European Geologist Journal 59

Static Digital Modelling of Stratified Geological Objects: Technological Applications

 

by Oleksandr Lobasov 1, Larysa Bosevska 1,2

1 Ukrainian Association of Geologists

2  Ukrainian Salt Research Institute

Contact: lara.bosevskaya@gmail.com

Abstract

In modern multi-purpose information systems supporting all aspects of geological activities, including the use and protection of the geological environment, structural-lithological modelling plays an increasingly significant role. However, modelling layered sedimentary objects with complicated stratification remains a complex task requiring a specialised approach. This paper proposes a statistical method for modelling the lithology of sedimentary strata based on non-stationary Markov matrices using lithological sections from wells. The model developed according to this principle synthesises the most probable lithological section at any given point, providing sufficiently accurate interval estimates of parameters for detailed object mapping and differentiated assessment of its resources. This technology has been implemented in a GIS environment and has been successfully tested on real geological objects.

Cite as: Lobasov, O., & Bosevska, L. (2025). Static Digital Modelling of Stratified Geological Objects: Technological Applications. European Geologist, 59. https://doi.org/10.5281/zenodo.16442336

Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 International License.

1. Introduction

Structural-lithological modelling of sedimentary strata has emerged as an independent area of geological modelling, aimed at the comprehensive utilisation and conservation of subsurface resources using modern computer (digital) technologies [1].

Layered sedimentary strata within stratigraphic units undergo significant temporal and spatial variations due to changes in sedimentation conditions and paleogeographic factors. As a result, sedimentary formations are composed of a series of lenticular and thinning layers of varying composition, often with gradual transitions from one lithological type to another. Consequently, modelling the lithological content of sedimentary bodies, even those that are well stratified, remains a non-trivial task requiring a specialised approach.

For practical applications, an accurate representation of the spatial distribution of lithological rock types is essential, given its relevance to various subsurface resource management and environmental issues. Addressing these questions through static digital modelling has been the focus of numerous studies, which have elucidated the objectives pursued by constructing diverse digital models of geological objects at different scales [2–6].

The purpose of this work is to present a validated technology for constructing digital structural-lithological models of layered geological objects using mathematical methods. These methods provide sufficiently accurate interval estimates of geological object parameters for detailed object mapping and differential resource assessment.

The results of this innovative technology have already been tested on several objects, with examples of visualisations presented in this study.

2. Materials and Methods

The object of static geological modelling is a spatially defined portion of the geological environment represented by layered sedimentary strata. Typically, in the vertical section, the selected geological object is constrained by stratigraphic, lithological, and/or tectonic boundaries. Horizontally, it may coincide with a tectonic structure, the area of distribution of a specific stratigraphic (or lithological) unit, a sedimentary formation, or be delineated by an arbitrary contour chosen to address specific practical tasks.

The object may consist of one or several stratigraphic subdivisions that are paragenetically interconnected, sharing common boundaries or separated spatially.

Depending on the research objective, a range of properties (lithological, facies, porosity-permeability, geochemical, hydrochemical, etc.) are used to describe the object. The focus here is on the structural-lithological model.

A static digital model is an idealised representation of the object that reflects its essential properties. The structural-lithological model represents a continuous image of the geological object, in contrast to primary (factual) data, which discretely represent the object.

There are several key requirements for a mathematical model:

  • The model must satisfactorily describe known data about the object;
  • The model must predict the properties of the object in areas lacking primary information;
  • The model should not contain purely mathematical artefacts (extremes) that are absent in the original data and/or do not correspond to theoretical expectations regarding the nature of this type of object.

2.1. Construction of the Morphological Model of a Layered Object (External and Internal Structures of the Object)

The construction of the structural model of an object is a necessary step before developing the lithological model. The technology for its construction is quite complex and should be discussed in more detail in a specialised work. In general terms, the strategy for construction follows this step-by-step scheme:

  1. Construction of the digital model of the Object’s External Boundaries
  2. Construction of Models for the Thickness of All Object Components (layers separated by lithological and stratigraphic boundaries)

The establishment of internal boundaries is the result of algebraic operations on the models constructed in the first two stages.

