# Estimation of absolute permeability using artificial neural networks (multilayer perceptrons) based on well logs and laboratory data from Silurian and Ordovician deposits in SE Poland

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## Abstract

Permeability is a property of rocks which refers to the ability of fluids to flow through each substance. It depends on several factors as pore shape and diameter. Also the presence and type of clay has a large influence on the permeability value. Permeability can be measured on rock sample in the laboratory by injecting fluid through the rock under known condition, but this provides only point information. Due to the dependence of the parameter on many factors, the deterministic estimation of permeability based on laboratory measurement and well logs is problematic. Many empirical methods for determining permeability are available in the literature and interpretation systems. An interesting approach to the problem is the use of artificial neural networks based on laboratory measurement and modern, high-resolution logging tools. The authors decided to use MLP artificial neural networks, which allow permeability estimation and can be used both in the test well and applied to neighbouring wells. The network was checked in several variants. Obtained results show the legitimacy of using artificial neural networks in the issue of estimating permeability. However, they also show limitations resulting from the lack of accurate data or influence of geological setting and processes.

## Keywords

Permeability Well logs Artificial neural networks## Abbreviations

- ANN
Artificial neural network

- MLP
Multilayer perceptron

- GR
Natural radioactivity, gamma ray log (API)

- RHOB
Bulk density, density log (g/cm

^{3})- NPHI
Neutron porosity in limestone unit, neutron porosity log (frac)

- DT
Sonic transit time, sonic log (μs/ft)

- LLD
Electrical resistivity, deep resistivity laterolog (\({\Omega }\;{\text{m}}\))

- LLS
Electrical resistivity, shallow resistivity laterolog (\({\Omega }\;{\text{m}}\))

- MSFL
Electrical resistivity, micro-spherically focused resistivity log (\({\Omega }\;{\text{m}}\))

- SP
Spontaneous potential log (mV)

- THOR
Thorium content, spectral gamma ray log (ppm)

- URAN
Uranium content, spectral gamma ray log (ppm)

- POTA
Potassium content, spectral gamma ray log (%)

- PE
Photoelectric factor, litho-density log (barn/electron)

- XRMI
Electrical resistivity, X-tended Range Micro Imager (\({\Omega }\;{\text{m}}\))

- VCL
Volume of clay, result of petrophysical interpretation of well logs (frac)

- VANH
Volume of anhydrite, result of petrophysical interpretation of well logs (frac)

- VDOL
Volume of dolomite, result of petrophysical interpretation of well logs (frac)

- VLIM
Volume of calcite, result of petrophysical interpretation of well logs (frac)

- VSAN
Volume of quartz, result of petrophysical interpretation of well logs (frac)

- VPYR
Volume of pyrite, result of petrophysical interpretation of well logs (frac)

- VTOC
Volume of total organic carbon, result of petrophysical interpretation of well logs (frac)

- PHI
Effective porosity, result of petrophysical interpretation of well logs (frac)

*K*Absolute permeability, result of petrophysical interpretation of well logs (mD)

- SWI
Irreducible water saturation, result of petrophysical interpretation of well logs (frac)

- PHSW
Volume of rock occupied by water, result of petrophysical interpretation of well logs (frac)

- K_LAB
Absolute permeability from laboratory measurements (mD)

- K_ANN
Absolute permeability from artificial neural network MLP 10-7-1 (mD)

- K_ANN1
Absolute permeability from artificial neural network MLP 12-9-1 (mD)

- K_ANN2
Absolute permeability from artificial neural network MLP 16-10-1 (mD)

- K_ANN3
Absolute permeability from artificial neural network MLP 9-6-1 (mD)

- K_ZAWISZA
Absolute permeability from Zawisza equation (mD)

- K_WYLLIE–ROSE
Absolute permeability from Wyllie–Rose equation (mD)

- Φ
Porosity from laboratory measurements (frac)

- FZI
Flow zone indicator (μm)

## Introduction

Permeability is one of the most difficult properties of rocks to estimate. It refers to the ability of fluids to flow through the substance. Permeability of a rock for oil, gas or water is a function of the absolute permeability and the fluid viscosity.

