Dendritic conduit simulation

Conduit System Fissured System Porous System Spring discharge Limit of significance

expectancies.   For   assessing   the   difference   between   the   “local”   and   the   average   values,   deviations   from  the  average  value  are  provided  in  per  cent.  Figure  4.5  shows  the  range  of  deviations  for  each   parameter.   For   the   reference   simulation   (Chapter   4.3.1),   the   difference   between  Af   and  Ef   at   the   observation  well  and  the  average  values  are  3.3%  and  27.3%  for  the  single  conduit  (configuration  1)   and  2.5%  and  12.2%  for  the  dendritic  simulation  (configuration  2).  The  smaller  diameter  but  more   evenly  distributed  dendritic  conduits  lead  to  a  more  homogeneous  distribution  of  groundwater  ages   and  even  more  so  of  groundwater  life  expectancies.  Since  life  expectancy  cannot  be  measured  in  the   field,  the  difference  between  the  age  values  is  the  more  important  parameter  which  lies  with  2–4%  

well  within  the  range  of  other  uncertainties  in  the  field.  

The  percentage  of  direct  recharge  rdir,  which  showed  the  largest  overall  influence  on  average  values,   has  the  largest  influence  on  the  spatial  distribution.  The  maximum  differences  between  the  average   age  and  the  age  in  the  observation  well  of  31.5%  for  configuration  1  and  89.5%  for  configuration  2   were  reached  for  high  rdir  values  above  90%.  For  high  rdir  values  the  newly  recharged  water  does  not   enter  the  fissured  system  evenly  distributed,  i.e.  as  diffuse  recharge  at  the  top,  but  indirectly  via  the   conduit   system   due   to   gradient   inversion.   Therefore,   lower   velocities   are   observed   in   the   fissured   system   and   the   average   age   increases.   For   configuration   2   the   observation   well   is   situated   significantly   closer   to   the   conduit   system   due   to   its   overall   higher   lateral   extent   (Figure   4.1).   It   receives  a  high  amount  of  young  water  from  the  conduit  system  which  leads  to  the  large  differences   in  groundwater  age  compared  to  the  average  age.  For  configuration  1,  the  ages  in  the  observation   well  tend  to  higher  values  compared  to  the  average  age  until  a  critical  value  of  95%  direct  recharge  is   reached  and  the  water  flux  from  the  conduit  system  is  high  enough  to  reach  the  observation  well.  

Further,  small  εf  values  of  5  m  have  a  relatively  high  influence  of  13.5%  for  configuration  2  and  8.9%  

for  configuration  1.  The  parameter  maf  can  lead  to  a  significant  homogenization  due  to  the  smaller   variety   of   different   paths   the   water   can   take.   For   small  maf   values,   the   differences   of  Af   and  Ef   between   the   observation   well   and   the   average   value   decrease   to   0.1%   and   2.3%,   respectively,   for   configuration  1.  For  configuration  2,  the  age  difference  also  decreases  to  0.1%.  The  difference  in  life   expectancies   increases   with   decreasing   aquifer   thickness,   however.   Since   the   observation   well   is   vertically  positioned  in  the  middle  of  the  aquifer,  it  lies  closer  to  the  aquifer  top  and  therefore  the   conduit  system  for  a  reduced  vertical  extent.  Therefore,  the  life  expectancy  is  reduced  more  strongly   in  the  observation  well.  For  a  thickness  of  10  m  the  water  in  the  observation  well  has  less  than  half   the  life  expectancy  of  the  average  groundwater  in  the  aquifer.  

As   can   be   expected,   the   groundwater   ages   at   the   spring   represent   a   significantly   better   approximation  of  the  average  values  than  the  individual  local  observation  well  age  measurements.  

For  the  reference  simulation  the  age  difference  is  only  0.7%  for  the  single  conduit  and  0.4%  for  the  

dendritic   conduit   configuration.   For   configuration   1   the   only   parameter   that   induces   a   significant   difference   of   over   2%   between   the   spring   and   the   average   age   value   is   the   percentage   of   direct   recharge   rdir   (Figure   4.5c).   The   dendritic   conduit   simulation   (configuration   2)   shows   higher   sensitivities  to  most  parameters.  The  two  porosities  θp  and  θf,  the  conduit  system  dispersivity  εc  and   the  porous-­‐fissured  exchange  coefficient  β  do  not  influence  the  spatial  distributions  for  either  setup.  

rdir   and   the   total   recharge  r   have   a   slightly   higher   influence   for   configuration   1   since   the   single   conduit  transmits  water  more  efficiently  (see  also  Chapter  4.3.3).  All  other  parameters  have  higher   influences   on   the   difference   between   average   fissured   system   age   and   spring   water   age   for   configuration   2.   Only   four   of   these   influences   are   significant,   however.   A   decrease   in   the   vertical   conduit   position  zc   or   in   the   fissured-­‐conduit   exchange   coefficient  α   leads   to   slightly   higher   spring   water   ages   and   slightly   lower   average   ages   in   the   fissured   system   (Chapter   4.3.2.1).   Decreasing   values  of  the  conduit  roughness  n  or  the  fissured  system  dispersivity  εf  both  increase  the  average  age   inside   the   fissured   system   and   the   spring   water   age   simultaneously.   Since   the   influence   of   both   parameters  on  the  average  ages  is  higher,  however,  they  also  increase  the  difference  between  spring   water  and  average  age  significantly.  

