pedophysics.pedophysical_models package

Submodules

pedophysics.pedophysical_models.bulk_ec module

pedophysics.pedophysical_models.bulk_ec.Fu(water, clay, por, wc, solid_ec, dry_ec, sat_ec, s=1, w=2)

Calculate the soil bulk real electrical conductivity using the Fu model and return

This is a volumetric mixing model that takes into account various soil properties such as clay content, porosity, and water content. It was exhaustively validated using several soil samples [1]. Reported R2 = 0.98

Parameters

waterarray_like

Soil volumetric water content [m**3/m**3].

clayarray_like

Soil clay content [g/g]*100.

por: array_like

Soil porosity [m**3/m**3].

wcarray_like

Soil water real electrical conductivity [S/m].

solid_ecarray_like

Soil solid real electrical conductivity [S/m].

dry_ecarray_like

Soil bulk real electrical conductivity at zero water content [S/m].

sat_ecarray_like

Soil bulk real electrical conductivity at saturation water content [S/m].

sfloat, optional

Phase exponent of the solid, default is 1.

wfloat, optional

Phase exponent of the water, default is 2.

Returns

array_like

The estimated bulk electrical conductivity [S/m].

Notes

The method uses default values for s and w, which are 1 and 2 respectively, but can be modified if necessary. Three different forms of the model are used depending on the soil data availability. The soil electrical conductivity of solid surfaces is calculated as in [1] using the formula of Doussan and Ruy (2009) [2]

References

[1]

Yongwei Fu, Robert Horton, Tusheng Ren, J.L. Heitman,

A general form of Archie’s model for estimating bulk soil electrical conductivity, Journal of Hydrology, Volume 597, 2021, 126160, ISSN 0022-1694, https://doi.org/10.1016/j.jhydrol.2021.126160. .. [2] Doussan, C., and Ruy, S. (2009), Prediction of unsaturated soil hydraulic conductivity with electrical conductivity, Water Resour. Res., 45, W10408, doi:10.1029/2008WR007309.

Example

>>> Fu(0.3, 30, 1.3, 2.65, 0.3, 0, np.nan, np.nan)
0.072626

pedophysics.pedophysical_models.bulk_ec.LongmireSmithEC(bulk_ec_dc, frequency_ec)

Calculate the soil bulk real electrical conductivity using the Longmire-Smith model and return

This is a semiempirical model that calculates the soil bulk real electrical conductivity at different electromagnetic frequencies [1].

Parameters

bulk_ec_dcarray_like

Soil bulk real direct current electrical conductivity [S/m].

frequency_ecarray_like

Frequency of electric conductivity measurement [Hz].

Returns

array_like

Soil bulk real electrical conductivity [S/m].

Notes

The Longmire-Smith equation uses a set of coefficients to account for the frequency-dependent dielectric dispersion. If all values in the bulk_ec_dc array are zero, the function returns 0.

Global Variables Used

epsilon_0float

The vacuum permittivity constant.

References

[1]

11. 19. Smith and C. L. Longmire, “A universal impedance for soils,”

Defense Nuclear Agency, Alexandria, VA, USA, Topical Report for Period Jul. 1 1975-Sep. 30 1975, 1975.

Example

>>> LongmireSmithEC(np.array([0.05, 0.10]), 130)
array([0.05153802, 0.10245936])

pedophysics.pedophysical_models.bulk_ec.Rhoades(water, wc, s_ec, E, F)

Calculate the soil bulk real electrical conductivity using the Rhoades model and return

This function estimates the bulk electrical conductivity of a soil-water mixture using the Rhoades equation [1]. The model combines the contributions of the water’s electrical conductivity and the soil’s electrical conductivity, adjusted by weighting factors based on the water content.

Parameters

waterarray_like

Soil volumetric water content [m**3/m**3].

wcarray_like

Soil water real electrical conductivity [S/m].

s_ecarray_like

Soil bulk real surface electrical conductivity [S/m].

Efloat

Weighting factor for the quadratic term of water content in the Rhoades equation.

Ffloat

Weighting factor for the linear term of water content in the Rhoades equation.

Returns

array_like

Soil bulk real electrical conductivity [S/m].

References

[1]

Rhoades, J.D., Raats, P.A.C. and Prather, R.J. (1976),

Effects of Liquid-phase Electrical Conductivity, Water Content, and Surface Conductivity on Bulk Soil Electrical Conductivity. Soil Science Society of America Journal, 40: 651-655. https://doi.org/10.2136/sssaj1976.03615995004000050017x

Example

>>> Rhoades(0.3, 0.5, 0.001, 1, 0.5)
0.121

pedophysics.pedophysical_models.bulk_ec.SheetsHendrickx(bulk_ec, temperature)

Calculate the soil bulk real electrical conductivity using the Sheets-Hendricks model and return

This function adjusts the apparent electrical conductivity (ECa) of soil to a standard temperature of 298.15 K (25°C). The adjustment is based on the Sheets-Hendricks model.

