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In this study, analytical models for predicting groundwater contamination in isotropic homogeneous porous formations are derived. The impact of dispersion diffusion coefficients is included in the solution of the advection-dispersion equation (ADE), subjected to transient (time-dependent) boundary conditions at the origin. A retardation factor zero-order production terms are included in the ADE. Analytical solutions are obtained using the Laplace Integral Transform Technique (LITT) the concept of linear isotherm. For illustration, analytical solutions for linearly space- time-dependent hydrodynamic dispersion coefficients along with molecular diffusion coefficients are presented. Analytical solutions are explored for the Peclet number. Numerical solutions are obtained by explicit finite difference methods are compared with analytical solutions. Numerical results are analysed for different types of geological porous formations i.e., aquifer aquitard. The accuracy of results is evaluated by the root mean square error (RMSE). Keywords: Solute; Advection; Diffusion; Dispersion; Aquifer; Finite Difference Method. INTRODUCTION Analytical solutions of the advection-dispersion solute transport equation remain useful for a large number of applications in water science engineering, hydrological science engineering, environmental science engineering etc. They are also useful to benchmark the numerical codes solution as a preliminary predictive tool in groundwater resource management. The bulk transport of solutes is dominated by advection dispersion with which diffusion is usually lumped. Longitudinal dispersion causes contaminant plumes to spread out in the direction of flow. The spreading is due to the heterogeneity of the medium, such as a distribution of pore sizes shapes. Dispersion which is perpendicular to the aquifer flow direction is called transverse dispersion. It is related to diffusion plays an important role in the remediation of contaminants. It helps dilute the concentration of contaminants mix them with reactive compounds. The combination of hydrodynamic dispersion molecular diffusion coefficients is considered to describe the solute transport (Batu, 2006; Sharma Reddy, 2004). The dominant process of solute transport is advection moving aqueous chemical species along with fluid flow. Most of the solute transport modeling begins with advective transport. The advection-dispersion equation describes the spatial temporal variation in solute concentration with specific initial boundary conditions. The governing equation known as the constant-parameter advection-dispersion equation may be derived for the case of steady unsteady flows. The traditional advection-dispersion equation represents a stard model to predict the solute concentration in an aquifer which is based on conservation of mass Fick's law of diffusion (Bear, 1972; Fried Combarnous, 1971). Scheidegger (1957) explored the dispersion theory based on a relationship between two parameters (i.e., dispersion coefficient seepage ) proposed two possible relationships that (i) the dispersion coefficient is proportional to seepage (ii) the dispersion coefficient is proportional to the square of seepage . A general theory of dispersion in porous media was explored by Scheidegger (1961). Later, the dispersion theory was generalized as dispersion coefficients proportional to the power of seepage where power ranges from 1 to 2 (Freeze Cherry, 1979). It has also been experimentally observed that the dispersion coefficient is directly proportional to the seepage with a power ranging from 1 to 1.2 (Ghosh Sharma, 2006). Ebach White (1958) described the time-dependent input concentration for longitudinal dispersion flow. Barry Sposito (1989) obtained a closed form solution to solute transport in a semi-infinite domain with an arbitrarily time-dependent dispersion coefficient arbitrary initial boundary flux conditions by variable transformation. They also computed a numerical solution using a simple scheme compared it with the analytical solution. The numerical scheme was unstable for large values of t . Ogata Banks (1961) presented a solution to a one-dimensional advection-dispersion equation in a homogeneous medium in which the impact of diffusion was not included. Gelhar et al. (1992) proposed that the dispersivity is often found to be scale-dependent. Huang et al. (1996) employed general analytical solutions for one-dimensional solute transport in heterogeneous porous media with linearly increasing dispersivity. Aral Liao (1996) studied the timedependent dispersion coefficient for contaminant transport. Basha El-Habel (1993) presented analytical solutions with one-dimensional linear, asymptotic exponential timedependent dispersivities in infinite domains. Huang et al. (2006) employed a parabolic distance-dependent dispersivity in a finite column. Guerrero Skaggs (2010) analysed the analytical solution for the advection-dispersion equation with the distance-dependent coefficient using the generalised integral transform technique (GITT). Chen et al. (2012) presented a novel method for solving analytically multi-species advectivedispersive transport equations sequentially coupled with firstorder decay reactions. Singh et al. (2012) proposed one-dimensional analytical solution with temporally dependent dispersion in a homogeneous semi-infinite porous formation. Guerrero et al. (2013) analysed the diffusion type problems with transient boundary conditions with Duhamel theorem. You Zhan (2013) studied the semi-analytical solution for solute transport in a finite column with linear asymptotic or exponen- tial distance-dependent