Pseudo Wells

This section contains documentation for the Pseudo Wells module.

Monte Carlo Simulations

stoneforge.pseudo_wells.monte_carlo_simulations.MCS_spacial_correlation(n, smooth_data, cov)

Determines n Monte Carlo Simulations (MCS) with spatial correlation for a given (Dvorkin et al.[1]).

Parameters:

• n (integer) – Number of simulations to be performed.

• smooth_data (array_like) – a smoothed version of the data under examination, or its general trend.

• cov (array_like) – Spatial symmetrical covariance matrix of the data.

Returns:

simulations – n Monte Carlo simulations with spatial correlation for a given property, each line of this matrix represents a different simulation.

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.analytical_variogram(distance, gama, initial_guess)

Fits the choosen analytical variogram function (model) to the experimental one Vugrin et al.[2], if no model is choosen, determines the best model to fit, comparing the Gaussian, Exponential and Spherical models Grana et al.[3], de Almeida[4].

Parameters:

• distance (array_like) – 1D array containing all the possible distances between a pair of points in the dataset.

• gama (array_like) – 1D experimental variogram of the data under examination.

• model (str, optional) –

  Analytical variogram model to be fitted. Should be one of:

  • ”exponential”: fits the exponential model

  • ”gaussian”: fits the gaussian model

  • ”spherical”: fits the spherical model

  • ”best-fit”: fits the three models above and verifies which one produces the smallest error.

• given (If not)

• "best-fit". (default method is)

• initial_guess (array_like) – Initial guess for the parameters of the variogram model. It should be a list containing the initial values for the correlation length, sill and nugget.

Return type:

array

Returns:

• modeled_variogram (array_like) – The variogram model that has been choosen, or the variogram model that fits the best the experimental one.

• coeficients (array_like) – The range, sill and nugget optimal values for the modeled variogram.

stoneforge.pseudo_wells.monte_carlo_simulations.cov_matrix(rho, var)

Determines the 1D spatial symmetrical covariance matrix from a modeled correlation function (Dvorkin et al.[1]).

Parameters:

• rho (array_like) – 1D modeled correlation function of the data under examination.

• var (float) – Variance (the square of the standard deviation) of the dataset.

Returns:

cov – Spatial symmetrical covariance matrix of the data.

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.experimental_correlation(data)

Determines the 1D experimental correlation function for a dataset by calculating the Pearson correlation coefficient for each possible separation of samples (Dvorkin et al.[1]).

Parameters:

data (array_like) – 1D dataset for which the experimental correlation function must be calculated.

Returns:

rho – 1D experimental correlation function of the data under examination.

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.experimental_variogram(data, rho)

Determines the 1D experimental variogram of a dataset (Dvorkin et al.[1]).

Parameters:

• data (array_like) – 1D dataset for which the experimental variogram must be calculated.

• rho (array_like) – 1D experimental correlation function of the data under examination.

Returns:

gama – 1D experimental variogram of the data under examination.

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.exponential_variogram_model(distance, correlation_length, sill, nugget=0)

Builds a variogram following the exponential model, using the correlation length, sill and nugget given (Journel and Huijbregts[5], Cressie[6], Chilés and Delfiner[7], Grana et al.[3]).

Parameters:

• distance (array_like) – 1D array containing all the possible distances between a pair of points in the dataset.

• correlation_length (float) – The range of the variogram, or the distance where it loses the correlation

• sill (float) – The maximum value of the variogram, it is equivalent to the variance of the data

• nugget (float) – The nugget effect, y value where the variogam begins

Returns:

rho – The variogram that follows the exponential model and has the given correlation length, sill and nugget

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.gamma_calc(data, depth, step=100)

Smooth the values of 1D data (Isaaks and Srivastava[8], Isaaks[9], Dvorkin et al.[1]).

Parameters:

• data (array_like) – 1D data samples From where the calculation will be made.

• depth (array_like) – 1D array containing the depth of each sample in the dataset.

• step (int, optional) – Step between samples, by default 100. This is the number of samples that will be skipped between each calculation of the correlation function.

Returns:

gamma – 1D array containing the calculated gamma values for each step.

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.gaussian_variogram_model(distance, correlation_length, sill, nugget=0)

Builds a variogram following the gaussian model, using the correlation length, sill and nugget given (Journel and Huijbregts[5], Chilés and Delfiner[7], Grana et al.[3]).

Parameters:

• distance (array_like) – 1D array containing all the possible distances between a pair of points in the dataset.

• correlation_length (float) – The range of the variogram, or the distance where it loses the correlation

• sill (float) – The maximum value of the variogram, it is equivalent to the variance of the data

• nugget (float) – The nugget effect, y value where the variogam begins

Returns:

rho – The variogram that follows the gaussian model and has the given correlation length, sill and nugget

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.modeled_correlation(gama, var)

Determines the 1D modeled correlation function from a variogram model (Dvorkin et al.[1]).

Parameters:

• gama (array_like) – 1D analytical variogram model.

• var (float) – variance (the square of the standard deviation) of the dataset.

Returns:

rho – 1D modeled correlation function of the data under examination.

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.p(n, data1, data2, smooth_data1, smooth_data2, cov)

Determines n Monte Carlo Simulations (MCS) using data1 and data2 as correlated variables (Dvorkin et al.[1]).

Parameters:

• n (integer) – Number of simulations to be performed.

• data1 (array_like) – A dataset that represents a given porperty, related to data1.

