Parameters affecting the occurrence of quicksand and the drying up of large diameter wells that gain water from the bottom: a case study from Iran Mazda Kompani-Zare & Nozar Samani & Siavash Behrooz-Koohenjani
Abstract Large diameter fully cased wells that gain water from the bottom are often dug in sandy and collapsible aquifers. They have cylindrical vertical walls lined with brick or concrete. The well bottom is partially filled with aquifer material through which the flow is vertically upward. When the vertical hydraulic gradient reaches a critical value, quicksand occurs and the well structure can be destroyed. Another difficulty encountered is drawdown in the wellbore and the drying up of the well. To overcome these problems, the flow around and beneath these wells is numerically simulated. The simulation results are used to investigate the effect of well and aquifer parameters on quicksand and drawdown. For practical purposes, the dimensionless drawdown-time and dimensionless vertical gradient-time curves are developed. It was found that the ratio of filling material thickness to well radius affects the shape of these type curves. The type curves may be used to predict the time after pumping commences when quicksand occurs and the well dries up. They are also useful to design the safe pumping rate and duration as well as the optimum well radius. These are demonstrated by analyzing the pumping test data from a case study in the arid Chah Kutah region, southern Iran.
Received: 15 January 2008 / Accepted: 15 December 2008 Published online: 27 January 2009 © Springer-Verlag 2009 M. Kompani-Zare Department of Desert Region Management, School of Agriculture, Shiraz University, Bajgah Ave., Shiraz, 71454, Iran N. Samani ()) : S. Behrooz-Koohenjani Department of Earth Sciences, College of Sciences, Shiraz University, Golestan Street, Shiraz, 71454, Iran e-mail:
[email protected] Tel.: +98-711-2282380 Fax: +98-711-2280926 Hydrogeology Journal (2009) 17: 1175–1187
Keywords Quicksand . Large diameter well . Inverse modeling . Groundwater flow . Iran
Introduction In many arid regions around the world the ground surface is covered by dune sand deposited by wind. These sediments are always thick and are the main source of groundwater. Sandy deposits carried and settled by wind are usually of fine sand grain size. The wells dug in these sandy aquifers are prone to collapse. Therefore, the wells are lined and sealed with brick, concrete or cemented stone walls. These wells usually gain water from their bottom. If the pumping rate exceeds a specific limit, quicksand occurs at the bottom. As quicksand occurs, a large amount of sand will gradually be extracted from the body of the aquifer around the well. The extraction of sand creates large holes and destroys the well structure. Quicksand causes land subsidence, filling of the wellbore (the physical hole inside the well casing) storage and abrasion of pipes and pumping utilities. Another problem in this kind of well is the drying up of the well. Because of the low hydraulic conductivity of fine sand, the drawdown in the wellbore is large, and the well dries up after a short period of pumping. The question here is, what is the safe pumping rate and duration of pumping to ensure that quicksand formation and drying up of the well does not occur? Quicksand and its controlling parameters have been studied before by Matthes (1953), Patchick (1966), Pastor (1981), Bello-Maldonado (1983), Bea and Aurora (1983), Garritty (1983) and Wang and Ma (1999). These studies are mainly concentrated on quicksand relating to dam construction and the foundations of large structures. Patchick (1966) investigated quicksand in water wells. His study is basically qualitative and he did not focus on the detailed simulation of flow into the wells. He suggested methods for drilling this type of well and, based on the average grain size and permeability of the associated sandy aquifers, he suggested an allowable entrance velocity to the well of 0.1 ft/s or (0.012 m/s). The quicksand phenomenon, which is known as “sand DOI 10.1007/s10040-008-0426-7
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production”, is considered to be one of the major problems that have perplexed the petroleum industry. Every year the petroleum industry spends millions of dollars in sand cleaning, repair problems related to sand production and lost revenues due to restricted production rates (Islam and George 1991) To prevent quicksand, a filter or screen can be used at the bottom of the well. However, because of the economic and technical problems of installing filters in such large diameter wells, it is preferable to find a way to control the quicksand rather than using filters. This paper investigates the effect of well dimensions and aquifer parameters in controlling the quicksand phenomenon. Well dimension parameters include wellbore diameter (2rw, where rw is the wellbore radius), well casing thickness and depth. Aquifer parameters include hydraulic conductivity (K), specific yield (Sy) and initial saturated thickness of the aquifer (b). By finding the effect of each parameter on the quicksand occurrence, a set of parameter values can be suggested for which quicksand does not occur. Flow to large diameter wells has been studied by many researchers (Singh and Gupta 1988; Balkhair 2002). However, large diameter wells gaining water only from the bottom have not yet been assessed. The unique property of the wells studied in this research is that the wellbore is partially filled with aquifer materials and the flow in the filled portion of the wellbore is vertically upward. The analytical solution for flow to these large diameter wells has not yet been found. Therefore, to determine the vertical hydraulic
gradient (iv), the main cause of quicksand, the flow to the well (Q) is simulated numerically. MODFLOW 2000 (Harbaugh et al. 2000) is a modular finite difference groundwater model code for simulation of groundwater flow. To simulate the radial flow around a well, MODFLOW 2000 may be embedded with methods such as those presented by Reilly and Harbaugh (1993) or Samani et al. (2004). Here the so-called LSM (log scaling method) presented by Samani et al. (2004) is utilized along with MODFLOW 2000. By simulation of the unsteady flow around and beneath the well, the spatial and temporal variation of the vertical hydraulic gradient at the bottom of the well and the drawdown in the wellbore (s) can be determined. The emphasis is on delineating how aquifer and well characteristics affect pumping-induced drawdown, so that field practitioners will be able to plan the safe pumping rate and duration of pumping and to design a well free of quicksand. A comparison to the case of a finite diameter partially penetrating well with well storage (Moench 1997) is made and the effective parameters controlling the quicksand and well drying up phenomena are investigated.