The thickness models of the internal layers of the object (hi) should be normalised according to the following formula 1:

Hi =    *  (S1 – Sn), i = 1, n – 1 (1)

where:

S1, Sn – the upper and lower boundaries of the object, respectively;

hi – the thickness model of the i-th layer,

Hi – the normalised thickness of the i-th layer, n – the number of internal and external boundaries of the object.

2.2. Construction of the Lithological Model of the Object

In general, a lithological section represents a fine interlayering of several lithological types. It is impossible to trace each layer in the inter-well space directly. Changes in lithological types within a well section occur in a patterned way, determined by the oscillatory nature of sedimentation conditions. This means that the transition from one type to another depends on previous transitions, with each transition having a certain probability. The mathematical models for such and similar natural processes are Markov chains of various orders, with the theory covered in [7]. Methodological aspects of applying Markov chains in geology, particularly for modelling lithological sections, have been examined by J. Harbaugh and G. Bonham-Carter [8]. This method was first introduced to geological problems in 1949 by A.B. Vistelius [9]. Since then, Markov chains have been applied by many researchers to study rhythmic patterns within sedimentary formations [11–14, et al.].

For modelling lithological sequences, we propose a first-order Markov chain, in which the probability of a current transition depends only on the immediately preceding transition and is independent of earlier transitions. The advantages of a first-order Markov chain are its straightforward software implementation, clear geological interpretation, and extensive use in geological applications.

We will consider the external boundaries of the object synchronous. Any surface obtained by linear interpolation between them will also be synchronous over the entire definition area.

The section of some well or any other point within the object may be presented by a probability matrix of lithological transition:

{ Pij },    i = 1, N;  j = 1, N (2)

or

P11  … P1j  … P1N

P =    Pi1  …  Pij  …  PiN

PN1  …  PNj … PNN

 (2а)

where:

N – the number of the lithological types, Pij  – the transition probability from the i-th lithological type into j-th one.

At the first stage, based on wells core data, we obtain a series of Markov matrices representing the sedimentation model at a specific spatial point. The algorithm for constructing the lithological transition probability matrix involves the following steps:

  1. Construction of the Matrix of Lithological Transition Counts (Matrix A)

A11  … A1j  … A1N

A =    Ai1  …  Aij  …  AiN

AN1  …  ANj … ANN

 (2b)

The matrix A is square and asymmetrical. Each matrix element Aij represents the number of transitions from lithological type i to type j within the studied depth interval [H1, Hn].

The matrix A is constructed based on the lithological section of the well. Initially, it is set to zero. Let K be the number of steps. Then depth step (the thickness of an elementary layer) at the point (x, y) will be: m(x, y) = [Sn(x, y) – S1(x, y)] / K. Starting from the upper boundary of the studied interval, we proceed downwards through the well section at a specified depth step m. At each depth step k within the well lithological section, we determine the lithological type number 𝑗 corresponding to the current depth Hk+1 = Hk + k*m.

Next, we register the transition from type 𝑖 to type 𝑗:

Aij = Aij + 1, (3)

The thickness 𝐿j of lithological type j in the studied interval is calculated as:

Lj  = m *  ij, i = 1,N (4)

The lithological coefficient for type j, denoted 𝐶j, is calculated as:

Cj  =  Lj /  ,   j = 1,N (5)
  1. The transition probability matrix P = {Pij}:
Pij  =  Aij  /  ij,  j = 1,N (6)

If we equate the interval [H1, Hn] to the entire thickness of the object section [S1, Sn], we obtain a generalised lithological characteristic of the strata, assuming constant formation conditions. Generally, sedimentation conditions vary, leading to a non-stationary Markov process as a model for sedimentation. From a geological perspective, if feasible, or purely formally, interval [S1, Sn] may be divided into a few smaller intervals [H1, Hn] with homogeneous sedimentation conditions. Each interval has its respective matrices A, P, and lithological coefficient vector C. The depth-dependent (time-dependent) variations of A, P, and C characterise changes in the sedimentation regime during the formation of the sedimentary layers near the well.

