rockphypy.EM

Module Contents

Classes

EM

classical bounds and inclusion models.
class rockphypy.EM.EM[source]

classical bounds and inclusion models.

static VRH(volumes, M)[source]

Computes Voigt, Reuss, and Hill Average Moduli Estimate.

Parameters:

  • volumes (list or array-like) – volumetric fractions of N phases

  • M (list or array-like) – elastic modulus of the N phase.

Returns:

float – M_v: Voigt average M_r: Reuss average M_0: Hill average

static cripor(K0, G0, phi, phic)[source]

Critical porosity model according to Nur’s modified Voigt average.

Parameters:

  • K0 (float or array-like) – mineral bulk modulus in GPa

  • G0 (float or array-like) – mineral shear modulus in GPa

  • phi (float or array-like) – porosity in frac

  • phic (float) – critical porosity in frac

Returns:

float or array-like – K_dry,G_dry (GPa): dry elastic moduli of the framework

static cripor_reuss(M0, Mf, phic, den=False)[source]

In the suspension domain, the effective bulk and shear moduli of the rock can be estimated by using the Reuss (isostress) average.

Parameters:

  • M0 (float) – The solid phase modulus or density

  • Mf (float) – The pore filled phase modulus or density

  • phic (float) – critical porosity

  • den (bool, optional) – If False: compute the reuss average for effective modulus of two mixing phases. If true, compute avearge density using mass balance, which corresponds to voigt average. Defaults to False.

Returns:

float or array-like – M (GPa/g.cc): average modulus or average density

References

  • Section 7.1 Rock physics handbook 2nd edition

static HS(f, K1, K2, G1, G2, bound='upper')[source]

Compute effective moduli of two-phase composite using hashin-strikmann bounds.

Parameters:

  • f (float) – 0-1, volume fraction of stiff material

  • K1 (float or array-like) – bulk modulus of stiff phase

  • K2 (float or array-like) – bulk modulus of soft phase

  • G1 (float or array-like) – shear modulus of stiff phase

  • G2 (float or array-like) – shear modulus of soft phase

  • bound (str, optional) – upper bound or lower bound. Defaults to ‘upper’.

Returns:

float or array-like – K, G (GPa): effective moduli of two-phase composite

static Eshelby_Cheng(K, G, phi, alpha, Kf, mat=False)[source]

Compute the effective anisotropic moduli of a cracked isotropic rock with single set fracture using Eshelby–Cheng Model.

Parameters:

  • K (float) – bulk modulus of the isotropic matrix GPa

  • G (float) – shear modulus of the isotropic matrix GPa

  • phi (float) – (crack) porosity

  • alpha (float) – aspect ratio of crack

  • Kf (float) – bulk modulus of the fluid. For dry cracks use fluid bulk modulus 0

  • mat (bool, optional) – If true: the output is in matrix form, otherwise is numpy array. Defaults to False.

Returns:

1d or 2d array – C_eff: effective moduli of cracked, transversely isotropic rocks

References

  • section 4.14 in The Rock Physics Handbook

static hudson(K, G, Ki, Gi, alpha, crd, order=1, axis=3)[source]

Hudson’s effective crack model assuming weak inclusion for media with single crack set with all normals aligned along 1 or 3-axis. First and Second order corrections are both implemented. Notice that the second order correction has limitation. See Cheng (1993).

Parameters:

  • K (float) – bulk modulus of isotropic background

  • G (float) – shear modulus of isotropic background

  • Ki (float) – bulk modulus of the inclusion material. For dry cracks: Ki=0

  • Gi (float) – shear modulus of the inclusion material

  • alpha (float) – crack aspect ratio

  • crd (float) – crack density

  • order (int, optional) –

    approximation order.

    1: Hudson’s model with first order correction. 2: Hudson’s model with first order correction. Defaults to 1.

  • axis (int, optional) –

    axis of symmetry.

