ANALYSIS OF UNSTEADY FLOW TO A LARGE-DIAMETER WELL USING DISCRETE KERNEL APPROACH

Abstract

Large-diameter wells are extensively used in many parts of the world. The low cost and simplicity of their construction and operation are the main reasons for their extensive use. Another important advantage of these wells is that thay are suitable for shallow aquifers with low transmissivity. In India and in other South Asian countries, people have been using large-diameter wells tapping mostly the phreatic and in some areas, the shallow semi-confined aquifers near to the surface since ancient times. Dug wells continue to be the primary source of groundwater in rural India. As reported by Ghosh (1987), of the total 14.84 million approximate number of wells in India about 67 psrcent are dug wells with large-diameter. Accounting for well storage, Papadopulos and Cooper (1967) have analysed unsteady flow to a large-diameter well, which taps an aquifer of infinite areal extent. The solution has been obtained by integral transform technique. Results for drawdown in the piezometric surface due to continuous pumping at constant rate have been presented by them. Since then many investigators have contributed to this field. For aquifer with low transmissivity, it may so happen that more water may flow from the aquifer to the well during recovery phase than during pumping. In such hydrogeological condition the analysis of flow to a large-diameter well during recovery is quite important. Most of the analyses of flow to a large-diameter well made so far, are based on the assumption that the pumping rate is independent of drawdown at the well. However, if a centrifugal pump is used for abstraction of water from the well, it is not possible to pump at a constant rate independent of the drawdown at the well. Another assumption, that the aquifer is of infinite areal extent, may not be valid for hard rock areas. Considering these facts and limitations analysis of unsteady flow to a large-diameter well has been

carried out in the present thesis by discrete kernel approach. The discrete kernel coefficients are the response of a linear system to a unit pulse excita tion. In the discrete kernel approach, the time parameter is discretised by uniform time-steps; the excitation and the response are assumed to be piece wise constants within each time-step; the response of the linear system to a time-depeniet,l excitation is predicted making use of the discrete kernel coefficients. Desired accuracy in the results can be achieved with selection of appropriate time-step size. The methodology provides tractable solution. In order to have a better understanding of the flow mechanism associated with the large-diameter wells in different hydrogeological and physical conditions, the following analyses have been carried out in the present thesis : (i) Analysis of flow to a large-diameter well during the recovery period. (ii) Analysis of unsteady flow to a large-diameter well due to abstraction that varies linearly with drawdown at the well. (iii) Analysis of flow to a large-diameter observation well due to pumping of a large-diameter production well. (iv) Analysis of unsteady flow to a large-diameter well experiencing well loss . (v) Analysis of flow to a large-diameter well in a finite aquifer. Analysis of Flow to a Large-Diameter Well During the Recovery Period Analysis of flow to a large-diameter well during pumping has been carried out by several researchers. Foremost among the solutions is that of Papadopu los and Cooper (1967), who have presented the type curves for estimating aquifer parameters. The evaluation of aquifer response by Papadopulos and

Cooper's method requires numerical integration of an improper integral involving Bessel's function. The numerical integration therefore involves large computa tions. Although a unique value of transmissivity can be obtained with the type curves given by Papadopulos and Cooper, the evaluation of storage coeffi cient from a short duration pump test data is questionable. According to Papadopulos and Cooper, for accurate determination of storage coefficient, 2 the well should be pumped beyond the time t = 25 r /T, where r and T are the radius of the well casing and aquifer transmissivity respectively. In case of aquifer with low transmissivity, it may not be possible to pump upto the required time as the well may go dry due to abstraction from well storage during pumping. Under such circumstances, evaluation of aquifer parameters with the help of recovery data is appropriate. In the present thesis analysis of unsteady flow to a large-diameter well both during pumping and recovery periods has been done using discrete kernel approach. A family of type curves has been presented for different durations of pumping. These type curves provide a fairly accurate means of determining aquifer para meters from data of pump tests conducted in large-diameter wells. The reple nishment of well storage at various times after the cessation of pumping has been estimated. The sensitivity of the solution to the time time-step size has been studied. Analysis of Unsteady Flow to a Large-Diameter Well due to Abstraction that Varies Linearly with Drawdown at the Well It has been found that if a centrifugal pump is used for abstraction from a dug well, there is a gradual decline in discharge because the height of water stored above the footvalve of the pump declines with pumping. The variation in discharge rate with time in several dug wells in basaltic terrains have been investigated by Athavale et al. (1983). It has been