Study site The study site is in the Chah Kutah region, in Bushire province, southwest Iran (Fig. 1). Chah Kutah in Persian means “shallow well”. This region is famous for its shallow and large diameter water wells which supply
Fig. 1 The location of the study area and the selected wells in Chah Kutah region, the selected water wells based on their owners’ names are shown as: GSH (Gholamali Shojaiee), MZ (Mahmood Zangeneh), and AMZ (Abdol Majid Zangeneh) Hydrogeology Journal (2009) 17: 1175–1187
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this region are shown in Fig. 1. The three wells selected for this study are highlighted based on their owners' names, they are GSH (Gholamali Shojaiee), MZ (Mahmood Zangeneh), and AMZ (Abdol Majid Zangeneh) wells. Because of its intermediate depth and radius, the MZ well is selected as a typical representative well for the simulation of flow.
Well structure and flow pattern
Fig. 2 A large diameter well with a brick wall. The well wall is extended above the ground surface
irrigation water (Fars Water Affair Organization 1998) and gain water from the bottom. The region is covered by dune sand in the northeast (Fig. 1). Further to the northeast, the underlying Bakhtyari Formation crops out, which consists of cemented conglomerate (James and Wynd 1965). The remainder of the region is covered by alluvium which consists of sand, silt and clay. The water table depth varies from 4 to 7 m. The thickness of the sandy aquifer is 30 to 40 m (Fars Water Affair Organization 1998). The average discharge rate from the water wells is about 4–5e–3 m3/s (Fars Water Affair Organization 2002). The locations of some of the wells in
The water wells in Chah Kutah region are large diameter wells. Their diameters are more than 2 m and sometimes as large as 5–6 m (Fig. 2). The aquifer material in this region is fine loose sand; therefore the wells are cased and stabilized by cemented stone pieces or bricks or by precast reinforced concrete cylinders creating an impervious well wall (Fig. 2). Their depths range from 6 to 13 m (Fars Water Affair Organization 2002). The schematic cross section of a typical well is illustrated in Fig. 3. The water enters the well from the bottom only. The bottom of the well is partly filled with aquifer material (Fig. 3), where the vertical upward flow of water may create quicksand. Quicksand happens when the hydraulic gradient of the flow is higher than the critical hydraulic gradient of the aquifer materials. The well dimensions and the aquifer parameters which may control the occurrence of quicksand are illustrated in Fig. 3. The parameters are the well wall thickness, the wellbore radius, the initial water-table elevation, the depth of aquifer bottom and the filling material thickness.
Fig. 3 Schematic diagram of a large diameter well gaining water from its bottom (ATB is the aquifer material thickness beneath the well bottom) Hydrogeology Journal (2009) 17: 1175–1187
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Fig. 4 Quicksand condition; h is hydraulic head above the bottom of excavation and d is the sand layer thickness (after Capper and Cassie 1978)
Quicksand When the upward seepage pressure due to an upward flow of water in sand exceeds the downward effective pressure due to its buoyant weight then quicksand will occur. Figure 4 shows an example of how quicksand may be encountered at the bottom of an excavation. The downward effective pressure at depth d is (γ–γw)d, where γ is the saturated weight density of the soil and γw is the weight density of water. The upward pressure is obtained by multiplying the hydraulic head h by the weight density of water, i.e. hγw. When the upward pressure exerted on the particles of sand is equal to the downward pressure, a critical condition ensues. If the seepage pressure increases beyond this critical point the particles will be suspended and quicksand conditions occur. hg w ¼ ðg g w Þ d
ð1Þ
h g gw ¼ d gw
ð2Þ
This is known as the critical hydraulic gradient, ic Replacing γ by ½ðGs þ eÞ=ð1 þ eÞw, Eq. (2) becomes (Capper and Cassie 1978; Terzaghi et al. 1996): ic ¼
Gs 1 1þe
ð3Þ
where Gs is the aquifer solid material density (mass of solid/volume of solid) and e is the aquifer material void ratio (volume of voids/volume of solid). Theoretically, the size of the sand grains has no effect, though it is well known that quicksand is of more frequent occurrence in fine sands (Capper and Cassie 1978). In general, fine sediments with a loose state of packing, of uniform grain size, having a high void ratio and a correspondingly low critical hydraulic gradient are susceptible to quicksand. The critical gradient can be determined in the laboratory by the creation of upward flow with different vertical gradients in a cylinder filled with the soil sample. The Hydrogeology Journal (2009) 17: 1175–1187
critical gradients obtained by laboratory methods for the selected samples are in the range of 0.79–1.4.
