# An iterative inversion of Dual Induction Tool logs from thin-bedded sandy–shaly formations of the Carpathian Foredeep using a modified simulated annealing method

## Abstract

In thin-bedded sandy–shaly Miocene formations of the Carpathian Foredeep, the main source of errors in gas saturation evaluation is the underestimation of resistivity of thin, hydrocarbon-bearing beds, which is the result of the low vertical resolution of induction logging tools. This problem is especially visible in older boreholes drilled in times where the Dual Induction Tool (DIT) was the primary induction tool used for determining the formation resistivity, and in shallowest depth intervals of newer boreholes where the DIT was used instead of newer array tools for cost-saving reasons. In this paper, we show how a global inversion algorithm was used to improve the vertical resolution of DIT logs. Our implementation of an iterative inversion utilizes a one-dimensional formation model, vertical response functions of the DIT, and a modified simulated annealing algorithm to determine the true vertical distribution of the formation resistivity. The algorithm was tested on resistivity logs recorded in a borehole drilled in the Carpathian Foredeep in Poland, where the DIT and the High-Resolution Array Induction (HRAI) tool were run in the same depth interval.

## Keywords

Well logs Dual Induction Tool Thin-bed problem Iterative inversion Simulated annealing## Introduction

In thin-bedded reservoirs, a vertical range of logging tools investigation volume is usually larger than the thickness of individual thin beds. Therefore, standard logging tools generally do not allow the direct measurement of physical properties of individual thin beds and in some cases even cannot detect individual beds themselves (Zorski 1987). The lower limit of bed thicknesses below which the thin-bed problem begins to be significant is defined by the vertical resolution of the deep resistivity tool used during the evaluation process since the deep resistivity is a key log in evaluating reservoir hydrocarbons (Passey et al. 2006).

In Poland, the thin-bed problem exists in sandy–shaly Miocene formations of the Carpathian Foredeep, one of the most important petroleum provinces in Poland. In these formations, the main source of errors in gas saturation evaluation is the underestimation of resistivity of thin, hydrocarbon-bearing beds, which is the result of the low vertical resolution of induction tools (Zorski 2009). Two induction tools commonly used in the Carpathian Foredeep are the Dual Induction Tool (DIT) and the High-Resolution Array Induction (HRAI) tool. The DIT was introduced in 1962 and provides two resistivity logs, medium and deep, at two radial depths of investigation (30 and 60 in.) with the vertical resolution around 5–8 ft (Anderson 2001). The HRAI tool was introduced in 2000 and provides resistivity logs at six radial depths of investigation (10, 20, 30, 60, 90, and 120 in.) with 1, 2, and 4 ft vertical resolution (Beste et al. 2000). The vertical resolution of logs provided by the DIT is significantly lower than the vertical resolution of logs provided by the HRAI tool. Therefore, the thin-bed problem is especially visible in older boreholes drilled in times where the DIT was the primary induction tool used for determining the formation resistivity, and in shallowest depth intervals of newer boreholes where the DIT was used instead of the HRAI tool for cost-saving reasons.

In this paper, we show how a global inversion algorithm was used to improve the vertical resolution of DIT logs. Our implementation of the inversion algorithm utilizes a one-dimensional formation model, vertical response functions of the DIT, and a modified simulated annealing algorithm to determine the true vertical distribution of the formation resistivity. To better deal with the nature of the problem, a probability of selecting model parameters to modification was changed from typical for simulated annealing random choice to weighted random choice. This modification allows the algorithm to focus on problematic depth intervals and results in faster optimization of the model of the true vertical distribution of the formation resistivity.

The algorithm was tested on resistivity logs recorded in a borehole drilled in the Carpathian Foredeep in Poland, where the DIT and the HRAI tool were run in the same depth interval.

## Iterative inversion

The iterative inversion is the main inversion method used in well logging applications (Passey et al. 2006). In this approach, no attempt is made to reverse physical processes occurring during well logging. Instead, the method utilizes an iterative forward modeling procedure to find a formation model that best explains measured data.

