Oup Vanduijn 1992

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Transcript of Oup Vanduijn 1992


    By C. J. VAN DUIJN

    (Department of Mathematics, Delft University of Technology, Julianalaan 132,2628 BL Delft, The Netherlands)

    and L. A. PELETIER

    {Mathematical Institute, Leiden University, The Netherlands)

    [Received 7 June 1990. Revise 10 March 1991]

    SUMMARYIn this paper we investigate the boundary layer which arises between the parallel flow

    of fresh and salt groundwater as occurs in vertical aquifers. It is caused by small moleculardiffusion and transversal dispersion brought about by the porous environment. Aself-similar transformation leads to a variant of the Falkner-Skan equation. Existence,uniqueness and various properties of the solution, such as monotonicity and asymptoticbehaviour, are established and the influence of the ratio of molecular diffusion totransversal dispersion is analysed. Some numerical results are presented.

    1. Introduction

    CONSIDER the two-dimensional flow of an incompressible fluid through ahomogeneous and isotropic porous medium. The fluid has a constant dynamicviscosity /i > 0 and a variable density p, and the porous medium is characterizedby a constant porosity 0.

    A typical example of such a situation arises in the flow of fresh and saltgroundwater in a two-dimensional vertical aquifer (1, 2, 3). In this applicationone has for the densities pf and p, of the fresh and of the salt groundwater,respectively:



    and mass balance:

    CD + div F = 0.dt


    In these equations, q = (q{,q2) = (, v) denotes the specific discharge of thefluid, p the fluid pressure and g the acceleration of gravity. In equation (1.4),the vector F denotes the mass flux. It is given by

    where D =

    F = pg D grad p, (1.5)

    is the hydrodynamic dispersion matrix defined by (1, p. 612)

    Du = {aT\q\ + Dmo[}dIJ

    (aL - (1.6)

    Here aT and aL are positive constants with aT < aL. They are called thetransversal and the longitudinal dispersion length and they describe the diffusiondue to the mechanical dispersion caused by the randomness of the structure ofthe porous matrix. Further, Dmol is a positive constant representing the moleculardiffusion in the fluid. Finally 5U denotes the Kronecker delta and | | the Euclideannorm in R2.

    In this paper we study a particular flow problem, which was recentlyinvestigated by Uffink (4, 5). In Fig. 1 below we sketch the situation and explainsome notation. In the region {.x < 0, z > 0} we suppose that fresh water (p = pf)has a uniform specific discharge u = U, v = 0 along an impervious boundary{x < 0, z = 0]. In the region {.v > 0} we suppose that fresh water is present farabove the plane {z = 0} and salt water far below.

    In the absence of diffusion or dispersion (D = 0 in (1.5)), an abrupt transitionor interface exists between the fresh and the salt water. It is well known thatthis leads to a jump in the tangential component of the specific discharge along

    Fresh water

    Salt water

    FIG. 1. Geometry of the problem


    the interface, the shear flow (1, 6), of magnitude

    (qf-qs). i=(p,- pf)g-sin p. (1.7)

    Here qf and q, denote the specific discharge just above and below the interfaceand f is the tangential unit vector along the interface pointing in the directionof positive x. Thus, if we look for a flow configuration in which the salt wateris stagnant, we must require that

    U = (P, ~ pf)g - sin p. (1.8)

    This leads to a fresh-salt interface which coincides with the half-plane {x > 0,z > 0 } .

    In general, when dispersion is present, a transition zone will develop whichwill be thin if the coefficients a.T, aL and Dmo, in the dispersion matrix D aresmall. In (4) Uffink presents a boundary-layer model to describe the steadymixing of the two fluids in this transition zone. Below we give the main stepsof the derivation.

    To eliminate the pressure from Darcy's law (1.3) we take the curl. This yields

    curll - q - pg I = 0,\K )

    or, written out in detail,

    du dv K [dp dp \+ -g s i n / ? - cos p = 0. 1.9)

    cz ox /< \oz ox )

    The system of equations consisting of (1.2), (1.4) to (1.6) and (1.9) describes themixing of the fluids.

    To investigate the development of the transition zone, a singular perturbationmethod is used (7). Thus we get


    and omitting the primes again for convenience, (1.9) yields

    dz dx \dz dx


    U \i

    If we now let b > 0 and use (1.8), we obtain

    du dp + = 0.dz dz

    Choosing u = 0 deep in the salt-water region, where p = 1, we obtain uponintegration

    u + p=\. (1.11)

    This yields u = 1 (or u = U in the original variables) in the fresh-water region,far from the transition layer.

