Modelling of pumping from heterogeneous unsaturated-saturated porous media M. Mavroulidou & R.I. Woods Email: M.Mavroulidou@surrey.ac.uk; R. Woods@surrey.ac.uk Abstract Practising civil engineers often use analytical solutions to estimate pumping rates and drawdowns associated with dewatering systems. These solutions are usually based on assumptions of predominantly horizontal flow as well as of homogeneous permeability and compressibility of the soil. The disadvantage of these methods is that they are based on the assumptions of saturated flow, despite the fact that after intensive pumping, more permeable layers of an multiaquifer system might desaturate. Engineers need more versatile modelling tools if they are to design successful dewatering systems. For this reason, a finite element code which takes into account soil desaturation has been developed by the writers. The present paper investigates the problem of well drying when water is continually abstracted for a certain time period in a multi-layered aquifer system. The analyses focus on the influence of heterogeneity induced by the soil layering. 1. Introduction Pumping over a period of time can cause a water table to form followed by the progressive development of suctions. Nevertheless, a standard approach to modelling the dewatering of soils by practising geotechnical engineers is to use the well known analogy of heat flow in a heat conducting medium to the flow of water through soil, based on the assumption of confined, saturated flow. Existing geotechnical computer codes that attempt to model moving water tables often assume saturated soils as well and can only have a limited application to well hydraulic
500 Hydraulic Engineering Software problems. These models neglect all water retained by and flowing through the unsaturated zone and assume instantaneous release of water when fall of the water table occurs. Moreover, simplified curves are used in these models for both hydraulic conductivity and moisture capacity terms, applied to any type of material (Desai and Li [3], Bathe et al. [1], Hsi et al [4]). It is known however, that the validity of numerical results involving unsaturated soils depends on the choice of appropriate unsaturated hydraulic coefficients for each soil. To model more realistically well pumping that might involve desaturation of the soil, the writers developed a finite element code in which hydraulic coefficients of a number of soils vary with suction according to experimental results found in the literature. The pumping problem considered here is relevant to well drying when water is continually abstracted over a certain time period in a multi-layered aquifer system. The purpose of these analyses is to focus on the effects of the material properties (sand, silt, clay) and to understand the mechanisms involved when multi-aquifer systems progressively desaturate. The simplified aquifers assumed here are sample aquifers used to demonstrate a numerical effect via parametric studies, rather than actual aquifers. The results can however show the general trends that actual aquifer materials will present upon pumping or drainage. 2. Pumping Analyses 2.1 Proposed Numerical Model The governing equation for transient flow through an unsaturatedsaturated soil system is: ir) + 2 = <-S, where h is the total head, C is the moisture capacity term, defined as C(\ )- ^, Ss is the elastic storage coefficient, 0 is the volumetric cty moisture content and n is the porosity. K%, Ky and K% are the
Hydraulic Engineering Software 501 permeabilities in the x, y and z directions respectively, given as functions of the pressure head \\i and Q are point sources or sinks. It should be noted that for steady seepage where dhdi =0, the right hand term of eqn(l) vanishes. For the derivation of eqn(l) validity of Darcy's law in both the saturated and the unsaturated zones has been assumed throughout the analysis. The dependence of both the permeability as well as the capacitance terms on the head variations causes eqn (1) to be strongly non-linear. For the linearization of the equation a Picard iterative scheme was adopted and a backward Euler finite-difference scheme was used for time integration. The non-linear relationship of volumetric water content and hydraulic conductivity to pressure head in the unsaturated zone, is expressed respectively by: + aiv (2)., 1 4- a2v The moisture capacity term C(v )= - is obtained by differentiation 3v of eqn(2) with respect to the pressure head. The parameters a,,nj and &2,n2 involved in these relationships are back-fitted from experimental data. In the present analyses, experimental water characteristic curves and permeability curves for a silty sand (given by Vauclin et al. [6]), Touchet silt (Brooks and Corey [2]) and Jurong clay (provided by Leong and Rahardjo [5]) have been used to represent the behaviour of three different soil types (see Figures la and Ib). 2.2 Geometry, boundary conditions and material properties The problem considered here, is relevant to pumping from a grid of wells 100m deep and spaced 1km apart, pumping an aquifer 200m deep. The layered aquifers analysed, this consist of an upper 50m layer of impermeable material and a 150m layer of more permeable material, which is being de watered. An initial hydrostatic state of pore pressure has been assumed throughout the aquifer depth. A single well of the grid, which is remote from the edge is being analysed herein. The mesh used in the analysis extends up to 500m horizontally (i.e. the mid-space
