Hsi, J. P., Carter, J. P. & Small, J. C. (1994). GLotechnique 44, NO. 3, 381-396 Surface subsidence and drawdown of the water table due to pumping J. P. HSI,* J. P. CARTER* and J. C. SMALL* A fully coupled numerical method is presented for L’article presente une methode numerique per- the analysis of the pumping of groundwater, taking mettant d’analyser le pompage des eaux souter- into account the drawdown of the water table. The raines en prenant en compte le rabattement de la surface subsidence due to pumping is first studied surface libre de la nappe. La subsidence en surface assuming that the water table is not lowered as the du sol, due au pompage, est etudiee en supposant groundwater is withdrawn. This solution is then que la surface piizometrique ne baisse pas au fur et compared with the solution for the case in which a mesure que l’eau est pomp&. Cette solution est the drawdown of the water table is considered. ensuite comparee a la solution obtenue avec un Extensive parametric studies were carried out to rabattement de la surface piezometrique. Des provide non-dimensionalized design charts for etudes parametriques extensives ont ete rCa1isee.s surface subsidence and drawdown of the water pour obtenides abaques adimensionnels donnant la table due to pumping. subsidence de la surface du sol et le rabattement de KEYWORDS: consolidation; finite elements; ground- la nappe dis au pompage. water; pore pressures; settlement. INTRODUCTION deformation. Solutions for the pore-pressure It is a common practice to extract water from the changes obtained from such analyses have gener- ground for domestic, agricultural or industrial ally been used to predict surface settlements in uses or to lower the groundwater level for con- conjunction with a deformation analysis, but such struction work. Many historical cases have shown analyses are uncoupled, since the pore-pressure that such an exercise may cause significant and deformation analyses are carried out separa- surface subsidence (e.g. Scott, 1978; Delfranche, tely (e.g. Gambolati & Freeze, 1973; Gambolati, 1978; Premchitt, 1979; Harada & Yamanouchi, Gatto & Freeze, 1974; Harada & Yamanouchi, 1983; Hsi & Small, 1992a). The reason for this is 1983). that when the groundwater is withdrawn, the Some approaches for the prediction of surface pore-water pressure reduces; also, the water table subsidence due to pumping consider only the is lowered and so there is an increase in effective effect of reductions in pore pressure in the soil for stress, causing consolidation of the layer of soil. the case where the phreatic surface remains This surface subsidence is of interest in geotech- unchanged (Booker & Carter, 1984, 1986a, 1986b, nical engineering, as excessive settlement in the 1987; Small & Booker, 1984; Booker, Carter & vicinity of a pumped well may cause damage to Small, 1985; Kanok-Nukulchai & Chau, 1990). surface structures or render them unserviceable. These solutions have generally been obtained by Therefore, an accurate prediction of ground set- analytic or semi-analytic techniques. However, tlement is sometimes crucial when groundwater is pumping will often draw down the water table, pumped. significantly affecting the distribution of the pore- Some solutions have been obtained for the pre- water pressure, and so these analyses of sub- diction of how the groundwater table will fall as a sidence may not be satisfactory in practice. result of pumping (e.g. Herbert, 1968; Taylor & In this Paper a fully coupled numerical solu- Luthin, 1969; Neuman & Witherspoon, 1971; tion to the pumping problem is obtained. The Rushton & Redshaw, 1979). However, these solu- method takes account of both the effects of the tions do not include predictions of the surface pore-pressure changes on the deformation of the soil and the effects of the deformation of the soil on the pore pressures, as well as the drawdown of the water table. Better predictions of surface sub- sidence are expected with the proposed coupled Manuscript received 22 March 1993; accepted 28 June 1993. method because the effects of pumping are more Discussion on this paper closes 1 December 1994; for accurately simulated; for the uncoupled methods further details see p. ii. already mentioned, soil deformation and draw- * University of Sydney, Australia. down of the water table are calculated separately. 