Steady interface flow towards the coast¶

Mark Bakker, TU Delft, The Netherlands

In [1]:

# these are needed so that the notebook can also be run on Python 2
from __future__ import division, print_function
from pylab import *
%matplotlib notebook
from semi_interface import *

Interface flow towards the coast¶

Flow is confined below the land and semi-confined below the sea. The leaky sea bottom extends a distance $L$ below the sea. The freshwater is moving and the saltwater is at rest. Input variables:

k: hydraulic conductivity [m/d]
H: aquifer thickness [m]
c: resistance of leaky layer [d]
grad: absolute value of head gradient towards coast upstream of interface toe
rhof: density of fresh water [kg/m3]
rhos: density of salt water [kg/m3]
Ls: length of leaky sea bottom below sea (may also be inf) [m]
ztop: elevation of top of aquifer with respect to datum [m]
sealevel: elevation of sealevel with respect to datum [m]

Case I of Bakker (2006)¶

In [2]:

sc1 = SemiCoast(k=10, H=10, c=100, grad=0.0005, 
                rhof=1000, rhos=1025, Ls=inf, 
                ztop=0, sealevel=0)
print('toe of interface at:', sc1.toe())
print('tip of interface at:', sc1.tip())
sc1.plot(xmin=-300, xmax=300);

toe of interface at: -211.68452838
tip of interface at: 153.261886479

Case II of Bakker (2006)¶

In [3]:

sc2 = SemiCoast(k=10, H=10, c=100, grad=0.00375, 
                rhof=1000, rhos=1025, Ls=inf, 
                ztop=0, sealevel=0)
print('toe of interface at:', sc2.toe())
print('tip of interface at:', sc2.tip())
sc2.plot(xmin=-300, xmax=300);

toe of interface at: 53.6836486682
tip of interface at: 298.632622947

Case III of Bakker (2006)¶

In [4]:

sc3 = SemiCoast(k=10, H=10, c=100, grad=0.0005, 
                rhof=1000, rhos=1025, Ls=80, 
                ztop=0, sealevel=0)
print('toe of interface at:', sc3.toe())
print('tip of interface at:', sc3.tip())
sc3.plot(xmin=-300, xmax=300);

toe of interface at: [-212.41139369]
tip of interface at: 80.0

Case IV of Bakker (2006)¶

In [5]:

sc4 = SemiCoast(k=10, H=10, c=100, grad=0.00375, 
                rhof=1000, rhos=1025, Ls=150, 
                ztop=0, sealevel=0)
print('toe of interface at:', sc4.toe())
print('tip of interface at:', sc4.tip())
sc4.plot(xmin=-300, xmax=300);

toe of interface at: 50.3243452459
tip of interface at: 150.0

Specification of upstream head (onshore, so $x<0$)¶

Rather than specifying the gradient upstream of the toe, the head h is specified at some point x in the aquifer below the land (so x is negative).

In [6]:

schead = SemiCoastHead(k=10, H=10, c=100, x=-1000, h=1, 
                       rhof=1000, rhos=1025, Ls=1000, 
                       ztop=0, sealevel=0)
print('toe of interface at:', schead.toe())
print('tip of interface at:', schead.tip())
print('location of h=1 (should be -1000)', schead.onshorex(h=1))
print('computed gradient:', schead.grad)
schead.plot();

toe of interface at: [-103.86297425]
tip of interface at: [ 181.97278682]
location of h=1 (should be -1000) [-1000.]
computed gradient: [ 0.00083693]

Figure for three different sealevels¶

In [7]:

schead = SemiCoastHead(k=10, H=10, c=100, x=-1000, h=0.25, 
                       rhof=1000, rhos=1025, Ls=1000, 
                       ztop=0, sealevel=0, label='h=0.25')
schead.plot(xmin=-500, xmax=200)
schead = SemiCoastHead(k=10, H=10, c=100, x=-1000, h=0.5, 
                       rhof=1000, rhos=1025, Ls=1000, 
                       ztop=0, sealevel=0, label='h=0.5')
schead.plot(newfig=False);
schead = SemiCoastHead(k=10, H=10, c=100, x=-1000, h=1, 
                       rhof=1000, rhos=1025, Ls=1000, 
                       ztop=0, sealevel=0, label='h=1')
schead.plot(newfig=False);
legend(loc='best');

In [ ]: