ldots . + s_{n} } \right) \, \hfill \\ & \quad {-} \, \left( {s_{1} \exp \left( { - 2.67 \, T_{1}^{0.75} } \right) \, + \, s_{2} \exp \left( { - 2.67 \, T_{2}^{0.75} } \right) \, + \, \ldots . \, + \, s_{n} \exp \left( { - 2.67 \, T_{n}^{0.75} } \right)} \right) \Rightarrow \hfill \\ \end{aligned}$$

(21)

$$\begin{aligned} & \ln \left( {1 - U\left( t \right)} \right) \\ & \quad = \ln [(s_{1} \exp ( - 2.67T_{1}^{0.75} ) + s_{2} \exp ( - 2.67T_{2}^{0.75} ) + \cdots + s_{n} \exp ( - 2.67T_{n}^{0.75} ))/s_{t} ] \Rightarrow \\ \end{aligned}$$

(22)

$$\begin{aligned} & \ln \left( {\exp \left( { - 2.67 \, T_{av}^{0.75} } \right)} \right) \, \\ & \quad = \ln [(s_{1} \exp ( - 2.67T_{1}^{0.75} ) + s_{2} \exp ( - 2.67T_{2}^{0.75} ) + \cdots + s_{n} \exp ( - 2.67T_{n}^{0.75} ))/s_{t} ] \Rightarrow \\ \end{aligned}$$

(23)

$$\begin{aligned} & T_{av} = \{ \ln [(s_{1} \exp ( - 2.67T_{1}^{0.75} ) + s_{2} \exp ( - 2.67T_{2}^{0.75} ) + \cdots + s_{n} \exp ( - 2.67T_{n}^{0.75} ))/s_{t} ]/( - 2.67)\}^{1/0.75} \\ \end{aligned}$$

(24a)

With

$$T_{i} = \, 4 \, \left( {t - t_{cr} \left( t \right)} \right) \, \left[ {c_{v - i} /D_{cc - i}^{2} } \right]$$

(24b)

But

$$T_{av} = \, 4 \, \left( {t - t_{cr} \left( t \right)} \right) \, \left[ {c_{v - av} /D_{cc}^{2} } \right]$$

(25)

Thus

$$D_{cc} = \, \left\{ {4 \, \left( {t - t_{cr} \left( t \right)} \right) \, c_{v - av} /T_{av} } \right\}^{0.5}$$

(26)

Substitution of T _av from (24) in (26) gives Eq. (11d).

Appendix 2: Back-analyses procedure estimating the factors A1 and B1 of Eq. (19)

Equation (19) predicts that Δ_tοs (t) equals:

$$\begin{aligned} \Delta_{to} s(t) & = A1\{ [(D - 0.5\{ a_{cw} /a_{c} \} h_{w - m} (t))]log[1 + \gamma_{w} h_{w - m} (t)(D - 0.5h_{w - m} (t))/(D\sigma^{\prime}_{v - av} )] \\ & \qquad [1 - \exp( - 2.67*B1(t - t_{{cr({\mathbf{t}})}} )^{.75} )] \\ & \quad - [(D - 0.5\{ a_{cw} /a_{c} \} h_{w - m} (to)]\log[1 + \gamma_{w} h_{w - m} (t_{o} ) (D - 0.5h_{w - m} (t_{o} ))/(D\sigma^{\prime}_{v - av} )]\\ & \qquad [1 - \exp( - 2.67*B1(t - t_{{{{cr}}({{to}})}} )^{.75} )]\} \\ \end{aligned}$$

(27)

When two pairs of settlement (Δ_tοs (t₁), (Δ_tοs (t₂)) for water table change heights (h_w−m (t1), h_w−m (t2)) exist at two different times (t1, t2), in the region of interest, Eq. (19) can be expressed as

$$\begin{aligned} \Delta_{to} s(t1)/\Delta_{to} s(t2) = \{ [(D - 0.5\{ a_{cw} /a_{c} \} h_{w - m} (t1)]\quad \log [1 + \gamma_{w} h_{w - m} (t1)(D - 0.5h_{w - m} (t1))/(D\sigma^{\prime}_{{v_{ - av} }} )[1 - \exp ( - 2.67B1(t_{1} - t_{cr(t1)} )^{0.75} )] - Ao\} / \{ [(D - 0.5\{ a_{cw} /a_{c} \} h_{w - m} (t2)](\log [1 + \gamma_{w} h_{w - m} (t2)(D - 0.5h_{w - m} (t2))/(D\sigma^{\prime}_{{v -_{av} }} )/[1 - \exp (2.67B1(t_{2} - t_{cr(t2)} )^{0.75} )] - Ao\} \\ \end{aligned}$$

(28a)

where

$$\begin{aligned} Ao & = [(D - 0.5\{ a_{cw} /a_{c} \} h_{w - m} (to)]\log [1 + \gamma_{w} h_{w - m} (t_{o} )(D - 0.5h_{w - m} (t_{o} ))/(D\sigma^{\prime}_{{v -_{av} }} )] \\ & \quad [1 - \exp ( - 2.67*B1(t_{0} - t_{cr} (to))^{.75} )] \\ \end{aligned}$$

(28b)

From Eq. (28), the factor B1 can be obtained numerically as:

$$f\left( {B1} \right) \, = \, 0$$

(29a)

where:

$$\begin{aligned} f(B1) = \{ \Delta_{to} s(t1)/\Delta_{to} s(t2)\} \{ [(D - 0.5\{ a_{cw} /a_{c} \} h_{w - m} (t2)](\log [1 + \gamma_{w} h_{w - m} (t2)(D - 0.5h_{w - m} (t2)/(D\sigma_{{v^{\prime}_{av} }} )/[1 - \exp (2.67B1(t_{2} - t_{cr} (t2))^{0.75} )] - \, Ao\} - \{ [(D - 0.5\{ a_{cw} /a_{c} \} h_{w - m} (t1)\} ]\log [1 + \gamma_{w} h_{w - m} (t1)(D - 0.5h_{w - m} (t1)/(D\sigma_{{v^{\prime}_{av} }} )) [1 - \exp ( - 2.67B1(t_{1} - t_{cr} (t1))^{0.75} )] - Ao\} \\ \end{aligned}$$

(29b)

Once the factor B1 is obtained, the factor A1 can be estimated from Eq. (27) as:

$$\begin{aligned} & A1 = \Delta_{to} s(t1)/\{ [1 - \exp ( - 2.67B1(t_{1} - t_{cr} (t1))^{0.75} )]* \\ & \quad [(D - 0.5\{ a_{cw} /a_{c} \} h_{w - m} (t1)]*\log [1 + (\gamma_{w} h_{w - m} (t1)*(D - 0.5h_{w - m} (t1)))/(D*\sigma_{voav} )] - Ao\} \\ \end{aligned}$$

(30)