    1: crack normals aligned along 1-axis, output HTI 3: crack normals aligned along 3-axis, output VTI Defaults to 3

Returns:

2d array – C_eff: effective moduli in 6x6 matrix form.

static hudson_rand(K, G, Ki, Gi, alpha, crd)[source]

Hudson’s crack model of a material containing randomly oriented inclusions. The model results agree with the consistent results of Budiansky and O’Connell (1976).

Parameters:

  • K (float or array-like) – bulk modulus of isotropic background

  • G (float or array-like) – shear modulus of isotropic background

  • Ki (float) – bulk modulus of the inclusion material. For dry cracks: Ki=0

  • Gi (float) – shear modulus of the inclusion material, for fluid, Gi=0

  • alpha (float) – crack aspect ratio

  • crd (float) – crack density

Returns:

float or array-like – K_eff, G_eff (GPa): effective moduli of the medium with randomly oriented inclusions

static hudson_ortho(K, G, Ki, Gi, alpha, crd)[source]

Hudson’s first order effective crack model assuming weak inclusion for media with three crack sets with normals aligned along 1 2, and 3-axis respectively. Model is valid for small crack density and aspect ratios.

Parameters:

  • K (float) – bulk modulus of isotropic background

  • G (float) – shear modulus of isotropic background

  • Ki (float) – bulk modulus of the inclusion material. For dry cracks: Ki=0

  • Gi (float) – shear modulus of the inclusion material, for fluid, Gi=0

  • alpha (nd array with size 3) – [alpha1, alpha2,alpha3] aspect ratios of three crack sets

  • crd (nd array with size 3) – [crd1, crd2, crd3] crack densities of three crack sets

Returns:

2d array – C_eff: effective moduli in 6x6 matrix form.

static hudson_cone(K, G, Ki, Gi, alpha, crd, theta)[source]

Hudson’s first order effective crack model assuming weak inclusion for media with crack normals randomly distributed at a fixed angle from the TI symmetry axis 3 forming a cone;

Parameters:

  • K (float) – bulk modulus of isotropic background

  • G (float) – shear modulus of isotropic background

  • Ki (float) – bulk modulus of the inclusion material. For dry cracks: Ki=0

  • Gi (float) – shear modulus of the inclusion material, for fluid, Gi=0

  • alpha (float) – aspect ratios of crack sets

  • crd (float) – total crack density

  • theta (float) – the fixed angle between the crack normam and the symmetry axis x3. degree unit.

Returns:

2d array – C_eff: effective moduli of TI medium in 6x6 matrix form.

static Berryman_sc(K, G, X, Alpha)[source]

Effective elastic moduli for multi-component composite using Berryman’s Consistent (Coherent Potential Approximation) method.See also: PQ_vectorize, Berryman_func

Parameters:

  • K (array-like) – 1d array of bulk moduli of N constituent phases, [K1,K2,…Kn]

  • G (array-like) – 1d array of shear moduli of N constituent phases, [G1,G2,…Gn]

  • X (array-like) – 1d array of volume fractions of N constituent phases, [x1,…xn], Sum(X) = 1.

  • Alpha (array-like) – aspect ratios of N constituent phases. Note that α <1 for oblate spheroids and α > 1 for prolate spheroids, α = 1 for spherical pores,[α1,α2…αn]

Returns:

array-like – K_sc,G_sc: Effective bulk and shear moduli of the composite

static PQ_vectorize(Km, Gm, Ki, Gi, alpha)[source]

compute geometric strain concentration factors P and Q for prolate and oblate spheroids according to Berymann (1980).See also: Berryman_sc, Berryman_func

Parameters:

  • Km (float) – bulk modulus of matrix phase. For Berryman SC approach, this corresponds to the effective moduli of the composite.

  • Gm (float) – shear modulus of matrix phase. For Berryman SC approach, this corresponds to the effective moduli of the composite.

