Pyritization in Stone-Building Materials Modeling of Geochemical Interaction

Abstract

Stone-building materials, despite their natural origin, must be tested for the concentration of trace elements necessary to assess their impact on the environment and humans. In addition to basic research determining their mineral composition and structural and textural features, it is important to analyze the geochemical interactions between the material matrix and the concentration of elements that have a negative impact on the surrounding natural environment and our health. In the presented study, mineralogical and geochemical studies were carried out on the Carpathian sandstones. It was shown that the studied sandstones are represented by lithic wackes and sublithic arenites. Rocks subject to the secondary process of sulfide mineralization were observed among sublithic arenites. Pyrite in the studied geomaterials took various forms. A detailed geochemical analysis was carried out in the material in which iron sulfides acted as a binder. The research was aimed at identifying possible variations in the concentration of elements, with a particular emphasis on the contact between the silica and mineralized phases. The assessment of the geochemical interaction of iron sulfides with silica at a successively enlarged measurement was carried out using the Mamdani–Assilian fuzzy inference model.

1. Introduction

Sandstones are a material commonly used in construction. In southern Poland, they are a frequent raw material used for regulating rivers and streams. Regulating structures made of stones, which include, among others, spurs, dams, weirs and gabion baskets with a wide range of applications, should be ecological. The performed regulatory work should be aimed at maintaining the natural conditions in the watercourse bed, with the hydrological regime unchanged. In order to cope with such difficult conditions of the exploitation of stone water structures, appropriate-quality rock material should be selected for their construction. Basic guidelines as to the types and number of tests are standardized, but there is no obligation to follow any standard guidelines [1,2]. The selection of a specific type of rock from a technical and ecological point of view must always be preceded by the necessary tests to assess its quality and suitability. In order to determine the possible direction of using sandstone as a building material, it is very important to perform basic mineralogical and geochemical research. In particular, these tests are important in the case of the use of sandstone as a stone for hydrotechnical works—in the form of large boulders or coarse-grained aggregate—used in structures subject to the continuous influence of water flowing at different velocities, waves or currents. Larger and heavier boulders can be built into marine or oceanic structures, where they will resist displacement by the movement of water. On the other hand, smaller and lighter stones will be suitable for slowly flowing rivers and streams or for filling gabion baskets.

Previous studies of Carpathian sandstones show that the mineral composition as well as the structural and textural features favor the deterioration of these rocks, which occurs as a result of reducing their firmness, among other reasons. Most of the sandstones are porous rocks, which makes them susceptible to weathering [3,4,5]. In sandstones, there are changes in their internal structure and physical and mechanical properties [6]. The sandstones are characterized by a significant petrographic differentiation, which results in a wide range of technical parameters. The most common binder of sandstones is silica, but often, it can also be calcium carbonate and clay minerals, which are not very resistant to water. There are also sandstones subject to secondary mineralization processes, where iron sulfides may be the binder [7]. This type of rock is particularly important in terms of the ecological assessment of building materials used as materials in hydrotechnical structures. In this case, the tests must include an analysis of the composition of the concentration of harmful trace elements that, after oxidation, may be released and significantly affect the water quality.

Pyrite (FeS₂), when it comes from coal mining, is treated as waste, and its disposal culminates in environmental problems for the coalfields. Increasing the purity of pyritic tailings and increasing its iron disulfide content strengthens the possibility of turning it into a value-added by-product [8]. Therefore, knowledge of their occurrence and overall mobility is essential to ensure adequate environmental protection measures.

The system of the assessment of materials used in construction must take into account their impact on the natural environment. This assessment requires a detailed analysis of the internal structure of the material and the external factors affecting the possible release of hazardous substances. Not only must composites based on harmful materials be tested, but natural stone materials must also be tested due to the possible presence of harmful components, even of natural origin. A material analysis must include the evaluation of the material matrix in terms of chemical, physical and mechanical properties. It is important not only in terms of industry but, above all, in terms of ecology. The impact of external factors and, in particular, the water environment (precipitation, aggressive groundwater and other aggressive liquids) may release additional amounts of components originally associated with the structure of the material. In many countries, work is underway to establish an environmental assessment system for materials used in construction. In these works, attention is paid to the leaching of chemical elements from recycled building aggregates and stone waste materials used in road construction [9,10]. Apart from the necessity to perform detailed laboratory tests of building materials, a balanced statistical analysis of the obtained results is also important. During comprehensive studies of the concentration of elements or chemical compounds present in stone-building materials, attempts are made to determine the concentration of elements using statistical methods [11]. When performing laboratory tests, we often have readings with a small number of results; in this case, it is very important to use significant non-quantitative knowledge about the system in order to obtain identifiable models. This algorithm was used in geological research [12] and the leachability of elements from building materials [13]. Such studies were carried out in the context of the leachability of strontium and barium [14]. In the presented work, the Mamdanie-go-Assilian model [15] was adopted for the preparation of the system concept. The choice of the Mamdan-go-Assilian fuzzy inference model was dictated by its capabilities in the context of the specificity and nature of geological research, such as a small number of empirical measurements and the possibility of obtaining analytical results without the need to determine the physical form of the dependence of the input quantities.