381 382 HSI, CARTER AND SMALL In this coupled analysis, flow is considered in (a) the soil is saturated (at least beneath the water both the saturated and unsaturated zones. table) Reduced flow in the unsaturated zone is simu- (b) the stresses are in equilibrium lated by reducing the permeability based on a (c) the effective stresses and strains obey Hooke’s simplified linear permeability-pore pressure rela- law tion. Deformation of the soil is calculated for the (d) the flow is governed by Darcy’s law entire soil domain, including both saturated and (e) the pore water and soil solids are incompress- unsaturated zones. However, when soil becomes ible unsaturated as the water table drops, pore (f) volume change within the soil corresponds to suction is usually generated in this region. In the expulsion of an equivalent volume of pore reality, this suction may develop only up to a fluid small limit value, depending on the type of soil. (g) ;FalFmations of the soil skeleton are To account for this effect, in the analysis the pore suction must be limited to a specified upper bound. In this way, the soil deformation in the In practice, it is often necessary to solve Biot’s unsaturated zone can be calculated more realisti- equations numerically. Some numerical cally. The permeability-pore pressure relation approaches have been developed by Sandhu & and the limiting pore suction can be selected from Wilson (1969), Christian & Boehmer (1970) experimental data for any particular soil (see Hwang et al. (1971), Small et al. (1976) and Hsi & Bouwer, 1964); however, typical values are used Small (1992b). In this particular problem involv- in the charts in this Paper. ing drawdown of the water table, the time march- Examples are given here for two separate cases : ing scheme (a integration rule) proposed by Small where the water table remains fixed and where it et al. (1976) and the iterative incremental form drops as a result of pumping. Numerical solutions proposed by Hsi & Small (1992b) were used in for the case of no drawdown are verified by pre- the solution procedure. vious analytic solutions (Booker & Carter, 1984) The residual flow procedure proposed by Desai and are then compared with solutions that allow & Li (1983) and Bathe et al. (1982) is used to the water table to be lowered. Non- account for the drawdown of the water table. The dimensionalized design charts are included to concept of this procedure is that when the water allow an estimation of ground surface deforma- table falls from a previous position to the present tion and drawdown of the water table likely to be one, the water initially stored in the pores of the caused by pumping of groundwater. soil in the zone between these two positions is released, and this amount of water needs to be imposed as a flow across the present free surface. COUPLED PROBLEM When this is done, a new location of the free The fully coupled numerical method, which has surface, which is also a zero pore-pressure been developed for the analysis of ground surface contour, can be determined. The essence of this subsidence due to pumping considering draw- solution procedure is that the finite element mesh down of the water table, involves theories of con- remains unchanged and the free surface is allowed solidation (Biot, 1941, 1956; Sandhu & Wilson, to pass through the elements. The terms water 1969; Christian & Boehmer, 1970; Hwang, Mor- table and free surface in this Paper both refer to genstern & Murray, 1971; Small, Booker & the surface along which the pore-water pressure Davis, 1976), transient unconfined seepage (Desai, equals the atmospheric pressure. Following the 1976; Bathe, Sonnad & Domigan, 1982; Desai & usual convention, atmospheric pressure is Li, 1983) and pumping of groundwater (Taylor & adopted as zero for the pore-pressure measure- Luthin, 1969; Neuman & Witherspoon, 1971; ment. Rushton & Redshaw, 1979; Booker & Carter, 1984, 1986a, 1986b, 1987). This particular coupled problem has been studied by Hsi & Small (1992a, 1992b, 1992c, 1992d) who also give details of the Governing equations development of the theory Hsi & Small (1992b) The finite element equations used to obtain an and Hsi (1992). approximate solution to the coupled problem of pumping of groundwater involving drawdown of the water table can be written as Theory K -YJT A&“’ The analysis for a consolidating soil used in --Y,L - y,( 1 - cc)At@ - y, GFS A/r*“’ = ’ this study is based on Biot’s equations (Biot, I[ 1 1941, 1956), which assume that (1) SURFACE SUBSIDENCE DUE TO PUMPING 383 where Along the free surface, the total water head is equal to the elevation head; this can be written as Af (0 yw crAt@h,* + y,(l - cc)AtcDR$,;” r= (2) ‘* - ‘a~_ = f N,(<> r~)(hj* - h,,,j) = 0 (4) --y LS + y LW”f+bl j=l +yo GFs(;:& -wh,*) + y,,, AtQ where m is the number of nodes attached to an All the terms used here are defined in Appendix 1 element; N,(t, q) is the shape function for the and Notation: details of their derivation are given head at node j, hi* and h,,, j are the total water by Hsi & Small (1992b) and Hsi (1992). head and elevation head at node j, and (& q) are Equations (1) and (2) can be solved for nodal local co-ordinates. From equation (4) the local displacements 6 and total water heads /I*. As the co-ordinates (<, r~) along the free surface can be free-surface boundary in the pumping problem found, and hence the global co-ordinates (x, y) (see Fig.