  • Ki (array-like) – 1d array of bulk moduli of N constituent phases, [K1,K2,…Kn]

  • Gi (array-like) – 1d array of shear moduli of N constituent phases, [G1,G2,…Gn]

  • alpha (array-like) – aspect ratios of N constituent phases. Note that α <1 for oblate spheroids and α > 1 for prolate spheroids, α = 1 for spherical pores,[α1,α2…αn]

Returns:

array-like – P,Q (array): geometric strain concentration factors, [P1,,,Pn],[Q1,,,Qn]

static Berryman_func(params, K, G, X, Alpha)[source]

Form the system of equastions to solve. See 4.11.14 and 4.11.15 in Rock physics handbook 2020. See also: Berryman_sc

Parameters:

  • params – Parameters to solve, K_sc, G_sc

  • K (array) – 1d array of bulk moduli of N constituent phases, [K1,K2,…Kn]

  • G (array) – 1d array of shear moduli of N constituent phases, [G1,G2,…Gn]

  • X (array) – 1d array of volume fractions of N constituent phases, [x1,…xn]

  • Alpha (array) – aspect ratios of N constituent phases. Note that α <1 for oblate spheroids and α > 1 for prolate spheroids, α = 1 for spherical pores,[α1,α2…αn]

Returns:

equation – Eqs to be solved

static Swiss_cheese(Ks, Gs, phi)[source]

Compute effective elastic moduli via “Swiss cheese” model with spherical pores. “Swiss cheese” model assumes a dilute distribution of spherical inclusions embedded in an * unbounded * homogenous solid. It takes the “noninteracting assumption” in which all cavities (pores) are independent so that their contributions can be added.

Parameters:

  • Ks (float or array-like) – Bulk modulus of matrix in GPa

  • Gs (float or array-like) – Shear modulus of matrix in GPa

  • phi (float or array-like) – porosity

Returns:

float or array-like – Kdry,Gdry (GPa): effective elastic moduli

static SC(phi, Ks, Gs, iter_n)[source]

Self-Consistent(SC) model with spherical pores considering the critical porosity and the interaction effect between inclusions.

Parameters:

  • phi (float or array-like) – porosity in frac, note that phi.shape== Ks.shape

  • Ks (float) – bulk modulus of matrix phase in GPa

  • Gs (float) – shear modulus of matrix phase in GPa

  • iter_n (int) – iterations, necessary iterations increases as f increases.

Returns:

float or array-like – K_eff,G_eff (GPa): effective elastic moduli

static Dilute_crack(Ks, Gs, cd)[source]

The non-iteracting randomly oriented crack model.

Parameters:

  • Ks (float) – bulk modulus of uncracked medium in GPa

  • Gs (float) – shear modulus of uncracked medium in GPa

  • cd (float or array-like) – crack density

Returns:

float or array-like – K_eff,G_eff (GPa): effective elastic moduli

static OConnell_Budiansky(K0, G0, crd)[source]

O’Connell and Budiansky (1974) presented equations for effective bulk and shear moduli of a cracked medium with randomly oriented dry penny-shaped cracks (in the limiting case when the aspect ratio α goes to 0)

Parameters:

  • K0 (float) – bulk modulus of background medium

  • G0 (float) – shear modulus of background medium

  • crd (float or array-like) – crack density

Returns:

float or array-like – K_dry,G_dry: dry elastic moduli of cracked medium

static OConnell_Budiansky_fl(K0, G0, Kfl, crd, alpha)[source]

Saturated effective elastic moduli using the O’Connell and Budiansky Consistent (SC) formulations under the constraints of small aspect ratio cracks with soft-fluid saturation.

Parameters:

  • K0 (float) – bulk modulus of background medium

  • G0 (float) – shear modulus of background medium

  • Kfl (float) – bulk modulus of soft fluid inclusion, e.g gas

  • crd (float) – crack density

  • alpha (float) – aspect ratio

Returns:

  • float – K_sat,G_sat: elastic moduli of cracked background fully saturated by soft fluid.