Parameters

bulk_ecarray-like

Soil bulk real electrical conductivity [S/m]

temperaturearray-like

Soil bulk temperature [K]

Returns

array_like

Soil bulk real electrical conductivity temperature corrected [S/m]

Notes

The Sheets-Hendricks model applies a temperature correction factor to adjust the apparent electrical conductivity to a standard temperature of 25°C. This correction is particularly important in precision agriculture and soil science studies where temperature fluctuations can significantly affect conductivity measurements.

Example

>>> SheetsHendrickxEC(np.array([1.2, 2.5]), 20)
array([0.13352103, 0.27816881])

pedophysics.pedophysical_models.bulk_ec.WunderlichEC(water, ec_init, wat_init, wc, Lw)

Calculate the soil bulk real electrical conductivity using the Wunderlich model and return

This is a effective medium model that uses a differential approach to compute the electrical conductivity based on the initial conditions and the water content [1]. Reported RMSE = 0.0055

Parameters

waterarray_like

Soil volumetric water content [m**3/m**3].

ec_initfloat

Initial soil bulk real electrical conductivity [S/m].

wat_initfloat

Initial soil volumetric water content [m**3/m**3].

wcarray_like

Soil water real electrical conductivity [S/m].

Lwfloat

Soil scalar depolarization factor of water aggregates (effective medium theory) [-]

Returns

array_like

The estimated soil bulk real electrical conductivity [S/m].

Notes

The Wunderlich model is a differential model and the approach used here employs a simple while loop with a step of 0.01 until the differential fraction reaches 1. The method takes into account the initial water content, initial electrical conductivity, and weighting factors to determine the bulk electrical conductivity.

References

[1]

Tina Wunderlich, Hauke Petersen, Said Attia al Hagrey, Wolfgang Rabbel;

Pedophysical Models for Resistivity and Permittivity of Partially Water-Saturated Soils. Vadose Zone Journal 2013;; 12 (4): vzj2013.01.0023. doi: https://doi.org/10.2136/vzj2013.01.0023

Example

>>> WunderlichEC(0.3, 0.08, 0.05, 0.50, 0.01)
0.182634

pedophysics.pedophysical_models.bulk_perm module

pedophysics.pedophysical_models.bulk_perm.Hilhorst(bulk_ec, water_ec, water_perm, offset_perm)

Calculate the soil bulk real relative dielectric permittivity using the Hilhorst model and return

This function calculates the soil bulk real relative dielectric permittivity of a soil-water mixture based on Hilhorst’s model. The relation connects the bulk electrical conductivity of the mixture with the permittivity of the water phase and an offset for the permittivity.

Parameters

bulk_ecarray_like

Soil bulk real relative dielectric permittivity [-].

water_ecarray_like

Soil water real electrical conductivity [S/m].

water_permarray_like

Soil water phase real dielectric permittivity [-].

offset_permarray_like

Soil bulk real relative dielectric permittivity when soil bulk real electrical conductivity is zero [-].

Returns

array_like

Soil bulk real relative dielectric permittivity [-].

References

[1]

Hilhorst, M.A. (2000), A Pore Water Conductivity Sensor.

Soil Sci. Soc. Am. J., 64: 1922-1925. https://doi.org/10.2136/sssaj2000.6461922x

Example

>>> Hilhorst(0.05, 0.5, 80, 4)
12.0

pedophysics.pedophysical_models.bulk_perm.LR(water, por, ap, sp, wp, alpha)

Calculate the soil bulk real relative dielectric permittivity using the Lichtenecker and Rother model and return

This function computes the bulk real relative dielectric permittivity of a soil mixture using the volumetric mixing model of Lichtenecker and Rother [1] (LR). The model incorporates the water content, bulk and particle densities, and permittivities of air, solid, and water.

Parameters

waterarray_like

Soil volumetric water content [m**3/m**3].

por: array_like

Soil porosity [m**3/m**3].

aparray_like

Soil air real relative dielectric permittivity phase [-].

sparray_like

Soil solid real relative dielectric permittivity phase [-].

wparray_like

Soil water phase real dielectric permittivity [-].

alphafloat

Soil alpha exponent as defined in volumetric mixing theory [-].