The exact structure of the iterative inversion algorithm depends primarily on a formulation of a forward problem and a method used to find the formation model that best explains measured data. Model parameters may be modified manually by the analyst until a satisfactory qualitative or quantitative fit between synthetic and measured data is obtained, but usually local or global optimization methods are used to find the formation model that minimalizes the value of an objective function (a quantitative measure of the difference between synthetic and measured data) (Passey et al. 2006; Sen and Stoffa 2013).

## Forward problem

The shape of the vertical response function depends on the physics and geometry of the measurement. In case of many logging tools (including the induction logging tools), the shape of the vertical response function varies depending on the properties of the geological formation. This nonlinearity is often omitted, and a constant vertical response function designed for an “average” set of geologic conditions is used to approximate the vertical response function across a wider range of conditions (Passey et al. 2006).

## Simulated annealing

The simulated annealing algorithm (Kirkpatrick et al. 1983) is based on the analogy between the simulation of annealing of solids and the problem of solving large combinatorial optimization problems. The algorithm can be viewed as a sequence of Metropolis algorithms (Metropolis et al. 1953) adapted to generate sequences of configurations of a combinatorial optimization problem and evaluated at a sequence of decreasing values of the temperature (control parameter without physical meaning) (van Laarhoven and Aarts 1987).

The process is repeated until the algorithm reaches the final temperature value or the acceptable objective function value (Kirkpatrick et al. 1983; van Laarhoven and Aarts 1987).

The cooling schedule is designed in such a way that the acceptance probability of modification, which increases the value of the objective function, decreases from a value close to 1 near the initial temperature value to 0 when the temperature approaches the final value. This allows the algorithm to make use of benefits of maximally explorative and minimally exploitive random walk algorithm and minimally explorative and maximally exploitive hill climbing algorithm (Kirkpatrick et al. 1983; van Laarhoven and Aarts 1987; Weise 2011).

The simulated annealing optimization method was previously used in well logging applications by Runge and Runge (1991), Szucs and Civan (1996), and Dobróka and Szabó (2001, 2011, and 2015).

## Structure of the algorithm

### Input data

measured well log data \((\varvec{d})\),

an appropriately discretized logging tool vertical response function \((\varvec{v})\),

an initial temperature value \((T_{0} )\),

a temperature change coefficient \((\Delta T)\),

a final temperature value \((T_{n} )\),

an acceptable value of the global objective function \(( {E^{{( \min)}} })\),

a parameter which controls the number of iterations per temperate value \(\left( {N_{T} } \right)\), and

parameters which control the size of modifications \((a,b,c).\)

### Initialization of the optimization procedure

the current formation model dataset: \(\left\{ {\begin{array}{*{20}c} {\varvec{m}^{{\left( \varvec{c} \right)}} = \varvec{m}^{\left( 0 \right)} } \\ {\varvec{d}^{{\left( \varvec{c} \right)}} = \varvec{d}^{\left( 0 \right)} } \\ {\varvec{E}^{{\left( \varvec{c} \right)}} = \varvec{E}^{\left( 0 \right)} } \\ \end{array} } \right.,\) and

the best formation model dataset: \(\left\{ {\begin{array}{*{20}c} {\varvec{m}^{{\left( \varvec{b} \right)}} = \varvec{m}^{\left( 0 \right)} } \\ {\varvec{d}^{{\left( \varvec{b} \right)}} = \varvec{d}^{\left( 0 \right)} } \\ {E^{\left( b \right)} = E^{\left( 0 \right)} } \\ \end{array} } \right..\)

### Optimization procedure

For each temperature value \(T_{i}\), a sequence of iterations is performed. In every iteration step, the algorithm tries to modify the value of a single model parameter.

### Optimization sequence

#### Initialization of the optimization sequence

This mechanism is absent in the standard simulated annealing algorithm and was added to adjust the optimization procedure to the specific character of the forward problem. Different parts of a geological formation may present a different degree of complexity. As a consequence, the different parts of the formation model may reach acceptable values of the objective function after a different number of iterations. The mechanism allows the algorithm to focus on these specific depth intervals where differences between measured data and synthetic data are the largest. This approach, in comparison with the random choice selection, which is typical for the simulated annealing, allows the algorithm to reduce the value of the objective function more quickly.