Interpreting the transition probability matrices provides insight into the characteristics of the section and is relatively straightforward. Typical sections and their corresponding Markov matrices are shown below.

Large values of “self-transitions” with small values for transitions to other lithological types indicate the section interval comprises a small number of thick lithological layers (Fig. 1).

Figure 1:  Interbedding of thick lithological layers.

If the probability of transitions i to j, j to k, k to i is significantly higher than that of other transitions, we are dealing with series i j k – i j k  – i j k  – i j k  – i j k   (Fig. 2):

Figure 2: Lithoseries.

The sequence of these series can vary within the cross-section, and comparing different series allows us to understand the dynamics of the sedimentation regime.

Using the transition probability matrix and a random number generator, it is possible to construct a synthetic well cross-section with the average and integral characteristics that reflect the actual cross-section (Fig. 3).

Figure 3: Construction of a synthetic well section using the stationary Markov matrix.

A synthetic well section constructed using a non-stationary Markov matrix is presented in Fig. 4.

Figure 4: Construction of a synthetic well section using a non-stationary Markov matrix.

The matrix can also be generated not based on the actual well section but on theoretical, generalised considerations of the temporal changes in sedimentation conditions. For example, it may rely on a composite structural-lithological section of a specific geological object. The differences in transition probability matrices across various wells within the study area reflect the differences in area sedimentation conditions. These differences can be due to changes in the set of lithological types or in the order of their succession in the section.

The sedimentation dynamics, recorded in the transition probability matrices, can be mapped. Mapping the transition probability matrices involves a smooth interpolation of matrix element values for each interval with constant sedimentation conditions within the object boundaries. For example, if three lithological types and four intervals with stable sedimentation conditions are observed in the section, the matrices in whole will contain 3*3*4=36 elements, generating a trend for each element (a total of 100 trends).

Several methods commonly used in geoformation systems were tested for interpolating matrix elements: polynomial interpolation, distance-weighted methods, kriging, and splines. All these methods require a substantial amount of data to construct adequate numerical models. The main challenge associated with their application is the need to store a large number of trend surfaces.

We employed the method of linear interpolation over a triangulated area [16]. This method operates with a minimal amount of data (formally, three initial points are sufficient) and helps avoid unnecessary extrema in the resulting trends. It also eliminates the need to store trends for individual matrix elements, as the required calculations are performed on demand. A disadvantage of the method is the spatial limitation of the model to the convex region defined by the triangulation.

By interpolating matrix elements within stratified horizons across the study area, we derive a sedimentation model for the entire object. Thus, by obtaining the spatial distribution of the transition probability matrices, it is possible to generate a matrix and obtain the most probable synthetic lithological section of a well at any arbitrary point within the geological object.    In essence, this yields a 3D lithological model that allows for accurate predictions of lithological characteristics and rock properties within any selected block. This model also enables the mapping of blocks with specified properties, calculating the lithological composition of a block, and estimating mineral reserves corresponding to specific lithological rock types, etc.

3. Results and discussion

Using the mathematical modelling framework described above, numerous digital models have already been constructed for sedimentary objects of various types. A few examples are given below.

The foundational data for the Kosmach object are sourced from the factographic database and relevant literature detailing the geological characteristics of this object [9, 10]. The construction process strictly followed the outlined methodology, resulting in illustrative and informative visualisations that demonstrate the model’s high level of detail and effectiveness (Fig. 5, 6, 7).

Figure 5: Visualisation of the digital structural-lithological model of the Kosmach area of the Rosilna potash deposit.