Returns

array_like

The estimated soil bulk real relative dielectric permittivity [-]

References

[1]

Lichtenecker, K., and K. Rother. 1931.

Die Herleitung des logarithmischen Mischungsgesetzes aus allgemeinen Prinzipien der staionären Strömung. Phys. Z. 32:255-260.

Example

>>> LR(0.3, 1.3, 2.65, 1, 4, 80, 0.5)
15.006

pedophysics.pedophysical_models.bulk_perm.LR_MV(water, por, ap, sp, wp, CEC)

Calculate the soil bulk real relative dielectric permittivity using the Lichtenecker-Rother model modified by Mendoza-Veirana and return

This function computes the bulk real relative dielectric permittivity of a soil mixture using the volumetric mixing model of Lichtenecker and Rother [1], and Mendoza Veirana [2] alpha correction model (LR_MV). The model incorporates the water content, bulk and particle densities, and permittivities of air, solid, and water, as well as the soil cation exchange capacity. Reported R2 = 0.94

Parameters

waterarray_like

Soil volumetric water content [m**3/m**3].

por: array_like

Soil porosity [m**3/m**3].

aparray_like

Soil air real relative dielectric permittivity phase [-].

sparray_like

Soil solid real relative dielectric permittivity phase [-].

wparray_like

Soil water phase real dielectric permittivity [-].

CECarray_like

Soil cation exchange capacity [meq/100g].

Returns

array_like

The estimated soil bulk real relative dielectric permittivity [-]

References

[1]

Lichtenecker, K., and K. Rother. 1931.

Die Herleitung des logarithmischen Mischungsgesetzes aus allgemeinen Prinzipien der staionären Strömung. Phys. Z. 32:255-260. .. [2] Gaston Mendoza Veirana, Jeroen Verhegge, Wim Cornelis, Philippe De Smedt, Soil dielectric permittivity modelling for 50-MHz instrumentation, Geoderma, Volume 438, 2023, 116624, ISSN 0016-7061, https://doi.org/10.1016/j.geoderma.2023.116624.

Example

>>> LR_MV(0.3, 1.3, 2.65, 1, 4, 80, 20)
28.577  

pedophysics.pedophysical_models.bulk_perm.LR_W(water, por, ap, sp, wp, clay)

Calculate the soil bulk real relative dielectric permittivity using the Lichtenecker and Rother model modified by Wunderlich and return

This function computes the bulk real relative dielectric permittivity of a soil mixture using the volumetric mixing model of Lichtenecker and Rother [1], and Wunderlich [2] alpha correction model (LR_W). The model incorporates the water content, bulk and particle densities, and permittivities of air, solid, and water, as well as the soil clay content.

Parameters

waterarray_like

Soil volumetric water content [m**3/m**3].

por: array_like

Soil porosity [m**3/m**3].

aparray_like

Soil air real relative dielectric permittivity phase [-].

sparray_like

Soil solid real relative dielectric permittivity phase [-].

wparray_like

Soil water phase real dielectric permittivity [-].

clayarray_like

Soil clay content [g/g]*100

Returns

array_like

The estimated soil bulk real relative dielectric permittivity [-]

References

[1]

Lichtenecker, K., and K. Rother. 1931.

Die Herleitung des logarithmischen Mischungsgesetzes aus allgemeinen Prinzipien der staionären Strömung. Phys. Z. 32:255-260. .. [2] Tina Wunderlich, Hauke Petersen, Said Attia al Hagrey, Wolfgang Rabbel; Pedophysical Models for Resistivity and Permittivity of Partially Water-Saturated Soils. Vadose Zone Journal 2013;; 12 (4): vzj2013.01.0023. doi: https://doi.org/10.2136/vzj2013.01.0023

Example

>>> LR_W(0.3, 1.3, 2.65, 1, 4, 80, 20)
17.505 

pedophysics.pedophysical_models.bulk_perm.LongmireSmithP(bulk_ec_dc, bulk_perm_inf, frequency_perm)

Calculate the soil bulk real relative dielectric permittivity using the Longmire-Smith model and return

This is a semiempirical model that calculates the soil bulk real relative dielectric permittivity at different electromagnetic frequencies [1].

Parameters

bulk_ec_dcarray_like

Soil bulk real direct current electrical conductivity [-].

bulk_perm_infarray_like

Soil bulk real relative permittivity at infinite frequency [-].

frequency_permarray_like

Frequency of dielectric permittivity measurement [Hz].

Returns

array_like

Soil bulk real relative dielectric permittivity [-].

Notes

The Longmire-Smith equation uses a set of coefficients to account for the frequency-dependent dielectric dispersion. If all values in the bulk_ec_dc array are zero, the function returns ‘bulk_perm_inf’.