Numerical simulation of vertical upward gradient beneath the well In order to predict when quicksand occurs or when the wellbore will dry up, it is necessary to study the flow pattern around and beneath the well during pumping. There is no analytical solution for the accurate prediction of flow around and beneath large diameter wells, especially those that gain water only from the bottom. In this study, the flow around the well is numerically simulated by MODFLOW 2000 (Harbaugh et al. 2000), equipped with the so-called LSM (Samani et al. 2004). Through simulation of flow around and beneath the well, the drawdown in the wellbore, the vertical gradient at the bottom of the well, and its temporal and spatial variations are determined. The simulations are done for the representative well, MZ, whose parameters are given in Table 1. Figure 5 presents schematically the well radial cross Table 1 Aquifer and well parameters used in the flow simulation Aquifer parameters Hydraulic conductivity (m/s) Specific yield Specific storage (1/m) Hydraulic conductivity (m/s) Aquifer bottom elevation (m) Initial water table elevation (m) Ground surface elevation (m) Well parameters Wellbore radius (m) Discharge from the well (m3/s) Elevation of well wall bottom (m) Elevation of well bottom (m) Well wall thickness (m) Model parameters Number of layers Number of columns Number of rows Number of stress periods Number of time steps Time steps multiplier Time length (s)
1.2e – 3 0.4 1.032e 3 1.2e + 6 80 94.6 1.0e + 2 1.1 4.71e – 3 90.7 92.2 1.65e – 1 49 68 1 1 4.5e + 3 1.0008558 2.16e + 4
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The simulation results The flow around the well, with the parameters given in Table 1, is simulated. The temporal and spatial variation of the vertical gradient beneath the well and the drawdown in the wellbore are determined. By knowing the vertical gradient beneath the well and the drawdown in the wellbore, the time at which quicksand occurs and the time when the wellbore dries up can be predicted. The results of the simulation are presented in the following sections.
The spatial variation of hydraulic head contours
Fig. 5 Cell network in the well radial cross-section used in the simulation of flow by MODFLOW 2000 and the log scaling method (LSM)
section and the cell network used in the simulation. To reduce the numerical error due to flow concentration and high hydraulic gradient in the vicinity of the well and at its bottom, the expansion factor of 1.4 (Barrash and Dougherty 1997) is used in the cell dimensions in radial and vertical directions. The cell widths at the casing edge in radial and vertical directions are 0.001 m and increase gradually away from the well with the expansion factor of 1.4 (Fig. 5). Also, the constant head boundary is set at a distance of about 450 m from the well center. In order to consider the effect of wellbore storage in the simulation, for the cells located in the wellbore the specific yield is set to 1 and the hydraulic conductivity is set to 1.2e + 6 m/s.
For the well with the parameters given in Table 1, the simulation is carried out. The hydraulic head contours which developed after 6 h (2.16e + 4 s) from the start of pumping are drawn in Fig. 6a. The hydraulic head contours are perpendicular to the well centerline and to the well casing. As can be seen in Fig. 6a, the hydraulic head contours are dense inside the well and widen gradually outwards from the well. The initial head in the aquifer in this simulation is 94.6 m. Figure 6a reveals that about 1.6 m or 73% of the total head difference (94.6 – 92.4=2.2 m) occurs through the filled part of the wellbore. Since the head in the wellbore is 92.4 and the elevation of the well bottom is 92.2 m, this means that the wellbore has not been dried up.
The spatial variation of the vertical gradient
The head difference per unit length of flow path in the vertical direction may be called the vertical gradient. Figure 6b illustrates the variations of vertical gradient of flow for the same case just simulated (see section The spatial variation of hydraulic head contours). In this figure,
Fig. 6 a The simulated hydraulic head contours, in meters, for the flow around the well and in the aquifer with the parameters given in Table 1. b The simulated hydraulic gradient zones for the flow around the well with the parameters given in Table 1 Hydrogeology Journal (2009) 17: 1175–1187
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the variations of vertical gradient are illustrated in different grey shading; as in Fig. 6a, the results are for 6 h of pumping. The white color in the figure illustrates zero or negative vertical upward gradients and the black color indicates the regions with the highest upward vertical gradient. As can be seen, the maximum vertical upward gradient is at the inner edge of the well casing. The simulation reveals that the vertical gradient at this point is always a maximum regardless of pumping time. This point is named as “the critical point”. On a circular plan view of the well bottom the critical points form a critical ring parallel to the well periphery. The critical ring is the favorable loci for the initiation of quicksand. To prevent quicksand the vertical gradient at the critical ring should be controlled.
Radial variation of vertical gradient The variation of the vertical upward gradient along the well radius is plotted in Fig. 7. The figure shows that the vertical gradient increases from the well center towards the well casing. It also illustrates that the increase in vertical gradient is gradual at the central part of the well and increases suddenly close to the well casing. For a well radius of 1.1 m, the vertical gradient increases from 0.8 to 1.2 from the center of well to 1 m radius. Then it changes from 1.2 to 6.7 along the radial distance interval of 1– 1.1 m from the well center.