  • References – ———-

  • - O’Connell and Budiansky, (1974)

static OC_R_funcs(params, crd, nu_0, w)[source]

Form the system of equastions to solve. Given crack density and w, solve for the D and nu_eff simulaneously using equations 23 and 25 in O’Connell and Budiansky, (1974)

Parameters:

  • params – Parameters to solve

  • crd (float) – crack density

  • nu_0 (float) – Poisson’s ratio of background medium

  • w (float) – softness indicator of fluid filled crack, w=Kfl/alpha/K0, soft fluid saturation is w is the order of 1

Returns:

equation – eqs to be solved

static PQ(Km, Gm, Ki, Gi, alpha)[source]

compute geometric strain concentration factors P and Q for prolate and oblate spheroids according to Berymann (1980). See also PQ_vectorize

Parameters:

  • Km (float) – Bulk modulus of matrix phase

  • Gm (float) – Shear modulus of matrix phase

  • Ki (float) – Bulk modulus of inclusion phase

  • Gi (float) – Shear modulus of inclusion phase

  • alpha (float) – aspect ratio of the inclusion. Note that α <1 for oblate spheroids and α > 1 for prolate spheroids

Returns:

float – P,Q (unitless): geometric strain concentration factors

static DEM(y, t, params)[source]

ODE solver tutorial: https://physics.nyu.edu/pine/pymanual/html/chap9/chap9_scipy.html.

static Berryman_DEM(Km, Gm, Ki, Gi, alpha, phi)[source]

Compute elastic moduli of two-phase composites by incrementally adding inclusions of one phase (phase 2) to the matrix phase using Berryman DEM theory

Parameters:

  • Km (float) – host mineral bulk modulus

  • Gm (float) – host mineral shear modulus

  • Ki (float) – bulk modulus of inclusion

  • Gi (float) – shear modulus of inclusion

  • alpha (float) – aspect ratio of the inclusion phase

  • phi (float) – desired fraction occupied by the inclusion

static SC_dilute(Km, Gm, Ki, Gi, f, mode)[source]

Elastic solids with elastic micro-inclusions. Random distribution of dilute spherical micro-inclusions in a two phase composite.

Parameters:

  • Km (float) – bulk modulus of matrix

  • Gm (float) – shear modulus of matrix

  • Ki (float) – bulk modulus of inclusion

  • Gi (float) – shear modulus of inclusion

  • f (float or array) – volume fraction of inclusion phases

  • mode (string) – ‘stress’ if macro stress is prescribed. ‘strain’ if macro strain is prescribed.

References

  1. Nemat-Nasser and M. Hori (book) : Micromechanics: Overall Properties of Heterogeneous Materials. Sec 8

Returns:

float or array – K, G: effective moduli of the composite

static SC_flex(f, iter_n, Km, Ki, Gm, Gi)[source]

iteratively solving self consistent model for a two phase compposite consisting random distribution of spherical inclusion, not limited to pore.

Parameters:

  • f (float or array) – volumetric fraction, f.shape== Km.shape

  • iter_n (int) – iterations, necessary iterations increases as f increases.

  • Km (float) – bulk modulus of matrix phase

  • Ki (float) – bulk modulus of inclusion phase

  • Gm (float) – shear modulus of matrix phase

  • Gi (float) – shear modulus of inclusion phase

  • Reference –

  • --------- –

  • (book) (S. Nemat-Nasser and M. Hori) –

Returns:

float or array – K_eff, G_eff (GPa): Effective elastic moduli

static MT_average(f, Kmat, Gmat, K1, G1, K2, G2)[source]

Compute Two-phase composite without matrix using modified Mori-Takana Scheme according to Iwakuma 2003, one of the inhomogeneities must be considered as a matrix in the limiting model.

Parameters:

  • f (float or array) – Volume fraction of matrix/inhomogeneity 1. f1=1-f2, (1-f) can be regarded as pseudo crack density.

  • Kmat (float) – Bulk modulus of matrix/inhomogeneity 1

  • Gmat (float) – shear modulus of matrix/ inhomogeneity 1

  • K1 (float) – Bulk modulus of inhomogeneity 1

  • G1 (float) – shear modulus of inhomogeneity 1

  • K2 (float) – Bulk modulus of inhomogeneity 2

  • G2 (float) – shear modulus of inhomogeneity 2

Returns:

float or array – K_ave, G_ave [GPa]: MT average bulk and shear modulus