Groundwater fluctuations in heterogeneous coastal leaky aquifer systems

Transcription

1 Hydrol Earth Syst Sci,, 89 8, 00 wwwhydrol-earth-syst-scinet//89/00/ doi:09/hess Authors) 00 CC Attribution 0 License Hydrology Earth System Sciences Groundwater fluctuations in heterogeneous coastal leaky aquifer systems M-H Chuang, C-S Huang, G-H Li, H-D Yeh Department of Urban Planning Disaster Management, Ming-Chuan University, Taoyuan, Taiwan Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan Received: June 00 Published in Hydrol Earth Syst Sci Discuss: 9 July 00 Revised: October 00 Accepted: October 00 Published: October 00 Abstract In the past, the coastal leaky aquifer system, including two aquifers an aquitard between them, was commonly assumed to be homogeneous of infinite extent in the horizontal direction The leaky aquifer system may however be heterogeneous of finite extent due to variations in depositional post depositional processes In this paper, the leaky aquifer system is divided into several horizontal regions for the heterogeneous aquitard underlying aquifer A one-dimensional analytical model is developed for describing the head fluctuation in such a heterogeneous leaky aquifer system The hydraulic head of the upper unconfined aquifer is assumed constant It is found that both the length location of the discontinuous aquitards presented in the coastal area have significant effects on the amplitude phase shift of the head fluctuation in the lower aquifer In addition, the influence of the formation heterogeneity on the spatial head distribution is also investigated Introduction The groundwater near the coast may fluctuate with periodical tides Such a phenemenon is an interesting practical issue for hydrogeologists The coastal leaky aquifer is referred to an aquifer system consisting of an upper unconfined aquifer bounded from below by one or more lower aquifers Chen Jiao, 999) Many researches revealed that the head fluctuation in the unconfined aquifer due to tides is significantly damped by its storage eg, Millham Howes, 99; Chen Jiao, 999) White Roberts 99) indicated that the farthest distance that tidal propagation can reach, defined as the intrusion distance, is usually not over Correspondence to: H-D Yeh hdyeh@mailnctuedutw) 0 m if the head fluctuation is heavily damped The water table is therefore considered to maintain constant Several previous researches indicated that the effect of the leakage on the groundwater in coastal leaky aquifer systems is significant Jiao Tang 999) developed an analytical solution for describing the head fluctuation in coastal confined aquifers, where the aquitard infinitely extends from the coastal line to the inl They indicated that the leakage diminishes the amplitude of the head fluctuation the intrusion distance Li Jiao 00) presented an analytical solution for describing leaky aquifer systems with considering the leakage the effect of aquitard storage They found that both the leakage the effect of aquitard storage on the head fluctuation in the lower aquifer are negligible when the aquitard storage is small the storage ratio of the aquitard to the underlying aquifer is less than 0 Guo et al 00) developed an analytical solution for describing the groundwater head response to tidal fluctuation in a coastal aquifer consisting of two zones with different hydraulic properties The solution was used to investigate the behaviors of amplitude phase shift of the head fluctuation in the lower aquifer In addition, the solution was also used to estimate the aquifer parameters in a real coastal two-zone aquifer The variations in depositional post depositional processes over a very long period of time may result in heterogeneous hydraulic conductivity nonuniform thickness of the aquitard Cherry et al, 00) thus produce various amounts of leakage to the lower aquifer The objective of this paper is to develop a mathematical model for describing the head fluctuation in the lower aquifer due to the tidal effect in a coastal leaky aquifer system This system is divided into several horizontal regions for the heterogeneous aquitard or lower heterogeneous aquifer The head of the upper unconfined aquifer is assumed constant The analytical solution of the model is developed by the direct Fourier method The effects of the length location for a discontinuous aquitard on the amplitude phase shift of the head fluctuation in Published by Copernicus Publications on behalf of the European Geosciences Union