    Similarly, we substitute (1.10) into equations (1.2) and (1.4) to (1.6) and letS -> 0. We then find the following set of boundary-layer equations for steady flow:



    duTx +

    uBpU dx

    u = '.


    1 -







    denotes the ratio between molecular diffusion and transversal mechanicaldispersion. The value of A depends on the type of soil: for homogeneous finegrained soils it can be large but for heterogeneous soils it is very small. Notethat, using (1.12)3 to eliminate p, (1.12)2 can be written as

    du du d f,. , ., du,

    At this point it is worthwhile to compare equations (1.12) with the Prandtlboundary-layer equations (7). The additional term in (1.12)4 is the degeneratediffusion term (|u|u2)r, where the subscript z denotes differentiation with respect


    to z. The appearance of this term enables us to consider solutions in the limitas k -0.

    We consider (1.12) in the half-space

    Cl = {(x, z):x > 0, - o o < z < oo}.

    Along the boundary x = 0 we require that

    (1.13)0 for z < 0

    and far into the fresh-water region we impose the condition that

    v(x, z)-0 as z > oo for every x > 0. (1-14)

    As with the classical Prandtl boundary layer in the wake of a flat plate, weconstruct the solution of (1.12) to (1.14) as a similarity solution. We introducethe stream function ip(x,z):

    dip diuu = and v =

    dz dx

    and write


    " = f'(i) an(J v =

    where the primes denote differentiation with respect to r\. This shows that (1.12),is satisfied for any / e C2(U). Substitution into (1.12)4 yields for / the ordinarydifferential equation

    (2?.f + \f'\f')" + ff" = 0. (1.15)

    Note that this equation resembles the Blasius equation (8); however, it has theadditional term ( | / ' | / ' ) " .

    The conditions (1.13) at the boundary x = 0 are satisfied if

    if t] -* co,

    and the condition (1.14) at z = +oo yields

    if'(i)-f(i)^O as// +oD. (1.17)

    In order to ensure that v(x, z) remains bounded for each x > 0, also as z - oo,we replace (1.16)2 by the stronger condition

    a s r / - > - c o . (1.18)


    Here the number a is a priori not known but needs to be determined as partof the solution. We shall show in section 4 that t]f'( t]) -> Oasr; -> oo. Hence

    al im v(x, z)= - .

    Thus, the number a is directly related to the amount of salt drawn into themixing zone. Therefore it is important to have some quantitative informationabout a.

    In this paper it is assumed that

    a for all rjeU. (1.19)

    In section 2 we show that this condition implies that solutions are monotonicso that u = / ' ^ 0, ruling out the possibility of back flow.

    To study the problem posed by (1.15) to (1.19), we reduce (1.15) in section2 to a second-order nonlinear diffusion equation, which we study in section 3.In particular, existence and uniqueness of a solution is established for any / ^ 0,subject to appropriate boundary conditions. This solution is then used in section4 to obtain the unique solution (a,f) of (1.15) to (1.19).

    In addition to proving the existence and uniqueness of a solution we derivea number of qualitative properties of the function f{t]) and the limiting valuea, in particular the dependence of the latter on L For / we find that

    f(ij) > t] for co < i] < oo

    and(a) if/ > 0, then

    f{rj)>a, / ' ( r / ) > 0 , f"(r])>0 for x < r / < x ;

    (b) if /. = 0, then there exists a constant a < 0 such that

    /(>;) = a f o r x < ^ 7 ^ a


    f{r\) > a, f'{tj) > 0, f"(rj) > 0 for a < t] < x .

    A numerical computation shows that % = 2-4130.About the asymptotic behaviour of f{rj) as r\ -> x and r\ -> x (or a) we

    show that

    e - i ' ' ^ + ; i >) as i\ -> x ,


    f(n) = a + O(eM' / - - x if ;. > 0

    l\a\(n-i)2 +O((>] - a ) 3 ) a s >, - a if ;. = 0.

    For a we find that it depends continuously and monotonically on / and that


    it tends to oo as k -* oo. Specifically, writing a = a{k) we prove that (i) forevery kl, k2 0 there exists a constant C > 0 such that

    (ii) for every ku k2 > 0,

    A, > k2=>a{kl)


    where as R. By a solution of (2.1) to (2.3) we shall mean a number a and afunction / : R -> [a, oo) such that (i) both / ' and (g(f'))' are locally absolutelycontinuous on R, (ii) equation (2.1) is satisfied almost everywhere and (iii) (2.2)and (2.3) hold.

    Suppose that / is a solution of (2.1) to (2.3). Then we can define the set /on which / ' > 0:

    I = {r1sR:f'(t])>0}.

    In view of the condition