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Supporting Information
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for
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DISCONTINUOUS STEADY-STATE ANALYTICAL SOLUTIONS OF THE
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BOUSSINESQ EQUATION AND THEIR REPRESENTATION BY MODFLOW
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6
by
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Jacob Zaidel
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Supporting Information consists of the following two sections. First section, Appendix
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A, presents analytical solutions corresponding to groundwater flow over a multi-step
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aquifer base. Second section, Appendix B, describes the comparison between the
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derived discontinuous 1D analytical solution and MODFLOW results, obtained for a 2D
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model, directly accounting for both horizontal and vertical flow components.
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APPENDIX A: Discontinuous Analytical Solutions over Multi-Step Boundary
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Case 1: Specified Heads Boundary Condition
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As it was mentioned in the main section of the manuscript, there is no closed form
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analytical solution of the Boussinesq equation subject to a specified heads boundary
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condition for a general case of an aquifer base represented by multiple steps. However,
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one series of particular analytical solutions, corresponding to discontinuous water table
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distributions can be derived analytically. These solutions correspond to a case of
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equally spaced steps with a uniform decline of their elevations between the inflow and
1
1
the
outflow
boundaries
(Figure
2b;
main
section
of
the
manuscript):
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l  li  li 1  L N ; b  bi  bi 1  b1 N  1; i  1,..., N  1 .
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With this additional assumption, for the discontinuous over each riser solution to occur
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(Curve 3, Figure 2b; main section of the manuscript), the following condition should be
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satisfied:
H1  b1 2  H 22  b 2 ……………………………………………..(S1)
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In this case, the upstream values of water table elevation over each riser are equal to
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the corresponding step elevation, i.e. hi  bi . The downstream values of water table
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elevation over each riser, with an exception of the last one ( i  N  1 ), are calculated as
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following:
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hi  bi 1  H1  b1 , i  1,..., N  2 …………………………………………(S2)
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The downstream value of the water table elevation at the last riser ( i  N  1 ) is
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calculated as:

hN 1  H1  b1   H 22
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
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…………………………………………..(S3)
Note that:

in case of N=2 and l1  L 2 , Eq.(S3) is equivalent to Eq. (5), presented in the main
section of the manuscript; and
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2

irrespective of H1, H2 and b1 condition (S1) becomes violated (i.e. no
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discontinuous solutions can be formed) for large enough number of steps
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since b approaches zero in this case. This conclusion can be explained as
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following. Under the utilized assumptions, as the number of steps increases, the
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stepping aquifer base approaches the straight line. The water table, represented
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1
by a continuous concave down curve, cannot cross the underlying impervious
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base, represented by a straight line sloping in the flow direction.
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Solution described above corresponds to the water table being discontinuous over each
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riser as long as condition (S1) is satisfied. Increasing the water table elevation at the
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outflow boundary (H2) results in the violation of this condition. As a result, the water
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table close to the outflow boundary becomes continuous (Curve 2, Figure 2b main
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section of the manuscript). Increasing H2 (and/or H1) further will eventually result in a
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fully continuous solution everywhere between the inflow and outflow boundaries (Curve
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1, Figure 2b; main section of the manuscript).
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The following procedure describes how the analytical solution of the Boussinesq
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equation is constructed as long as a discontinuous water table configuration exists at
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least over the first riser, closest to the inflow boundary.
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First, assuming the existence of discontinuous water table configuration over the
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boundary between first and second steps (i,e., above the first stair riser), calculate the
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flux as:
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q  K ( H 1  b1 ) 2 / 2l …………………………………………………………………(S4)
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Second, based on the continuity of flux condition, calculate the downstream value of the
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water table elevation at the last riser ( hN 1 ) , using Eq. (S3). Third, compare hN 1 ,
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obtained from Eq. (S3) with bN 1  b . Water table configuration further upstream from
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the last step will be discontinuous with hN 1  bN 1 if hN 1  bN 1 . This case, corresponding
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1
to the solution discontinuous over each riser, has been already described above. Water
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table configuration over the last riser becomes continuous if hN 1  bN 1 . In this case,
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hN 1  hN 1  hN 1 and continuity of flux condition can be used to calculate hN 2 :
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
hN 2  bN 1  2ql / K  (hN 1  bN 1 ) 2

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………………………………………………(S5)
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Fourth, moving in the upstream direction, compare hN 2 , obtained from Eq. (S5) with
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elevation of the next step, i.e. bN  2  2b . Water table configuration further upstream
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from this step will be discontinuous with
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configuration
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hN  2  hN  2  hN  2 and continuity of flux condition can be used to calculate h N 3 :
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over

this
riser
hN 2  bN  2 if hN 2  bN 2 . Water
becomes continuous if hN 2  bN 2 .
hN 3  bN 2  2ql / K  (hN 2  bN 2 ) 2

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In
table
this
case
………………………………………………(S6)
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The last step of the procedure outlined above, can be repeated to calculate water table
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elevations at the remaining risers (i.e. hN 4 , hN 4 , hN 5 , hN 5 ,..., h1 , h1 ), as well as the
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distribution of heads between them. Performing these calculations one more time over
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the first step at the inflow boundary ( x  l1 ) provides a verification of the assumption that
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water table brakes down (i.e. becomes discontinuous) over the first riser, i.e. h0,
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calculated based on the equation similar to (S6), should be equal to H1, in this case.
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Case 2: Specified Recharge Rate and Constant Head at the Outflow Boundary
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In this case solution of Equation (1) is assumed to be subject to the following boundary
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conditions: dh dx  0 at x=0 and h( L)  H 2 . The following analytical solution describes the
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water table configuration over the mulpti-step base corresponding to these boundary
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conditions and specified recharge rate(s):
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hN  j  bN  j 1 
1/ 2


 iN  j

2
 

Ri (li  li 1 )  RN  j 1 (lN  j 1  lN  j )l N  j 1  lN  j K 1 
 hN  j 1  bN  j 1  2


 i 1






………..(S7a)
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


 bN  j
h , h
……………………………………………………………...(S7b)
hN  j   N  j N  j

bN  j , hN  j  bN  j
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where j  1,2,..., N ; hN  H 2 ; and Ri is a piecewise constant over each step net recharge
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rate. Construction of the solution based on Equations (S7a) and (S7b) stars from j=1,
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corresponding to the step closest to the outflow boundary, and progresses upstream, in
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the direction of a ‘no-flow’ boundary (x=0). Note that condition hN  j  hN  j (equal heads
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on the upstream and downstream sides of a riser) corresponds to the continuous water
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table over the boundary between two adjacent steps, while hN  j  bN  j represents the
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water table jump between them.
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