2 80 M-H Chuang et al: Groundwater fluctuations in heterogeneous coastal leaky aquifer systems Fig Schematic diagram of a heterogeneous coastal leaky aquifer system Figure Schematic diagram of a heterogeneous coastal leaky aquifer system the lower aquifer are investigated The present solution is applied to simulate the head distribution for the coastal aquifer system in the Chek Lap Kok CLK) airport, Hong Kong Jiao Tang, 999) the results are compared with the observed data taken at this airport In addition, the influence of formation heterogeneity on the spatial head distribution in the lower aquifer is also examined Methods Conceptual model Figure shows a coastal leaky aquifer system in which the aquitard underlying aquifer are heterogeneous in the horizontal direction The system is divided to N different horizontal regions the formation materials of the aquitard underlying aquifer in each region are homogeneous The origin of the coordinates in the x direction is located at the coastal line The distance measured from the origin to the interface between regions is d Similarly, the distance measured from the origin to the interface between regions n n is d n The mean sea level MSL) is chosen as reference datum Consider that the unconfined aquifer may store a large amount of water which can damp the tidal effect so that the water table fluctuation in the unconfined aquifer is negligible in comparison with that in the lower one Therefore, the water table is assumed to be the MSL The governing equation describing the head distributions in these regions is expressed as S n h n t = T n h n x L n h n ) where n is an integer from,, N; h n, T n S n are the hydraulic head, transmissivity, storativity for the lower aquifer in n-th region, respectively; L n is the leakage for the aquitard in n-th region The tidal boundary is expressed as where A ω are the amplitude frequency of the tide, respectively The remote boundary condition is h N x,t) lim = 0 ) x x where h N is the hydraulic head for the farthest lower aquifer from the coastal line The continuities of the head flux required at the interfaces d n are respectively h n x,t) = h n+ x,t) at x = d n ) T n h n x,t) x = T n+ h n+ x,t) x at x = d n ) Solution for heterogeneous aquifer systems Based on the direct Fourier method, the solutions for describing the head distributions in the regions demonstrated in Fig can be expressed as [ h n x,t) = Re A X n x) e iω t] ) where Re means the real part of the complex expression i is equal to Substituting Eq ) into Eq ) eliminating the exponential terms can obtain ordinary differential equations in terms of X n x) Substituting Eq ) into Eq ) with n = eliminating the exponential terms acquire the boundary condition expressed as X 0) = Similarly, the equation lim X N/ x = 0 can be obtained from Eqs ) x ) The continuity requirements ie, X n = X n+ T n X n / x = T n+ X n+ / x) at x = d n can also be acquired from Eqs ) ) in a similar manner The general solutions for X n x) can then be solved as X n x) = c n e λ nx +c n e λ nx with 7) h 0, t) = Acosω t) ) λ n = a n un i) 8) Hydrol Earth Syst Sci,, 89 8, 00 wwwhydrol-earth-syst-scinet//89/00/

3 M-H Chuang et al: Groundwater fluctuations in heterogeneous coastal leaky aquifer systems 8 where the parameter a n = ωs n /T n is a reciprocal of the decay length for the n-th lower aquifer u n = L n /ωs n is the dimensionless leakage The coefficients, c n c n, for n-th lower aquifer can then be determined by the boundary conditions X 0) = lim X N/ x = 0 as well as x the continuity requirements X n = X n+ T n X n / x = T n+ X n+ / x The equations for solving c n c n are expressed in matrix form as c c D N N c N c c c N with I I D = B C E F G H N N N 0 = 0 N 9), 0) I = [ ] N, ) G = [ ] N, ) H = [ ] N, ) B = E = e λ d e λ d e λ d e λ d e λ N d N e λ N d N e λ N d N e λn d N N ) N Tλe λd Tλe λd Tλe λd Tλe λd TN λn e λn dn TN λn e λn dn TN λn e λn dn TNλN e λndn N ) N ) ) where both C F are N ) N matrixes