Potash reserves in cross-section: Specific reserves (reserves per unit area, m³/m²). 

Figure 6: Visualisation of the digital structural-lithological model of the Kosmach area of the Rosilna potash deposit.

Geological cross-section along Line 1-1 (look at figure 5).

Figure 7: Visualisation of the digital structural-lithological model of the Kosmach area of the Rosilna potash deposit.

Geological cross-section along Line 2-2 (look at figure 5).

Another example demonstrates the model obtained using well core data from the Dnipro-Donetsk Depression. The object mainly consists of terrigenous rocks (argillites and sandstones) and corresponds stratigraphically to the Lower and Upper epochs of the Carboniferous period (C1–C2). The real lithological sections of the wells are shown in Fig. 8, while the object’s roof is presented in Fig. 9.

Figure 8: Initial real well sections of some wells within the object boundaries.

Figure 9: Dnipro-Donetsk basin. Srebnenska depression (middle part). The upper boundary of the object (C2 roof).

The cross section along the line 1-1 you can see on figure 10.

Figure 10: Geological cross-section along line 1-1 (look at figure 8).

The next example demonstrates the model of the object with more complex layered structure, also obtained with well core data. The object is located on the northwestern shelf of the Black Sea (22 km southwest of the Tarkhankut Peninsula), is mostly composed of marls and limestones, and corresponds stratigraphically to the Lower and Middle epochs of the Cretaceous period (K1–K2). The real lithological sections of the wells are shown in Fig. 11, while the object’s roof is presented in Fig. 12.

Figure 11: Initial real well sections of some wells within the object boundaries.

Figure 12: Northern-West shelf of Black sea. The upper boundary of the object (K2 roof).

The cross section along the line 1-1 you can see on figure 13.

Figure 13: Geological cross-section along line 1-1 (look at the figure 10).

The method imposes certain requirements on the input data. The outer boundaries of the object must be synchronous, and the object must not contain unconformity surfaces or tectonic faults within its boundaries.

As demonstrated in this section, the modelling technique performs sufficiently well for layered sedimentary formations (Figures 8–10), in areas with heterogeneous composition (Figures 11–13), and in regions affected by intense tectonic activity (Figures 5–7).

4. Conclusions

The described above mathematical method for constructing lithological models offers a systematic approach to modelling geological objects of complex structure, using well core data. The method is based on a non-stationary first-order Markov chain to synthesise probable lithological sections.

Unlike previous studies, we extend the application of Markov chains to construct a lithological model of a spatial geological object. The sedimentation conditions recorded by the Markov matrices in the object’s wells provide a discrete model of its formation. By interpolating the matrix elements, we are able to generate a matrix and synthesise a lithological section at any point within the object’s boundaries, thus producing a continuous spatial model.

Markov chains are applied both to layered sections, which clearly exhibit the Markov property, and to sections of heterogeneous composition that lack visible patterns.

By capturing the essential lithological properties and spatial relationships within sedimentary formations, this method enables an interval-based assessment of parameters, which is critical for creating detailed structural-lithological models.

This approach is particularly advantageous for geological settings where lithological layers exhibit significant variability. Although the model does not provide point estimates, it offers sufficiently accurate interval estimates that are adequate for solving typical geological problems of varying complexity. The theoretical framework of the method does not assume any specific distribution laws nor impose other conditions on the object, making it applicable to a wide range of geological settings.

The described technology has been implemented in the ArcView 3.x environment as a specialised system that integrates a spatial database of innovative structure with tools for model construction, processing, visualisation, mapping, and resource estimation. The technology has been tested on various classes of sedimentary geological objects and has proven effective in addressing issues related to the efficient management of natural resources, especially in the context of the “green transition”.