dispersivities time-dependent sources. Zamani Bombardelli (2014) explored analytical solutions to the ADR (advection-dispersion-reaction) equation from which one can know the spatio-temporal changes in flow field dispersivity. Van Genuchten et al. (2013) presented a series of one- multi-dimensional solutions of the stard equilibrium advection-dispersion equation with without terms accounting for zero-order production first-order decay, which proved useful for simplified analyses of contaminant transport in surface water, for mathematical verification of more comprehensive numerical transport models where the isotherm concept was not used. In this study, we derived two novel analytical solutions for advection-dispersion equations with the impact of space- time-dependent dispersion with diffusion. Both equations were solved analytically, subjected to linear adsorption coefficient in the solid liquid phases. First, they were solved with respect to the space-dependent dispersion in longitudinal direction of flow with diffusion. The asymptotic dispersivity was considered which increased steadily with respect to space from the input source. The domain was initially not solute free i.e., the geological formation was not clean which means some initial background concentration existed. It was represented by a linear expression of the initial source with zero order production term which affected the solute concentration. One end of the domain, the time-dependent decreasing source was injected along with the uniform concentration of the initial source. The solute mass ejected from the other end of the domain so the flux was supposed to be zero. The concept of dispersion directly proportional to the seepage , was employed (Freeze Cherry, 1979). The Peclet number was used in the characteristic form with the space-dependent dispersivity which also affected the domain. A new time variable was used to define the old time in the longitudinal direction. A new transformation was used to reduce the advection term. Second, we considered the time-dependent dispersion with diffusion. Here, we used the concept of dispersion as directly proportional to the ratio of the initial seepage with the corresponding mean travel time. The dispersivity was defined for the positive time interval with respect to diffusion, i.e., the reactive solute was activated from the origin of source in the domain travel along the longitudinal direction of flow. The objective of this work was to derive an analytical solution with the help of Laplace Integral Transform Technique (LITT) for predicting contaminant concentration with distance time. Numerical simulation for three different types of expressions (i.e., exponentially decreasing, sinusoidally varying, algebraic sigmoid) with the corresponding mean travel time was done. The analytical solutions were analysed for the aquifer aquitard formations. In general, the transmission capacity is low in aquitard as compared to the aquifer. So, the rate of solute concentration was high in the aquitard as compared to the aquifer. The mean travel time Peclet number were also compared. MATHEMATICAL FORMULATION OF THE PROBLEM Considering a system in which non-reactive solute transport is primarily one-dimensional, i.e., solute concentrations are horizontally vertically well-mixed so that concentrations vary only in the longitudinal or downstream direction. The effects of dispersion are constant or the time-dependent with respect to distance; the solutes are conservative in the unsteady field. Given these assumptions, the conservation of mass yields the constant parameter advection-dispersion equation (ADE) as c 1 - n F c + = D - uc - c t n t x x (1) where c [ML3] is the volume averaged dispersing solute concentration in the liquid phase, F [ML3]is the volume averaged dispersing solute concentration in the solid phase, D [L2T1]is the longitudinal dispersion coefficient (i.e. representing longitudinal dispersion), u [LT1]is the unsteady uniform downward pore seepage , x [L] is the length in longitudinal direction of flow , t [T] is time, [T1] is the first order decay rate coefficient for solute production in the liquid phase, n is the porosity of the different geological formations. A linear adsorption coefficient in terms of the solid liquid phases is considered as F = Kd c where K d is the distribution coefficient. Using equation (2) in equation (1), we have R c c = D - uc - c t x x 1- n Kd n (2) (3) where R = 1 + (4) is the retardation factor. The retardation commonly describes a reversible sorption process, including adsorption ion exchange. Sorption is experimentally defined by the measured quantity of a solute that can be sorbed by a particular sediment, soil or rock. Initially the aquifer is assumed to be contaminated (i.e., some initial background concentration exists in the aquifer) it is represented by a linear combination of some kind of initial concentration the zero order production term with seepage . The mathematical expression for the assumption can be written as c ( x,0 ) = ci + , x > 0, t = 0 (5) where ci [ML3] is the initial background concentration, [ML3T1] is the zero order production rate coefficient for solute production in the liquid phase. Solute transport is also affected by the boundary condition chosen as input time-dependent source concentration with initial background concentration at the origin i.e., at x = 0 . To solve ADE given in equation (1), an inlet boundary condition is prescribed as c ( 0, t ) = ci + c0 exp ( -t ) t > 0, x = 0 (6) where [T1] is the decay rate constant. At the other end of the aquifer, solute transport may not be affected therefore, an outlet boundary condition is