Permeability is influenced, the same as porosity, by many depositional and diagenetic factors. The most important factors that depend on the permeability of the rock is shape, pore diameter and pore connection (Schön 2011). Moreover, among the depositional factors affecting permeability, size and sorting of the grains should be mentioned (Beard and Weyl 1973; Bloch 1991; Lucia 1995). Coarse and well-sorted grains ensure better flows in the reservoir. Diagenetic factors such as compaction and cementation cause a reduction in permeability. Also, the type of minerals, which build the rock, affects this parameter. The presence of quartz increases the absolute permeability, despite the smaller porosity, while clay minerals due to their properties and distribution in rock significantly affect its reduction (Neuzil 1994).

The permeability is related to the productivity of the rock formation. Occurrence of oil or gas saturation in low-permeability reservoirs requires additional activities such as hydraulic fracturing or drilling of horizontal wells to obtain the hydrocarbon flow. Determination of the absolute permeability is crucial in order to properly identify the reservoir parameters and determine the profitability of possible hydrocarbon production.

Permeability can be estimated in many different methods. The most reliable method is laboratory measurements, which mainly are based on injecting fluid to the core sample under known condition (Tiab and Donaldson 2000; Jarzyna and Puskarczyk 2009) and are carried out on liquid permeameters. Laboratory measurements provide only point information and are mainly used to calibrate deterministic calculations based on well logging (Iturrarán-Viveros and Parra 2014; Wawrzyniak-Guz 2016).

where \(K\)—absolute permeability, \(\phi_{{\text{e}}}\)—effective porosity, \(F_{{\text{s}}}\)—shape factor, \(\tau\)—tortuosity, \(S_{{{\text{gv}}}}\)—surface area per unit grain volume.

where \(a\)—regional factor, \(\phi\)—porosity, \(S_{{{\text{wir}}}}\)—irreducible water saturation.

An interesting approach to the problem of permeability determination is the use of artificial neural networks based on laboratory measurement on core samples, modern, high-resolution logging tools or results of their qualitative interpretation.

## Method

The precursor to the emergence of artificial neural networks was the development of the model of the neuron in the human brain and explained the mechanism of memorizing information via the biological network in the 1940s by McCulloch and Pitts (1943) (Tadeusiewicz 1992). The first designed and constructed neural network was the perceptron developed by Rosenblatt (1958). Initially, neural networks have not gained much interest because the use of single-layer networks is limited. Further work demonstrating that multilayer, nonlinear networks have unlimited possibilities caused a significant increase in their use. Neural networks are widely used also in geophysical and petrophysical problems (Huang et al. 1996; Aminzade and De Groot 2006; Bhatt and Helle 2002; Sudakov et al. 2019).

Artificial neural networks (ANNs) with the ability to reproduce complicated functions are used to reduce interference, classification or prediction of data and parameters. Neural networks have the ability to learn, memorize and generalize calculations based on the training data set. They cope very well with inconsistent, distorted data. ANN are stable and resistant to damage. Calculations using the network in relation to the structure are very efficient, even operating on large data sets.

During the design of neural networks, the most important stages are the selection process and the learning stage, because they project on the network and obtained results. Romeo (1994) distinguishes three main causes resulting in bad network performance: bad network configuration, algorithm suspension in the minimum and wrong learning set. The key moment is choosing the number of neurons in the hidden layer. There are no clearly defined rules for their number. However, the number of layers should not be too large because it can cause the network to be overly adapted to the test data. The best method to select the right number of layers is trial and error, starting with a small number of layers. In the case of selecting a training sample, it should be ensured that they are as representative as possible and possibly free from measurement errors. Avoiding the above-mentioned errors, it is possible to design networks that are capable of solving complex problems and predicting parameters whose relationship with measurements is not easy to describe using simple mathematical functions. One of such parameters used in petrophysics is the absolute permeability.

where \(N\)—number of cases used for learning, \(y_{i}\)—network prediction result, \(t_{i}\)—measured parameter value.

Presented analysis was performed in Statistica software (version 13) using Multilayer Perceptron algorithm with exponential activation function for the hidden and output neurons (StatSoft 2011).

