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FormationModel.py
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FormationModel.py
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__author__ = 'XSheng'
__doc__ = '''
This module is the interface of formation model.
'''
import numpy as np
from scipy.special import iv,kn # bessel functions
from numpy import sqrt, real
from cmath import tanh, cosh, sinh
def coth(x):
return cosh(x)/sinh(x)
I1 = lambda x: iv(1, real(x))
I0 = lambda x: iv(0, real(x))
K1 = lambda x: kn(1, real(x))
K0 = lambda x: kn(0, real(x))
class FormationModel():
def __init__(self, name, BC ):
"""
Name = Model Name
BC = "ConstantPressure" or "Closed"
"""
self.ModelName = name # Model Name
self.BC = BC # Model Boundary Condition, fixed pressure
def SphericalLaplace(self,u, R_D, C_D, S):
"""
laplace solution of ideal one-layer spherical reservoir model with constant pressure boundary
u is the laplace variable
R_D is the dimensionless reservoir radius
C_D is the dimensionless storage factor
S is skin factor
"""
sqrt_u = np.sqrt(u)
sqrt_uRd = sqrt_u*R_D
if self.BC == "ConstantPressure":
fu = sqrt_u*(K1(sqrt_u)*I0(sqrt_uRd) + K0(sqrt_uRd)*I1(sqrt_u)) /\
(K1(sqrt_u)*I0(sqrt_uRd) - K0(sqrt_uRd)*I0(sqrt_u))
elif self.BC == "Closed":
fu = sqrt_u*(K1(sqrt_u)*I0(sqrt_uRd) - K0(sqrt_uRd)*I1(sqrt_u)) /\
(K1(sqrt_u)*I0(sqrt_uRd) + K0(sqrt_uRd)*I0(sqrt_u))
pwd = 1/u*(1+S*fu)/(fu+C_D*u*(1+S*fu))
return pwd
def InfiniteSize(self, u, C_D, S):
"""
u is the laplace variable
C_D is the dimensionless storage factor
S is skin factor
"""
sqrt_u = np.sqrt(u)
pwd = 1/u*(K0(sqrt_u) + S*sqrt_u*K1(sqrt_u))/\
(sqrt_u*K1(sqrt_u) + C_D*u*
(K0(sqrt_u)+S*sqrt_u*K1(sqrt_u)))
return pwd
def InfiniteSizeLineSource(self, u, C_D, S):
"""
Simplified line source solution
u is the laplace variable
C_D is the dimensionless storage factor
S is skin factor
"""
sqrt_u = sqrt(u)
pwd = 1/u*(K0(sqrt_u) + S)/(1+C_D*u*(K0(sqrt_u)+S))
return pwd
def func_f_steady(self, _omega, _lambda, u):
return (_omega*(1-_omega)*u + _lambda)/((1-_omega)*u + _lambda)
def func_f_plate(self, _omega, _lambda, u):
return _omega + sqrt(_lambda*(1-_omega)/3/u)*tanh(sqrt(3*(1-_omega)*u/_lambda))
def func_f_spherical(self, _omega, _lambda, u):
tmp = 15*(1-_omega)*u/_lambda
return _omega + 1.0/5*_lambda/u*(sqrt(tmp)*coth(sqrt(tmp)) -1)
def func_f_cylinder(self, _omega, _lambda, u):
"""
f(u) = \omega +2*\sqrt((1-\omega)*\lambda/(15u)) * I_1(\sqrt((1-\omega)*\lambda/(15u)))/I_0(\sqrt((1-\omega)*\lambda/(15u)))
"""
tmp = (1-_omega)*_lambda/15/u
return _omega + 2*sqrt(tmp)*I1(tmp)/I0(tmp)
def InfiniteSizeDualPoro(self, f, u, c_D, S):
return self.InfiniteSize(f(u)*u, c_D, S)
def test():
from InverseLaplace import InverseLaplace
import matplotlib.pyplot as plt
RadialFormation = FormationModel("Infinite Radial", "ConstantPressure")
# infinite reservoir line source solution
F = lambda u: RadialFormation.InfiniteSizeLineSource(u, 1.0, 1.0)
invLap = InverseLaplace(F)
t = np.arange(0.01, 10*np.pi, 0.1)
ft = invLap.Stehfest(t)
# infinite reservoir spherical solution
Fcp = lambda u:RadialFormation.SphericalLaplace(u, 10.0, 1.0, 1.0)
invLapcp = InverseLaplace(Fcp)
ftcp = invLapcp.Stehfest(t)
# infinite reservoir
Fls = lambda u:RadialFormation.InfiniteSize(u, 1.0, 1.0)
invLapls = InverseLaplace(Fls)
ftls = invLapls.Stehfest(t)
ftls_dehoog = np.vectorize(invLapls.DeHoog)(t)
# infinite dual porosity reservoir
_omega = 0.5
_lambda = 0.15 # lambda value has a great impact on latter period pressure response
f = lambda u: RadialFormation.func_f_cylinder(_omega, _lambda, u)
Fdp = lambda u: RadialFormation.InfiniteSizeDualPoro(f, u, 1.0, 1.0 )
invLapDp = InverseLaplace(Fdp)
ftdp = invLapDp.Stehfest(t)
ft_dehoog = np.zeros_like(t)
DeHoog = np.vectorize(invLapDp.DeHoog)
ft_dehoog = DeHoog(t)
#for i in range(0, len(t), 1):
# ft_dehoog[i] = invLapDp.DeHoog(t[i])
plt.plot(#t, ft,
#t, ftcp,
#t, ftls,
t, ftls_dehoog,
#t, ftdp,
t, ft_dehoog
)
plt.show()
if __name__=="__main__":
test()