prescribed as a no flux boundary condition, i.e. c = 0 , x x (7) Analytical solution for space-dependent dispersion with diffusion Now applying equation (11) in equation (10) we have Let = 0 f ( mt ) where f ( mt ) is non-dimensional, m [T1] is the flow resistance coefficient. Here D0 [L2T1] is the initial longitudinal dispersion coefficient, u0 [LT1] is the initial seepage , 0 [T ] is the initial first order decay term, 0 [ML3T1] is the initial zero-order production term. Equation (3) can be written as 1 u = u0 f ( mt ) , D = D0 f ( mt ) , = 0 f ( mt ) c T * = qxu0 + D* c ) xc - u (1 - q ) x - c (12) Applying the following non-dimensional parameters C= D D c xu T *u0 2 , X = 0, T = , 1 = 0 2 0 , 1 = 0 0 c0 D0 D0 u0 c0u02 (13) in equation (12), one can get as R C 2C C = D1 2 - A1 - 1C T X X (14) R c c = D0 - u0c - 0c f ( mt ) t x x Further we introduce a new time variable (Crank, 1975) (8) D1 = (9) 0u0 D0 T = f ( mt ) dt D* 1 x , A1 = 1 - , Pe = , 0 = qx D0 Pe 0 (15) Using equation (9) in equation (8) we obtain R c T where 0 is asymptotic dispersivity. Now the initial boundary conditions are expressed in terms of non-dimensional parameters as C ( X ,0 ) = C ( 0, T ) = C =0, X ci + 1X , c0 c D0 - u0c - 0c x x (10) X > 0, T = 0 (16) (17) (18) The dispersion coefficient is expressed as a combination of hydrodynamic dispersion diffusion as D0 = qxu0 + D* ci D + 1 - *T , X = 0, T > 0 , where * = 20 c0 u0 X , T 0 (11) We used the transformation where q is the non-dimensional constant parameter growing rate of dispersivity w. r. t. distance, D* [L2T1] is the molecular diffusion coefficient. In most of the previous works, the impact of diffusion is not considered because it does not vary significantly for different soil contaminant combinations ranges from 1×109 to 2×109m2/sec (Mitchell, 1976). Later, it was found that even a small value for the diffusion coefficient may also have significant impact on solute transport in porous media, especially for fine-grained geologic materials (Gillham Cherry, 1982). A 1 A2 C ( X , T ) = K ( X , T ) exp 1 - 1 + 1 T , 2 D1 R 4 D1 (19) to eliminate the convective terms form of equation (14) where K ( X , T ) is another dependent variable. Using equation (19) for reducing the advective term from equation (14) analytical solution obtained subject to initial boundary conditions given in equations (16) (18) by applying the Laplace transform technique, the solution can be written as ci ci * + 1 F ( X , T ) - G ( X , T ) - H ( X , T ) c0 c0 A 1 A2 C ( X ,T ) = exp 1 - 1 + 1 T A1 A12 T A 2 D1 R 4 D1 c A - X - + 1 1 I ( X ,T ) + i + 1 X - 1 1 T e 2 D1 4 D1R R R c0 1 R 1 R 1 R R where F ( X , T ) = exp T - X erfc 2 D T X - T + exp T + D X erfc 2 D T X + T 2 D1 1 1 1 (20) (21a) G ( X ,T ) = 1 R R 1 R X exp T - X erfc X - T 2 T - 2 DT D1 D1 4 1 1 R R 1 R X exp T + X erfc X + T + 2 T + 2 DT D1 D1 4 1 (21b) A2 A2 1 R 1 R 1 A A T A A T H ( X , T ) = exp 1 T - 1 X erfc X- 1 X+ 1 + exp 1 T + 1 X erfc 2 2 D1 2 D1R 2 D1 2 D1R 2 D1T 2 D1T 4 D1R 4 D1R I ( X ,T ) = A2 1 R D1 A1T R A A T X exp 1 T - 1 X erfc X- 1 - 2 DT 4D R 2 A1 D1R 2 D1 2 D1R D1 1 1 (21c) A2 1 R D1 A1T R A A T + + X exp 1 T + 1 X erfc X+ 1 2 DT 4D R DR D1 2 A1 1 2 D1 2 D1R 1 1 (21d) 1 A12 + 1 R 4 D1 (21e) The value of maximum dispersivity Q for the timedependent diffusion in the semi-infinite domain is considered as follows: Analytical solution for time-dependent dispersion with diffusion In this case, we use the concept of dispersion which is directly proportional to the initial seepage inversely proportional to the mean travel time of the solute transported f ( mt ) f ( mt ) , D = D0 , through the aquifer. Let u = u0 K1 K1 K1 ; t > 0 Q = f ( mt ) 0 ; t<0 Using equation (25) in the equation (24), we have R f ( mt ) 2c c * = Q + D* 2 - u0 - 0c K1 x x c (26) (27) time * 0 f * 0 ( mt ) 1 K1 0 f ( mt ) K1 where K1 is the mean travel For the positive value of time the equation (27) reduces to R 2c c * = 1 + D* 2 - u0 - 0c x x c [T ]is the initial first order decay rate coefficient for solute production in the liquid phase, * [ML3T1] is the 0 initial zero-order production term. Now equation (3) can be written as R K1 c c * = D0 - u0c - 0c f ( mt ) t x x (22) i.e., R = D*1 2c x - u0 c * - 0c x (28) Again using the non-dimensional variable we have Further, using the new time-dependent variable C= (23) c xu *u02 *D *D , X = 0, = , 2 = 0 2 0 , 2 = 0 0 c0 D0 D0 u0 c0u02 (29) f ( mt ) dt K1 Equation (28) reduces to in the above equation (22), we get as follows: R c * = c * D0 - u0c - 0c x x D* C 2C C = D11 2 - - 2C , where D11 = 1 D0 X X (30) (24) The corresponding initial boundary conditions in terms of the non-dimensional variable, can be written as C ( X ,0 ) = ci + 2 X , X > 0, = 0 c0 where D0 is the dispersion coefficients defined in terms of the linear time-dependent expression D0 = Q f ( mt ) + D* K1 (25) (31) C ( 0, ) = C =0, X ci D + 1 - * , X = 0, > 0 , where * = 20 c0 u0 X , 0 (32) where Q is the maximum dispersivity, K1 is the mean travel time, D is the molecular diffusion coefficients, f ( mt ) is (33) the generalised case of the time-dependent function which describes the diffusivity in the whole domain. Since the mean travel distance for the dispersivity is directly proportional to the mean travel time in the longitudinal flow with a constant field, the two formulations for the diffusivity are considered in the equilibrium nonequilibrium forms. Further using the following transformation 1 1 1 C ( X , ) = K ( X , ) exp X- + 2 R 4 D11 2 D11 (34) from the above equations (30) (33) by applying the same procedure as discussed earlier, we get the solution as ci ci * + 1 F1 ( X , ) - G1 ( X , ) - H1 ( X , ) c0 1 1 1 c0 + 2 exp C ( X , ) = X- 