#### Single iteration of the optimization sequence

The value of the part of Eq. (17) inside the bracket changes from 1 in the initial temperature to 0 in the final temperature. Therefore, the value of the standard deviation of the normal distribution, from which the perturbation value \(\Delta m\) is randomly chosen, is lowered with the advance of the procedure. Parameters \(a\),\(b\) and \(c\) in Eqs. (17) and (18) allow additional control of the size of modifications.

If the modification is accepted, the current model dataset is actualized. The parameter \(m_{j}^{\left( c \right)}\) is substituted by the parameter \(m_{j}^{{\left( {n_{j} } \right)}}\), and parts of vectors \(\varvec{d}^{{\left( \varvec{c} \right)}}\) and \(\varvec{E}^{{\left( \varvec{c} \right)}}\) affected by the modification are substituted by vectors \(\varvec{d}^{{\left( {n_{j} } \right)}}\) and \(\varvec{E}^{{\left( {\varvec{n}_{\varvec{j}} } \right)}} .\)

#### Finalization of the optimization sequence

### Finalization of the optimization procedure

The optimization sequence is iteratively repeated until the algorithm reaches the acceptable value of the global objective function \(( {E^{(b)} \le E^{{(\min)}} })\) or the final temperature value \(\left( {T_{i} \le T_{n} } \right).\)

## Application of inversion algorithm to DIT logs

The algorithm was tested on resistivity logs recorded in a borehole drilled in the Carpathian Foredeep in Poland, where the DIT and the HRAI tool were run in the same depth interval.

The borehole penetrates a multi-horizon gas field located within thin-bedded sandy–shaly Miocene deposits. The depth interval selected for the test is located within one of gas horizons encountered in the well. Top 8 m of the selected depth interval was cored. Rock samples indicate that the sedimentary formation in the cored interval consists of mudstones, siltstones, shales, and sandstones. The thicknesses of individual layers within cored interval range from millimeters within heterolithic complexes to around 70 cm in case of relatively thick sandstone layers located at the bottom of the cored interval.

Values of inversion parameters

Parameter | Value |
---|---|

\(T_{0}\) | 0.0001 |

\(\Delta_{T}\) | 0.9 |

\(T_{n}\) | 0.000000001 |

\(E^{{\left( {\hbox{min} } \right)}}\) | 0 |

\(N_{T}\) | 200 |

\(a\) | 0.00025 |

\(b\) | 0.5 |

\(c\) | 0.0001 |

Results of inversion are also very repeatable. The maximal difference in the value of the specific model parameters within 50 independent runs of the algorithm ranges from 0.0231 to 0.0942 Ωm with the mean value at 0.0483 Ωm in case of models obtained from the ILM log, and from 0.0389 to 0.1797 Ωm with the mean value at 0.0936 Ωm in case of models obtained from the ILD log. Taking into account the mean values of model parameters, the maximal difference in value of specific model parameters within 50 independent runs of the algorithm accounts from 1.08 to 2.81% of the mean values of these parameters with the mean value at 1.89% in case of models obtained from the ILM log, and from 1.49 to 5.91% of the mean values of these parameters with the mean value at 3.53% in case of models obtained from the ILD log.

## Summary

The paper presents the use of the global inversion algorithm to improve the vertical resolution of DIT logs. The algorithm was tested on resistivity logs recorded in the borehole drilled in the Carpathian Foredeep in Poland, where the DIT and the HRAI tool were run in the same depth interval. Resistivity models computed on the basis of DIT logs very closely follow HRAI logs with a similar depth of investigation and 1 ft vertical resolution. The modification introduced to the mechanism of selecting model parameters to modification allowed the algorithm to reduce the objective function value more quickly and obtain a lower final value of the objective function in comparison with the unmodified algorithm.

## Notes

### Acknowledgements

This paper was presented at the CAGG 2019 Conference “Challenges in Applied Geology and Geophysics” organized at the AGH University of Science and Technology, Krakow, Poland, 10–13 September, 2019 and 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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