Geological Background

The term Carpathian sandstones includes sandstone deposits occurring in the Carpathian Orogen, from the Cretaceous to the Miocene. These rocks were formed as a result of deep-sea flysch sedimentation, diagenetic processes and significant tectonic disturbances. In the study, there were analyzed rocks represented by litofacial varieties of Krosno and Istebnia sandstones.

The Krosno sandstones occupy quite large areas of flysch formations of the Silesian, Skole and Magura units and also occur in the central part of the Carpathian depression and in the Dukla fold formations, as shown in Figure 1. There are three levels in the Krosno sandstones: lower, middle and upper. In the lower level, lumpy sandstones are noticeable, which occur in thick beds. These rocks easily weather and disintegrate, and there are no clay inserts in them. In contrast to the sandstones from the middle level, where thin beds of fine-grained, very compact sandstones with a large number of shale inserts are formed, the rocks of the upper level are of a shale nature and are of little exploitation importance. The discussed sandstones occur within the Silesian Nappe and represent the Upper Oligocene deposits [16,17,18,19]. The sedimentary environment of the Krosno layers in the lower part of the profile was marked as the central part of the deep-sea cone, and in the upper part, it was marked as the abyssal plane [20].

The petrography of these sandstones was the subject of the research works [21,22,23,24,25,26,27,28], which described them mainly as gray-blue, fine-grained arenites and sub arcose, sublithic greywacke with a random or laminated structure. These deposits are characterized by an abundant carbonate–silica–clay binder, in which dolomite has a significant share. In the profiles, there mostly are fine-grained sandstones with randomly distributed grains of a coarser fraction.

They are composed mainly of quartz grains in a proportion of over 35% and lithoclasts in an amount of about 14%. Feldspar and mica are found in minor, several-percent shares. Glauconite, organic matter and pyrite are found occasionally. The remaining percentage is the binder. Among the carbonate components, sparite predominates over micrite [29].

The Istebna sandstones of the Lower Campanian-Maastrichtian belong to the Silesian nappe [30,31], as shown in Figure 1. They are a product of deep-sea silicoclastic sedimentation in the diverse and rapidly changing deposition environments of the channels and their outlet zones of the central cone and within the non-canalized silicoclastic apron [32,33,34]. In the works, they are described as gray-greenish or gray-yellow, subarcoses and sublithic arenites with a silica binder. They are strongly differentiated in terms of graining; they correspond to fine, medium and coarse sand fractions [6,22,25,27,28,34].

2. Materials and Methods

The research methodology used in the work covered a wide analytical spectrum. In the course of the research, the following were used:

–

scanning electron microscopy (SEM) combined with an X-ray energy dispersion microscope (EDS). This made it possible to analyze the chemical composition of the samples in the micro-areas. For this purpose, an (SEM) FEI Quanta 200 FEG electron microscope equipped with an X-ray spectrometer (EDX Genesis) and a backscattered electron detector (BSE) was used. The tests were carried out on fresh fractures of piece samples, sputtered with elemental carbon.

–

Microscopic observations in reflected light were carried out using a Motic Panthera TEC POL trin polarizing microscope. The micrographs were taken using a high-sensitivity microscope camera with an sCMOS matrix and a shutter type Globar Shutter of Motic Pro-S5 Lite. This allowed the mineral composition to be determined. The observations were made on rock preparations impregnated with epoxy resin. Preparations were covered with a coverslip with dimensions of 24 × 46 mm. In addition, during the course of observation with polarized reflected and transmitted light, an optical waveguide, a quarter-waveguide and a quartz wedge were used.

–

Mercury porosimetry was used to identify textural characteristics. The effective porosity, the specific surface area of the pore space and the average pore diameter were determined [35].

–

Pearson’s linear correlation coefficient for two variables was determined on the basis of [36].

