Mixtures of colliery spoils and fluidized bed bottom ash: strength and swelling behavior under compressive stress in isolated conditions

The amount and variety of waste materials used in civil engineering are steadily increasing due to the depletion of natural resources and the implementation of environmental policies. A new mixture of colliery spoils and fluidized bed bottom ash was recently developed in Poland, which proved to be prone to excessive, long-term swelling, potentially damaging structures such as road embankments. Numerous papers discuss the heaving mechanisms in spoils or fluidized bed ash. However, no reports analyzing the expansion of their mixture seem to be available. The purpose of this paper is to fill this void. The experiment involved measurements of linear swelling of mixture samples stored inside CBR test molds and isolated from moisture loss/gain. Material expansion is caused by the reactivity of the fluidized bed ash; its content in the mix correlates positively with the final swelling strain. At the same time, colliery spoils provide an environment particularly conducive to the phenomenon’s occurrence. A linear swelling strain of 0.3–3.2% was observed after 6 months (disregarding the first 24 h of rapid volume growth). The expansion of mixtures was also measured under constant compressive stress of 0–100 kPa to better represent their actual working conditions. Based on the results, it is proposed to adapt a mathematical model developed for anhydrite-rich rock to describe the load-swelling relationship in mixtures of colliery spoils with fluidized bed bottom ash. Such a model can be potentially used, e.g., in numerical analyses, to assess the impact of the material’s expansion on embankments.


Introduction
The depletion of natural aggregate deposits worldwide (Bisht 2022), including in Poland (Kozioł 2019), generates a growing demand for alternative aggregates, particularly those of waste origin (Meininger and Stokowski 2011). Unburnt colliery spoils (also referred to as coal shale, mine tailings, or minestone), a by-product of hard coal mining, belong to this category. This material is available in an exceptionally high supply in mining regions; its annual production in Poland amounts to about 30 million tons, approximately 24% of the total weight of waste produced (GUS 2020). This material has been used for many decades, among other things, in constructing flood protection dikes (Laan et al. 1984) and waste impoundment embankments in coal basins (Rainbow et al. 1987). Presently, spoils are also used for road embankments under certain conditions (Kettle 1983;Toussaint 1984;Cadierno et al. 2014). The main benefits of such applications are high availability (regionally) and low price, but also specific geotechnical properties beneficial for embankment construction (Skarżyńska 1995;Cadierno et al. 2014;Duszyński et al. 2018;Suescum-Morales et al. 2019). Its wide use is essential for implementing the circular economy concept (Kledyński et al. 2020). On the other hand, it is an exceptionally heterogeneous material with a complex and highly variable composition, excessive moisture content, susceptible to, inter alia, grain disintegration, spontaneous combustion, and leaching of pollutants (Kettle 1983;Skarżyńska 1995;Cadierno et al. 2014).
To address these adverse properties, some researchers have proposed modification of the spoils, e.g., by supplementing fine-grained fractions or stabilizing the material using a binder (Kettle 1983;Rainbow et al. 1983;Thomas et al. 1987;Suescum-Morales et al. 2019). In 2011, a Polish mining company began the production of ash-stabilized colliery spoils approved for use in road embankments and flood protection dikes. According to Szymkiewicz et al. (2012), the proposed modification method achieved grain size and moisture content correction, increased strength, reduced filtration coefficient, and lowered solubility. The binder used was fluidized bed combustion bottom ash, a coal-fired power plant waste produced in Poland in significant quantities, and an estimated 2.5 million tons per year (Kledyński et al. 2020). The resulting mixture, composed entirely of industrial by-products, is of considerable ecological and economic value and thus highly attractive for the construction of embankments, especially in mining regions.