1 1 R 4 D11 2 D11 + 2 I X , + ci + X - 2 T e- 2 D11 X - 4 D11R ) 2 R 1( R c0 1 1 1 R R R R where F1 ( X , ) = exp - X erfc X - + exp + X erfc X + 2 D 2 D 2 D11 D11 11 11 (35) (36a) G1 ( X , ) = 1 R X exp - 2 - D11 4 1 R + X exp + 2 + D11 4 1 R X erfc X - 2 D D11 11 1 R R X erfc X + 2 D D11 11 (36b) 1 1 R 1 1 X erfc X- - exp 2 D 2 D11 2 D11R 11 4 D11R 1 H1 ( X , ) = 2 1 1 R 1 1 + exp 4 D R + 2 D X erfc 2 D X + 2 D R 11 11 11 11 I1 ( X , ) = 1 D11R 1 R R 1 1 X exp X erfc X- - - 2 D D11 2 D11R 2 D11 2 D11R 11 4 D11R (36c) 1 D11R 1 R R 1 1 X exp X erfc X+ + + + 2 D D R D11 2 2 D11 2 D11R 11 4 D11R 11 (36d) 1 1 + 2 R 4 D11 (36e) Using equation (37) in equation (14) we have R C = D1 1 - X ' T NUMERICAL SOLUTION 2C X '2 - D2 1 - X ' C ) X - C , ' 1 (38) Numerical dispersion was first quantified by Lantz (1971). The second-order error was investigated through the truncated Taylor series approximation by using the explicit finite difference method to solve one-dimensional advection dispersion equation (Chaudhari, 1971). Numerical studies explored the effect of numerical dispersion (De Smedt Wierenga, 1977; Dudley et al., 1991; van Genuchten Gray, 1978). The root mean square was used to calculate the average error at each nodal point of the grid (Roberts Selim, 1984). Moldrup et al. (1994) presented an explicit finite-difference model based on an unsteady one-dimensional solute transport equation. In general, the solution of the transport equation can be obtained by numerical integration. The objective of this study is to validate the analytical solution with a numerical solution for solute transport in a homogeneous porous medium with space- time-dependent dispersion diffusion. Numerical solution for space-dependent dispersion diffusion where D2 = D1 + A1 . Initial boundary conditions are transformed as follows: C X ' ,0 = C ( 0, T ) = C X ' ci 1 ' + 1 log ; X > 0, T = 0 c0 1- X ' (39) (40a) ci + 1 - *T ; X ' = 0, T > 0 c0 = 0 ; X ' = 1, T > 0 (40b) In order to solve the advective-dispersive equation by an explicit finite difference method, the semi-infinite medium is changed into a finite medium by using the following transformation: X ' = 1 - exp ( - X ) (37) The one-dimensional solute transport equation represents a partial differential equation of parabolic type in which, finite difference technique is commonly used to obtain numerical solution. An explicit finite difference technique is commonly used, even though it requires extended computing time because of its restrictive stability criteria. The finite difference technique is derived by using Taylor's expansion (Carnahan et al., 1969; Mickley et al., 1957) in which partial derivatives are approximated hence, solutions can be obtained by using the first, second or higher order accurate methods. Here, we have used the general form of an explicit finite difference approximation with forward time central space forward difference scheme in equation (38) together with initial boundary conditions given in equations (39), 40(a) to 40(b) which are approximated, respectively, as follows: 2 D T Ci , j +1 = 1 - 1 T Ci , j + 1 1 - X i ' Ci +1, j - 2Ci , j + Ci -1, j R R X '2 D T - 2 1 - X i ' Ci +1, j - Ci -1, j ( 41) R 2X ' Numerical solution for time-dependent dispersion with diffusion Ci ,0 = C0, j 1 ci + 1 log ; 1- X ' c0 i Applying the transformation defined in equation (37), the time-dependent diffusion equation (30) expressed in finite domain is written as follows: R C = D11 1 - X ' i>0 (42) (43a) (43b) c = i + 1 - *T j ; c0 2C X '2 - D12 1 - X ' j>0 j>0 C ) X - C ' 2 (47) CM , j = CM -1, j where i is the longitudinal space subscript, j gives different time levels for which the solutions are approximated, X ' is the increment in the longitudinal direction, T is the time increment in equation (41). The space domain X ' time domain T are discretized by a rectangular grid of point X i, T j with mesh-size X T , respectively: where D12 = 1 + D11 . The corresponding initial boundary conditions are written as follows: C X ' ,0 = C ( 0, ) = ci 1 + 2 log ; c0 1- X ' X ' > 0, = 0 (48) (49a) (49b) ci + 1 - * ; c0 X ' = 0, > 0 C X ' = 0 ; X ' = 1, > 0 X 'i = X 'i -1 + X ' , i = 1, 2,..., M , X '0 = 0, X ' = 0.01 j T j = T j -1 + T , = 1,2,..., I , T0 = 0, T = 0.0001 The contaminant concentration at a point for space X 'i with j th subinterval of time T is defined by Ci , j . Similarly, we have used the general form of the explicit finite difference approximation with forward time central space forward difference scheme applied in equation (47) together with the initial boundary conditions given in equations (48), 49(a) to 49(b) which are approximated, respectively, as follows: 2 D Ci , j +1 = 1 - 2 Ci , j 11 1 - X i ' Ci +1, j - 2Ci , j + Ci -1, j R R X '2 D - 12 1 - X i ' Ci +1, j - Ci -1, j ( 50 ) R 2 X ' Stability analysis A finite difference method is convergent if the discretization error approaches zero, as the mesh sizes in time spatial domain i.e., T X ' , respectively, tend toward zero (Carnahan et al.,1969). Here, we used the forward difference in time for the first order derivative with respect to time which contains the first order accuracy. The stability condition of the finite difference scheme is determined using the matrix method proposed by Smith (1978) this technique was used by Notodarmojo et al. (1991). The finite difference scheme of the governing partial differential equation can be rewritten as follows: Ci ,0 = 1 ci + 2 log 1- X ' c0 i ci + 1 - * j c0 i>0 (51) C0, j = j>0 (52a) (52b) Ci , j +1 = ( + ) Ci -1, j + ( - 2 ) Ci , j + ( - ) Ci +1, j where = 1 - D1 T D T 2 R X '2 R 2 X ' R In the matrix form, it can be written as follows: T , = j +1 (44) CM , j = CM -1, j j>0 where superscript j refers to time; subscript i refers to space, is the time increment, X ' is the space increment in equation (50). [C ] = A[C ] (45) The space domain X ' time domain are discretized by a rectangular grid of points X i, j with mesh-size X , respectively: X 'i = X 'i -1 + X ' , i = 1, 2,..., M , X '0 = 0, X ' = 0.05 j j = j -1 + , = 1,2,..., I , 0 = 0, = 0.0001 The contaminant concentration at a point for the space X 'i with j th subinterval of time denoted by Ci , j . Stability analysis where matrix A contains all the constants. The difference equation is stable, if the eigenvalues of A must have modulus values less than or equal to unity, i.e., a 1 , where a is the eigenvalue of the matrix A . On applying the Gerschgorin circle method to determine the bounds of eigenvalues of the matrix A , the stability criterion for the time step is obtained as follows: T 1 1 2 D1 + 2 2 R R X ' (46) In a similar approach used earlier, the finite difference scheme of the governing partial differential equation (50) can be rewritten as follows: Ci, j +1 = ( 1 + 1 ) Ci -1, j + (1 - 21 ) Ci, j + ( 1 - 1 ) Ci +1, j where 1 = 1 - D D 1 = 12 , 1 = 11 R 2X ' R X '2 R In matrix form, it can be written as follows: (53) 3) Algebraic sigmoid: (54) u = u0 mt ( mt ) + K12 u02 mD0 ( mt )2 + K12 - K1 [C ] j +1 = B [C ] j where matrix B contains all the constants. On applying the same Gerschgorin circle method to determine the bounds of eigenvalues of matrix B , the stability criterion for the time step is obtained as follows: 1 (55) 2 2 D11 + 2 2 R RX ' Accuracy To assess the accuracy of the solution, we compare the numerical solution to the analytical one. The accuracy is the degree of similarity of concentration values of the analytical result to that of the numerical result. We found that the solutions were convergent, if the stability criterion was fulfilled. Towler Yang (1979) adopted the criterion of comparison which was more systematic consistent: (1) the RMSE (Root-Mean-Square Error) (2) the absolute maximum error between the analytical solution the numerical solution at all grid points. For testing the accuracy of the solutions obtained in the present work, we used RMSE which is one of the most commonly used measures to check the accuracy. The root mean square was used to calculate the average error at each point which is defined as follows: RMSE = 1 N where m is the flow resistance coefficient K1 is the mean travel time corresponding to dispersion diffusion. The first two have been used by Aral Liao (1996) Singh et al. (2009) without an impact of mean travel time the last one is based on the properties of an algebraic sigmoid function which includes the error function. The sinusoidal form of the pattern represents the nature of groundwater contamination in the tropical region in which the fluctuation behavior of groundwater recharge is shown. But the exponentially decreasing form of shows a decreasing nature of contaminant concentration (Jain et al., 2007; Singh Singh, 2001). The algebraic sigmoid form includes the error function. It starts to progress from a small beginning, accelerates in the rainy season, then reaches a limit over a period of time. For graphical representation of the analytical solution given in equation (20), the set of input values was given as ci = 0.01, c0 = 1.0, 0 = 0.0001 year 1, u0 = 0.01 km/year, D0 = 0.1 km 2 /year, = 0.001, 0 = 0.0007, K1 = 0.07, kd = 0.25, Pe = 0.7, 0 = 2.0 km, D* = 0.000015 km 2 /year, i =1 (56) where C = Canalytical - Cnumerical . Here, the number of data denoted by N the difference between analytical numerical values is denoted by C . NUMERICAL SIMULATION DISCUSSION This work considers the above derived solution of scaledependent time-dependent dispersion with diffusion. The sensitivities to different parameters included in these solutions were explored. For the sake of simplicity, the contaminant behaviour predicted in the different geological formations considered three different types of patterns: 1) exponentially decreasing form, 2) sinusoidally varying form 3) algebraic sigmoid form. These patterns usually exist in the rainy season reaches up to a certain limit over the period of time. Mathematically, the expressions their corresponding new time variable are derived as follows: 1) Exponential decreasing: u = u0 exp ( - K1mt ) n = 0.32 gravel, 0.55 clay, m = 0.04 year -1 The non-dimensional length of the aquifer was considered in the range 0 < X 0.1 . The concentration pattern is depicted for 1, 1.5 2 years, respectively. The source was assumed to be activated from 1 year in the domain of geological formation. The effect of source contaminant was predicted for the regular interval of six months. The effects of the various parameters are shown by the graph. The different patterns are also revealed through Fig. 1 to Fig. 3. The distribution pattern of the contaminant concentration was high in the aquitard compared to the aquifer for the same time interval ultimately it went to its minimum harmless concentration with distance. The exponentially decreasing form of with mean time in the direction of flow i.e., longitudinal direction, is described in Fig. 4. Fig. 5 shows for the sinusoidally varying form of the pattern, where the distribution pattern of the concentration decreases with decreasing values of dispersivity Q . The dispersivity effect with respect to concentration levels is predicted up to a distance of 1.0 km in the domain. From Fig. 5, it is clear that the concentration u02 1 - e- K1mt K1mD0 Sinusoidal 2) form: u = u0 4 K1 + m cos ( K1mt ) 2000 Time u0 D0 1 4 K1t + sin ( K1mt ) K1 Fig. 1. Exponentially decreasing pattern. 