reported by them that the discharge rate may be either a linear or a nonlinear function of the drawdown. In the present study unsteady flow to a largediameter well induced by a drawdown-dependent time-variant pumping has been analysed using discrete kernel approach. A linear relationship between pumping rate and drawdown at the well has been assumed to hold good. Tractable analytical expressions have been derived for determining the aquifer contribution, well storage contribution and drawdown at any point in the aquifer. It is shown that with an average pumping rate, it will not be possible to simulate the drawdown and aquifer response that would evolve due to drawdowndependent time-dependemi pumping of a large-diameter well. Analysis of Flow to a Large-Diameter Observation Well due to Pumping of a Large-Diameter Production Well A large-diameter well can also serve as an observation well if a pumping test is conducted in a production well of negligible diameter. Storage associa ted with large-diameter production or observation well modifies and causes delay in the aquifer response. Barker (1984), has identified that, if both the production well and the observation well have storages, a tractable solu tion for the drawdown at any point in the aquifer is yet to be known. In the present study a generalised discrete kernel approach has been described to analyse the combined effect of the production and the observation well storages on drawdown at any point in the aquifer during pumping and recovery phases of a pumping test . The nondimensional time-drawdown graphs have been presented for four different combinations of production and observation wells located at a distance, r apart which may or may not have storage. The contribution of observation well storage to the aquifer during pumping and the replenishment of observation well storage during recovery have been presented both for different distances between the production and observation

wells and for different radii of well casings. It has been verified that the drawdown in an observation well with negligible storage due to pumping in a large-diameter well is same if the roles of the wells are reversed. It is seen that the influence of the observation well storage on drawdown at the production well during recovery is more pronounced than during abstrac tion phase. The production well storage controls the drawdown at the produc tion well during pumping irrespective of the observation well storage. Analysis of Unsteady Flow to a Large-Diameter Well Experiencing Well Loss The concept of step-drawdown test in a water well was first presented by Jacob (1947) as a means to separate the components of drawdown pertai ning to laminar and turbulent flow regimes. Jacob assumed that the laminar component is directly proportional to the discharge rate and that the turbul ent component is a second-order function of well discharge. This assumption is widely used in practice. Since then significant contributions were made by several investigators towards the development of the techniques for collec tion and analysis of the step drawdown test data to find the flow components and aquifer parameters. Although many researchers have dealt with step drawdown test and estimation of well losses, no attempt was made to take into account of the well storage. In the present study unsteady flow to a large-diameter well in a confined aquifer has been analysed taking into account the well losses. The effect of well storage on well loss component and on the specific drawdown has been .investigated. It is found that, if well storage effect is accounted for, the variation of specific drawdown with pumping rate is nonlinear. However, for small and large pumping rates, the variation tends to be linear. The well loss component can be greately reduced by providing well storage.

Analysis of Flow to a Large-Diameter Well in a Finite Aquifer In hard rock areas, the weathered and the fractured zones form an aquifer. Therefore, the aquifers in a hard rock area are likely to be of finite areal extent and the hydrologic boundary is likely to be a no-flow boundary. In the present thesis, using discrete kernel approach, unsteady flow to a large-diameter well located at the centre of a finite aquifer of circular shape has been analysed during pumping and recovery phases. The nondimensional time-drawdown graphs at specific locations in the aquifer have been presented. The recovery characteristics of well storage has also been analysed. It is found that well storage contribution is little affected by the presence of the barrier boundary where as the drawdown characteristics during pumping as well as during recovery are influenced significantly by the barrier boundary. It is shown that various problems of unsteady flow to a large-diameter well in a homogeneous isotropic and confined aquifer during pumping as well s during recovery, can be solved with ease by discrete kernel approach. The solutions obtained by discrete kernel approach are tractable for numerical computations.