Solutes in an aquifer are transported by two separate

Chapter 11, Problem 27

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Solutes in an aquifer are transported by two separate mechanisms. The process by whicha solute is transported by the bulk motion of the flowing groundwater is called advection.In addition, the solute is spread by small-scale fluctuations in the groundwater velocityalong the tortuous flow paths within individual pores, a process called mechanical dispersion.The one-dimensional form of the advectiondispersion equation for a nonreactivedissolved solute in a saturated, homogeneous, isotropic porous medium under steady,uniform flow isct + vcx = Dcxx, 0 < x < L, t > 0, (i)where c(x, t)is the concentration of the solute, v is the average linear groundwater velocity,D is the coefficient of hydrodynamic dispersion, and L is the length of the aquifer. Supposethat the boundary conditions arec(0, t) = 0, cx(L, t) = 0, t > 0 (ii)and that the initial condition isc(x, 0) = f(x), 0 < x < L, (iii)where f(x) is the given initial concentration of the solute.(a) Assume that c(x, t) = X(x)T(t), use the method of separation of variables, and findthe equations satisfied by X(x) and T(t), respectively. Show that the problem for X(x) canbe written in the SturmLiouville form[p(x)X]+ r(x)X = 0, 0 < x < L, (iv)X(0) = 0, X(L) = 0, (v)where p(x) = r(x) = exp(vx/D). Hence the eigenvalues are real, and the eigenfunctionsare orthogonal with respect to the weight function r(x).(b) Let 2 = (v2/4D2). Show that the eigenfunctions areXn(x) = evx/2D sinnx, (vi)where n satisfies the equationtanL = 2D/v. (vii)(c) Show graphically that Eq. (vii) has an infinite sequence of positive roots and thatn = (2n 1)/2L for large n(d) Show thatL0r(x)X2n (x) dx = L2 + v4D2nsin2 nL.(e) Find a formal solution of the problem (i), (ii), (iii) in terms of a series of theeigenfunctions Xn(x).(f) Let v = 1, D = 0.5, L = 10, and f(x) = (x 3), where is the Dirac delta5 function.Using the solution found in part (e), plot c(x, t) versus x for several values of t, such ast = 0.5, 1, 3, 6, and 10.Also plot c(x, t) versust for several values of x. Note that the numberof terms that are needed to obtain an accurate plot depends strongly on the values of tand x.(g) Describe in a few words how the solution evolves as time advances.