However, due to its unique but not fully recognized properties, some specific problems and risks are associated with utilizing this novel anthropogenic material. Both components of the mixture are susceptible to volumetric expansion. The spoils are prone to at least three different physical and chemical swelling mechanisms -absorption of water on soil particles, swelling of clay minerals (such as montmorillonite), and, predominantly, oxidation of pyrite and the subsequent hydration and crystallization of sulfates (often driven by microorganisms) (Quigley et al. 1973;Grattan-Bellew and Eden 1975;Caldwell et al. 1984;Bérubé et al. 1986;Hawkins and Pinches 1987;Pye and Miller 1990). Fluidized bed combustion ash can also expand due to its strong chemical reactivity related to the presence of sulfates (Rajczyk 2012;He et al. 2021). Combining these two industrial by-products generates a potentially cumulative risk of substantial swelling. Such suspicion was confirmed in preliminary studies of the long-term expansion of spoils-ash mixtures conducted at the Warsaw University of Technology (Kledyński et al. 2017), in which the material swelled by up to 1.7% in volume over nine months.
An important factor driving the soil swelling behavior is access to moisture; for this reason, typical standardized soil expansion testing methods assume that the material is submerged in water (EN 13286-47:2012, EN 13286-49:2004 (PKN 1998(PKN , 2007(PKN , 2012. The typically expected test duration is in the order of a few days. Meanwhile, several sources report that expansion in spoils can be reduced after water immersion (Quigley and Vogan 1970;Bérubé et al. 1986) and that swelling occurs in spoils-ash mixtures without access to additional water and over several months (Kledyński et al. 2017).
Examples of road pavement failures occurring on embankments incorporating fluidized bed ashes are described, among others, in Deschamps (1998) and Yoon et al. (2007). These and other sources point to their high calcium sulfate content as a critical factor causing volumetric expansion (Hunter 1988;Dermatas 1995;Weir et al. 2014). Anhydrite (CaSO 4 ) in contact with water hydrates into dihydrate (CaSO 4 ·2H 2 O) with an increase in the molar volume of about 61% (Kurdowski 2010). Soils containing gypsum, aluminates, and calcium oxide (all found in fluidized bed ashes) provide perfect conditions for the synthesis of ettringite if pore water and an elevated pH are present (Wang et al. 2019). The transformation of anhydrite into gypsum and ettringite involves joining water molecules, in the latter case, as many as 26. At the same time, there is a Fig. 1 Swelling test results of anhydrite-containing rock under various compressive stress; approximation of the relation by a linear-logarithmic model. Based on Wittke-Gattermann and Wittke (2004) change in the molar mass and an increase in the product's volume relative to the reaction's substrates by about 137% (if water is not taken into account). Moreover, Lea and Desch (1970), Maher et al. (2011), andHawkins (2012) report that the observed swelling values are higher than expected from the sole difference in the volume of substrates and products of chemical reactions. Ettringite is an expansive mineral, extensively discussed in the literature on cement concrete, forming needle-shaped crystals that, when growing, can lead to swelling and gradual fracturing of the hardened composite material due to excessive tensile stress (Dermatas 1995;Collepardi 2003;Scherer 2004;Pavoine et al. 2012;Wang et al. 2019). This effect is known as crystallization pressure, which occurs when crystal growth is confined in the material, i.e., inside voids of sufficiently small size (Taylor et al. 2001;Scherer 2004).
Similar swelling is observed in rocks containing anhydrite (CaSO 4 ) under sudden water inflow conditions, which is relevant, for example, for tunnel engineering (Wittke 2003). Wittke-Gattermann and Wittke (2004) citing (Grob 1972;Wittke 1978;Paul 1986;Kiehl 1990) described a model linking the linear swelling strain of the rock to the compressive stress counteracting the material's expansion. According to cited studies, this relationship has, in a given direction, the character shown in Fig. 1 (under the oedometer test). Above a specific value (σ 1 ), the compressive stress begins to reduce the swelling strain, and there is also a stress value significant enough to stop the expansion completely (σ 0 ). In between the two values, the load-swelling relationship is best approximated by a logarithmic curve. Understanding this relationship is crucial to realistically model the behavior of a swelling soil medium, e.g., using software based on the finite element method (Wittke-Gattermann and Wittke 2004). The referenced mathematical relation could potentially be used to model the swelling behavior of colliery spoils and fluidized bed bottom ash mixtures.