identical to matrixes B E, respectively, except having a minus sign before the exponent in the exponential terms Based on Cramer s rule, the coefficients c n c n can be determined as c n = det D n det D ) c n = det D N+n det D 7) where det is an abbreviation for determinant of the matrix; D n D N+n can be obtained by replacing the n-th N +n)-th columns of matrix D, respectively, with a column matrix that the top element is one the others are zero Define the variable a Xn b Xn representing the real imaginary parts of the result calculated from Eq 7), respectively, thus one can have X n x) = a Xn +ib Xn Substituting X n x) = a Xn + ib Xn e iωt = cosωt) isinωt) into Eq ) taking the real part of Eq ) lead to the result in terms of cosine function as h n x,t) = A c n cosω t φ n ) 8) with c n = a X n +b X n 9) φ n = arccos axn c n ) 0) where φ n is the phase shift c n is the amplitude coefficient Special cases Jiao Tang 999) presented an analytical solution for describing the head fluctuation in a coastal leaky aquifer system Both the aquifer aquitard in their system are homogeneous of infinite extent from the coastal line Their solution 999, Eq ) can be obtained from the present solution for N = written, in our notation, as h x,t) = A c cosω t φ ) ) where c = exp φ = a { a [ u + ) 0 +u ] } x ) a [ u + ) ] x ) 0 +u In addition, Guo et al 00) presented an analytical solution to describe the head fluctuations in a coastal aquifer system consisting of two zones which have different hydraulic properties in the horizontal direction When N = u = u = 0, the present solution reduces to Guo et al s 00) solution 00, Eq 7a for the first aquifer Eq 7b for the second aquifer) which is expressed, in our notation, as h x,t) = A c cosω t φ ) ) wwwhydrol-earth-syst-scinet//89/00/ Hydrol Earth Syst Sci,, 89 8, 00

4 8 M-H Chuang et al: Groundwater fluctuations in heterogeneous coastal leaky aquifer systems h x,t) = A c cosω t φ ) ) where c = ax +bx, c = ax +bx, ) ) ) ax ax φ = arccos, φ = arccos c c a X = µ a +ν a χ b X = σ b +τ b χ, b X = µ b +ν b χ, a X = σ a +τ a, 7) χ ) u a = a T a T cos[a x d )] 8) e ) a d x) +e a d +x) ν a = cosa x) 9) [a T a T ) e a x +a T +a T ) e a d x) ] [ ) µ b = a T e a x e d a e ax sin[a d x)] 0) ) ] + e d a e a x sina x) +a T a T sina x) e ) a d x) +e a x [ ) ν b = a T e a x e d a e a d +x) sin[a d x)] ) ) ] + e d a e a x sina x) σ a = a T a T a T )cos[d a +a d x)] ) e d a +a d x) τ a = a T a T +a T )cos[d a +a x d )] ) e d a +a d x) σ b = a T a T +a T )sin[d a +a d x)] ) e d a +a d x) τ a = a T a T +a T )sin[d a +a x d )] ) e d a +a d x) χ = a T a T ) +a T +a T e d a ) ) +e d a cosd a ) a T a T Accordingly, the solutions derived by Jiao Tang 999) Guo et al 00) can be considered as our special cases Figure The curves of normalized amplitude of the head fluctuation versus dimensionless inl distance for u =, u = 0, a = a = 0 Fig The curves of normalized amplitude m -, various of the aquitard headlengths fluctuation versus dimensionless inl distance for u =, u = 0, a = a = 0 m, various aquitard lengths Results discussion The effect of aquitard length Consider the case that the leaky aquifer system with a homogeneous lower aquifer, overlain by a heterogeneous aquitard, can be divided into two regions ie, regions ) The aquitard in region has semi-permeable with leakage u = a finite length d while the aquitard in region is impermeable u = 0) of infinite extent in the x direction The purpose of this case is to investigate the effect of aquitard length on the amplitude phase shift of the head fluctuation in the lower aquifer Figure shows the curves of normalized amplitude of the head fluctuation versus dimensionless inl distance for a = a = 0 m aquitard length d of 0, 0, 0, 00, 80 m The normalized amplitude is defined as the ratio of the amplitude of the head fluctuation to that of the tide The data represented by the open circle is obtained by the present solution with no leakage ie, u = u = 0) The curves shown in Fig indicate that a larger d has a smaller normalized amplitude a shorter tidal intrusion distance In addition, the present solution approaches Jiao Tang s999) solution for u = a = 0 m when d goes large say d = 80 m) This is because the aquitard length is larger than the tidal intrusion distance Figure shows the curves of phase shift of the head fluctuation versus dimensionless inl distance for u =, u = 0, a = a = 0 m at d = 0, 0, 00, 80 m The phase shift increases with the inl distance for various Hydrol Earth Syst Sci,, 89 8, 00 wwwhydrol-earth-syst-scinet//89/00/