Global Variables Used

epsilon_0float

The vacuum permittivity constant.

References

[1]

11. 19. Smith and C. L. Longmire, “A universal impedance for soils,”

Defense Nuclear Agency, Alexandria, VA, USA, Topical Report for Period Jul. 1 1975-Sep. 30 1975, 1975.

Example

>>> LongmireSmithP(0.1, 5, 50e6)
23.328

pedophysics.pedophysical_models.bulk_perm.WunderlichP(water, perm_init, wat_init, wp, Lw)

Calculate the soil bulk real relative dielectric permittivity using the Wunderlich model and return

This is a effective medium model that uses a differential approach to compute the dielectric permittivity based on the initial conditions and the water content [1]. Reported RMSE = 0.009

Parameters

waterarray_like

Soil volumetric water content [m**3/m**3].

perm_initfloat

Initial soil bulk real relative dielectric permittivity [-].

wat_initfloat

Initial soil volumetric water content [m**3/m**3].

wparray_like

Soil water phase real dielectric permittivity [-].

Lwfloat

Soil scalar depolarization factor of water aggregates (effective medium theory) [-]

Returns

array_like

The estimated soil bulk real relative dielectric permittivity [-]

Notes

The Wunderlich model is a differential model and the approach used here employs a simple while loop with a step of 0.01 until the differential fraction reaches 1. The method takes into account the initial water content, initial relative dielectric permittivity, and weighting factors to determine the bulk real relative dielectric permittivity.

References

[1]

Tina Wunderlich, Hauke Petersen, Said Attia al Hagrey, Wolfgang Rabbel;

Pedophysical Models for Resistivity and Permittivity of Partially Water-Saturated Soils. Vadose Zone Journal 2013;; 12 (4): vzj2013.01.0023. doi: https://doi.org/10.2136/vzj2013.01.0023

Example

>>> WunderlichP(0.3, 7, 0.05, 80, 0.01)
24.591

pedophysics.pedophysical_models.water module

pedophysics.pedophysical_models.water.LR(bp, por, ap, sp, wp, alpha)

Calculate the soil volumetric water content using the Lichtenecker and Rother model.

This function computes the volumetric water content of a soil mixture using the volumetric mixing model of Lichtenecker and Rother [1] (LR). The model incorporates the bulk dielectric permittivity, bulk and particle densities, and permittivities of air, solid, and water.

Parameters

bparray_like

Soil bulk real relative dielectric permittivity [-]

por: array_like

Soil porosity [m**3/m**3].

aparray_like

Soil air real relative dielectric permittivity phase [-].

sparray_like

Soil solid real relative dielectric permittivity phase [-].

wparray_like

Soil water phase real dielectric permittivity [-].

alphafloat

Soil alpha exponent as defined in volumetric mixing theory [-].

Returns

array_like

Soil volumetric water content [m**3/m**3].

References

[1]

Lichtenecker, K., and K. Rother. 1931.

Die Herleitung des logarithmischen Mischungsgesetzes aus allgemeinen Prinzipien der staionären Strömung. Phys. Z. 32:255-260.

Example

>>> LR(10, 1.3, 2.65, 1, 4, 80, 0.5)
0.210

pedophysics.pedophysical_models.water.LR_MV(bp, por, ap, sp, wp, CEC)

Calculate the soil volumetric water content using the Lichtenecker and Rother model modified by Mendoza-Veirana and return

This function computes the volumetric water content of a soil mixture using the volumetric mixing model of Lichtenecker and Rother [1], and Mendoza Veirana [2] alpha correction model (LR_MV). The model incorporates the bulk dielectric permittivity, bulk and particle densities, and permittivities of air, solid, and water, as well as the soil cation exchange capacity. Reported R2 = 0.94

Parameters

bparray_like

Soil bulk real relative dielectric permittivity [-]

por: array_like

Soil porosity [m**3/m**3].

aparray_like

Soil air real relative dielectric permittivity phase [-].

sparray_like

Soil solid real relative dielectric permittivity phase [-].

wparray_like

Soil water phase real dielectric permittivity [-].

CECarray_like

Soil cation exchange capacity [meq/100g].

Returns

array_like

Soil volumetric water content [m**3/m**3].

References

[1]

Lichtenecker, K., and K. Rother. 1931.

Die Herleitung des logarithmischen Mischungsgesetzes aus allgemeinen Prinzipien der staionären Strömung. Phys. Z. 32:255-260. .. [2] Gaston Mendoza Veirana, Jeroen Verhegge, Wim Cornelis, Philippe De Smedt, Soil dielectric permittivity modelling for 50-MHz instrumentation, Geoderma, Volume 438, 2023, 116624, ISSN 0016-7061, https://doi.org/10.1016/j.geoderma.2023.116624.

