One-dimensional springflow model for time variant recharge

Transcription

1 Hydrological Sciences-Journal-des Sciences Hydrologiques, 42(3) June One-dimensional springflow model for time variant recharge A. K. BHAR & G. C. MISHRA National Institute of Hydrology, Jalvigyan Bhawan, Roorkee , India Abstract The linear mathematical model for springflow suggested by Bear (1979) can simulate springflow for an initial instantaneous recharge. A springflow model has been developed, using the Bear model and Duhamel's approach, which can simulate springflow for time variant recharge. The suggested model can also be used to compute the time variant recharge to the springflow domain from a given springflow time series. The inverse problem, which contains linear recharge terms and nonlinear depletion terms, has been solved using the Newton-Raphson method for solving a set of nonlinear equations. The model has been tested to compute recharge for Kirkgoz spring, a first magnitude karst spring in the Mediterranean region of Turkey. The estimated annual recharge computed by the model on a monthly basis compared well with the annual recharge which had been estimated (Korkmaz, 1990) using the Bear model. Modélisation à une dimension du débit d'une source résultant d'une recharge en régime transitoire Résumé Le modèle linéaire du débit d'une source proposé par Bear (1979) permet de simuler ce débit pour une recharge initiale instantanée. En nous appuyant sur le modèle de Bear et sur l'approche de Duhamel, nous avons élaboré un modèle susceptible de simuler le débit d'une source résultant d'une recharge en régime transitoire. Le modèle proposé peut également être utilisé pour calculer la recharge transitoire à partir d'une série chronologique du débit de la source. Ce problème inverse, qui comporte des termes linéaires dépendant de la recharge et des termes non-linéaires dépendant du tarissement, conduit à un système d'équations nonlinéaires résolu par la méthode de Newton-Raphson. Le modèle a été testé pour calculer la recharge de la source de Kirkgoz, très importante source karstique de Turquie située sur la bordure méditéranéenne de ce pays. La recharge annuelle estimée par le modèle au pas de temps mensuel est très comparable à celle estimée (Korkmaz, 1990) s'appuyant sur le modèle de Bear. INTRODUCTION Bear (1979) suggested two simple conceptual springflow models. The first is a tank model to analyse the unsteady flow of a spring during the recession period after instantaneous recharge has occurred. In the second model, the flow domain is hydrologically decomposed into two parts: (a) a recharge zone; and (b) a transmission zone. The model is based on the assumption that the springflow is linearly proportional to the dynamic storage in the springflow domain. This latter Bear model is a one-parameter model. The model parameter reflects the geometric and hydraulic characteristics of the springflow domain except the storativity of the transmission zone. As the recharge to a springflow domain is a time variant process, the Bear model has been adapted in this paper to simulate springflow for a time variant Open for discussion until 1 December 1997

2 382 A. K. Bhar & G. C. Mishra recharge. Also, the adapted Bear model has been used to find the depletion time and the monthly recharge of a first magnitude spring in Turkey (Korkmaz, 1990). STATEMENT OF THE PROBLEM Let h 2 be the initial level of the groundwater table which is equal to the level of the spring threshold (Fig. 1). Let the spring be inactive before the onset of recharge. The springflow domain is conceptualized to consist of two parts: (a) a recharge zone; and (b) a transmission zone. The length of the transmission zone is /. A time varying recharge, r(t), through the entire recharge area commences at time t = 0 due to which the groundwater table in the recharge zone rises to h(t). Consequently, immediately after the onset of recharge, springflow emerges from the spout of the spring of width W s (Fig. 1). It is required to find the variation of the springflow with time. ACTUAL RECHARGE RATE ASSUMED RECHARGE RATE Fig. 1 The proposed model.