0.0088 0.0086 0.0084 0 1000 2000 Time 3000 4000 5000 Fig. 2. Sinusoidally varying pattern. 0.00999998 0.00999997 0.00999996 0.00999995 0 1000 2000 Time 3000 4000 5000 levels increase with increasing dispersivity of the solute. The algebraic sigmoid form of depicts the contaminant concentration for different values of the mean travel time in the longitudinal direction is revealed in Fig. 6. The contaminant concentration increases with decreasing values of the mean travel time. The mean travel time is particularly evaluated from the linear combination of dispersion diffusion in the porous formation with longitudinal flow direction. So, if the value of the solute in the longitudinal direction slightly increases or decreases the mean travel time changes accordingly. The little variation for the source concentration is also observed only due to the time dependent source concentration taken into consideration at the origin of domain. Different types of the expression were considered for the pictorial representation of the obtained analytical solution given in equation (35) for the time dependent dispersion. exp ( - K1mt ) 1) Exponential decreasing: u = u0 K1 1 - e- K1mt K12mD0 u02 2) Sinusoidal 4K1 + m cos ( K1mt ) form: u = u0 K1 Fig. 3. Algebraic sigmoid pattern. 1 0.9 0.8 0.7 Concentration 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.07 0.08 0.09 0.1 2 year 1.5 year 1 year Medium(n) Line Gravel(0.32) Solid Clay(0.55) Dotted u0 2 4 K12t + sin ( K1mt ) D0 K12 3) Algebraic sigmoid: u = u0 1 K1 2 2 ( mt ) + K1 mt u0 2 ( mt )2 + K12 - K1 mD0 K1 For graphical representation, the one-dimensional analytical solution of the homogeneous advection dispersion equation with the time-dependent dispersion given in equation (35) was computed for the set of input data: ci = 0.01, c0 = 1.0, u0 = 0.01 km/year, D0 = 0.1 km 2 /year, = 0.001, * = 0.0007, kd = 0.25, * = 0.0001, K1 = 0.07, 0 0 D* = 0.000015 km2 /year, n = 0.32 gravel, 0.55 clay, m = 0.04 year -1 The length of the domain for the time-dependent dispersion was taken 0 X 0.1 . The source of contaminant was predicted for the same time intervals: 1, 1.5 2 years, respectively, as discussed earlier. From Fig. 7, it is clear that the concentration values near the source decrease with time in the aquitard as compared to the aquifer, but the trend is just reverse after some distance say 0.01. This may happen due to the inclusion of mean travel time in the expression. It is also depicted that the concentration values decrease with distance in both the geological formations. The level of contaminant concentration is observed slightly higher in the time-dependent dispersion as compared to space-dependent dispersion. This can be shown through Fig. 4 Fig. 7. The contaminant concentration values are different for different values of zero order production term in the sinusoidal form of . The fluctuation of the source concentration in the domain is observed due to time-dependent source input concentration. The concentration level for the different time intervals decreased near the source approximately up to some distance but after damping, it showed the reverse pattern Fig. 4. The concentration distribution pattern for space dependent dispersion with exponentially decreasing . 1 0.9 0.8 0.7 Concentration 0.6 0.5 0.4 0.3 0.2 0.1 0 1.5 year 2 year Dispersivity Line 2.0 Solid 6.0 Dotted 1 year Fig. 5. The concentration distribution pattern for space dependent dispersion with sinusoidal form of in aquifer (i.e., gravel) with respect to dispersivity. 1 0.9 0.8 0.7 1.5 year Concentration 1 year Mean Travel Line Time 0.07 Solid 0.01 Dotted Medium(n) Gravel(0.32) Clay(0.55) Line Solid Dotted 0.9 Concentration 1 year 1.5 year 2 year 2 year Fig. 6. The concentration distribution pattern for space dependent dispersion with algebraic sigmoid form of in aquitard (i.e., clay) with respect to mean travel time. 1 0.98 0.96 0.94 Concentration 0.92 1.5 year 0.9 2 year 0.88 0.86 0.84 0.82 1 year Medium(n) Line Gravel(0.32) Solid Clay(0.55) Dotted Fig. 9. The concentration distribution pattern for the algebraic sigmoid form of in aquifer (i.e., Gravel) aquitard (i.e., Clay). 1 0.9 2 year 0.8 0.7 1.5 year Concentration 0.6 0.5 0.4 0.3 0.2 0.1 1 year Retradation Factor 5.0 1.0 Line Solid Dotted Fig. 7. The concentration distribution pattern for time dependent dispersion with exponentially decreasing form of in aquifer (i.e., gravel) aquitard (i.e., clay). Fig. 10. The concentration distribution patterns for the constant dispersion with diffusion with respect to retardation factors. 1 Dispersivity(Q) Line 7.0 Solid 1.0 Dotted Zero Order Production 0.0007 0.0001 Line Solid Dotted 0.9 0.8 0.7 Concentration 0.94 Concentration 0.92 0.9 0.88 1 year 1.5 year 2 year 1 year 1.5 year 2 year 0.05 Distane Fig. 8. The concentration distribution pattern for sinusoidal form of in aquifer (i.e., gravel) with respect to the zero order production term. Fig. 11. The concentration distribution pattern for the linear dispersion with diffusion with respect to the dispersivity. i.e., the concentration level slightly increased with increasing time period beyond some distance. Then slowly the concentration level became stable with respect to distance. As we increased the value of zero order production term, the concentration values increased with their respective time, but ultimately the concentration level became stable after some distance. So, the fluctuation nature in the tropical region is described for the sinusoidal form of the pattern in Fig.8. In Fig.9, we have discussed the algebraic sigmoid form of the pattern. We observed that the solute concentration increased in the aquitard where the transfer rate of the liquid was low as compared to the aquifer. So, the level of contaminant concentration pattern was minimum in the aquifer as compared to the aquitard. 