Fig. 8 Variation of vertical gradient versus time for points located at different radii from the well center. These points are located at the well wall bottom elevation. The parameters of the simulation are given in Table 1
The variation of vertical gradient over time is important in finding the duration through which the vertical gradient is lower than the critical gradient. To study the variations of the vertical gradient with respect to time, the changes in vertical gradient versus time at four points located at different radial distances from the well center and on the horizontal plane passing through the critical ring are drawn in Fig. 8. The four points are located at radii of 0.002, 0.6, 1.09 and 1.1 m. The figure illustrates that for
all points the increase in vertical gradient with time is rapid early in pumping and gradually decreases reaching steady state conditions later on. For the points located at radial distances of 0.002 and 0.6 m, the vertical gradient is lower relative to that at larger radii, i.e. 1.09 and 1.1 m later on in pumping. Figure 8 further demonstrates that at radial distances from 1.09 to 1.10 m the vertical gradient rises from 2.8 to 6.71 at the steady-state condition. To generalize the spatial variations of vertical gradient during time (t), the dimensionless instantaneous vertical gradient (iviD=iv A K/Q) versus dimensionless radius (rD = r/rw) is plotted for different values of dimensionless time (tD =4 Kt/rw; Fig. 9). Note that r is the radial distance, A is the circular cross section area of the well, and other parameters are as defined previously. The dimensionless parameters iviD, rD and tD and dependent parameters are defined in Table 2. i viD is the ratio between the instantaneous vertical gradient and the average vertical gradient at the bottom of the well. The average vertical gradient at the bottom of the well is defined as Q/KA, in
Fig. 7 Vertical gradient versus radial distance at the well wall bottom elevation for a well with parameters given in Table 1 after 6 h of pumping
Fig. 9 Variation of instantaneous dimensionless vertical gradient iviD versus dimensionless radius rD for different dimensionless time tD. rw is wellbore radius and Qp is pumping rate
Variation of vertical gradient with time
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1181 Table 2 The dimensional and the dimensionless parameters Parameter name
Symbol The equation
Dimensionless vertical gradient Vertical gradient Vertical head difference (m) Vertical distance in the numerical grid (m) Wellbore circular cross section area (m2) Hydraulic conductivity of aquifer (m/s) Pumping flow rate (m3/s) Dimensionless time Wellbore radius (m) Specific yield Transmissivity (m2/s) Time (s) Dimensionless drawdown Drawdown (m) Aquifer initial saturated thickness (m) Relative dimensionless vertical gradient Instantaneous flow rate to the well (m3/s) Dimensionless radius Radius from the center of the well (m) Well filling material thickness to wellbore radius ratio
ivD iv Δhv ΔLv A K Qp tD rw Sy T t sD s b iviD Q rD r α
ivD ¼ iv QApK Dhv DLv
rw2 t tD ¼ 4K rw
T = Kb s K rw sD ¼ 4 Q p K iviD ¼ iv A Q
rD ¼ rrw
α = WF/rw (WF given in Table 3)
which the instantaneous total flow rate to the well from the aquifer, Q, is determined by summing the vertical flow through the cells located at the bottom of the well. As can be seen in Fig. 9, the pattern of changes in iviD, along dimensionless radius rD remains constant during the dimensionless time tD, the same as the pattern displayed in Fig. 7. Note that the curves in Fig. 9 are drawn for two wells with the parameter values given in Table 1. For the first well rw =1.1 m and Qp =4.71e – 3 m3/s and for the second well rw =3.0 m and Qp =1.57e – 3 m3/s. The plotted curves consist of nine and three dimensionless times for the first well and second well, respectively. As can be seen, the curves of both wells coincide. In other words, Fig. 9 demonstrates that the spatial and temporal variations of iviD at the bottom of the well are unique. In this study, the vertical gradient at the critical ring and the drawdown in the wellbore are of most concern. In the next sections the variation of these two parameters with time will be illustrated in the form of dimensionless type curves. Also, the effects of the well dimensions and the aquifer parameters on the type curves are investigated.
Vertical gradient and drawdown type curves In this section, two type curves are developed. The first is the dimensionless vertical gradient at the critical ring ivD, versus dimensionless time tD. The second is the dimensionless drawdown in the wellbore sD, versus dimensionless time. The dimensionless parameters ivD, sD and tD correspond to the dimensional parameters iv, s and t and derived by the formulas given in Table 2. To generate the type curves, the well and aquifer parameter default values given in Table 1 are used. The sensitivity of both curves to the well and aquifer parameters were checked. The well parameters are the material filling thickness (WF), the well wall thickness (WWT) and the wellbore radius (rw). The aquifer parameters are the pumping rate from the well (Qp), the hydraulic conductivity of the aquifer (K), the specific yield of the aquifer material (Sy), the aquifer material thickness beneath the well bottom (ATB), and the initial water table elevation (IWT). To check the sensitivity of the type curves to the above parameters, the type curves are drawn for the parameter values given in Table 3. By drawing and comparing the type curves with different aquifer and well parameter values, it was found that the shape of the type curves is only sensitive to material filling thickness (WF) and the wellbore radius (rw) and is insensitive to all other parameters. The type curves for both dimensionless vertical gradient and the drawdown versus dimensionless time are therefore plotted selecting the ratio between the material filling thickness and the wellbore radius, α = WF/rw as variable (Fig. 10a, b). In Fig. 10a, the type curves of dimensionless vertical gradient versus time are drawn for different values of α in logarithmic scale. This figure illustrates that the type curves have discrepancies in the early stages of pumping and they converge together through time and follow the same trend along a common asymptote in the later stages. These curves are parallel with a unit slope in the early stages and this is because of the effect of the wellbore storage. The curves also illustrate that in the early and intermediate stages smaller α values result in higher values for the dimensionless vertical gradient. The dimensionless curves of drawdown versus time for different values of α are drawn in logarithmic scale in Fig. 10b. The figure shows that the type curves merge, in the early stages of pumping, into a unit slope straight line
Table 3 Parameter values used in the simulations Parameter Name
Symbol
Parameter values
Pumping rate (m3/s) Hydraulic conductivity (m/s) Specific yield Depth of aquifer bottom (m) Initial water table elevation (m) Filling material thickness (m) Well wall thickness (m) Wellbore radius (m)
QP K SY DAB IWT WF WWT rw
5.15e–3 3.6e–3 0.1 10.7 94.6 1.5 0.165 1.1
4.71e – 3 1.2e – 3 0.2 7.7 96.6 3.5 0.5 2
4.08e – 3 1.2e – 4 0.3 4.7 102.6 9.5 0.75 3
3.46e – 3 1.2e – 5 0.4
2.83e – 3
1.0
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less vertical gradient will occur for a given material filling thickness and well radius.