3 One-dimensional springflow model for time variant recharge 383 ASSUMPTIONS The following assumptions are made: (a) as implied in the Bear model, the flow in the recharge zone is in the vertical direction, and the flow in the transmission zone is in the horizontal direction. The Dupuit-Forchheimer assumptions are taken to apply in the transmission zone. (b) springflow occurs immediately after a recharge; and (c) in the transmission zone, the unsteady state flow is equivalent to a succession of steady state flows. The assumption (b) implies that there is no storage effect in the transmission zone and that the flow in the transmission zone is pressure flow. ANALYSIS First, an expression for springflow has been derived for a unit impulse recharge. Using the response of the spring-aquifer system to a unit impulse recharge, the response to a unit pulse perturbation has been derived. The springflow for a time varying input has been obtained using the convolution technique and the unit pulse response coefficients. Let an impulse recharge quantity N per unit area occur at time t = 0. Let the spring be inactive during the time t < 0. Since it is assumed that there is no storage effect in the transmission zone, and also that the unsteady flow in the transmission zone is a succession of steady state flows, the flow at any section of the transmission zone is equal to the discharge, Q(t), of the spring. The depletion rate from the recharge zone, in turn, is equal to the springflow. A water balance equation for the recharge zone over the time period t to t + At is: l+al = -Ah <j\ A where Ah is the fall in the groundwater table in the recharge zone during the time interval At; A is the area of the recharge zone, with A = LW R, L the length of the recharge zone, W R the width of the recharge zone; and <j> x is the specific yield of the aquifer in the recharge zone. When At > 0, equation (1) reduces to:,, d/z Q(t) = -^A (2) According to Darcy's law, the flow in the transmission zone is given by: Q(t) = W s T\h(t)-h 2 }ll (3) where J is the average transmissivity of the aquifer. Combining equations (2) and (3): àh W S T àt Al(j) i h{t)-h 2 ) (4) (1)

4 384 A. K. Bhar & G. C. Mishra The water table height in the recharge zone at time t -> 0 for the impulse recharge of N per unit area at t = 0 is given by: h(o) = h 2 +N/0 i (5) The solution of the linear differential equation (4) for the initial condition given by equation (5) is: WsTl h(t) = h 2 +(N/fa)e W < (6) Substituting this expression for hit) into equation (3) and defining the term A10 1 /(W S T) = T O, Q(t)={w s TN/(l<f> l )} e^ = [w s TAN/(Al^))e~" r " (7) = Rh^" 1 " where R = AN and R is the total recharge occurring instantaneously at t = 0. The springflow at time t = 0 is: 2(0) = R/T 0 (8) Thus the flow Qif) can be written as: Q{t) = <2(0)e^> (9) Equation (9) is the Bear equation for springflow. The parameter, r 0, is the depletion time and is equal to Al(j>J{W s T) as derived by Bear. Equation (4) being linear, the Duhamel principle can be applied to derive the expression for the discharge of a spring due to a variable recharge. Let kit) be defined as the flow of a spring due to a unit impulse recharge through the entire recharge area, A, taking place at t = 0. Therefore, k(t) = Qif) when R = 1. Hence, putting R = 1 in equation (7): k(t) = (VT 0 )e~ i,t " (10) For a varying recharge rate, rit), the corresponding springflow rate, qif), can be expressed as: q{l)=\k{t-x)r{x)àx (11) o Let the time span be discretized by uniform time steps of size At, Thus, t = nat where «is a positive integer. The springflow rate at the end of nth time step is: AI q(nat) = \\k(nat-t)r(t)dt+ J^(nzl/-r)r(r)dr yai (Y-\)AI 2 AI 0 AI jk(nat - r)r(r)dr +... jk(nat - r)r(r)dr nai (II~\)AI (12)

5 One-dimensional spring/low model for lime variant recharge 385 Let the recharge be represented by a train of pulses. Let R(f) be the total recharge during the yùi time step. Hence the recharge rate, r(r), during the 7th time step is R{y)IAt. Assuming that the recharge rate is constant during a time step, but varies from one time step to another, equation (12) simplifies to: qinat) = ]T R {ï) IAt \k{nat - r) dr VU = ^R(r)/At l(vt 0 y aât ' T) ' T " dr (r-i)ai =1L R {ry* )(At/T 0 y A,^+t)^ dv (13) Equation (13) can be further simplified in terms of discrete kernel coefficients in the following manner. Let a discrete kernel coefficient S(At, n) be defined as: -, AI s(at,n) = \k{nat-t)dt (14) At 0 S(At, n) is the response of the spring-aquifer system due to a unit pulse perturbation imparted during the first time step. Incorporating the expression for k(t) in equation (14): -, AI ô(at,n) = l(l/t 0 )e- { " A '- TVT "ât (15) At Let T = vat and dr = Atdu. Making these substitutions in equation (15) and integrating: 1 1 ô(at,n) = l(at/t 0 )e- M "- u) ' T ''do At -(AtlT oy na " T «e^' /r du (16) 0 = {_ e -»^r. +e -(»-l)^r J /id/ In terms of discrete kernel coefficients, equation (13) simplifies to: 11 q{nat) = Y J R{r)s(At,n^y + \) (17) If the depletion time, r 0, is in month units, and At is in month units, the springflow

