Volume/Height relationship in wells with square cross-section

In [1]:

    

using Polynomials, DataFrames, Gadfly
plen = Poly([2.67, 0.96 / 11.43])  # polynomial representing size of square


    

    




┌ Info: Loading DataFrames support into Gadfly.jl
└ @ Gadfly /home/bates/.julia/packages/Gadfly/09PWZ/src/mapping.jl:228






    
Out[1]:




2.67 + 0.08398950131233596∙x

In [2]:

    

parea = plen * plen


    

    
Out[2]:




7.1289 + 0.448503937007874∙x + 0.007054236330694883∙x^2

In [3]:

    

pvol = polyint(parea)


    

    
Out[3]:




7.1289∙x + 0.224251968503937∙x^2 + 0.002351412110231628∙x^3

In [4]:

    

H = 0.:0.01:11.43;
d = DataFrame(H=H);
d[:V] = polyval(pvol,d[:H]);
first(d, 6)


    

    




┌ Warning: `head(df::AbstractDataFrame)` is deprecated, use `first(df, 6)` instead.
│   caller = top-level scope at In[4]:4
└ @ Core In[4]:4






    
Out[4]:




6 rows × 2 columns
H
V
Float64
Float64
1
0.0
0.0
2
0.01
0.0713114
3
0.02
0.142668
4
0.03
0.214069
5
0.04
0.285515
6
0.05
0.357006

In [5]:

    

last(d, 6)


    

    
Out[5]:




6 rows × 2 columns
H
V
Float64
Float64
1
11.38
113.634
2
11.39
113.765
3
11.4
113.897
4
11.41
114.029
5
11.42
114.16
6
11.43
114.292

In [6]:

    

plot(d,x="H",y="V",Geom.line)


    

    
Out[6]:

In [7]:

    

real(roots(pvol - 10.)[1])


    

    
Out[7]:





-48.357046148247086

In [8]:

    

real(roots(pvol -12.)[1])


    

    
Out[8]:





-48.48516975908005

In [9]:

    

real(roots(pvol - 20.)[1])


    

    
Out[9]:





-48.97899098468442

In [10]:

    

pradius = Poly([2.25, (0.25)/(11.47)])


    

    
Out[10]:




2.25 + 0.021795989537925022∙x

In [11]:

    

polyval(pradius,[0.,11.47])


    

    
Out[11]:





2-element Array{Float64,1}:
 2.25
 2.5 

In [12]:

    

parea = π * pradius * pradius


    

    
Out[12]:




15.904312808798327 + 0.3081335427452936∙x + 0.001492461216435598∙x^2

In [13]:

    

polyval(parea,[0.,11.47])


    

    
Out[13]:





2-element Array{Float64,1}:
 15.904312808798327
 19.634954084936208

In [14]:

    

pvol = polyint(parea)


    

    
Out[14]:




15.904312808798327∙x + 0.1540667713726468∙x^2 + 0.0004974870721451993∙x^3

In [15]:

    

H = collect(0.:0.01:12.);


    

In [16]:

    

plot(x=H,y=polyval(pvol,H),Geom.line,
Guide.XLabel("Height above well bottom (mm)"),
Guide.YLabel("Volume (cubic mm)"))


    

    
Out[16]:

In [17]:

    

real(roots(pvol - 50.)[1])


    

    
Out[17]:





-156.37132054994754

In [18]:

    

real(roots(pvol - 100.)[1])


    

    
Out[18]:





-157.81466509468657

In [19]:

    

real(roots(pvol - 150.)[1])


    

    
Out[19]:





-159.18546710697348

In [21]:

    

H = 0.0:0.1:12.0;
[H polyval(pvol,H)]


    

    
Out[21]:





121×2 Array{Float64,2}:
  0.0    0.0    
  0.1    1.59197
  0.2    3.18703
  0.3    4.78517
  0.4    6.38641
  0.5    7.99074
  0.6    9.59816
  0.7   11.2087 
  0.8   12.8223 
  0.9   14.439  
  1.0   16.0589 
  1.1   17.6818 
  1.2   19.3079 
  ⋮             
 10.9  192.306  
 11.0  194.252  
 11.1  196.201  
 11.2  198.153  
 11.3  200.109  
 11.4  202.069  
 11.5  204.032  
 11.6  205.998  
 11.7  207.967  
 11.8  209.941  
 11.9  211.917  
 12.0  213.897