1 Supporting Information 2 for 3 DISCONTINUOUS STEADY-STATE ANALYTICAL SOLUTIONS OF THE 4 BOUSSINESQ EQUATION AND THEIR REPRESENTATION BY MODFLOW 5 6 by 7 Jacob Zaidel 8 9 Supporting Information consists of the following two sections. First section, Appendix 10 A, presents analytical solutions corresponding to groundwater flow over a multi-step 11 aquifer base. Second section, Appendix B, describes the comparison between the 12 derived discontinuous 1D analytical solution and MODFLOW results, obtained for a 2D 13 model, directly accounting for both horizontal and vertical flow components. 14 15 APPENDIX A: Discontinuous Analytical Solutions over Multi-Step Boundary 16 Case 1: Specified Heads Boundary Condition 17 As it was mentioned in the main section of the manuscript, there is no closed form 18 analytical solution of the Boussinesq equation subject to a specified heads boundary 19 condition for a general case of an aquifer base represented by multiple steps. However, 20 one series of particular analytical solutions, corresponding to discontinuous water table 21 distributions can be derived analytically. These solutions correspond to a case of 22 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): 2 l li li 1 L N ; b bi bi 1 b1 N 1; i 1,..., N 1 . 3 With this additional assumption, for the discontinuous over each riser solution to occur 4 (Curve 3, Figure 2b; main section of the manuscript), the following condition should be 5 satisfied: H1 b1 2 H 22 b 2 ……………………………………………..(S1) 6 7 In this case, the upstream values of water table elevation over each riser are equal to 8 the corresponding step elevation, i.e. hi bi . The downstream values of water table 9 elevation over each riser, with an exception of the last one ( i N 1 ), are calculated as 10 following: 11 hi bi 1 H1 b1 , i 1,..., N 2 …………………………………………(S2) 12 The downstream value of the water table elevation at the last riser ( i N 1 ) is 13 calculated as: hN 1 H1 b1 H 22 14 15 16 12 …………………………………………..(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 17 18 2 irrespective of H1, H2 and b1 condition (S1) becomes violated (i.e. no 19 discontinuous solutions can be formed) for large enough number of steps 20 since b approaches zero in this case. This conclusion can be explained as 21 following. Under the utilized assumptions, as the number of steps increases, the 22 stepping aquifer base approaches the straight line. The water table, represented 2 1 by a continuous concave down curve, cannot cross the underlying impervious 2 base, represented by a straight line sloping in the flow direction. 3 Solution described above corresponds to the water table being discontinuous over each 4 riser as long as condition (S1) is satisfied. Increasing the water table elevation at the 5 outflow boundary (H2) results in the violation of this condition. As a result, the water 6 table close to the outflow boundary becomes continuous (Curve 2, Figure 2b main 7 section of the manuscript). Increasing H2 (and/or H1) further will eventually result in a 8 fully continuous solution everywhere between the inflow and outflow boundaries (Curve 9 1, Figure 2b; main section of the manuscript). 10 11 The following procedure describes how the analytical solution of the Boussinesq 12 equation is constructed as long as a discontinuous water table configuration exists at 13 least over the first riser, closest to the inflow boundary. 14 15 First, assuming the existence of discontinuous water table configuration over the 16 boundary between first and second steps (i,e., above the first stair riser), calculate the 17 flux as: 18 q K ( H 1 b1 ) 2 / 2l …………………………………………………………………(S4) 19 Second, based on the continuity of flux condition, calculate the downstream value of the 20 water table elevation at the last riser ( hN 1 ) , using Eq. (S3). Third, compare hN 1 , 21 obtained from Eq. (S3) with bN 1 b . Water table configuration further upstream from 22 the last step will be discontinuous with hN 1 bN 1 if hN 1 bN 1 . This case, corresponding 3 1 to the solution discontinuous over each riser, has been already described above. Water 2 table configuration over the last riser becomes continuous if hN 1 bN 1 . In this case, 3 hN 1 hN 1 hN 1 and continuity of flux condition can be used to calculate hN 2 : 4 hN 2 bN 1 2ql / K (hN 1 bN 1 ) 2 12 ………………………………………………(S5) 5 Fourth, moving in the upstream direction, compare hN 2 , obtained from Eq. (S5) with 6 elevation of the next step, i.e. bN 2 2b . Water table configuration further upstream 7 from this step will be discontinuous with 8 configuration 9 hN 2 hN 2 hN 2 and continuity of flux condition can be used to calculate h N 3 : 10 over this riser hN 2 bN 2 if hN 2 bN 2 . Water becomes continuous if hN 2 bN 2 . hN 3 bN 2 2ql / K (hN 2 bN 2 ) 2 12 In table this case ………………………………………………(S6) 11 The last step of the procedure outlined above, can be repeated to calculate water table 12 elevations at the remaining risers (i.e. hN 4 , hN 4 , hN 5 , hN 5 ,..., h1 , h1 ), as well as the 13 distribution of heads between them. Performing these calculations one more time over 14 the first step at the inflow boundary ( x l1 ) provides a verification of the assumption that 15 water table brakes down (i.e. becomes discontinuous) over the first riser, i.e. h0, 16 calculated based on the equation similar to (S6), should be equal to H1, in this case. 17 18 Case 2: Specified Recharge Rate and Constant Head at the Outflow Boundary 19 In this case solution of Equation (1) is assumed to be subject to the following boundary 20 conditions: dh dx 0 at x=0 and h( L) H 2 . The following analytical solution describes the 4 1 water table configuration over the mulpti-step base corresponding to these boundary 2 conditions and specified recharge rate(s): 3 4 hN j bN j 1 1/ 2 iN 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) 5 6 bN j h , h ……………………………………………………………...(S7b) hN j N j N j bN j , hN j bN j 7 8 where j 1,2,..., N ; hN H 2 ; and Ri is a piecewise constant over each step net recharge 9 rate. Construction of the solution based on Equations (S7a) and (S7b) stars from j=1, 10 corresponding to the step closest to the outflow boundary, and progresses upstream, in 11 the direction of a ‘no-flow’ boundary (x=0). Note that condition hN j hN j (equal heads 12 on the upstream and downstream sides of a riser) corresponds to the continuous water 13 table over the boundary between two adjacent steps, while hN j bN j represents the 14 water table jump between them. 15 5