The assessment of the geochemical interaction of iron sulfides with silica with a successively enlarged measurement was carried out using the fuzzy inference model. The choice of the method of analysis was made due to the lack of sharp boundaries between the values of the variables under consideration. One of the advantages of using fuzzy models compared to the more frequently used mathematical models, as described in the works [14,37,38,39], is the lack of the requirement to have a large database of results with the structure, including empirical results.

Additionally, information about the described system or phenomenon may not be precise. Another advantage of using fuzzy models is their ability to obtain a response between the input and output, even when this relationship is difficult to describe with a mathematical model. Fuzzy inference is based on the logic of actions on fuzzy sets and additional expert knowledge on the studied phenomenon. An example is the content of the ions of elements in a rock sample.

In this study, three linguistic variables were used: small, average and good. On the other hand, fuzzy sets were assigned to the values of linguistic variables. The relationships between linguistic variables are fuzzy conditionals. Figure 2 shows a general scheme of the fuzzy inference of the element concentrations—Si, S and Fe, as input quantities of the model—content in the empirically tested sublithic arenite rock samples.

The model was used to determine the relationships between the input quantities (Si, S and Fe contents of arenite [%]) for successive and deeper points of the cross-section on the value of the output value in the form of an inference result. The results were obtained thanks to the use of the decision block. Due to the nature of the analyzed data, models were used in the decision block. It is justified by the necessity to evaluate rock samples for different Si, S and Fe contents in increasingly deeper parts of the sample’s cross-section, which results in the lack of sharp boundaries between the values of the variables. The calculations were made thanks to the Mamdan–Assilian model, as part of the add-on “Fuzzy Logic Toolbox”, operating in the Matlab environment. The analysis was carried out in version 18b of the Matlab program.

3. Results

3.1. Mineralogical Characteristics of the Studied Material

The conducted analyses showed that the tested material is representative of lithologically and genetically diverse sandstones in

Table 1. Quantitative mineral composition of the studied Carpathian sandstones (own research).

Component	Krosno Sandstones [vol %]			Istebna Sandstones [vol %]
	Min	Max	Average	Min	Max	Average
Quartz	22.5	62.2	51.3	15.8	65.4	49.7
Feldspar	0.5	13.7	5.5	1.9	27	15.6
Mica	-	7.8	6.15	-	15.3	1
Glauconite	-	6.75	0.45	-	4.3	0.5
Rock grain	-	13.5	14.5	-	9.3	6.8
Heavy minerals	0.2	2.2	1	0.3	1.5	0.8
Pyrite	-	1.5	0.5	-	2.2	1.3
Silica binder	1.0	9.5	7.2	0.5	3.5	2.2
Clay binder	0.8	7.6	5.5	0.5	5.4	4.7
Carbonate binder	1.6	43.5	26.78	-	-	-
Iron binder	-	-	-	-	1.5	0.7
Fine-grained binder	-	4.5	1.5	15.7	19.8	20.2

. The microscopic analysis of the terrigenous components that make up the skeleton of the studied rocks is in accordance with the classification according to F.J. Pettijohna et al. [40], who state that the examined sandstones of the Krosno strata in the studied case represent lytic wackes, while the sandstones of the Istebna strata can be classified as sublithic arenites.

The main mineral component of the studied rocks is quartz, the content of which ranges from 51.3 [vol.%] in the Krosno sandstones and from 49.7 [vol.%] in the Istebna sandstones (Table 1). In the studied rocks, apart from monocrystalline quartz, we can find polycrystalline types. Based on the research with the use of cathodolumination analysis, it was found [41] that the origin of quartz grains in the Carpathian sandstones is diverse. Grains of metamorphic, igneous and volcanic origin were determined in this type of material. The grain size of the quartz and their degree of rounding also vary. We distinguish grains of coarser psamite fractions, characterized by a better degree of rounding than the grain of smaller fractions. The second components in terms of quantity are feldspars, among which we can most often distinguish potassium feldspar, represented by orthoclase, microcline and pertyite. They are preserved to varying degrees. Grains under the secondary mineralization process—mainly argillitization—were found. In the microscopic image, worm-like forms of kaolinite are visible, resulting from the transformation of feldspars. Micas range from about 1 to about 6 vol.%. and are represented by both muscovite and biotite. Some of the muscovite lamellae transform into hydromuscovite, and opacitization processes have been observed on the surface of biotytes. Among the skeletal components, we can also find lithoclasts, represented by crumbs: igneous rocks, quartzites and mudstones. They are most often large in size and quite well preserved. Accessory minerals are represented by zircon, tourmalines, apatites and iron oxides. Among the discussed minerals in sandstones, iron sulfides are of particular interest. Initially, the presence of these compounds was found macroscopically, most often in the fracture zones and in the form of a coat with a characteristic metallic sheen with a green coloration. Research has shown that they are represented by pyrite. It varies in shape and size. Observations using a polarizing microscope have shown that pyrite is in the form of separate small crumbs, irregularly distributed in the sandstone. This sulfide is noticed in the pore spaces and around the quartz grains in an amorphous form (Figure 3). Using a scanning microscope, the morphological diversity of this sulfide was demonstrated. Cryptocrystalline pyrite, framboidal and full-crystalline (euhedral) grains were encountered (Figure 4a). Due to the ecological assessment of the tested materials, the authors’ attention was drawn to sandstone, in which a strong sulfide mineralization process was observed. In the sublithic arenite, the binder in the form of iron sulfides took the form of a massive filling mass in Figure 4b.

