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Permeability Layering

A number of functions used to help characterize heterogeneity in permeability layering and/or create permeability array consistant with observed levels of heterogeneity and observed average permeability.

Heteogeneity is characterized via the Lorenz coefficient, and implemented via one of two methods, both of which can be described by a single Beta value;
  • Exponential ('EXP')
  • Langmuir ('LANG')

Background on the approach taken as well as the derivation for the Exponential implementation can be found in this LinkedIn article,

For the Langmuir formulation:
  • At a specific cumulative Phi.h -> Kh = Phi.h * VL / (Phi.h + PL)
  • Integrating this, subtracting 0.5 and doubling -> Lorenz = (VL - PL * VL * Ln(VL) + PL * VL * Ln(PL) - 0.5) * 2
  • Where PL = 1 / Beta and VL = 1 / Beta + 1

pyrestoolbox.lorenz2b

lorenz2b(lorenz, lrnz_method = 'EXP') -> float

Returns the Beta value consistent with the Lorenz coefficient given, and implementation method selected

Inputs
Parameter Type Description
lorenz float Lorenz coefficient (0 < lorenz < 1)
lrnz_method float Implementation method. Can be either 'EXP' or 'LANG'. Default is 'EXP'

Examples:

>>> from pyrestoolbox import pyrestoolbox as rtb
>>> rtb.lorenz2b(0.75, lrnz_method = 'LANG')
16.139518537603912

>>> rtb.lorenz2b(0.75)
7.978108090962671

pyrestoolbox.lorenzfromb ======================

lorenzfromb(B: float, lrnz_method: str = 'EXP') -> float

Returns the Lorenz coefficient consistent with the Beta value given, and implementation method selected

Inputs
Parameter Type Description
B float Beta value (B > 0)
lrnz_method float Implementation method. Can be either 'EXP' or 'LANG'. Default is 'EXP'

Examples:

>>> rtb.lorenzfromb(16.139518537603912, lrnz_method = 'LANG')
0.750000182307895

>>> rtb.lorenzfromb(7.978108090962671)
0.7500000108799212

pyrestoolbox.lorenz_from_flow_fraction ======================

lorenz_from_flow_fraction(kh_frac, phih_frac, lrnz_method= 'EXP') -> float

Returns the Lorenz coefficient consistent with observed best flow fraction from a phi_h fraction

Inputs
Parameter Type Description
kh_frac float The cumulative flow fraction of the best contributing flow unit ( 0 < kh_frac < 1 )
phih_frac float The cumulative porosity thickness fraction of the best contributing flow unit ( phih_frac < kh_frac )
lrnz_method float Implementation method. Can be either 'EXP' or 'LANG'. Default is 'EXP'

Examples:

60% of the observed flow comes from 15% of the net thickness .. code-block:: python

>>> lorenz = rtb.lorenz_from_flow_fraction(kh_frac=0.6, phih_frac=0.15) >>> lorenz 0.6759312029093838

pyrestoolbox.lorenz_2_flow_frac ======================

lorenz_2_flow_frac(lorenz, phih_frac, lrnz_method = 'EXP', B = -1) -> float

Returns expected flow fraction from the best phi_h fraction, with a specified Lorenz coefficient.

If B is left default, then it will be calculated. If B is explictly specified > 0, then it will be used instead of the provided lorenz coefficient so as to eliminate repetitive solving for B.

Inputs
Parameter Type Description
lorenz float Lorenz coefficient (0 < lorenz < 1). If B is provided, will ignore this parameter to be more efficient. If not, will calculate B from this parameter.
phih_frac float The cumulative porosity thickness fraction of the best contributing flow unit ( 0 < phih_frac < 1 )
lrnz_method float Implementation method. Can be either 'EXP' or 'LANG'. Default is 'EXP'
B float Beta value (B > 0). Will calculate if only lorenz variable defined

Examples:

>>> rtb.lorenz_2_flow_frac(lorenz=0.6759312029093838, phih_frac=0.15)
0.6000001346893536

pyrestoolbox.lorenz_2_layers ======================

lorenz_2_layers(lorenz, k_avg, nlayers = 1, shuffle = False, lrnz_method = 'EXP', B = -1, phi_h_fracs = []) -> np.ndarray

Returns np.array of permeability values honoring a specified average permeability (assuming equal thickness layers unless list of phi_h_fracs is provided), with degree of heterogeneity consistant with specified Lorenz coefficient and method

If B is left default, then it will be calculated. If B is explictly specified > 0, then it will be used instead of the provided lorenz coefficient so as to eliminate repetitive solving for B.

Inputs
Parameter Type Description
lorenz float Lorenz coefficient (0 < lorenz < 1). If B is provided, will ignore this parameter to be more efficient. If not, will calculate B from this parameter.
k_avg float The thickness weighted average permeability of all the layers - Sum(kh) / h
nlayers int The number of permeability layers desired (>1 needed unless a list of phi_h_fracs is supplied)
shuffle bool Boolean flag to determine whether to return the permeability array in decreasing order (False), or random order (True). Default False
lrnz_method float Implementation method. Can be either 'EXP' or 'LANG'. Default is 'EXP'
B float Beta value (B > 0). Will calculate if only lorenz variable defined
phi_h_fracs list Optional ability to specify a sorted list of phi_h fractions to calculate permeabilities for. If this list does not add to unity, then one additional layer permeability will be returned. The list needs to be in sorted order of best flow capacity to worst. If list adds to more than 1, it will be normalized

Examples:

>>> rtb.lorenz_2_layers(lorenz = 0.67, nlayers = 5, k_avg = 10, shuffle = True)
array([10.58944038,  0.29499066, 34.9323596 ,  3.21009656,  0.9731128 ])

>>> rtb.lorenz_2_layers(lorenz = 0.67, k_avg = 10, phi_h_fracs=[0.05, 0.5])
array([51.72990694, 14.12556056,  0.77938749])