#!/usr/bin/env python
# -*-coding:utf-8 -*-
import numpy as np
[docs]class Empirical:
""" Empirical relations that widely applied
"""
[docs] @staticmethod
def krief(phi, Kg,Gg):
"""Compute porous background elastic constants as a function of porosity according to Krief model.
Parameters
----------
phi : float or array
porosity in porous rock
Kg : float
grain bulk modulus
Gg : float
grain shear modulus
Returns
-------
float or array
lamda,G,K
"""
a0= 1-(1-phi)**(3/(1-phi))
G = (1-a0)*Gg
K = (1-a0)*Kg
lamda= K-2*G/3
return lamda,G,K
[docs] @staticmethod
def esti_VS(Vp, vsh):
"""Estimate, using the Greenberg-Castagna empirical relations, the shearwave velocity in a brine-saturated shaly sandstone with vp
since we only assume two minearl phases: so L=2, X1= 1-vsh, X2= vsh
Parameters
----------
Vp : float or array-like
compressional velocities, m/s
vsh : float or array-like
shale volume
Returns
-------
float or array-like
vs (m/s): estimated shear wave velocities
References
----------
- handbook of rock physics P516
"""
Vs_sand= 0.80416*Vp/1000 - 0.85588 # km/s
Vs_shale= 0.76969*Vp/1000 - 0.86735 #km/s
Vs_arith= (1-vsh)*Vs_sand+ vsh*Vs_shale
Vs_harm=( (1-vsh)/Vs_sand +vsh/Vs_shale )**-1
Vs= 0.5*(Vs_arith+Vs_harm)
return Vs*1000
[docs] @staticmethod
def han(phi,C):
"""Han (1986) found empirical regressions relating ultrasonic (laboratory) velocities to porosity and clay content.effective pressure is 20Mpa
Parameters
----------
phi : float or array-like
porosity
C : float or array-like
clay volume fraction
Returns
-------
float or array-like
P and S wave velocities
"""
vp=5.59-6.93*phi-2.81*C
vs=3.52-4.91*phi-1.89*C
return vp*1000,vs*1000
########################################################## to do
[docs] @staticmethod
def ehrenberg(Z):
"""porosity reference trend for Norwegian Sea sandstone. Note that the functional form of the porosity model is not published in Ehrenberg (1990). It is obtained by linear regression of the digitized data point from the original plot in the paper.
Parameters
----------
Z : float or array
burial depth below see floor in Km
References
----------
Ehrenberg, S., 1990, Relationship between diagenesis and reservoir quality in sandstones of the Garn Formation, Haltenbanken, mid-Norwegian continental shelf: AAPG bulletin, 74, no. 10, 1538
Returns
-------
float or array
porosity
"""
Z= Z/1000
phi = -0.092200747*Z+0.4802422
return phi
[docs] @staticmethod
def yu_segment_trend(Z):
"""Reference trend for Norwegian sea normally buried clean sandstones
Parameters
----------
Z : float or array
burial depth below see floor in m
Returns
-------
float or array
P wave velocities
"""
V1=1707.998+0.66*Z[Z<=2630]
V2=1200.84+0.853*Z[Z>2630]
V= np.hstack((V1,V2))
return V
[docs] @staticmethod
def ramm_porosity(Z, HB=True):
"""porosity reference trend according to Ramm & Bjørlykke (1994)
Parameters
----------
Z : float or array
burial depth wrt. sea floor in m
HB : bool, optional
if True: only show the regression line for halten bakken area porosity data False: The regression line for all porosity from north sea and norwegian sea, by default True
Returns
-------
float or array
porosity
"""
if HB==True:
phi= 46.4-0.0085*Z
else:
phi= 42.7-0.0069*Z
return phi/100
[docs] @staticmethod
def ramm_porosity_segment(Z):
"""segment porosity reference trend according to Ramm & Bjørlykke (1994) considering the mechanical and chemical compaction
