MATHEMATICAL MODELLING OF SPRINGFLOW

Abstract

A spring is a natural outlet for concentrated discharge of groundwater either on land surface or into a body of surface water. Springs have been used as dependable and ready source of water in many parts of the world. Springs could be of various sizes from small trickles to large streams under both water table and artesian conditions. An active spring can be treated as a flowing well with constant head. This feature could be used conveniently in the mathematical modelling of springflow. In the analysis of regional groundwater flow, a spring can serve as a boundary condition of Dirichlet type. The physical process of release of spring water from groundwater storage can be compared with the lower portion of the recession part of a flood hydrograph in a river and can be simulated by a linear reservoir. A linear reservoir is a conceptual reservoir in which outflow is linearly proportional to the storage. Combination of this postulation with continuity equation gives the equation for base flow, Q = Q k" t Or where k is the recession constant or depletion factor and is equal to exp(-l/r ). r is a parameter of the spring and is designated as the depletion time. Bear (1979) suggested a simple mathematical model to simulate the unsteady flow of a spring over the recession period for a lumped recharge. Bear model assumes a linear relationship between springflow and storage. A springflow model has been developed using Bear's model and the convolution technique for simulating springflow for a known time variant. recharge and aquifer parameters. However, the time yariant recharge is not known. The Newton-Raphson iterative method for solving non-linear equation has been used to compute the time variant recharge, and the model parameter, i.e., depletion time (r ) from the springflow. The model has been tested on three springs. The springs are: (i) Sulkovy Pramney springs, Czechoslovakia emerging from sandstone strata (a third magnitude spring) (ii) Kirkgoz spring, Turkey emerging from Karstic aquifer (a first magnitude spring) and (iii) White Rock spring, Nevada from perched waters in volcanics tuffs (a eighth magnitude spring). For the Kirkgoz spring, Turkey, the added up monthly recharge for each year matched with the annual rochniqo for c yonrn rompntod by nu nnrlior invent iqatoi uning lieai 'a model . In Bear's model, the logarithm plot of springflow with time during the period of recession follows a straight line. It is found that during the process of recession, the variation of logarithm of flow with time follows a straight line, provided the springflow domain is a closed one. A closed flow domain implies that nil the rnchnrgn will appear nn npringflow. II The Bear model assumes that an unsteady state is the succession of the steady stale conditions and there is no time lag between onset of recharge and emergence ol springflow at the spring's threshold. But, in case, the transmission zone of the spring in the flow domain is long and the hydraulic diffusivity is low, there would be a time lag between the onset of recharge and its appearance as springflow at the spring's threshold due to the storage and translation effect in the transmission zone. In order to simulate springflow for such a geohydrological system, a mathematical model has been developed considering an unsteady state for simulating springflow for a known time variant aquifer recharge. Starting from the basic solution given by Carslaw and Jaegar for flow in an aquifer of finite length and using convolution technique, the unit pulse response function coefficients for outflow due to unit recharge in the recharge zone has been obtained. Using the unit pulse response function coefficient and convolution technique, springflow has been computed for the time varying recharge. The storativity of the transmission zone reduces the magnitude of peak springflow and it causes delay in the appearance of peak springflow. When storage in the transmission zone is small, the springflows simulated by the two models compare well. With an initial guess of the range of values of the model parameters i.e.,0 (specific yield), 0 (storage coefficient), T (Transmissivity), LW /W (a linear dimension representing recharge R S area and spring width), 1 (length of transmission zone ), the time variant recharge and model parameters are computed by random III search technique. The recharge computed by the random jump technique compares well with those obtained by the Newton-Raphson technique. The Bear's model and the model with long transmission zone deal with one dimensional flow. However, the flow processes associated with springflow will be two dimensional. Therefore, using basic solution given by Hantush for the evolution of piezometric surface due to recharge from a rectangular basin, a two dimensional springflow model has been developed. The response of the spring for unit pulse recharge through the rectangular recharge zone of the spring has been obtained. Using these unit response function coefficients, springflow for any time variant recharge can be computed. The variation of logarithm of springflow with time during recession, does not follow a straight line. Only towards the latter part of recession, the variation is approximately linear. Using the random jump techniqeeand the springflow model for an open flow domain, recharge area, spring opening, distance of the spring from the recharge area, transmissivity and storativity of the transmission zone and the recharge have been estimated from observed springflow data. Since the domain is an open one, the recharge computed by the model which is based on Hantush's solution, is found higher than those computed using the model for a closed system.