3.2. Textural Features of the Sandstone Pore Space

The component distributions in sandstone, their morphology and the variable content and type of binder result from the conditions of sediment deposition and the diagenetic changes that they have undergone. These features influence the differentiation of the rock pores and their shape, degree of connectivity and spatial distribution as well as the size of the pores [42]. The textural parameters of the sandstones, determined with the use of mercury porosimetry, as presented in

Table 2. Results of the analysis of pore space measurements by mercury porosimetry (average value) (own research).

Parameter	Krosno Sandstones	Istebna Sandstones
Total porosity [%]	4.25	13.98
Average pore diameter DPOR [µm]	0.7	1.33
Specific surface SPOR [m2/g]	0.4	0.34
Effective porosity PPOR [%]	41	37

, show that the tested samples are characterized by typical porosity for the sandstones of this region. In the Istebna sandstones, it is 13.98%, and in the Krosno sandstones, where the share of the binder dominated by silica and carbonate cement is much higher, it amounts to 4.25%. The average pore diameter in the Istebna sandstone is 1.33, and in the Krosno sandstone, it is 0.7. The confirmation of microscopic observations is the analysis of the effective porosity. In the Istebna sandstone, it is about 37%, next to the regular and irregularly shaped pores, and it is 41% in the Krosno sandstone.

3.3. Geochemical Research

The geochemical studies of the Carpathian sandstones focused on the analysis of sublithic arenite covered by the sulfide mineralization process. The analyses were carried out in four areas; they included the grain skeleton and the binder in direct contact with silica. The research was aimed at identifying possible variations in the concentration of elements, with a particular emphasis on the contact between the silica and mineralized phases. It was shown that the highest concentration of sulfur and iron occurred in point 4 and amounted to S 52.70% wt. and Fe 43.48% wt., respectively (

Table 3. The content of individual tested elements in individual areas of the research sample (own research).

Element	Studied Area [wt.%]
	1	2	3	4
Al	0.075	0.035	0.061	-
Si	29.121	24.717	22.848	-
S	-	0.014	0.172	52.708
Ca	0.081	0.040	0.025	0.203
Mn	-	0.054	0.074	0.356
Fe	0.053	0.161	0.181	43.483
Cu	0.159	0.072	0.145	0.336
Zn	0.085	0.173	0.055	0.316
Cd	0.030	0.095	0.118	0.138

). In this area, the presence of Si and Al was not found, and the concentration of elements such as Mn, Cu and Zn was, on average, 0.3% wt. The Cd content was 0.13% wt. (Table 3). This elemental composition shows the geochemical nature of sulfide mineralization in the studied lithic arenite. The tests in the following points, along with the distance from the mineralized zone, indicate an increase in silica from 22.8% wt. up to 29.12% wt. in Table 3. There is an inversely proportional relationship between silica and iron sulfides. An increase in the Si content causes a decrease in the values of Fe and S.

It can also be noticed (Table 3) that, with the increase in pyrite sulfur (52.71 wt.%), the concentration of elements such as Cd (0.138 wt.%), Zn (0.32 wt.%), Cu (0.33 wt.%) and Mn (0.36 wt.%) increases. This dependence and the concentration of the determined elements are comparable and similar to the data in the literature. It has been shown that, in the crystalline forms of sulfides within sandstones, the content of trace elements is Cd (0.11 wt.%), Zn (0.41 wt.%), Cu (0.11 wt.%) and Mn (0.33 wt.%) [7].