## Materials

Research was made on data from three wells located on Lublin Syncline in Poland. The analysis covered the Silurian and Ordovician shale and mudstone formations, which are potential unconventional shale gas deposits. These formations are characterized by poor reservoir properties, in particular low permeability (Krakowska and Puskarczyk 2015). Due to the location of the wells within various geological regions, analogous formations are buried at various depths. The distance and complex tectonics of the area can affect the differences in petrophysical parameters between the layers.

List of available data. Symbols are explained in the section: list of symbols

Well | Well logs | Interpretation data | Laboratory data |
---|---|---|---|

A | GR, RHOB, NPHI, DT, LLD, LLS, MSFL, SP, THOR, URAN, POTA, PE, XRMI | VCL, VANH, VDOL, VLIM, VSAN, VPYR, VTOC, PHI, K, SWI, PHSW | K_LAB, Φ |

B | GR, RHOB, NPHI, DT, LLD, LLS, MSFL, SP, THOR, URAN, POTA, PE, XRMI | VCL, VDOL, VLIM, VSAN, PHI, K, SWI, PHSW | Φ |

C | GR, RHOB, NPHI, DT, LLD, LLS, MSFL, THOR, URAN, POTA, PE, XRMI | VCL, VDOL, VLIM, VSAN, PHI, K, SWI, PHSW | – |

## Results

The first, but very important step in the process of using neural networks is the appropriate selection of the data used. According to the assumption, the output data being tested, and training and validation data should be associated with the input data. In the case of creating neural networks, it is not entirely true to say that the more the better, because variables unrelated to the output data may cause deterioration of the network (StatSoft 2011).

Basic statistics of laboratory data

Well | Parameter | Number of samples | Mean | Median | Minimum | Maximum | Standard deviation |
---|---|---|---|---|---|---|---|

A | Total porosity (frac) | 117 | 0.051 | 0.049 | 0.0110 | 0.105 | 0.019 |

Permeability (mD) | 115 | 0.268 | 0.064 | 0.0001 | 4.596 | 0.536 | |

B | Total porosity (frac) | 301 | 0.045 | 0.036 | 0.0001 | 0.014 | 0.030 |

C | Total porosity (frac) | – | – | – | – | – | – |

Basic statistics of laboratory data indicate that the studied formations exhibit very poor reservoir parameters, in particular absolute permeability, which is typical for unconventional hydrocarbon resources, such as tight gas or shale gas (Xiao et al. 2014). For the training well A, the absolute permeability is very low, mostly not exceeding 0.01 mD, as evidenced by the median, i.e. the value that prevents the flow of hydrocarbons. The network learning process was carried out on the basis of trial and error, both checking the various functions of activation of input and output neurons and observing the behaviour of the error function for individual network dimensions. Network design was carried out using automated algorithms in which a random selection of data for learning, testing and validation sets was applied.

At the beginning, for the well A, the networks were designed based on well logs, for the whole available depth interval. In the second step, division into stratigraphic units (periods: Silurian and Ordovician) was implemented as qualitative input. In the third step, based on the same data, the division into stratigraphic units was provided with details, taking into consideration Polish informal stratigraphic units: Ludlow, Wenlock, Llandovery, Ashgill, Caradoc. The absolute permeabilities, which are equal to the minimum detected value (*K* = 0.0001 mD) in gas permeameter, were not considered, as not informative for the ANN. The total number of permeability values from laboratory measurements was equal to 109. Thanks to this procedure, three absolute permeabilities were obtained: K_ANN—no stratigraphic units were applied, K_ANN1—stratigraphic units were applied, and K_ANN2—informal stratigraphic units were applied.

Quality of artificial neural network

Artificial neural network name | Multilayer perceptron input-hidden-output | Quality of probe (determination coefficient) | ||
---|---|---|---|---|

Learning | Test | Validation | ||

ANN | MLP 10–7-1 | 0.86 | 0.79 | 0.47 |

ANN1 | MLP 12–9-1 | 0.81 | 0.74 | 0.43 |

ANN2 | MLP 16–10-1 | 0.82 | 0.85 | 0.54 |

Basic statistics of measured, estimated and calculated absolute permeabilities

Absolute permeability | Mean | Median | Standard deviation | Minimum | Maximum |
---|---|---|---|---|---|