Example

>>> LR_MV(10, 1.3, 2.65, 1, 4, 80, 20)
0.078

pedophysics.pedophysical_models.water.LR_W(bp, por, ap, sp, wp, clay)

Calculate the soil bulk real relative dielectric permittivity using the Lichtenecker and Rother model modified by Wunderlich and return

This function computes the volumetric water content of a soil mixture using the volumetric mixing model of Lichtenecker and Rother [1], and Wunderlich [2] alpha correction model (LR_W). The model incorporates the bulk dielectric permittivity, bulk and particle densities, and permittivities of air, solid, and water, as well as the soil clay content.

Parameters

bparray_like

Soil bulk real relative dielectric permittivity [-]

por: array_like

Soil porosity [m**3/m**3].

aparray_like

Soil air real relative dielectric permittivity phase [-].

sparray_like

Soil solid real relative dielectric permittivity phase [-].

wparray_like

Soil water phase real dielectric permittivity [-].

clayarray_like

Soil clay content [g/g]*100

Returns

array_like

Soil volumetric water content [m**3/m**3].

References

[1]

Lichtenecker, K., and K. Rother. 1931.

Die Herleitung des logarithmischen Mischungsgesetzes aus allgemeinen Prinzipien der staionären Strömung. Phys. Z. 32:255-260. .. [2] Tina Wunderlich, Hauke Petersen, Said Attia al Hagrey, Wolfgang Rabbel; Pedophysical Models for Resistivity and Permittivity of Partially Water-Saturated Soils. Vadose Zone Journal 2013;; 12 (4): vzj2013.01.0023. doi: https://doi.org/10.2136/vzj2013.01.0023

Example

>>> LR_W(10, 1.3, 2.65, 1, 4, 80, 20)
0.177

pedophysics.pedophysical_models.water_ec module

pedophysics.pedophysical_models.water_ec.SenGoode(T, C_f)

Calculate soil water real electrical conductivity using the Sen and Goode model and return

Based on the model proposed by Sen and Goode (1992 a, b) [1][2], this function estimates the oil water real electrical conductivity given the soil temperature and salinity of the bulk pore fluid.

Parameters

Tarray_like

Soil bulk temperature [K]

C_farray_like

Soil (NaCl) salinity of the bulk pore fluid [mol/L].

Returns

water_ecarray_like

Soil water real electrical conductivity [S/m].

References

[1]

Sen P and Goode P (1992a)

Influence of temperature on electrical conductivity on shaly sands. Geophysics 57: 89-96. .. [2] Sen P and Goode P (1992b) Errata, to: Influence of temperature on electrical conductivity of shaly sands. Geophysics 57: 1658.

Notes

The function uses specific coefficients based on the Sen and Goode model to compute the electrical conductivity of soil water, taking into account the temperature and salinity effects.

Example

>>> SenGoode(298.15, 0.01)
0.117822

pedophysics.pedophysical_models.water_perm module

pedophysics.pedophysical_models.water_perm.MalmbergMaryott(T)

Calculate soil water phase real dielectric permittivity using the Malmberg & Maryott model and return

This function utilizes the model proposed by Malmberg and Maryott (1956) [1] to estimate the soil water phase real dielectric permittivity based on a given soil temperature. Reported RMSE = 0.0046

Parameters

Tarray_like

Soil bulk temperature [K].

Returns

water_permarray_like

Soil water phase real dielectric permittivity [-]

References

[1]

Malmberg C and Maryott A (1956)

Dielectric constant of water from 0 °C to 100 °C. Journal of Research of the National Bureau of Standards 56(1): 1-8, Paper 2641.

Example

>>> MalmbergMaryott(298.15)
78.303

pedophysics.pedophysical_models.water_perm.Olhoeft(T, C_f)

Calculate soil water phase real dielectric permittivity using the Olhoeft (1986) model and return

Parameters

Tarray_like

Soil bulk temperature [K].

C_farray_like

Soil (NaCl) salinity of the bulk pore fluid [mol/L].

Returns

water_permarray_like

Soil water phase real dielectric permittivity [-]

References

[1]

Revil, A., Schwaeger, H., Cathles, L. M., and Manhardt, P. D. (1999),

Streaming potential in porous media: 2. Theory and application to geothermal systems, J. Geophys. Res., 104(B9), 20033-20048, doi:10.1029/1999JB900090.

Example

>>> Olhoeft(298.15, 0.1)
76.945

Module contents