• data2 (array_like) – A dataset that represents a given porperty, related to data2.

• smooth_data1 (array_like) – A smoothed version of the data1, or its general trend.

• smooth_data2 (array_like) – A smoothed version of the data2, or its general trend.

• cov (array_like) – Spatial symmetrical covariance matrix representing both data1 and data2.

Returns:

simulations – n Monte Carlo Simulations with correlated variables for data1 and n Monte Carlo Simulations with correlated variables for data2, in this order.

Return type:

array_like

stoneforge.pseudo_wells.monte_carlo_simulations.spherical_variogram_model(distance, correlation_length, sill, nugget=0)

Builds a variogram following the spherical model, using the correlation length, sill and nugget given (BURGESS and WEBSTER[10], Grana et al.[3]).

Parameters:

• distance (array_like) – 1D array containing all the possible distances between a pair of points in the dataset.

• correlation_length (float) – The range of the variogram, or the distance where it loses the correlation

• sill (float) – The maximum value of the variogram, it is equivalent to the variance of the data

• nugget (float) – The nugget effect, y value where the variogam begins

Returns:

rho – The variogram that follows the spherical model and has the given correlation length, sill and nugget

Return type:

array_like

class stoneforge.pseudo_wells.monte_carlo_simulations.variogram_model(dif, depth, step=100)

Bases: object

_summary_ A class to calculate and visualize the variogram of a dataset, using the exponential model. It can also normalize the data and calculate the variogram for normalized data. It provides methods to graph the variogram, calculate the variogram for a given depth, and normalize and denormalize the data (Isaaks and Srivastava[8], Isaaks[9], Dvorkin et al.[1]).

denormalization(data)

Denormalize the data using the min-max normalization method.

graph(correlation_length, sill=False, nugget=0.0)

Graph the variogram of the dataset using the exponential model.

Parameters:

• dif (array_like) – 1D data samples From where the calculation will be made.

• depth (array_like) – 1D array containing the depth of each sample in the dataset.

• step (int, optional) – Step between samples, by default 100. This is the number of samples that will be skipped between each calculation of the correlation function.

norm_graph(correlation_length, sill=False, nugget=0.0)

Graph the normalized variogram of the dataset using the exponential model.

norm_variography(depth=False)

Calculate the normalized variogram of the dataset using the exponential model.

normalization(data)

Normalize the data using the min-max normalization method.

variography(depth=False)

Calculate the variogram of the dataset using the exponential model.

References

[1] (1,2,3,4,5,6,7,8)

Jack Dvorkin, Mario A. Gutierrez, and Dario Grana. Seismic Reflections of Rock Properties. Cambridge University Press, 03 2014. ISBN 9780511843655. URL: https://www.cambridge.org/core/books/seismic-reflections-of-rock-properties/79680145E8B02CF09F1514A15121AFF9, doi:10.1017/cbo9780511843655.

[2]

K. W. Vugrin, L. P. Swiler, R. M. Roberts, N. J. Stucky‐Mack, and S. P. Sullivan. Confidence region estimation techniques for nonlinear regression in groundwater flow: three case studies. Water Resources Research, 03 2007. URL: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html, doi:10.1029/2005wr004804.

[3] (1,2,3,4)

Dario Grana, Tapan Mukerji, and Philippe Doyen. Seismic Reservoir Modeling: Theory, Examples, and Algorithms. Wiley, 04 2021. ISBN 9781119086215. URL: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119086215, doi:10.1002/9781119086215.

[4]

Ana Carolina Oliveira de Almeida. Geração de pseudo-perfis de propriedades petrofísicas e parâmetros elásticos de poços. Monography - Universidade Federal Fluminense (UFF) Niterói, Brazil, 2023. URL: http://www.geofisica.uff.br/sites/default/files/projetofinal/2023_ana_carolina_oliveira_de_almeida.pdf.

[5] (1,2)

A.G. Journel and C.J. Huijbregts. Mining Geostatistics. Academic Press, 1978. ISBN 9780123910509. URL: https://mmaelicke.github.io/scikit-gstat/reference/models.html.

[6]

Noel A. C. Cressie. Statistics for Spatial Data. Wiley, 09 1993. ISBN 9781119115151. URL: https://mmaelicke.github.io/scikit-gstat/reference/models.html, doi:10.1002/9781119115151.

[7] (1,2)

Jean‐Paul Chilés and Pierre Delfiner. Geostatistics: Modeling Spatial Uncertainty. Wiley, 03 1999. ISBN 9780470316993. URL: https://mmaelicke.github.io/scikit-gstat/reference/models.html, doi:10.1002/9780470316993.

[8] (1,2)

E. H. Isaaks and R. M. Srivastava. Applied Geostatistics. Oxford University Press, 1989. ISBN 9780195050134. URL: https://www.amazon.com/Introduction-Applied-Geostatistics-Edward-Isaaks/dp/0195050134.

[9] (1,2)

Edward Isaaks. What the heck is a variogram? 2013. URL: https://www.youtube.com/watch?v=SJLDlasDLEU (visited on 2025-05-30).

[10]

T. M. BURGESS and R. WEBSTER. Optimal interpolation and isarithmic mapping of soil properties: i the semi‐variogram and punctual kriging. Journal of Soil Science, 31(2):315–331, 6 1980. URL: https://mmaelicke.github.io/scikit-gstat/reference/models.html, doi:10.1111/j.1365-2389.1980.tb02084.x.