The numerical representation of the empirical results of the content of elements in successive and deeper areas of the sample, according to Figure 5, is presented in Table 3. The empirical percentage values presented in Table 3 were used to determine the linear correlation coefficient for the trace elements determined in the studied areas in

Table 4. Linear correlation coefficient for the trace elements determined in the studied areas (own research).

Al	Al
Fe	1.00	Fe
S	1.00	−0.97	S
Ca	1.00	1.00	1.00	Ca
Mn	1.00	1.00	0.99	0.90	Mn
Cu	1.00	0.77	1.00	0.70	0.97	Cu
Zn	1.00	1.00	1.00	0.84	0.91	0.60	Zn
Cd	1.00	−0.31	1.00	−0.60	0.12	−0.05	−0.17	Cd
Si	0.64	−0.97	−0.97	−0.82	−0.21	−0.90	−0.91	−0.80

and are also the input values of the fuzzy inference analysis. The process of fuzzy inference in this paper is based on assigning “linguistic” assignments to the values of the input and output variables. On the other hand, fuzzy sets were assigned to the linguistic values. The process of fuzzy inference was carried out in the Matlab environment within five stages: definition of the membership function, fuzzification, knowledge representation, inference and defuzzification (described in more detail in [15]).

The analysis of the linear correlation coefficient determined (Table 4) shows that the correlation value, e.g., Si, is negligible between such biocatalyst metals as Cu, Zn and Cd. A high correlation coefficient occurs between Si: Al = 0.64. Taking into account the value for the correlation coefficient Si (S = −0.97 and Fe = −0.97), it is easy to notice a very strong linear correlation relationship close to −1. An increase in the Si content will cause a decrease in the value of Fe and S. In this case, it can be shown to be inversely proportional. The weakest strength of the correlation relationships in relation to Si is in the notation Mn = −0.21.

3.4. The Process of Fuzzy Inference

The process of fuzzy inference, according to the works [43,44], was carried out in the following stages: adoption of the appurtenance function, fuzzification, assignment of rules, inference and defuzzification.

3.4.1. Functions of Membership, Fuzzy Inference

The model was built using two types of membership functions:

• Gauss curve

$$\mathrm{gaus}(\mathrm{t}. \mathsf{\sigma }. \mathrm{m})=e^{\frac{-{\left(t-m\right)}^2}{2{\sigma }^2}}$$

(1)

where:

• σ—standard deviation
• m—mean expected value
• t—independent variable.

2.

Sig-curve

$$\mathrm{sig}(\mathrm{t}. \mathrm{a}. \mathrm{c})=\frac{1}{1+e^{-a\left(t-c\right)}}$$

(2)

where:

• a—growth rate
• c—inflection point
• t—independent variable.

3.4.2. Fuzzification

The fuzzification stage, i.e., the conversion of the input variables to the fuzzy domain, was carried out by using two curves, expressed by the Formulas (1) and (2).

• Three input variables for the percentage of Si, S and Fe (linguistic variables are the input variables).
• One output variable Y (linguistic variable is the output variable).

The adopted functions, along with the ranges of variability of the input quantities (real and linguistic), are presented in

Table 5. Ranges of variation for input quantities.

C1-1.Si [%]	C1-1.S [%]	C1-1.Fe [%]	Y
Small: sig {−29.121; 0} High: sig {0; 29.121}	Small: sig {−0.014; 0} High: sig {0.014; 0}	Small: sig {−0.053; 0} High: sig {0.053; 0}	Good: sig {−10; 0.5} Average: gaus {0.2; 0.5} Small: sig {10; 0.5}

. The minimum and maximum values of all three input variables were taken from the experimental data recorded by the authors and are presented in Table 3.

The sinusoidal curves of the input variables determined according to Formula (2) of the linguistic variables belonging to the fuzzy inference model are presented in Figure 6, Figure 7 and Figure 8. The sinusoidal and Gaussian curves determined according to Formulas (1) and (2) of the linguistic variable membership in the fuzzy inference model are presented in Figure 9.

3.4.3. Knowledge Representation

Eight rules of the expert system were defined for each of the four analyzed samples of three input variables and one output variable. The rules are based on the assumption that the linguistic variable “good” is assigned to the phenomenon of the occurrence of each of the input variables at the maximum value, and the linguistic variable “small” is assigned when each of the input variables has the minimum value. On the other hand, in the case of the occurrence of different input linguistic variables, the output linguistic variable “average” was adopted.

Table 6. Application of the fuzzy inference rule.