5 M-H Chuang et al: Groundwater fluctuations in heterogeneous coastal leaky aquifer systems 8 Figure The curves of phase shift of the head fluctuation versus dimensionless inl Fig The curves of phase shift of the head fluctuation versus distance for u =, u = 0, a = a = 0 m -, various aquitard lengths dimensionless inl distance for u =, u = 0, a = a = 0 m, various aquitard lengths Figure The curves of normalized amplitude of the head fluctuation versus dimensionless Fig The curves of normalized amplitude of the head fluctuation versus inl distance dimensionless for u = 0, u inl =, a = distance a = 0 mfor -, u various = 0, values u = of, d a = a = 0 m, various values of d values of d, indicating that the response of groundwater in the lower aquifer to the tide becomes slow over a large inl distance The phase shift for the case of no leakage represented by the open circle is significantly larger than that of Jiao Tang s 999) solution represented by the solid circle, indicating that the leakage can diminish the phase shift of the head fluctuation in the lower aquifer The slopes of the curves with d = 0 m d = 0 m begin to close to that of Jiao Tang s 999) solution near x = m x = m, respectively, which are in fact fairly close to the end of the aquitard On the other h, these two curves match with the solid circle near the coastal line, reflecting the fact that they are all influenced by the leakage 7 Effect of aquitard location In contrast to the previous case, the aquitard in region with a length d is now treated as impermeable, ie, u = 0, while the aquitard in region has u = The aquitard lower aquifer in region are considered to be of infinite extent The purpose of this case is to study the effect of the value of d on the amplitude phase shift of the head fluctuation in the lower aquifer Figures demonstrate the curves of normalized amplitude phase shift, respectively, of the head fluctuation in the lower aquifer versus dimensionless inl distance for a = a = 0 m d = 0, 0, 00, or 0 m Both figures show that the curve with d = 0 m matches with the curve of no leakage, indicating that the aquitard resided far away from the costal line does not affect the amplitude phase shift In addition, the figures also 7 Figure The curves of phase shift of the head fluctuation versus dimensionless inl Fig The curves of phase shift of the head fluctuation versus dimensionless distance for u = 0, u inl =, a distance = a = 0 for m -, u various = 0, uvalues = of, d a = a = 0 m, various values of d show that those dashed lines approach the line with large d, indicating that the effects of the aquitard on the amplitude phase shift decrease with increasing d Once the d is larger than the tidal intrusion distance, the effects of the leakage on both amplitude phase shift become negligible 8 wwwhydrol-earth-syst-scinet//89/00/ Hydrol Earth Syst Sci,, 89 8, 00

6 8 M-H Chuang et al: Groundwater fluctuations in heterogeneous coastal leaky aquifer systems 7 Figure The spatial head distributions in the lower aquifer with heterogeneous trending Fig The spatial head distributions in the lower aquifer with heterogeneous aquitards for scenarios trending aquitards with forhomogeneous scenariosaquitards but having with different homogeneous conductivity values aquitards for scenarios but having at high-tide different period conductivity ω t = π ) values for scenarios at high-tide period ωt = π) Effect of heterogeneity in aquitard or aquifer Consider the case that a leaky aquifer system has a heterogeneous aquitard The aquitard is divided into three regions the hydraulic conductivity of the aquitard in each region is 9 homogeneous The lower aquifer is considered to be homogeneous in order to investigate the effect of aquitard heterogeneity on the aquifer head fluctuation Figure shows the spatial head distribution of the lower aquifer with a = a = a = 0 m for the aquitard having four different scenarios at the