6 386 A. K. Bhar & G. C. Mishra rate, qqiat), is in the unit of m 3 per month if all linear dimensions are in metres. An observed springflow at any time, n, consists of two parts: one part is the response to the recharge that has occurred since the time origin and the other part is the response to the perturbation prior to the time origin. The component q{nat) corresponds to the recharge taking place since the time origin. Thus, using the convolution technique and the Bear model, the springflow for a time variant recharge can be simulated. THE INVERSE PROBLEM In the inverse problem, the unknowns are the time varying recharge to the springflow domain and the depletion time, r 0. Using the measured discharge of the spring during a recharge period and at least one value of the springflow during the non-recharge period, the unknowns can be computed. Let K(t) be the unit step response function of a spring. The unit step response function, K{t), can be evaluated by integrating k(t), the response function for a unit impulse recharge, defined in equation (10). Hence: K{t)= \k{x)àx= (l/i 0 )exp(-x/x 0 )dt = l-e^" T " 0 0 or (18) K(nAt) = l-{exp(-«a?/x 0 )) = 1-/' where y = exp(-at/t 0 ) and At is the unit time step. Further, the discrete kernel coefficient, 8{At, n), can be expressed in terms of a unit step response function, K(nAt), as: Hence: ô(at, n) = [K(nAt) - K(nAt - At)]/At â(at, l) = K(At)IAt = (l-y)/at ô(at,l) = {K(2At)-K(At)}/At = (y^y 2 )/At (19) ô(at,n)={k(nat)-k(nat-at)}/at = (j/ M -y")at The component of the springflow, Q B, which is the response to the perturbation prior to the time origin is given by: Q (nat) = g(0) exp(- nat/r Q ) = q(0)y" (20) where Q(0) is the springflow at time t = 0. Hence:

7 One-dimensional spring/low model for time variant recharge 387 Q B {yai) = Q{0)y r where Q B (yat) is the component response of the springflow rate at t = yat to perturbation before time t = 0. Let Q(At), Q{2At), Q(3At),..., Q{(n - l)at}, Q{nAt) be the springflow rate and R{1), R(2), R(3),..., R(n - 1) the values of the corresponding recharge, which are unknowns. The recharge, R(n), is equal to zero. The springflow rate at the end of nth time step, Q{nAt), is given by: t d R{y)ô(At,n^y + l)+q(0)^"^ = Q(nAt) or Y J R{y)^t,n-y + ]) + Q(6)e" & " T " -g(»af) = 0 (22) or t l R{r){y"~ r -y'- r+l )/At + Q(0)y" -Q{nAt) = 0 y = \ Let r(y) R(y)IAt. The equations for each of the n observations are: r{l)(\-y) + Q{ )y^q{at) = 0 = f l [r(l),r(2),r(3),...,r(«-l),j,],{l)(y-y 2 ) + r(2)(l-y) + Q(0)y 2 ~Q(2At) = 0 = f 2 [r(l),r(2),r(3),..,,r(«-l),>>] (21) r(l)(y 2 -y 3 ) + r(2)(y-y 7 ) + r(3)(l-y)+q{0)y 3 ~Q(3At) = 0 (23) = f 3 [r(\),r(2),r{3),...,r(n-l),y] r(l){y'" ] -y") + r(2)(y"- 2 -y"' l ) + r(y)(y'" r -~y"~~ r+, ) r(n-\\y"- l ''- x) ~y" A "" 2) ) + Q(0)y" -Q{nAt) = 0 = f\[r(\),r{2),r{3),...,r(n-l),y] In equation (23), r(l), r(2), r(3),...,/ («- 1) and y are the «unknowns which are to be solved from the n equations. The procedure for solving a set of nonlinear equations by the Newton-Raphson iterative technique is well documented