No.	The Rules of Fuzzy Inference
1	If (Si [%] is high) and (S [%] is small) and (Fe [%] is small) then (Y1 is good)
2	If (Si [%] is high) and (S [%] is small) and (Fe [%] is high) then (Y1 is average)
3	If (Si [%] is high) and (S [%] is high) and (Fe [%] is small) then (Y1 is average)
4	If (Si [%] is high) and (S [%] is high) and (Fe [%] is high) then (Y1 is average)
5	If (Si [%] is small) and (S [%] is high) and (Fe [%] is small) then (Y1 is average)
6	If (Si [%] is small) and (S [%] is small) and (Fe [%] is high) then (Y1 is average)
7	If (Si [%] is small) and (S [%] is small) and (Fe [%] is small) then (Y1 is average)
8	If (Si [%] is small) and (S [%] is high) and (Fe [%] is high) then (Y1 is bad)

presents the fuzzy inference rules used.

3.4.4. Inference

The solution to the problem of the inference of the fuzzy changes in the percentage of Si, S and Fe from the sublithic arenite was obtained thanks to the Mamdi–Assilian model implemented in the Matlab environment. The analysis was carried out in accordance with the rules of fuzzy inference for the linguistic input variables presented in Table 6. The process of fuzzy inference itself was carried out in accordance with the scheme presented in Figure 10 and with the use of the linguistic operators presented in

Table 7. Linguistic operators used during the fuzzy inference process.

Linguistic Operator	Operator Use Case
Conjunction	Min
Alternative	Max
Implication	Min
Aggregation	Max

.

3.4.5. Defuzzification

The defuzzification stage carried out in the Matlab environment consisted of the refinement and conversion of fuzzy sets into the acute domain. The precise conversion of the sets was possible thanks to the determination of the mean values of the centers of gravity of the response surface in accordance with Formula (3).

$$\mathrm{Y}=\frac{{{\displaystyle \sum }}_{\mathrm{l}=1}^{\mathrm{M}}{\mathrm{c}}_{\mathrm{l}}·{\mathsf{\mu }}_{{\mathrm{F}}^{\left(\mathrm{l}\right)}\left({\mathrm{x}}^{\left(\mathrm{l}\right)}\right)}}{{{\displaystyle \sum }}_{\mathrm{l}=1}^{\mathrm{M}}{\mathsf{\mu }}_{{\mathrm{F}}^{\left(\mathrm{l}\right)}\left({\mathrm{x}}^{\left(\mathrm{l}\right)}\right)}}$$

(3)

where:

• cl—fuzzy set center.
• μF (l)—function of membership of fuzzy sets F (l) corresponding to a given input variable.

The obtained graphs of fuzzy inference carried out in the Matlab environment for subsequent, deeper points of the arenite cross-section are presented in Figure 11, Figure 12, Figure 13 and Figure 14.

The results of the defuzzification for all analyzed values of the elements content for the successive depths of the cross-section of the arenite sample and the input variables adopted in the analysis are presented in

Table 8. List of the element contents in subsequent test points of the arenite sample.

No	Content of Elements in the Tested Points [%]			Result of Defuzzification
[-]	Ci.Si	Ci.S	Ci.Fe	Yi
1	29.121	0.014	0.053	0.523
2	24.717	0.014	0.161	0.587
3	22.848	0.0172	0.181	0.688
4	22.848	52.708	43.483	0.523

.

The presented results of fuzzy inference after defuzzification are presented in Figure 11, Figure 12, Figure 13 and Figure 14 and Table 8. In each case under consideration, Yi takes a value greater than 0.5. This proves that the presented model may be considered “average”. This allows for a limited application of the above model, taking into account additional expert assessment of future analyses. An additional, significant result of the analyses carried out is the fact that the value of the defuzzification result changes along with the change in the measurement depth of the cross-section of the tested rock sample. The model obtained the greatest credibility for the measurement depth of “3” of the cross-section of the sample, the location of which is shown in Figure 5.

The values of the output linguistic variables obtained on the basis of the values of the input linguistic variables were obtained by using the centroid function. This allowed for the determination of the center of gravity of the surface. The obtained response surfaces of the level of satisfaction Y, in relation to the content of Si, S and Fe [%] in subsequent points of the tested sample, are presented in Figure 15.

The obtained results of the fuzzy inference of the resultant function Y of Si, S and Fe contents from the sublithic arenite using the Mamdan–Assilian model are presented in

Table 9. Summary of the results of the fuzzy inference of Si, S and Fe contents from the sublithic arenite for all considered depths of the sample cross-section.