Figure   4.5.   Differences   in   age   and   life   expectancy   between   the   fissured   system   average   and:   (a)   the   observation  well  for  the  single  conduit  simulation  (configuration  1)  (b)  the  observation  well  for  the  dendritic  

4.3.3 Transient age simulations – influence of recharge events

As  described  in  Chapter  4.2.2  a  theoretical  recharge  event  was  introduced  into  the  model.  Figure  4.6   shows  the  event  and  the  corresponding  groundwater  age  responses  in  the  fissured  system  and  at  the   spring.  Initially,  the  spring  water  age  represents  the  average  age  of  the  water  in  the  fissured  system.  

While  the  response  of  the  groundwater  age  at  the  spring  is  rapid  and  intense,  the  fissured  system   average   age   only   decreases   slightly   and   with   a   lag   time   of   several   days.   For   configuration   1   the   minimum  for  the  average  fissured  age  will  be  reached  after  26  days  and  for  configuration  2  after  27.4   days  (not  shown)  and  only  differ  from  the  initial  age  by  ca.  0.16  years.  The  influence  on  the  age  in  the   porous  system  (not  shown)  is  in  the  order  of  1x10^-­‐4  years  and  is  not  significant.  Diffusive  exchange   processes   are   too   slow   to   significantly   react   to   the   relatively   short-­‐term   recharge   event.   It   is   noticeable  that  the  spring  age  recovers  much  faster  back  to  average  values  than  the  spring  discharge.  

For  both   configurations,   the   spring   water   age  is  higher   than   the   fissured   system   average   after   the   simulation  period  of  20  days,  even  though  the  spring  water  age  is  still  ca.  0.1-­‐0.2  years  younger  than   at  the  beginning  of  the  recharge  event  (Figure  4.6).  This  is  due  to  a  slight  decrease  in  the  average  age   of  groundwater  in  the  fissured  system.  The  young  recharge  water  is  stored  in  the  upper  regions  of   the  aquifer,  while  the  conduits  drain  an  important  amount  of  water  from  greater  depths  (Figure  4.3).  

Therefore,  this  effect  is  stronger  for  the  single  conduit  than  for  the  dendritic  system.  

The   change   of   groundwater   ages   at   the   spring   is   significantly   less   pronounced   for   the   dendritic   conduit  than  for  the  single  conduit  configuration  (Figure  4.6b).  The  dendritic  conduit  pattern  results   in   flow   regimes   where   conduit   water   flows   into   the   fissured   system   for   head   gradient   conditions   directed  towards  the  fissured  system  immediately  after  direct  recharge  events.  The  direct  recharge   pulse  is  not  transmitted  completely  and  directly  to  the  spring,  as  is  the  case  for  the  single  conduit.  

This  effect  also  dampens  spring  discharge  but  this  dampening  is  comparatively  less  well  developed.  

The   variation   in   spring   water   age   for   the   dendritic   system   is   less   than   50%,   while   the   variation   in   discharge  is  almost  80%  of  that  of  the  single  conduit  configuration.  This  shows  the  relatively  large   amount  of  fissured  matrix  water  mobilized  by  the  new  recharge  mixed  into  the  spring  discharge.  

system,   the   amplitude   of   the   maximum   is   well   met,   but   the   timing   of   the   peak,   which   is   a   consequence  of  the  spatial  aquifer  extent,  is  not  reproduced  (Figure  4.7).  The  calibrated  parameter  η   (Eq.  4.9)  has  in  all  cases  the  value  1,  which  implies  that  the  piston  flow  component  of  the  model  is   switched   off   and   a   simple   exponential   model   would   have   given   the   same   goodness   of   fit.   The   simulated  average  transit  time  is  with  ca.  1.8  years  significantly  shorter  than  that  of  the  numerical   model  with  ca.  3.6  years.  The  exponential  curves  drop  below  those  of  the  numerical  model  after  ca.  

70  years  and  do,  therefore,  not  include  the  very  old  components  of  the  porous  system  significantly   contributing   to   longer   average   ages.   The   PFM   fails   in   all   cases   to   simulate   the   transit   time   curves   because  even  the  first  peak  could  not  be  described  by  piston  flow  behaviour.  The  fit  of  the  first  peak   can  possibly  be  improved  if  direct  recharge  is  considered  in  the  simulation.  

The  DM  fails  to  approximate  the  transit  time  curves  at  the  spring.  The  fit  is  improved  considerably  if   only  the  last  part  of  the  curve  is  considered  for  calibration,  i.e.  water  with  ages  higher  than  2.5  years   as   already   suggested   by   Maloszewski   et   al.   (2002).   Nevertheless,   average   transit   times   are   still   underestimated  by  more  than  one  year.  For  the  fissured  system  ages  monitored  at  the  observation   well,  the  statistical  fit  of  the  DM  is  with  a  difference  of  ca.  4×10^-­‐4  years  only  slightly  worse  than  that   of  the  EPM.  The  DM  can  nonetheless  be  viewed  as  superior  since  it  approximates  the  average  transit   time  at  the  observation  well  very  well  with  only  differences  of  0.06  and  0.09  years  for  the  dendritic   and  the  single  conduit  system,  respectively.  The  apparent  dispersion  parameter  PD  (Chapter  4.2.3)  is   in  both  cases  ca.  1.9  showing  a  high  importance  of  the  dispersive  component.  

The  transit  time  distribution  for  the  porous  system  monitored  at  the  observation  well  is  very  flat  and   broad  due  to  the  slow  diffusion  processes.  Both,  the  EPM  and  the  DM  estimate  the  average  transit   time  of  the  distributed  model  of  101.1  years  almost  correctly  with  a  difference  of  1.1  years  for  DM   and  only  0.3  years  for  EPM.