8 388 A. K. Bhar & G. C. Mishra (Carnahan, 1969) and is adopted here for finding the unknowns. Estimation of recharge and depletion time by the Newton-Raphson method The Bear springflow model has been applied to Kirkgoz Spring emerging from a karstified limestone aquifer (a first magnitude spring) near Antalya city in the Mediterranean region of Turkey. The monthly springflow data for the Kirkgoz Spring for the period October 1973 to May 1981 (92 months) are available (Korkmaz, 1990). It has been assumed that all the springflow data are free from any error. Since monthly springflow data are reported, the time step size has been taken as one month. From the springflow data, the slope, a, of the graph of the logarithm of springflow vs time for each time step is calculated using: a = [log ]0 Q(t + At)^log w Q(t)^At (24) The consecutive time steps in which the slopes do not change their sign and do not differ much, say less than about 20% or so, are the time steps of no recharge or no abstraction from the springflow domain. After ascertaining the period of no recharge and no abstraction from the springflow domain, the Newton-Raphson method for solving the set of nonlinear equations has been used to compute the time varying recharge and to estimate the depletion time, r 0, for the spring. The springflow values have been arranged so as to form a number of sets such that the last springflow value in each set corresponds to a period of no recharge and no abstraction. The selection of this last value of the springflow in a set is a critical task and recharge or abstraction must be zero at that time step. However, there may be other time steps in the set where the recharge or abstraction is zero. Eight sets of springflow values were formed from the available monthly springflow data for the period October 1973 to May The first value of springflow in a set of data has been used to estimate the component of springflow due to perturbations prior to the time origin. The last value of a set is taken as the first value of springflow for the succeeding set. Initial guesses for the depletion time, t 0, and the recharge are made. It has been found that the Newton-Raphson method performs well with any initial guess of the model parameter, r 0, and the recharge values. The iteration continued till the difference between two successive iterated values became less than The depletion time computed by all the eight sets for the spring were averaged and the recharge computed for the average depletion time. The average depletion time was 6.1 months. Korkmaz (1990) estimated the average depletion time to be 8.2 months. The recharge and depletion time computed by the Newton-Raphson method and the recharge computed with the average depletion time by the Bear model are presented in Table 1.

9 One-dimensional spring/low model for time variant recharge 389 Table 1 Observed springflow, recharge and depletion time computed by the Newton-Raphson method for the Kirkgoz Spring, Turkey. Time Observed springflow Slope (a) (month) (10 6 m 3 per month) (month -1 ) 69(7/79) (7/80) 81(7/80) (5/81) , , , , Pulse recharge estimated by: Newton-Raphson Bear model for average method T 0 = 6.1 month (10" m 3 ) (10 6 m 3 ) r 0 = 6.21 month r = 6.05 month Table 2 Annual recharge to the Kirkgoz Spring, Turkey. Water year Recharge estimated by the Korkmaz (1990) method on annual basis (10 6 m 3 ) Recharge estimated by the model on a monthly basis (10 6 m 3 ) For the Kirkgoz Spring, Turkey, the summations of monthly recharge for the water years 1974 to 1980 (October to September) computed by the model were compared with the annual recharge for these water years estimated by Korkmaz

10 390 A. K. Bhar & G. C. Mis lira (1990). It was found that the recharge values computed by the model were of comparable magnitude with the annual recharge values estimated by Korkmaz (1990) except for the water year The comparison is presented in Table 2. CONCLUSION Using the Bear model as the foundation and applying the convolution technique, the monthly recharge to a springflow domain and the depletion time (a model parameter) for a spring have been estimated using the Newton-Raphson method. The average depletion time which has been estimated as a solution of the inverse problem using springflow time series compared well with the average value of the depletion time estimated previously. Except for one year, the annual recharge for seven years computed earlier (Korkmaz, 1990) is in close agreement with the summation of the computed monthly recharges. REFERENCES Bear, J. (1979) Hydraulics of Ground Water. McGraw Hill, Israel. Carnahan, B. et al. (1969) Applied Numerical Methods. John Wiley & Sons Inc., New York. Korkmaz, N. (1990) The estimation of groundwater recharge from spring hydrographs. Hydrol. Sci. J. 35(2), Received 27 August 1996; accepted 18 November 1996