No	Content of Elements in the Tested Points [%]			Result of Defuzzification
[-]	Ci.Si	Ci.S	Ci.Fe	Yi
1	29.121	0.014	0.053	0.523
2	24.717	0.014	0.161	0.587
3	22.848	0.0172	0.181	0.688
4	22.848	52.708	43.483	0.523

.

The results of fuzzy inference for each of the considered depths of the cross-section of the sublithic arenite sample prove the “average” assessment of the model used. Therefore, with the parallel use of expert opinions from the conducted analyses, the Mamdanie–Assilian model can be used to assess the change in the Si, S and Fe contents in the considered range of input variables. The undeniable advantages of the model used are: the ease of expressing the interpretation of the obtained results, the stability of the model and the possibility of the limited interpolation of the output variable for input variables outside their accepted range.

4. Discussion

Mineralogical and geochemical studies of the rock raw materials used in construction are very important from the point of view of basic research. As previous works [45,46] have shown, mineralization processes have an impact on both the physical and mechanical parameters of rock materials, as well as their ecological assessment. It was found that, as a result of the transformation of siliceous mineral phases in the rock, its physical and mechanical properties change. With the increase in the crystallinity index (CI), the values of parameters such as air-dry compressive strength and apparent density increase. Their porosity decreases. As a result of the transition from opal -A to quartz, the water content, porosity and permeability in the rock are reduced. The increase in temperature, pressure and alkalinity and the presence of various catalysts accelerate the process of these transformations [47]. In the present work, from a geotechnical point of view, the pyritization process demonstrated must undoubtedly have an impact on the reduction in porosity. The recrystallization of iron sulfides in the tested material leads to the complete covering of free pores, as well as systems of cracks and micro-cracks. This process will have a secondary impact on the reservoir features of the studied rocks. Identification in rock’s raw materials and sulfide mineralization processes are also very important from the point of view of environmental protection. The environmental degradation effects of weathering pyrite are comparable to those of “acid rain” and anthropogenic sources. This process not only takes place in mining areas. Rock formations containing mainly pyrite which are devoid of natural buffering components, as a result of their oxidation, are often a source of acidic mine water H₂SO₄. Examples of the influence of pyrite weathering on water chemistry in Poland are: the “Purple lake” in Ridawy Janowskie (Sudety Mountains), the water reservoirs in the “Podwiśniówka” quarry and the acid reservoirs and puddles in the mining reservoir [48,49,50,51].

In mining areas, the oxidation and dissolution of sulfides occur more intensively. This is due to differences in hydrological conditions, the pace of weathering processes and the impact on the natural environment. In these places, there is an easier access of air to workings and heaps of waste rock and metal ores. Contact with air is facilitated by the larger specific (reactive) surface area of the sulfides. The diversified mineral composition of deposits and heaps is also important.

In the presence of oxygen and water, pyrite decomposes according to the reaction:

FeS₂ + 3,5O₂ + H₂O = Fe²⁺ + 2SO₄²⁻+ 2H⁺

(4)

If the water is not in direct contact with pyrite, then dissolved iron (II) sulfate (VI) is oxidized to (5), and the resulting iron (III) sulfate (VI) is subsequently hydrolyzed with the precipitation of iron (III) hydroxide (6):

Fe ²⁺ + 1/4O₂ + H⁺ = Fe³⁺ + ½H₂O

(5)

Fe³⁺ + 3H₂O = Fe(OH)₃ + 3H⁺

(6)

In sulfide deposits, H₂S is often formed as a result of the reaction of H₂SO₄ with minerals such as FeS pyrrhotite and ZnS sphalerite. H₂S is rapidly oxidized to native sulfur. In the presence of Fe³⁺, no transient sulfur oxoanions or sulfates are formed [52].

In the presence of Fe³⁺ ions, the pyrite decomposition reaction (7) is 2–3 times faster than that in the presence of oxygen [53] and, at the same time, more efficient, producing eight times more hydrogen ions [54]. It occurs at low pH values due to the low solubility of Fe²⁺ ions in neutral solutions.