The authors consider the further testing and development of the modelling technology to be their immediate priority.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

References

  1. Khrushchov, D.P.; Kovalchuk, M.S.; Remezova, O.O.; et al. Structural-Lithological Modelling of Sedimentary Formations; Interservis: Kyiv, Ukraine, 2017; 352 pp. ISBN 978-617-696-631-9. (in Russian)
  2. Khrushchov, D.P.; Lobasov, O.P.; Remezova, O.O.; et al. Digital structural-lithological models as an informational-analytical basis for decision-making in the exploitation and protection of mineral resources and the geological environment. In Proceedings of the 5th International Scientific-Practical Conference “Environmental Safety: Problems and Solutions”, Alushta, Ukraine, 7–11 September 2009; Raider: Kharkiv, Ukraine, Volume 2, pp. 60–65.
  3. Khrushchov, D.P.; Lobasov, O.P. Principles of developing digital structural-lithological models of sedimentary formation units. Geol. J. 2006, 2–3, 90–102. (in Russian)
  4. Khrushchov, D.P.; Lobasov, O.P.; Kovalchuk, M.S.; et al. Digital structural-lithological models of sedimentary formations: A tool of informational-analytical supervision for exploitation and protection of mineral resources and geological environment. In Abstracts of International Conference “GeoDarmstadt 2010 – Geosciences Secure the Future”, Darmstadt, Germany, 2010; Volume 68, pp. 306–307.
  5. Khrushchov, D.P.; Lobasov, O.P.; Heichenko, M.V.; et al. Structural-lithological models of prospective sedimentary formations. Miner. Resour. Ukr. 2010, 4, 39–44. (in Russian)
  6. Khrushchov, D.P.; Lobasov, O.P.; Kovalchuk, M.S.; Remezova, O.O.; Bosevska, L.P.; Kyrpach, Yu.V. Targeted expert systems of geological orientation. Geol. J. 2012, 2, 87–99.
  7. Main, H.; Osaki, S. Markov Decision Processes; Nauka: Moscow, Russia, 1977; 176 pp. (in Russian)
  8. Harbaugh, J.; Bonham-Carter, G. Computer Modelling in Geology; Rodionov, D.A., Ed.; Translated from English; Earth Sciences Series, Vol. 57; Mir: Moscow, Russia, 1974; 319 pp.
  9. Vistelius, A.B. On the mechanism of layer formation. Proc. USSR Acad. Sci. 1949, 65(2).
  10. Doveton, J.H. An application of Markov chain analysis to the Ayrshire Coal Measures succession. Scott. J. Geol. 1971, 7, 11–27.
  11. Ethier, V.G. Application of Markov analysis to the Banff Formation (Mississippian), Alberta. Math. Geol. 1975, 7, 47–61.
  12. Miall, A.D. Markov chain analysis applied to an ancient alluvial plain succession. Sedimentology 1973, 20, 347–364.
  13. Schwarzacher, W. The use of Markov chains in the study of sedimentary cycles. Math. Geol. 1969, 12, 213–234.
  14. Dolinsky, I.P.; Lobasov, O.P.; Ogarkov, B.M. Linear interpolation on a triangulated area as a method of construction of 2D parameter models in the absence of a priori information. Theor. Appl. Asp. Geoinformatics 2015, 12.
  15. Kyrpach, Yu. Geological structure of the southwestern part of the Precarpathian Foredeep (Rosilna–Dzvyniach–Starunia area) and the structural-lithological model of salt deposits. Collect. Sci. Works Inst. Geol. Sci. NAS Ukr. 2009, 2, 78–81. https://doi.org/10.30836/igs.2522-9753.2014.146870 (in Ukrainian)
  16. Kyrpach, Yu. The features of salt deposits geological structure and lithological composition in the central part of Boryslav-Pokutian zone (Ukrainian Carpathian Foredeep). Tectonics Stratigr. 2013, 135–143. https://doi.org/10.30836/igs.0375-7773.2013.94075

This article has been published in European Geologist Journal 59 – UN Sustainable Development Goals – where the geology lies