high-tide period ie, ωt = π) The aquitard is heterogeneous in the first two scenarios homogenous but with different values of hydraulic conductivity in the other two scenarios The normalized hydraulic head is defined as the ratio of the hydraulic head of the lower aquifer to the amplitude of the tide For scenario, the leakages in regions are chosen as u = 0, u = u =, respectively In contrast, the leakages in these three regions are selected as u =, u = u = 0 for scenario In addition, the leakages are chosen as u = u = u = for scenario u = u = u = 0 for scenario The head distribution curve of scenario is below that of scenario close to that of scenario, indicating that a large leakage occurring near the coastal line region ) significantly diminishes the hydraulic head On the other h, the curve of scenario in region is below that of scenario even when the leakage in region of scenario is the same as that of scenario This is because large leakages occurring in regions will influence the hydraulic head distributions in regions Figure 7 The spatial head distributions for the homogeneous aquifer heterogeneous Fig 7 The spatial head distributions for the homogeneous aquifer trending heterogeneous aquifer at ω t = trending π ω t = aquifer 0 π at ωt = π ωt = 0π Accordingly, the aquitard heterogeneity has significant effect on the hydraulic head distribution Alluvial fans formed at the base of mountains usually have coarser sediment at the upstream part of the fan finer sediment at the downstream part The formation of the coastal leaky aquifer in the alluvial fan may exhibit the phenomenon of trending heterogeneity Freeze Cherry, 979) The lower aquifer is divided into three regions for simulating trending heterogeneity ie, T = 0 m /day, 0 T = 0 m /day T = 00 m /day) The leakages in these three regions are chosen the same for assessing the effect of aquifer heterogeneity Figure 7 shows the spatial head distributions at ωt = π ωt = 0π for the homogeneous aquifer heterogeneous trending aquifer Following parameter values are used in the analyses: u = u = u =, S = S = S = 0, d = 00 m, d = 00 m ω = π/0 day The hydraulic conductivities are considered as T = T = T = 0 m /day for the homogeneous aquifer T = 0 m /day, T = 0 m /day T = 00 m /day for the trending aquifer The figure indicates that the trending aquifer has a smaller tidal intrusion distance than the homogeneous one This is because the region has a smaller hydraulic conductivity In addition, the slopes of the normalized head distribution are markedly different near x = 00 m 00 m because the hydraulic conductivities in these regions are different Obviously, the aquifer heterogeneity also has an impact on the hydraulic head distribution Comparison of present solution with observed data Figure 8 shows a geological cross section of the leaky aquifer system in the CLK airport, Hong Kong Jiao Tang, 999) The lower aquifer is homogeneous lies on the Hydrol Earth Syst Sci,, 89 8, 00 wwwhydrol-earth-syst-scinet//89/00/

7 M-H Chuang et al: Groundwater fluctuations in heterogeneous coastal leaky aquifer systems 8 Fig 8 A geological cross section in the CLK Airport, Hong Kong modified from Jiao Tang, 999) Figure 8 A geological cross section in the CLK Airport, Hong Kong modified from Jiao Tang, 999) bedrock composed of granite as the impermeable boundary The upper aquifer filled with gravel s is an unconfined aquifer The aquitard in between mainly consists of clay The vertical dashed line shown in the figure divides this aquifer system into two regions with different thicknesses The thickness of the aquitard in region is about a half of that in region ; therefore, the leakage in region is about two times that in region Jiao Tang 999) applied their solution to simulate the head fluctuation for the aquifer system in the CLK airport The observed well was located at 7 m distance from the coastal line The lower aquifer is considered to be homogeneous with a = 7 0 m The thickness of the aquitard is assumed constant in the horizontal direction the leakage is chosen as u = 98 0 The amplitude frequency of the tide are chosen as A = 08 m ω = π/0 day, respectively Figure 9 shows the observed data taken from the aquifer system in the CLK airport, Hong Kong