502 Hydraulic Engineering Software between two wells). Because of symmetry, the boundary conditions at the edge of the mesh are assumed to be no-flow conditions (i.e. impermeable boundary). It is assumed that the pumping is produced at a constant head, corresponding to an initial sudden drawdown of 90m in the well. This is represented by fixed head boundary conditions equal to the level of the water in the well, imposed on the corresponding nodes of the mesh. For the sake of simplicity, no lining of the well was assumed, i.e. there is a possibility of seepage face developing along any node of the well, up to the soil surface. To assess the effects of the material properties (sand, silt, clay) and the influence of the heterogeneity (layered soil) in the progression of the pore pressure development aquifer, two different sets of transient analyses have been made. These analyses consider layered aquifers consisting of an impermeable material overlying a permeable material. Thus, thefirstof these analyses represents a sandy aquifer overlain by a clayey layer (referred to as 'analysis CA'), while the second analysis represents a sandy aquifer overlain by a silty layer (referred to as 'analysis BA'). The results are represented in form of total heads at different distances away from the well. 2.3 Presentation and discussion of the results For the sake of brevity, only selected results are presented in this section. From the figures shown here (figures 2a-2c and 3a-3b), it can be seen that for all sets of analyses, as expected, at the initial time steps, the controlling parameter for the pumping evolution is the most permeable soil. The total head profiles in the impermeable material show a substantial head drop, consistent with the under-drainage of the impermeable layer. More precisely, the clay-sand profiles show that in 1 years time, the clay layer is still saturated and that most of the head drop is within the clay. There is indication of some horizontal flow in the clay -against what is usually assumed in aquitard-aquifer analyses-. The sands yielding capacity is already reduced as with respect to the profile of Sdays, and this shown by the total head profile that tends to assume a vertical position in the sand. After 73 years, the sand has already achieved steady state (not shown here). Conversely the pattern of flow in the clay layer is evolving at very slow pace and the total head difference between 248 and 2000 years, when the analyses stopped (the latter time level results are not shown here) is only a few meters of difference. The change of head is slower than in a homogeneous clay material (this analysis performed by the writers is not shown here), and this might be
Hydraulic Engineering Software 503 an indication that the desaturation of the sand is obstructing flow between the two layers. Indeed, at these suction levels, the hydraulic conductivity of the sand is lower than that of the clay (Fig. Ib). The BA (silt-sand) analyses results start off showing a typical behaviour of a relatively impermeable material (silt) overlying a more permeable material (sand), i.e. an inclined head profile in the silt (showing head drop inside the more impermeable layer and underdrainage of this layer) and a relatively horizontal total head profile in sand. Subsequently, however, the silt-sand layered material behaviour changes, and the profiles head uniformly towards the steady state (ie. become progressively vertical throughout both layers see Fig 3b) that is achieved after 248 years. This can also be explained by Fig la, showing that the total heads developed in the sand, correspond to low negative pressure heads (i.e. the range of negative pressure heads where the hydraulic conductivity of the sand is still higher than that of the silt). In subsequent times however, the hydraulic conductivity of the sand drops, and the silt-sand system starts acting as an homogenised material of similar hydraulic conductivity. One can also notice that there is horizontal flow developing in the silt layer at time level equal to 1 year. This is not what it is commonly assumed, i.e. that when a permeable material is overlain by an impermeable material (difference of at least two orders of magnitude in hydraulic conductivities) flow is vertical in the aquitard and horizontal in the aquifer. 