The subject of this article is the long-term swelling of mixtures of colliery spoils and fluidized bed combustion bottom ash under imposed constant loads in the range of 0-100 kPa. The purpose of the study is to provide a preliminary determination of the effect of mix composition and loading (compression) on the material's swelling strain. The experiment assumed storing samples in isolation from moisture inflow/loss. The test method determines the material's potential for expansion under conditions different from those typically applied in laboratory tests (water immersion). According to the authors, immersion is not particularly representative of a material embedded in a road embankment. It also does not necessarily provide the optimal conditions for the most significant long-term volume increase.

Materials and methods
The main component of the tested mixtures is unburnt coal shale (colliery spoils) from the Sobieski Mining Plant in Jaworzno, Poland. It was obtained in August 2018 directly from the mechanical coal enrichment plant after a wet coal extraction process. The second component of the mixture is bottom ash from the combustion of hard coal and coal sludge in circulating fluidized bed boilers. This dry-cooled material was obtained in August 2018 from Poland's Jaworzno II power plant. Selected properties of the spoils and ash are listed in Table 1, their appearance is shown in Figs. 2 and 3, and their oxide composition is listed in Table 2 (WDXRF test using a Philips PW 2400 spectrometer). The test mixtures were composed by combining the two raw materials with the optional addition of batch water. The samples were conceptually divided into two groups of recipes: • S group ("soil-like") -intergranular cavities mostly not filled with water or fine fractions, material's structure similar to stabilized soils. A portion of spoils of varied, regulated water content was mixed with an amount of bottom ash proportional to this moisture content, the latter component being treated as a moisture regulator of the mixture (maintaining a constant total water/ash ratio); • C group ("concrete-like") -cavities between coarse grains possibly tightly filled with ash-water paste, resulting in a material's structure similar to low-strength con-crete. Paste of different water/ash ratios was added to the water-saturated spoils, each time in equal volume, sufficient to fill the free spaces and separate the coarse grains (constant weight of the spoils portion, constant volume of paste in the samples).
The compositions of all test mixtures are characterized in Tables 3 and 4. The experiment included two additional control samples, prepared based on the S b recipe, except that ash/spoils were replaced entirely with natural aggregates of equivalent grain size distribution and with approximately equal total grain volume (K a /K b samples, respectively).
The experiment aims to investigate the factors influencing the swelling potential of the material with a composition close to optimum (to achieve high density and bearing capacity), as well as a number of mixtures with different component proportions. The recipe representative of the first type, S c (spoils:ash:water -1.000: 0.268:0.190) is similar to the one actually produced and built into embankments (spoils:ash:water -1.000:0.250:0.206) (Szymkiewicz et al. 2012). The other compositions in the experiment allow to investigate how deviations from the approximate optimum contents influence the strength and swelling potential of the mixtures.