1 0.9 0.8 0.7 Concentration Retradation Line Factor 5.0 Solid 1.0 Dotted 1.5 year 1 year 0.8 Concentration 1 year 1.5 year 0.6 Analytical Solution Exponential Decreasing Algebraic Sigmoid Numerical Solution Exponential Decreasing Algebraic Sigmoid 0 0.02 0.04 0.06 0.08 0.1 0.12 Distance 0.14 0.16 0.18 0.2 Line 2 year Fig. 12. The concentration distribution pattern for the exponential dispersion with diffusion with respect to the retardation factors. 1 0.9 0.8 0.7 Concentration 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.07 0.08 0.09 0.1 1.5 year 1 year Analytical Solution Exponential Decreasing Algebraic Sigmoid Numerical Solution Exponential Decreasing Algebraic Sigmoid Line Fig. 15. Comparision between the concentration distribution pattern of the analytical numerical solution for the gravel medium. 1 0.9 0.8 Concentration 0.7 0.6 0.5 0.4 1 year 1.5 year 2 year Solution Analytical Numerical Line Solid Dotted 0.25 Distance Fig. 13. Comparision between the concentration distribution pattern of the analytical numerical solution for the gravel medium. Fig. 16. Comparision between the concentration distribution pattern of the analytical numerical solution for the clay medium along sinusoidal pattern. 1 0.9 0.8 0.7 Concentration 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.07 1 year 1.5 year 2 year Solution Analytical Numerical LIne Solid Dotted 2) 3) Linear dispersion: f ( mt ) = t Exponential: u0 2t 2 2 K1D0 t f ( mt ) = K1 1 - exp - K1 t u02 K1t + exp - - 1 D0 K1 Fig. 14. Comparision between the concentration distribution pattern of the analytical numerical solution for the clay medium along sinusoidal pattern. The time-dependent dispersion for the concentration distribution pattern was described only for the asymptotic type of dispersion (Basha El-Habel, 1993). In view of this, we discussed the three different types of generalized cases for the time dependent function which are as follows: u 2K t 1) Constant dispersion: f (mt ) = K1 0 1 D0 The same input values are considered for the computation of the solution given in Eq. (35) with parameters like dispersivity Q = 1 , retardation factor R = 1 the diffusion coefficient D* = 0 which were taken by Basha El-Habel (1993). Fig. 10 depicts the contaminant concentration for the constant type of dispersion with the same value that was already used for the time-dependent dispersion. In Fig. 10, we also explore the contaminant concentration for the constant type of dispersion for different values of the retardation factor. From this, we observed that the values of contaminant concentration decreased at each of the positions compared with the concentration values with respect to the retardation factor used by Basha El-Habel (1993). In both cases, the concentration pattern attained its minimum concentration level after traveling some distance. The contaminant concentration pattern was also predicted for the linear exponential types of dispersion in Fig. 11 Fig. 12 with the mean travel time K1 = D* + Q K1 = 0.632Q + D* (Basha El-Habel, 1993), respectively. The contaminant concentration was predicted for the same set of data except R = 1, D* = 0, Q = 1.7 , as shown in Fig. 11. From this, we observed that the value of contaminant concentration decreased with increasing values of dispersivity with their respective time attained its minimum concentration level with respect to their respective distances. Due to the exponentially decreasing term present in the exponential dispersion, the value of the dispersivity slightly changed from 1 to 1.5 for their respective mean travel times. From Fig.12, we observe that the contaminant concentration values for the exponential type of dispersion also decreased compared to the value of retardation factor taken by Basha El-Habel (1993) at each of the positions. In Figs. (13) (14) the predicted concentration distribution pattern for the space-dependent diffusion is presented, whereas Figs. (15) (16) show the predicted concentration distribution pattern for the time-dependent diffusion. The numerical result is predicted for the same set of input values which is taken for the analytical result. Comparison of the analytical result to the numerical one with different patterns, the concentration distribution pattern for the gravel medium with average porosity (0.32) for different time duration, are shown in Fig. 13. The concentration values by the analytical solution for the exponential decreasing form of the pattern yielded decreasing values at each of the positions as compared to the numerical solution. The concentration values increased at each of the positions with increasing time duration for both solutions i.e., analytical numerical. Similar types of contaminant concentration were depicted for the algebraic sigmoid form of the pattern which showed the fluctuation pattern of groundwater after rainy seasons. From both the solutions, we observed that concentration values were higher in the case of the exponential decreasing pattern as compared to the algebraic sigmoid form of the for both time durations. The contaminant concentration decreased with distance increased with time. From Fig. 14 the minimum concentration values were observed at each of the positions in the case of numerical solution as compared to the analytical solution for the sinusoidal form of the pattern for the clay medium with the average porosity of 0.32. The concentration values increased with time decreased with distance for both the analytical numerical. Fig.15 follows the reverse nature of the concentration distribution for different