the present solution computed with the parameters d = 00 m, a = a = 7 0 m, ω = π/0 day, A = 08 m, u = 98 0 u = 0u This figure shows that the present solution is identical to Jiao Tang s 999) solution has a good agreement with the observed data Such a result can be attributed to the fact that both leakages in regions are very small the length of the thin aquitard exceeds the tidal intrusion distance ie, 0 m), which is estimated based on Eq 8) Concluding remarks Based on the direct Fourier method, a one-dimensional analytical model is developed for describing the head fluctuation due to the tidal effect in a coastal leaky aquifer system The aquitard underlying aquifer can be considered to be heterogeneous divided into several regions each region has a homogenous hydraulic conductivity The water table of the upper unconfined aquifer is assumed constant while the hydraulic head of the lower aquifer is subject to the effect of tidal fluctuation The solution developed by Jiao Tang 999) can be considered as a special case of the present solution when N = or the length of the first aquifer is larger than the tidal intrusion distance The present Figure 9 Predicted hydraulic heads by the present solution Jiao Tang s solution Fig 9 Predicted hydraulic heads by the present solution Jiao 999) Tang s the observed 999) data solution measured at the the CLK observed airport, Hong data Kong measured at the CLK airport, Hong Kong solution can reduce to Guo et al s 00) solution if N = u = u = 0 With considering the change in aquitard thickness, the present solution has a good agreement with the observed data from the aquifer system in the CLK Airport, Hong Kong Jiao Tang, 999) Based on the present solution, the effects of formation heterogeneity as well as the length location of the aquitard on the head fluctuation in the lower aquifer can be concluded as follows: The aquitard with a large leakage /or long length near the coastal line will diminish the amplitude of the head fluctuation in the lower aquifer The phase shift of the head fluctuation in the lower aquifer increases slowly with the inl distance for the case of large leakage dramatically for the case of no leakage A lower aquifer with trending heterogeneity in a coastal leaky aquifer system usually has a small tidal intrusion distance low head distribution Acknowledgements This study was supported by Aim for the Top University Plan of the National Chiao Tung University Ministry of Education, Taiwan the Taiwan National Science Council under the contract numbers NSC 98--E-007-0, NSC 99-NU-E , NSC 99--E MY The authors would like to thank the associate editor two anonymous reviewers for their valuable constructive comments Edited by: J Carrera wwwhydrol-earth-syst-scinet//89/00/ Hydrol Earth Syst Sci,, 89 8, 00

8 8 M-H Chuang et al: Groundwater fluctuations in heterogeneous coastal leaky aquifer systems References Chen, C X Jiao, J J: Numerical simulation of pumping tests in multilayer wells with non-darcian flow in the wellbore, Ground Water, 7), 7, 999 Cherry, J A, Parker, B L, Bradbury, K R, Eaton, T T, Gotkowitz, M B, Hart, D J, Borchardt, M A: Contaminant Transport through Aquitards: A State-of-the-Science Review, International Water Association, UK,, 00 Freeze, R A Cherry, J A: Groundwater, Prentice-Hall, Inc, New Jersey, 979 Guo, H, Jiao, J J, Li, H: Groundwater response to tidal fluctuation in a two-zone aquifer, J Hydrol, 8 ), 7, doi:00/jjhydrol009009, 00 Jiao, J J Tang, Z: An analytical solution of groundwater response to tidal fluctuation in a leaky confined aquifer, Water Resour Res, ), 77 7, 999 Li, H Jiao, J J: Analytical studies of groundwater-head fluctuation in a coastal confined aquifer overlain by a leaky layer with storage, Adv Water Resour, ), 7, 00 Millham, N P Howes, B L: A comparison of methods to determine K in a shallow coastal aquifer, Ground Water, ), 9 7, 99 White, J K Roberts, T O L: The significance of groundwater tidal fluctuations, Groundwater Problem in Urban Areas: Proceeding of the International Conference Organized by the Institution of Civil Engineers Held in London, June 99, edited by: Wilkinson, W B, Thomas Telford, London,, 99 Hydrol Earth Syst Sci,, 89 8, 00 wwwhydrol-earth-syst-scinet//89/00/