The preparation of the test samples emulated the engineering application of spoils-ash mixtures. The setup of the swelling measurements is based on PN-EN 13286-47 (PKN 2012) (samples isolated from moisture exchange), with certain modifications. A portion of dry ash (S group) or ash-water paste (C group) was added to moist spoils (in most cases saturated with

Mixture description/assumptions
S a "Soil-like": varied water content w s in spoils Ash content adjusted depending on w s Voids in the mix not tightly filled with ash and water (air void ratio > 15%) w s = 10.5% S b w s = 16.5% S c w s = 19.0% C a "Concrete-like": water-saturated spoils Voids in the mix mostly filled with ash-water paste (air void ratio approximately 2-6%) Greater ash and water content than S group Water/ash = 0.30 (in paste) C b Water/ash = 0.35 (in paste) C c Water/ash = 0.40 (in paste) K a Control sample -based on S b , but FBC bottom ash was replaced with natural sand of a similar particle size distribution K b Control sample -based on S b , but colliery spoils were replaced with an aggregate of a similar particle size distribution (quartz and basalt aggregate mix) water) in appropriate amounts and mixed until homogenization was achieved. A situation was reproduced in which the ash is an additive that rectifies the excessive natural moisture content of the spoils to achieve a water content close to the optimum for compaction of the material (S group). At the same time, the influence of the phenomena occurring shortly after contact between the spoils and water on the mixture swelling was reduced since it happened in advance relative to the forming of the test samples. Freshly prepared mixtures were placed in CBR steel molds ( Fig. 4) and compacted using a Proctor apparatus (rammer A, mold type B according to PN-EN 13286-2 (PKN 2010)); a rubber sealing pad was placed at the bottom of the mold. The top surface of the material was further leveled manually, and a steel plate with an adjustable stem was placed on it, the movement of which was observed with a displacement gauge mounted on a steel tripod attached to the mold. After compaction and setting up of the mold equipment, the samples were isolated by sealing them in plastic wrap; some were additionally placed in a dedicated custom-made apparatus allowing the application of a vertical compressive load of up to 100 kPa (Fig. 5). The load value is coded at the end of the sample code, expressed in kPa (with an accuracy of 5%). Swelling measurements exclude the first 24 h after the samples were compacted in molds and isolated (a mellowing period). The experiment focuses on long-term swelling, which is of greater practical importance than "immediate" swelling. In full-scale applications, at least a few hours pass between the time the components are initially mixed and them being placed and gradually compacted. Another reason for mellowing is to limit the influence of short-term expansion caused by water absorption and swelling of clay minerals on the observed strain. All the test samples are listed in Table 5, with their codes given.
Sample deformation monitoring continued for about 5 months, and the samples were stored at 20 ± 1 °C. After the monitoring period, 10.5-cm diameter cores were drilled from the hardened material for uniaxial compressive strength testing. The material's modulus of elasticity was roughly measured (only on two S c recipe specimens, after 60 days of curing, using a method similar to PN-EN 13286-43:2005(PKN 2005). The swelling measurements are presented as relative linear strain (elongation expressed in %), related to the original height of the sample.

Results and discussion
The material of all mix compositions, except for S a , had a cemented form at the end of the observation period, resembling low-strength concrete (UCS reaching 7.3 MPa for C a samples), due to the binding reactions of ash in contact with water. This allowed cores to be drilled for strength tests (Table 6) and the elastic modulus estimation (an approximate value of 150 MPa was measured, S c recipe, 60 days of curing). A cross-section through such a core sample is shown in Fig. 6. Table 4 Summary of compositions of tested mixtures per 1 kg of colliery spoils (dry mass) a Ash replaced with quartz sand with a similar particle size distribution (and almost identical particle density) b Spoils replaced with a mix of quartz and basalt aggregates with a similar particle size distribution (but a significant difference in particle density)  Figure 7 shows an example SEM image of the microstructure of the hardened material, revealing the reaction products that probably account for the swelling and binding. The image confirms the sulfate-related origin of the phenomenon described shortly in the Introduction.