Parameters
----------
Z : float or array
burial depth wrt. sea floor in m
Returns
-------
float or array
porosity
"""
phi1= 45*np.exp(-0.00025*Z)
phi2= 25-0.013*(Z-2500)
phi1[phi1>phi2]=phi2[phi2<phi1]
return phi1
[docs] @staticmethod
def empirical_StPeter(Pe, sample=1):
"""compute the Vp and Vs for st peter sandstone using the empirical relationship in the form of V= A+KPe-Be^(-DPe)
Parameters
----------
Pe : float or array
effective pressure in Kbar, 1kbar= 100Mpa
sample : int, optional
1-sample 1, phi= 0.205. 2-sample 2, phi= 0.187, by default 1
Returns
-------
float or array
Vp,Vs in km
"""
if sample==1:
Vp= 4.21+0.187*Pe-0.746*np.exp(-24*Pe)
Vs= 2.58+0.160*Pe-0.781*np.exp(-20*Pe)
elif sample ==2:
Vp= 4.55+0.298*Pe-0.8*np.exp(-19*Pe)
Vs= 2.83+0.195*Pe-0.741*np.exp(-16*Pe)
return Vp*1000, Vs*1000
[docs] @staticmethod
def Scherbaum(Z):
""" velocity depth trend for Lower and Middle Buntsandstein
Parameters
----------
Z : float or array
burial depth wrt. sea floor in m
References
----------
Scherbaum, F., 1982. Seismic velocities in sedimentary rocks—indicators of subsidence and uplift?. Geologische Rundschau, 71(2), pp.519-536.
Returns
-------
float or array
P wave velocities in m/s
"""
V = 2325+0.51*Z
return V
[docs] @staticmethod
def Sclater(phi):
"""Sclater-Christie exponential curve for sandstone
Parameters
----------
phi : float or array
porosity
Returns
-------
Z : float or array
depth wrt. sea floor in km.
"""
Z= 3.7*np.log(0.49/phi)
return Z
[docs] @staticmethod
def Storvoll(Z):
"""Storvoll velocity compaction trend. The trend is for shale and shaly sediments but also used for siliciclastic rock like sandstone
Parameters
----------
Z : float or array
depth wrt. sea floor in m.
Returns
-------
float or array
Vp meters per second
"""
V= 1/1.76*Z+2600
return V
[docs] @staticmethod
def Hillis(Z):
"""compaction trend for Bunter Sandstone in North sea
Parameters
----------
Z : float or array
depth below sea bed (in kilometers)
Returns
-------
float or array
Vp km/s
"""
transit= 135.9-20.22*Z/1000
V= 304.8*1000/transit
return V
[docs] @staticmethod
def Japsen(Z):
"""a segmented linear velocity–depth function, These equations are considered as approximation for bunter sandstone trend although they are originally for bunter shale. proposed by Japsen 1999
Parameters
----------
Z : float or array
depth below sea bed in m.
Returns
-------
float or array
Vp m/s
"""
V1=1550+0.6*Z[Z<1393]
V2= -400+2*Z[(Z>=1393)&(Z<2000)]
V3=2600+0.5*Z[(Z>=2000)&(Z<3500)]
V4=3475+0.25*Z[(Z>=3500)&(Z<5300)]
V= np.hstack((V1,V2,V3,V4))
return V
[docs] @staticmethod
def hjelstuen(Z):
"""Velocity-depth relationships for the Bjørna-Sørkapp margin deposits. note: the seismic velocities are not directly comparable with velocities from sonic logs (because of the different frequencies), and the velocity-depth profile of Hjelstuen et al. (1996) has not been corrected for uplift and erosion
Parameters
----------
Z : float or array
Z< 3.8km
Returns
-------
float or array
V: m/s
"""
V=1.87+0.55*(Z/1000) # convert Z to km
return V*1000
[docs] @staticmethod
def Cp(phi):
"""The coordination number n depends on porosity, as shown by
Murphy 1982.
Parameters
----------
phi : float or array
total porosity , for a porosity of 0.4, n=8.6
Returns
-------
float or array
coordination number
"""
return 20-34*phi+14*phi**2