K_ANN (mD) | 0.230 | 0.140 | 0.350 | 0.002 | 1.637 |

K_ANN1 (mD) | 0.275 | 0.199 | 0.297 | 0.002 | 1.637 |

K_ANN2 (mD) | 0.282 | 0.167 | 0.313 | 0.001 | 8.176 |

K_ZAWISZA | 0.709 | 0.256 | 1.563 | 0 | 40.504 |

K_WYLLIE–ROSE | 0.156 | 0.061 | 0.040 | 0 | 9.139 |

K_LAB | 0.268 | 0.064 | 0.536 | 0.001 | 4.596 |

Permeabilities from Zawisza (available from the interpretation data set) and Wyllie–Rose (calculated by authors) formula were estimated in order to check the quality of the ANN results. Results obtained for Wyllie–Rose equation were better but similar to the Zawisza equation. The determination coefficient for K_LAB and K_Wyllie–Rose is 0.26. During the analysis, an attempt was made to learn the network applied to the division created using XRMI tool measurements, based on resistivity contrast between layers. Nevertheless, due to the small number of samples, the obtained results were unsatisfactory.

Analysing the calculated statistics, the parameter variability with the depth and the qualitative fit between the estimated permeability (K_ANN, K_ANN1, K_ANN2) and the input permeability (K_LAB, K_ZAWISZA), it was decided that the best match appears in the networks created for the most detailed stratigraphy (K_ANN2), even if the calculated matching factors are not the best.

Basic statistic of absolute permeability (K_ANN3) estimated based on interpretation data and FZI parameter

Permeability K_ANN3 (mD) | ||||
---|---|---|---|---|

Mean | Median | Standard deviation | Minimum | Maximum |

0.176 | 0.078 | 0.270 | 0.001 | 2.614 |

The correlation coefficient between the absolute permeability from the artificial neural networks and the Zawisza equation will be lower than 0.5 for both wells. Nevertheless, due to the lack of data from laboratory measurements on core samples, it is not possible to accurately assess this parameter. In particular, in the case of absolute permeability from the Zawisza formula calculated for the well C, it seems to be overestimated in relation to the actual parameters for the formation in the studied area. Lack of success in the application of the best ANN for wells B and C is caused by the strong heterogeneity of reservoir parameters for Silurian and Ordovician deposits and active tectonic in the research area. Even small changes in the diagenesis process for the thin-layered mudstone and shale deposits reveal enormous change in the petrophysical parameters, such as effective porosity and absolute permeability.

## Conclusions

The presented work aimed at checking the legitimacy of using artificial neural networks to determine the absolute permeability parameter. Tests were carried out on data from Silurian and Ordovician shale and mudstone formations. These formations are characterized by poor reservoir parameters, such as effective porosity and absolute permeability, which additionally hampered the assumed task.

Attempts were made in providing the different sets of variables as the input data. The best results were obtained for well logs and the data from the well log interpretation, given as follows: RHOB, NPHI, DT, LLD, LLS, MSFL, THOR, URAN, POTA, PE, VCL, VANH, VDOL, VLIM, VSAN, VPYR, VTOC, PHI, K, SWI.

The best ANN was ANN2 with MLP 16-10-1 characteristic, using Polish informal stratigraphic units, as an input division. In both cases, learned networks for smaller depths intervals worked better. Determination coefficient for the relationship between the absolute permeabilities derived from ANN2 and from absolute permeability from gas permeameter is *R*^{2} = 0.72.

Unfortunately, due to the lack of data from laboratory measurements on core samples in the remaining B and C wells, it was impossible to make an unambiguous and objective assessment of their quality. A possible poor fit may result from the complicated tectonic structure of the research area. An attempt in learning the network using measurements from the XRMI tool, which seems to be an interesting issue, was made and failed also due to the small number of samples.

Artificial neural networks can be used in petrophysical analysis because they give the possibility to get the information in the form of the log, from larger amount of data and calibrated with the laboratory measurements. Calculations of ANN are fast and quite cheap in comparison with the measurements and give good results for the parameters with nonlinear characteristic such as permeability in reference to well logs.

## Notes

### Acknowledgements

Data are ownership of POGC, Warsaw, Poland. Paper was financially supported from the research subsidy nr 16.16.140.315 at the Faculty of Geology Geophysics and Environmental Protection of the AGH University of Science and Technology, Krakow, Poland, 2019.

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