The effect of mixture composition on swelling strain
Graphs illustrating the swelling of mixture samples of different compositions have similar shapes, resembling a logarithmic curve or a reflected and shifted exponential curve. The phenomenon is monotonically increasing with the expansion rate decreasing over time and expires approximately between 100 and 160 days of observation; this is a behavior typical for a gradually decaying chemical reaction in isolated samples and known from research on coal shale (Caldwell et al. 1984). The final swelling value differs significantly between samples of different compositions. The development of deformation over time of all no-load samples is summarized in Fig. 8; the strain values at three selected time points and compressive strength after demolding the samples are given in Table 6. The swelling test results of the spoils-ash mixture samples and the two control samples confirm that the long-term expansion of the composite material results mainly from reactions of the fluidized bed bottom ash with water. K a sample, composed of colliery spoils and medium sand, does not significantly change its volume in the experiment -it lacks sulfates present in the ash. However, swelling might occur in it, for example, under conditions of additional water supply or within a short time immediately after preparation (due to water absorption and clay mineral swelling in spoils -not tested here). In the K b sample (bottom ash mixed with quartz/basalt aggregate), swelling occurs with a strain value similar to S a samples (having very low water content) but several times lower than the other tested compositions. Thus, the study's results suggest an amplifying effect, causing mixtures of spoils with fluidized bed bottom ash to possess Fig. 5 Loading device: a CBR with sample material inside, b rigid post transferring the load onto a steel plate, c lever support in a wall, d lever arm, and e applied load Table 5 Plan of the experiment -twenty test samples with their codes given A blank cell means that a sample with such properties was not tested. Control samples K a and K b were tested only under no load a Two test samples -S b 50 and S b 100 -the same mixture recipe but made of a separate batch from S b 0 and S b 9. S b 50 sample observed for a reduced period of 53 days

Load (kPa)
Experiment plan -sample codes 0 S a 0 S b 0 S c 0 C a 0 C b 0 C c 0 4 S a 4 9 S a 9 S b 9 25 S c 25 C c 25 50 S b 50 a S c 50 C a 50 C b 50 C c 50 100 S b 100 a S c 100 C a 100 C b 100 certain physical or chemical characteristics that predispose them to a relatively intense and prolonged volumetric expansion over time. For example, spoils' exceptionally high water absorption provides a large reservoir of water needed for reactions. The susceptibility of its grains to disintegration can promote the release and migration of reactants in the medium. It also means there are no significant cohesion forces to counteract crystallization pressure inside the voids in coarse grains. Due to its rich composition and instability, the coal shale may provide additional substrates for key chemical reactions, such as aluminates or sulfates stemming from delayed pyrite oxidation (Grattan-Bellew and Eden 1975; Bérubé et al. 1986;Hawkins and Pinches 1997).

S group
In the S group, the no-load samples can be arranged according to increasing swelling strain in the order S a , S b , S c -consistent with their increasing water and ash content.
The most water-deficient mixtures (S a ) are characterized by an order of magnitude lower swelling strain (about 0.3% compared to 1.5-2.5% for S b and S c ). Ash accesses the moisture contained in the spoils and hydrates even if it is below the absorption capacity of the shale. No additional water is needed for swelling to occur, either at the time of mixture preparation or added later. However, increased initial water content has a strong positive effect on swelling rates. Test results on the S a samples indicate that combining relatively dry spoils (well below saturation) with a limited amount of ash is relatively safe, probably because of competition between the water demand of spoils and ash particles. However, questions should arise regarding the usability of such mixtures due to, among other things, the non-existent stabilization (binding) effect. Reduced moisture content may also contradict the common aim of achieving the best possible compaction of the mix. Increased ash dosage (to adjust for the higher initial water content of the spoils), particularly in the S c formulation, leads to a more dense material structure with higher strength and is likely suitable for more efficient compaction. The described characteristics are generally favorable when considering the material for embankment construction. The experiment suggests that such an effort to maximize the tightness of the material's structure might simultaneously create optimal conditions for swelling, increasing the risk associated with it. The strength of the mixtures is also positively correlated with their final strain (Fig. 9). Thus, the same reactions (synthesis of gypsum and ettringite) probably lead to depletion of the water content, an increase in compressive strength, and swelling.