Application of the type curves The type curves just derived (see section Vertical gradient and drawdown type curves) can be utilized to determine the hydraulic conductivity of the aquifer material, to define the safe pumping rate and duration and to design the well radius. In this section, the applicability of type curves to real field data for the determination of hydraulic conductivity is demonstrated and also how to design the safe pumping rate and duration as well as the well radius for predetermined design criteria.
Determining the hydraulic conductivity of aquifer material
corresponding to pure wellbore storage flow. Later, the curves diverge from each other and correspond to radial flow in the aquifer. Figure 10b also indicates that in the intermediate and late stages the curves with smaller values of α have smaller values of the dimensionless drawdown and are located below the curves with larger values of α. In order to examine the effect of α on the vertical gradient type curves of Fig. 10a, the dimensionless times tD and their corresponding α values on the curves are extracted for dimensionless vertical gradient ivD, equal to one. The extracted dimensionless times, tD, are plotted against their corresponding α values in Fig. 11. As can be seen in this figure, there is a linear relationship (with r2 = 0.998) between the dimensionless time and α values for a dimensionless vertical gradient of one. This means that by increasing the material filling thickness or decreasing the well radius the dimensionless time at which the unit vertical gradient is reached will increase. This curve is helpful in determining the time when the unit dimensionHydrogeology Journal (2009) 17: 1175–1187
7 6 5 4
tD
Fig. 10 a The type curves of the dimensionless vertical gradient ivD on the critical ring versus dimensionless time tD for different values of α (well filling material thickness to wellbore radius ratio). b The type curves of the dimensionless drawdown in the wellbore sD versus dimensionless time tD for different values of α (well filling material thickness to wellbore radius ratio)
The hydraulic conductivity of the aquifer can be determined by superimposing the drawdown versus time plot obtained by a pumping test on the type curves of Fig. 10b. Values of observed drawdown are plotted against values of time on logarithmic paper of the same size and scale as for the type curves. The observed drawdown-time data are superimposed on the type curve, keeping the coordinate axes of the two curves parallel, and adjusted until a position is found whereby most of the plotted points of the observed data fall on a segment of the type curve whose α value is equal to α = WF/rw of the well. Any convenient point is then selected, and the coordinates of this match point are recorded. With values of sD, tD, s and t obtained, K can then be determined from the equations in Table 2. For this study, the pumping test data for the MZ well, presented in Table 4, are plotted on logarithmic paper and superimposed on the type curve of Fig. 10b. The well radius (rw) is 1.1 m and the material filling thickness (WF) is 1.5 m. Therefore the drawdown-time curve should be matched with the type curve of α=1.4 (Fig. 12). The
3 2
+ 0.3372 tDD==0.692(WF/WR) 0.692 α + 0.3372
1
R r 2 ==0.9988 0.998
0 0
1
2
3
4
5
α
6
7
8
9
10
Fig. 11 Variation of dimensionless time tD versus α ratio for the unit dimensionless vertical gradient from the type curves of Fig. 9a; r2 is the coefficient of determination for the fitted linear equation DOI 10.1007/s10040-008-0426-7
1183 Table 4 Sample pumping test data from MZ well t(s)
s(m)
60 120 180 240 300 360 420 480 600 960 900 1,020 1,200 1,380 1,500 1,680 1,860 2,100 2,400 2,840 3,300 3,900
0.05 0.15 0.26 0.38 0.47 0.57 0.66 0.76 0.89 1.13 1.27 1.38 1.56 1.72 1.83 1.96 2.08 2.23 2.37 2.56 2.71 2.9
coordinates of the selected match point are: sD =2.42, tD = 2.58, s=1 m and t=600 s (Table 5, 2nd row). Substituting the coordinates of the match point into the following equations, already presented in Table 2, the hydraulic conductivity of the aquifer material is determined as follows: tD ¼
4t K 4 10 K )K ) 2:58 ¼ rw 1:1
¼ 1:18e 3m=s
s¼
ð4Þ
Q 4:34e 3 2:42 ) K sD ) 1 ¼ 4Krw 4 K 1:1
¼ 7:59e 4m=s
ð5Þ
As can be seen in Eqs. (4) and (5), the hydraulic conductivity of the aquifer can be determined based on s and sD parameters or based on t and tD parameters individually. By ignoring specific yield in Eqs. (4) and (5), the hydraulic conductivity of the aquifer can be determined by both equations. The calculated hydraulic conductivities calculated by the above two equations are somewhat different. This may be because of personal error in the collection of pumping test data and particularly due to a small variation of Qp during the test. The hydraulic conductivity of the aquifer may be taken as the average of these two values, and equal to 9.695e – 4 m/s.