FeS₂ + 14Fe³⁺ + 8H₂O = 15Fe²⁺ + 2SO₄²⁻ + 16H⁺

(7)

It must therefore be assumed that the oxidation of pyrite initiated by oxygen lowers the pH to about 4, which further leads to the activation of reaction (4). Oxygen is still required for F³⁺ supplementation (reaction (5)). Oxygen does not have to diffuse to the pyrite surface all the time, and Fe³⁺ can take its role. Complex chemical and microbiological processes occur on the surface layer of pyrite, which have thicknesses of up to several dozen nanometers [55]. The oxidation process of Fe²⁺ in acidic environments is very slow and is independent of pH [55]. The lack of a correlation between pH and SO₄²⁻ could be explained by the complex processes of the interaction between water and sediment, the transformation of secondary minerals, sorption and desorption and the presence of unstable intermediate forms of sulfur in water [56]. The rate of pyrite oxidation in natural conditions depends on the following factors: physicochemical and biological conditions, i.e., air and water flow, the bacterial growth rate, the habitat preferences of microorganisms, the presence of organic compounds, temperature gradients, the formation of secondary minerals, neutralization reactions and climatic conditions. In turn, the amount of acidic waters produced from pyrite depends on the rapid transport, the rate of convection and oxygen diffusion, the hydrological and climatic variables, the geomorphology of the area and the vegetation, mineral and structural composition of the waste. In the process of pyrite oxidation, the content of trace elements weakening the crystal network is important. The process of the decomposition of pyrite or marcasite is faster if these minerals form amorphous aggregates and contain significant isomorphic admixtures of arsenic [57]. The admixtures in the pirite of the isomorphic forms of Co and Ni can increase its resistance to oxidation. The presence of carbonates and aluminosilicates of different weather resistances and the inclusion of other minerals may accelerate the decomposition of parent minerals. Pyrite is subject to bacterial catalyzed oxidation [58,59]. Bacteria that slowly float in water, oxidize Fe²⁺ to Fe³⁺ and then decompose pyrite play a greater role than bacteria that adhere directly to its surface [60,61]. It is also recognized that pyrite can be successfully used as a redox proxy to evaluate the genetic model of the uranium deposits used in sandstone. The data suggest that, during the translation of the redox interface, changes in geochemical features that are well documented in the pyrite microscale are responsible for U mineralization [62,63].

5. Conclusions

In the mineralogical and geochemical studies carried out, particular attention was paid to the rock materials subject to the secondary mineralization process. In the varieties of Istebna sandstones and sublithic arenites, the process of pyritization was found. The recrystallization of iron sulfides in the tested material leads to the complete covering of free pores as well as systems of cracks and micro-cracks. This process will have a secondary impact on the reservoir features of the studied rocks by reducing the porosity.

From the preliminary observations, the geochemical affinity between the mineralized zone and the concentration of trace elements can be concluded. It was observed that, with the increase in pyrite sulfur content, the concentrations of Cu, Zn, Cd and Mn increase. The greater concentration of harmful elements in sulfides compared to that in the parent rocks is due to the possibility of the accumulation of these metals during pyrite growth. Such a relationship was emphasized in the works [7,64].

Accompanying minerals and parent rocks with buffering properties play an important role in reducing the negative effects of pyrite weathering. In this case, an inverse relationship was demonstrated between silica and iron sulfides. An increase in the Si content will cause a decrease in the Fe and S values. The pyrite matrix will be made of a weather-resistant quartz, which, in the absence of other sulfide minerals and buffering components, can act as a natural buffer zone. The structural and textural nature of pyrite, which, in the examined case, is in the form of a coarse-crystalline binder, will delay the process of its oxidation. Moreover, it was found that the forms of sulfides in the studied sandstones are massive and coarse-crystalline. These features will delay the process of the weathering of the mineralized zone and reduce the negative effects of iron sulfides on the natural environment.

The assessment of the geochemical interaction of iron sulfides with silica with a successively enlarged measurement was carried out using the fuzzy inference model. The presented results of the result function Y of changes in the content of Si, S and Fe in the sublithic arenite against the adopted empirical data prove the “average” assessment of the inference model for the experiments that were carried out. A “mean” result of the result variable Y was obtained for each tested point of the cross-section of the rock samples. The most favorable defuzzification result was obtained for sample No. 3. During the analyses, no anomalies or step changes in the value of the result function were noticed for the conducted research. Therefore, the obtained “average” result allows for a justified application of the Mamdanie–Assilian model to the content of Si, S and Fe in the sublithic arenite samples. Moreover, increasing of reliability of the expert result obtained from the model can be achieved by expanding the scope of the experimental data and by additionally comparing them with other independent analyses.

A benefit of using the fuzzy inference method described in this paper is the possibility of using it for various types of experiments. Another advantage of this model is the possibility of extending it with additional input variables. This will enable the prediction of the potential effects of the stabilization of changes in Si, S and Fe contents as a function of the location on the discrete grating of the cross-section of the sublithic arenite sample.