C group
Despite the relatively small variations in compositions in group C (see Table 4), markedly different values of swelling strain were observed; recipes ranked according to increasing water/ash ratio in the "paste" were C a , C b , and C c . The total swelling of the first of these (no-load sample) reached almost 3.3%, the greatest in the experiment. Including additional water in the mixes (increase in w/a) in the C b 0 and C c 0 samples did not increase the swelling -unlike in the S group. C group samples were characterized by a consistency and porosity similar to concrete mixes. This was coupled with a slight excess of water relative to ash, as the spoils added in saturated state did not compete for water. As a result, a negative correlation was observed between the mixture's water content and its swelling. This might be caused by an increase in the volume of pores available for expansive substances or a decrease in ash content, which contains the expansive sulfates. At the same time, as the w/a ratio increased, the strength of the material decreased, an observation typical of concrete technology. Again, it can be concluded that the same chemical reactions probably led to the increase in strength and swelling, but an excessive amount of water was achieved in the samples of the C group, limiting the effects of these reactions. This observation supports the need to test the mixtures in question not solely under complete immersion in water. The observed co-occurrence of the most significant strain and strength of the hardened material in the C a samples is probably simply due to the highest ash content in its composition. Hence, the conclusion that even a composition of the material that guarantees the greatest strength, similar to those of low-grade concretes, does not seem to ensure the formation of chemical bonds in the material that would effectively resist the tensile stress resulting from sulfate crystal expansion. The compressive stress values do not follow any particular trend depending on the load of the test samples. Either the load in the range of 0-100 kPa does not influence the strength of the hardened samples, or its influence is less significant compared to the random variations of test results on the material in the experiment. Fig. 9 The swelling strain after 60 days (no-load samples) plotted against the compressive strength after demolding the samples (age listed in Table 6). A "0" value of the strength of one of the S group specimens means that the material lacks cohesion and the test cannot be performed Based on the above discussion and correlation analysis, the conclusion is that the swelling potential of the mixtures in the experiment (also other properties, such as strength) is most effectively predicted by the ash content in their composition (Fig. 10), and secondly by the initial water content (Fig. 11). The first relationship is positive in both groups of samples; the latter is of the opposite nature in both groups. Ash, similar to cement, has a certain water demand required for complete hydration; water is also a medium in which the reactants migrate and crystallization occurs. Thus, a relatively high water content, in the range of 15-20%, is required for intense swelling, but values above a certain threshold cause a reduction in expansion, possibly due to an increased volume or size of voids in the material, allowing more freedom for sulfate crystallization.
The air void ratio is generally not strongly correlated with the swelling strain, though within just the S group, a likely negative correlation is present. The relative differences in swelling strain between samples do not change significantly over time. However, minor discrepancies are noticeable in the final fading phase of expansion, possibly due to some unintended differences in curing conditions. For that reason, the graphs provided show strain values 60 days after molding of the samples, an arbitrarily chosen point in time.
A diversity of factors influence the phenomenon, while the properties of the two constituent materials are highly variable. Thus, only a preliminary assessment of the swelling risk of spoils-ash mixtures can be based on the presented experiment. Swelling tests conducted on a specific material, composed, compacted, and mixed under conditions that mimic the designed application conditions as closely as possible can provide more reliable information on the swelling risk.
For example, the "PKEW" mixture applied for road and water engineering embankments is characterized by an ash/ spoils mass ratio of 0.25 and a moisture content at compaction of 0.165 (Szymkiewicz et al. 2012), which is similar to the tested S c composition (Table 4), whose linear swelling in the test exceeded 2.3%. The manufacturer of the mentioned mixture declares a swelling of 0.04%, presumably tested according to PN-EN 13286-47 and immersed in water (PKN 2012). The discrepancy may be due to natural differences in the properties of the spoils and ash used or to a significant difference in the test methodology, potentially related to curing conditions. For example, the restrictions on swelling of materials in road embankments allow no more than 2% strain according to the Polish standard PN-S-02205:1998(PKN 1998. Thus, the possibility of swelling of spoils-ash mixtures, which for almost all tested samples was in the range of 1.7-3.2% linear strain (Table 6), should not be neglected, especially since these are underestimated values disregarding the first 24 h of rapid volume growth. For comparison, Kwiecień et al. (2017) estimate that swelling of the order of 0.16-0.28% of the material embedded in the road embankment could have been the cause of the failure of an expansion joint in an overpass abutment.