Determining the safe pumping rate, safe pumping duration and the optimum well radius
To find any one of the safe pumping rate, the safe pumping duration or the optimum well radius by knowing the other two parameters, the occurrence of both the quicksand phenomenon and the drying up of the well have to be checked. Based on the type curves presented in Fig. 10a, the safe pumping rate for which no quicksand occurs can be calculated; and from the type curves of Fig. 10b, the safe pumping rate for which no drying up of the well occurs can be estimated. Of these two determined safe pumping rates, the lower value is the allowable safe pumping rate for which neither quicksand nor drying up occur. The calculations are given in the following section.
Determining the safe pumping rate for a given pumping duration and well radius Based on the pumping test data given in Table 4, it is assumed the safe pumping rate for a pumping duration of 1 h (t=1 h) during which the pumping well does not dry up is to be determined. The initial depth of water in the wellbore is 1.2 m. Therefore the allowable drawdown in the wellbore is 1.2 m. Putting rw =1.1 m, K=9.695e – 4 m/s
Fig. 12 Matched pumping test data on the drawdown type curve with α=1.4 Hydrogeology Journal (2009) 17: 1175–1187
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1184 Table 5 The parameter sets for the different design conditions Design parameter
Hydraulic conductivity Safe pumping rate, no drying up Safe pumping rate, no quicksand Safe pumping duration, no drying up Safe pumping duration, no quicksand Optimum wellbore radius, no drying up Optimum wellbore radius, no quicksand
Initial depth of water in wellbore (m) 1.2 1.2 1.2
tD
sD
ivD
2.58 12.69a 12.69a 3.82 0.87a 25.3 20.77
2.42 6.10 3.216a
a
4.97
0.737a
7.32
t (s)
s (m)
K (m/s)
600 3,600 3,600 1,083.5a 247.6a 21,600
1 1.2
9.69e 9.69e 9.69e 9.69e 9.69e 9.69e 9.69e
7.40
1.2
– – – – – – –
4a 4 4 4 4 4 4
rw (m)
Qp (m3/s)
1.1 1.1 1.1 1.1 1.1 3.31a 3.82a
4.34e 2.63e 7.40e 5.00e 5.00e 6.00e 6.00e
– – – – – – –
3 3a 4a 3 3 3 3
In all examples, α=1.4 and the critical gradient is 1 a Calculated parameter values
and t=1 h in Eq. (4), the dimensionless pumping duration tD, is determined to be equal to 12.69 (see Table 5, 3rd row). From Fig. 10b, the corresponding sD value for tD = 12.69 is 6.10. Therefore the maximum safe pumping rate (Qp ) is determined from Eq. (5) and is equal to 2.63e – 3 m3/s (Table 5, 3rd row). To then determine the maximum pumping rate at which no quicksand occurs, it is assumed that the critical gradient of the aquifer material is 1. For tD =12.96 from Fig. 10a, ivD =4.97. The maximum safe pumping rate, Qp, for which no quicksand occurs is determined by the following equation already presented in Table 2: ivD
iv A K 1 1:12 9:695e 4 ¼ ) 4:97 ¼ Qp Qp ) Qp ¼ 7:4e 4m3 s
ð6Þ
Comparing the pumping rates calculated by Eq. (4) (Qp = 2.63e – 3 m3/s) and Eq. (6) (Qp =7.40e – 4 m3/s), the allowable discharge rate for which no drying up and no quicksand will occur during 1 hour of pumping is 7.4e – 4 m3/s (see also Table 5, 4th row ).
Determining the safe pumping duration for a given pumping rate and well radius Again based on the pumping test data presented in Table 4, it is assumed the safe pumping duration with a pumping rate of 5e – 3 m3/s and a well radius of 1.1 m is to be determined. The first step is to determine the time at which the well dries up. As previously mentioned, the initial water elevation in the wellbore or the allowable drawdown is 1.2 m. Using Eq. (5), the dimensionless drawdown (sD) is determined equal to 3.216. From Fig. 10b, for sD =3.216, tD =3.82. Therefore the time (t) at which the allowable drawdown occurs is calculated by Eq. (4) as equal to 1,083.5 s (0.3 h), see Table 5, 5th row. The second step is to determine the time when the vertical gradient on the critical ring reaches the critical Hydrogeology Journal (2009) 17: 1175–1187
value of 1 with the given pumping rate (Q=5e – 3 m3/s) and the known well radius (rw =1.1 m). The corresponding dimensionless vertical gradient (ivD) for the unit vertical gradient (iv =1) is calculated from Eq. (6) equal to 0.737. From Fig. 10a, for ivD =0.737, the corresponding value of tD is 0.873. The duration at which the critical gradient occurs is therefore determined by Eq. (4) as equal to 247.6 s (see also Table 5, 6th row). As a result the allowable pumping duration with no quicksand and no drying up of the well is the shorter duration that is 247.6 s.