3. Conclusions When unsaturated flow is involved, the response of soil due to pumping may diverge from commonly assumed behaviour. Results from the present analyses results suggest that in layered soils subject to desaturation, an inversion of the conditions might occur, such that the coarse material becomes less permeable than the fine-grained material. Therefore, if desaturation is expected to occur, further research might be needed in order to assess the validity of simplifying assumptions commonly used to model multi-layered aquifer systems, e.g. that the system will show mainly horizontal flow in the permeable material and vertical flow in the aquitard material, provided these two materials differ by two orders of magnitude. It has been shown here that this assumption may not hold true throughout a transient pumping analysis when the soil layers are allowed to desaturate. Computer codes such as that developed by the writers show promise for a more realistic modelling of pumping
504 Hydraulic Engineering Software involving desaturation of the soil layers, provided reliable soil data exist for the assumed hydraulic property variations in the unsaturated zone. References [1] Bathe, K.J., Sonnad, V., Domigan, P., "Some experiences using finite element methods for fluid flow problems", Proceedings of the 4th conference on finite element methods in water resources. Hannover, West Germany, June 1982, pp.9.3-9.16. [2] Brooks R.H. & Corey A. T., Properties of Porous Media Affecting Fluid Flow, ASCE Journal of the Irrigation and Drainage Division. June 1966 (IR 2), pp. 61-88. [3] Desai, C.S., Li, G.C., "A Residual Flow Procedure with Application for Free Surface Flow in Porous Media", Advances in Water Resources. 6, 1983, pp. 27-35. [4] Hsi J.P., Carter J.P. and Small J.C., Surface subsidence and drawdown of the water table due to pumping, Geotechnique 44 (3), pp. 381-396, 1994, [5] Leong B.C. & Rahardjo H., Typical Soil-Water Characteristic Curves for two Residual Soils from Granitic and Sedimentary Formations, Irst International Conference on Unsaturated Soils. Vol. 2, Balkema, RotterdamEcole Nationale des Fonts et Chaussees, Paris, pp519-524, 1995. [6] Vauclin M., Khanji D., Vachaud G., Etude experimental et numerique du drainage et de la recharge des nappes a surface libre, avec prise en compte de la zone non saturee, Journal de Mecanique. Voll5(2),pp.307-348, 1976.
Silty Sand (Vauolin?l al., 1976) Touchet Silly loam (Brooks & Corey, 1966) Jurong cloy fleong & Raherdjo, 1995) Hydraulic Engineering Software 505 O OJ OT o O P 2-22 -17-12 -7-2 PRESSURE HEAD (m) Comparative hydraulic conductivities of three soils 0.50 Silly Sand (Vauclm et al,, 1976) Touchet Silly loam (Brooks & Corey, 1968) Jurong clay (Leong & Rahardjo, 1995) 0.00 0 5 10 15 pressure head (m) Comparative volumetric contents for three soils (sand, silt, clay) Figures la) and Ib) Hydraulic properties of the three soils assumed here
506 Hydraulic Engineering Software \ ': 4 JA 5 1 Tata head * - Tota head AAAAA Tota head -MI Tota1 head ^ ; ^ ' ^Jj 1 7 1 Tr ; j 3! 2 i.4 a I 2rn from the at 20m from the at 50m from the well at 200m from the well 0-0,0 60.0 100,0 150.0 Total heed (m) 1 1 200.0 Variation of total heads with depth clay-sand layered aquifer aquitard system, Ume=B days a-ar*-**. at 2jn from the ve]] at 20m from the well at 50m from the well at 200m from the well )\ ( \ ^ $ * \-~~ <**** * o - 0.0 50.0 100.0 160.0 Total heed (m) 200,0 Variation of total heads with depth clay-aand layered aquifer-aquitard system, time=j year
Hydraulic Engineering Software 507 ' '' *** -«* al 2ni from the well al 20m from the well at 50m from the well at 200m from the veil 1 Hfi - 1 fia - 1 9ft 1 nn QA f ^^ X y nri ^ A - 0,0 50,0 100,0 150,0 200,0 Total heed (nj) Variation of total heads with time clay-sand layered aquifer-a quitard system, time=248 years Figures 2a)-2c) Presentation of selected results a clay-sand aquitard-aquifer system 220 200-4 (% _ ^^ 1 9 A - <L> o flrt - An ~~*-~ *+.+**> mm-mm-m r 1 ^^ ii % ;i 5; 1f i j 1T at 2m from the wel] at 20m from the well al 50m From the well at 200m from the well 0-0,0 60.0 100,0 150.0 (m) Variation of total heads with depth silt sand layered aquifer, time=l year 1 200,0
508 Hydraulic Engineering Software nnfi *****.._... at Sin from the well al 20m from the well at 50m from the well al 200m from the well 160 s : 120 - rs *-* I & o : o : 80-60 - 40-20 -,i \# j 0-0,0 50.0 100,0 150,0 (m) Variation of total heads with time silt-sand layered aquifer, time=73 years 1 f-^ 200,0 *-* * *****. -.^ at 2rn from the well al 20m from the well al 50m from the well al 200m from the well <u» 0-0.0 50.0 100,0 150,0 200,0 Tolal heed (ni) Variation of total heads with time silt-sand layered aquifer, time=24b years Figures 3a)-3c) Presentation of selected results for a silt-sand aquitardaquifer system