velocities in the case of analytical numerical solutions of time-dependent diffusion as in Fig.13. Analytical solutions yielded high concentration values as compared to the numerical solution for both exponentially decreasing algebraic sigmoid forms of the pattern. In the case of t = 1.5 years, the numerical solution for exponentially decreasing forms initially produces less concentration values than did the analytical solution. However, after some distance the concentration values were observed to be approximately the same for both the solutions. We also observed from Fig. 15 that the concentration values were higher for the exponentially decreasing form of than for the algebraic sigmoid form of the pattern for both the solutions. The concentration distribution for the sinusoidal form of the pattern is depicted in Fig. 16. We observed that initially the source of concentration was the same for the entire duration of time, but in the first year the numerical solution gave lower concentration values than did the analytical one after some distance it gave the reverse pattern. For the next 1.5 2 years the concentration distribution pattern followed the reverse pattern. In the respective years the numerical solution gave higher concentration values than did the analytical solution. Also the numerical solutions followed the uniform nature of the decreasing concentration pattern after covering some distance. Finally from Fig. 16, it is observed that at the end of the domain the concentration values were higher in the case of numerical solution than for the analytical solution. RMSE of clay gravel media for the fixed time duration i.e., 2 years are tabulated for the space-dependent dispersion with diffusion, as shown in Tables 1(a) 1(b), respectively. However, RMSE of the same is tabulated for the timedependent dispersion with diffusion, as shown in Tables 2(a) 2(b), respectively. Tables 1(a) 1(b) are tabulated for the fixed space time increments i.e., X ' = 0.01, T = 0.0001 , however, Tables 2(a) 2(b) are tabulated for the fixed space time increments, i.e., X ' = 0.05, = 0.0001 . It is observed from Table1 (a) that the RMSE value is minimum for the exponentially decreasing sinusoidal forms unlike the algebraic sigmoid form of the patterns of the clay medium. Similar observations were made for the gravel medium with a fixed time duration of 2 years. From Table 1 (a) 1 (b), it is clear that the RMSE values were lower in the aquifer (i.e., gravel) medium than in the aquitard (i.e., clay) medium for different respective patterns. From Table 2 (a) (b) the RMSE values were minimum for the algebraic sigmoid but maximum for the exponentially Table 1(a). RMSE of the clay medium for average porosity (0.55) with 2 years duration for scale dependent dispersion with diffusion. Exponential decreasing Analytical result Numerical result 0.4391 0.4407 0.1289 0.1267 0.0344 0.0326 0.0196 0.0186 0.0199 0.0178 0.00179 Sinusoidal form Analytical result Numerical result 0.4209 0.4295 0.1150 0.1181 0.0302 0.0301 0.0190 0.0183 0.0199 0.0178 0.00420 Algebraic sigmoid Analytical result Numerical result 0.2559 0.2852 0.0343 0.0418 0.0164 0.0169 0.0179 0.0177 0.0199 0.0182 0.01354 Distance 0.02 0.04 0.06 0.08 0.10 RMSE Table 1(b). RMSE of the gravel medium for average porosity (0.32) with 2 years duration for scale dependent dispersion with diffusion. Distance 0.02 0.04 0.06 0.08 0.10 RMSE Exponential decreasing Analytical result Numerical result 0.3844 0.3857 0.0903 0.0895 0.0239 0.0236 0.0183 0.0178 0.0199 0.0179 0.00115 Sinusoidal form Analytical result Numerical result 0.3655 0.3741 0.0791 0.0827 0.0217 0.0223 0.0181 0.0177 0.0199 0.0179 0.00427 Algebraic sigmoid Analytical result Numerical result 0.2018 0.2307 0.0229 0.0287 0.0160 0.0163 0.0179 0.0177 0.0199 0.0183 0.01320 Table 2(a). RMSE of the clay medium for average porosity (0.55) with 2 years duration for time dependent dispersion with diffusion. Distance 0.1 0.2 0.3 0.4 0.5 RMSE Exponential decreasing Analytical result Numerical result 0.8934 0.9213 0.7951 0.8587 0.7004 0.8012 0.6113 0.7600 0.5296 0.7487 0.13046 Sinusoidal form Analytical result Numerical result 0.8899 0.9145 0.7870 0.8427 0.6883 0.7766 0.5960 0.7294 0.5120 0.7165 0.11926 Algebraic sigmoid Analytical result Numerical result 0.8469 0.8384 0.6976 0.6745 0.5612 0.5246 0.4429 0.4179 0.3455 0.3887 0.02978 Table 2(b). RMSE of the gravel medium for average porosity (0.32) with 2 years duration for time dependent dispersion with diffusion. Distance 0.1 0.2 0.3 0.4 0.5 RMSE Exponential decreasing Analytical result Numerical result 0.8822 0.8946 0.7699 0.8041 0.6631 0.7209 0.5645 0.6613 0.4761 0.6454 0.09240 Sinusoidal form Analytical result Numerical result 0.8778 0.8867 0.7606 0.7858 0.6495 0.6930 0.5477 0.6266 0.4574 0.6089 0.07972 Algebraic sigmoid Analytical result Numerical result 0.8270 0.8086 0.6592 0.6144 0.5098 0.4381 0.3853 0.3137 0.2882 0.2809 0.05033 decreasing form of the pattern. The aquifer (i.e., gravel) had minimum values of RMSE other than the aquitard (i.e. clay) medium for different respective patterns except the algebraic sigmoid form. The RMSE was minimum for clay as compared to gravel for the algebraic sigmoid form of the pattern. So, the RMSE values were close for the gravel medium for both space- time-dependent diffusion (except algebraic sigmoid). For space-dependent diffusion RMSE was very close for the exponential decreasing , whereas in the case of time-dependent diffusion result the closest values were found for the algebraic sigmoid form of the pattern. SUMMARY CONCLUSION
Journal of Hydrology and Hydromechanics – de Gruyter
Published: Jun 1, 2017
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