Designing the well radius for a given pumping rate and duration
It is assumed that a pumping rate of 6e – 3 m3/s for a period of 6 h (2.16e + 4 s) is required. The well filling material thickness is 1.5 m and the allowable drawdown is 1.2 m. The problem is to design a well radius for which no drying up of the well occurs. The allowable dimensionless drawdown can be determined in terms of the well radius rw from Eq. (5) as sD =2.436 rw, based on the parameters given in Table 5. The dimensionless duration at which the allowable drawdown occurs can also be written in terms of the well radius rw as tD =83.76/rw using Eq. (4) and the parameter values given in Table 5. Combining the relationships sD = 2.436 rw and tD =83.76/rw results in: sD ¼ 204:04=tD
ð7Þ
which is plotted in Fig. 13a as a straight line. The plotted equation in Fig. 13a, illustrates that wells with different radius (1.1–4.7 m) can satisfy the above-mentioned required pumping rate and duration. The plotted points with different values of well radius can cross several type curves if they are plotted on the type curves of Fig. 10b. It is assumed here that the material filling thickness to well radius ratio is α=1.4. By adding the type curve with α= 1.4 from Fig. 10b, 11, 12 and 13a, it can be seen that the plotted points and the type curve intersect each other at a point. The coordinates of the intersection point satisfy the required pumping rate and duration. The coordinates of the intersection point in Fig. 13a are tD =25.3 and sD = DOI 10.1007/s10040-008-0426-7
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a
condition of no quicksand, the optimum well radius rw is calculated as equal to 3.82 and 0.25 m, respectively. Of the two radii calculated above, the larger value is considered as the optimum designed well radius that is 3.82 m (Table 5, 8th row). At this radius, the discharge rate of 6e – 3 m3/s can be pumped from the well for a 6-h period without the well being dried up and quicksand occurring.
Type curve with α = 1.4 sD = 204.04 / t D
Comparing the model with the solution for partially penetrating wells with wellbore storage
1.E+02
vD
1.E+01
Type curve with α = 1.4 WR1WF1 Eq.s ivD = (15) 3556and .97 /(16) t D2
1.E+00
1.E-01 1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
tD Fig. 13 a Drawdown type curve with α=1.4 and the plotted points determined by Eq. (7) (i.e. sD =204.04/tD) for different rw values. b Vertical gradient type curve with α=1.4 and 2 the plotted points determined by Eq. (8) , i.e. ivD ¼ 3556:97 tD ) for different rw values
7.32. By putting one of the obtained coordinates in Eqs. (4) or (5) the optimum well radius rw is determined as equal to 3.31 m (Table 5, 7th row). The well radius may also be designed for the given pumping rate and duration while preventing the occurrence of quicksand. Assuming that the critical gradient is 1 and using Eqs. (5) and (4) the dimensionless vertical gradient and the dimensionless time at which the critical gradient will occur can be written in terms of rw, as ivD=0.507rw2 and tD =83.76/rw, respectively. Combining these two relationships gives ð8Þ ivD ¼ 3556:97 tD2 Equation 8 is now plotted for the tD and ivD values corresponding to rw =1.1–4.7 m on the type curve of Fig. 10a with α=1.4 and presented as Fig. 13b. The optimum well radius can be determined based on the coordinates of the intersection point of the two curves in Fig. 13b. At the intersection point tD =20.77 and ivD =7.40. By substituting these values in Eqs. (4) and (5), for the Hydrogeology Journal (2009) 17: 1175–1187
For the purpose of validating the solution, the theoretical responses for a finite diameter partially penetrating well with wellbore storage according to Moench (1997), are compared with the numerical solution developed here for a large diameter well gaining water from the bottom. A comparison is made between these two solutions for the wells with the parameter values given in Table 1. The dimensionless drawdown versus dimensionless time curves for the both models are plotted in Fig. 14. The Moench solution is plotted for 5 and 74% penetration and annotated as “Moench 5%” and “Moench 74%” respectively. The curve labelled as “Quicksand” is the authors’ solution for the large diameter well gaining water from the bottom. For the Moench solution the well is screened from an elevation of 90.7 to 90.0 m for 5% penetration and from 90.7 to 80.0 m for 74% penetration. From Fig. 14, it is observed that the three curves follow the same trend. Due to wellbore storage, in the early stages of pumping, the three curves follow a unit slope straight line as expected. Later, the curves correspond to radial flow, when the effect of wellbore storage has subsided and the flow is radial in the aquifer. Figure 14 also demonstrates that the lower the penetration the higher is the drawdown. The reason that the drawdown in a large diameter well gaining water from the bottom is higher than that of the
1.E+00
Moench (1997), 74% Moench (1997), 5% Quicksand
s (m)
b
1.E-02
1.E-04 1.E-01
1.E+01
1.E+03
1.E+05
t (s) Fig. 14 Comparison of drawdown-time curve for a large diameter well gaining water from its bottom (Quicksand), with that of partially penetrating wells with wellbore storage (Moench 1997) for 5 and 74% penetration DOI 10.1007/s10040-008-0426-7
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Moench solution with 5% penetration is because of dominant vertical flow and energy loss along the well filling materials and along the aquifer material beneath the well.
Results and discussion In this work, the parameters affecting drying up and the occurrence of quicksand in large diameter wells gaining water only from the bottom are studied. These wells have a cased cylindrical vertical wall. They are dug in dune sands, which are loose and collapsible materials. The wellbore is partly filled with the aquifer material. During pumping vertical upward flow through the filling materials is established. Dune sand can easily be liquefied due to vertical upward flow. When the flow rate and the vertical upward hydraulic gradient reach a critical value, quicksand occurs. In addition, the well may dry up due to the low hydraulic conductivity of the aquifer material. Quicksand causes destruction of the well and pumping facilities. In the study area, the critical gradient of the aquifer material determined by laboratory methods ranges from 0.72 to 1.4. The flow around the well and the vertical upward flow beneath the well are numerically simulated by MODFLOW 2000 (Harbaugh et al. 2000) equipped with LSM (Samani et al. 2004). By simulation of the flow around and beneath the well the spatial and temporal variations of the vertical gradient and the temporal variations of the drawdown at the wellbore are determined. The simulation illustrates that as pumping continues the vertical gradient beneath the well and the drawdown at the wellbore increases. The changes in the early stage of pumping are rapid and the rate of change decreases gradually through time until reaching a steady state in the later stage. The simulation reveals that because of wellbore storage, a part of the pumping flow rate is supplied by the aquifer and the rest of the flow is gained from the wellbore storage. The portion of flow supplied from the aquifer is zero early on and increases gradually as pumping continues. In other words, as pumping proceeds the contribution of wellbore storage to provide the well discharge reduces and that of the aquifer increases, so that in the later stages of pumping the bulk of well discharge is supplied by the aquifer. The vertical gradient beneath the well increases with time while the drawdown rate in the wellbore decreases. The vertical gradient increases gradually from the center of the well towards the wall in a radial direction. It is found that the maximum vertical gradient is established close to the inner edge of the well casing bottom on the so-called “critical points”. In the circular plan view of the well bottom the critical points form a “critical ring” parallel to the well inner periphery. The rate of increase in vertical gradient along the radius of the well grows suddenly close to the well casing with the highest value at the critical ring. At the critical ring, the vertical gradient is about 7.5 times the average vertical gradient at the Hydrogeology Journal (2009) 17: 1175–1187
bottom of the wellbore at all times during pumping. Therefore, quicksand will begin initially at the critical ring. To predict the quicksand and the drying up phenomena, the vertical gradient at the critical ring and the drawdown at the wellbore versus time are drawn on a log–log scale for wells with different sets of parameters. The sensitivity of these curves to aquifer and well parameters demonstrates that among all the parameters considered the material filling thickness and the well radius affect the curves. The effective parameters are selected as variable and two sets of type curves were generated (Fig. 10a,b). The first type curve is the dimensionless vertical gradient versus dimensionless time (Fig. 10a) and the second is the dimensionless drawdown versus dimensionless time (Fig. 10b). Curves in each set are differentiated by the ratio of material filling thickness to the well radius. To determine the flow behavior in large diameter wells gaining water from the bottom and also to predict quicksand and drying up phenomena in these wells, the dimensionless type curves developed here for the vertical gradient at the critical ring and the drawdown at the wellbore can be used. These curves may be utilized to determine the hydraulic conductivity of the aquifer material, safe pumping rate, safe pumping duration and the optimum well radius. For the safe pumping rate and duration and the optimum well radius, quicksand and drying up phenomena will not occur. The generated drawdown-time type curve is validated by comparing it with the type curve of a finite diameter partially penetrating well with wellbore storage due to Moench (1997). The comparison reveals that a large drawdown in the large diameter well gaining water from the bottom is a result of vertical upward flow and further energy loss through the materials filling the bottom part of the well.
Conclusions Where fine sand materials are subjected to an upward force due to an upward flow of water in excess of their buoyant weight, quicksand occurs. A typical situation is in large diameter wells dug in aquifers consisting of dune sand and gaining water from the bottom. These wells are also low yielding and dry up after a short period of pumping. To prevent the quicksand and drying up phenomena, flow to such wells is numerically simulated. The results of the simulation are presented as the temporal and spatial variation of vertical gradient, the dimensionless drawdown-time curve and dimensionless gradienttime curves. The highest vertical gradient, the cause of quicksand formation, develops in the so-called “critical ring”—a circular zone close to the inner edge of the well casing. The position of the critical ring is independent of time; however its magnitude is controlled by the ratio of material filling thickness to the well radius. The dimensionless type curves are therefore generated as a function of this ratio. The dimensionless drawdown-time curves merge in the early stages of pumping into a unit slope DOI 10.1007/s10040-008-0426-7
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straight line corresponding to pure wellbore storage flow. In the later stages, the curves diverge from each other and correspond to radial flow in the aquifer. This flow regime is shown to be comparable to the flow regime of a finite diameter partially penetrating well with well storage (Moench 1997). The dimensionless gradient-time curves are parallel with a unit slope early in pumping and they converge together through time and follow the same trend along a common asymptote in the later stages. The generated dimensionless curve can be used by field practitioners to plan the safe pumping rate and duration and to design the optimum well radius for which the quicksand and drying up phenomena will not occur. The probability of quicksand and drying up occurring is lower for a larger well radius, but economical considerations encourage designing the minimum possible well radius. Acknowledgements The financial support of Fars Water Affair Organization, contract no. 19483/151, and the support of Shiraz University are hereby acknowledged. The authors thank the officers of Fars Water Affair Organization, Mr. Nejati and Mr. Shakeri for their kind co-operation and comments on the first draft of the manuscript. Useful comments and suggestions given by the editor and two anonymous reviewers are appreciated.
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DOI 10.1007/s10040-008-0426-7