The effect of loading on swelling strain
Figures 12 (S group samples) and 13 (C group samples) show the samples' swelling strain development over time. The shape of curves is similar regardless of loading, while the final value of the swelling strain varies significantly. Under loads in the range of 25-100 kPa, each tested mixture composition experienced an apparent reduction in strain value. Such a reduction does not occur in the range of about 0-9 kPa, or its little effect is unnoticed compared to the random differences between the test samples.
To help analyze the effect of the load value on swelling among different mixture compositions, the strain values were converted to a relative measure and presented in a summary chart (Fig. 14). It is equal to the ratio of the strain of the loaded sample to that of the corresponding no-load sample. It was calculated at arbitrary times of 30, 60, and 140 days since the sample preparation (a value of 1 means no reduction, and a value of 0 is a complete halt in swelling).
As mentioned earlier, loads up to about 9 kPa (S a and S b compositions) have no significant effect on the swelling Fig. 12 Swelling curves of S group samples (various loading). Samples made according to one mixture recipe (but with differing loads) are marked with one color rate or its final value under the conditions of the experiment. In the remaining sample groups, the load values were increased, and a significant reduction in swelling was observed under stress in the range of 25-100 kPa. A load of 25 kPa reduces the total strain by as much as 40% (S c composition). Increasing the stress to 50 and 100 kPa leads to strain reductions of about 40-50% and 50-60%, respectively; i.e., additional loading leads to a progressively lower strain reduction. The C c sample group (with the highest water content) deviates slightly from the rest, with a noticeably smaller reduction in swelling -by 23% at 25 kPa loading and 37% at 50 kPa. The S b group of samples is subject to uncertainty due to the preparation of S b 50 and S b 100 samples from a separate mixture batch and was omitted in the next part of the article as less representative.
Based on the graphs in Fig. 14, the relative strain reduction is virtually the same at each of the selected moments, i.e., after 30, 60, and 140 days of observation, so for the sake of this experiment, it can be considered constant over time. The collected data suggest a noticeable reduction in the swelling strain of spoils-ash mixtures is achieved at a certain stress value between 9 and 25 kPa. The experiment does not directly show at what pressure the swelling becomes negligibly small (e.g., reduced by 90%). Further predictions related to the load-swelling strain relationship are presented further in the article.

Modeling the relation between swelling strain and loading
Based on the presented results, an approximate model of the relationship between swelling strain and compressive stress can be proposed for the spoils-ash mixture (load-swelling model). Two basic mechanical parameters are needed: elastic modulus E = 150 MPa (based on the preliminary test performed in the experiment) and Poisson's number ν = 0.35 (assumed roughly after (Suescum-Morales et al. 2019)). The proposed model is also based on the extrapolation of load values well beyond the 0-100-kPa interval assumed in the experiment.
The analyzed relationship is entirely based on the rock swelling model mentioned in the introduction to the article (Huder and Amberg 1970;Grob 1972;Wittke-Gattermann and Wittke 2004). It assumes that the relationship of swelling strain to opposing compressive stress is well approximated by a logarithmic curve (tested in an oedometer, i.e., in a uniaxial state of strain, the same as in the CBR form). The envelope of results illustrated in the graphs in Fig. 14   Fig. 13 Swelling curves of C group samples (various loading). Samples made according to one mixture recipe (but with differing loads) are marked with one color (for S c , C a , C b , and C c recipes) is presented below in the logarithmic scale. The base strain of the non-loaded sample is arbitrarily assumed as 0.0235, equal to the test result on sample S c 0, whose recipe is most similar to the mixtures used in practical applications (Szymkiewicz et al. 2012) (and for which the modulus of elasticity was determined).
The vertical axis represents values of the "final" swelling strain (after 140 days of testing), obtained by multiplying the base value of 0.0235 by the corresponding reduction factor, as shown in Fig. 14 (for example, a factor equal to 1.0; i.e., no reduction, was assigned to a load of 9 kPa). The described graph is shown in Fig. 15a. The line of minimum reduction fits the logarithmic model well, while the line of maximum reduction is noticeably different. Due to the variations in the effect of loading on the swelling of mixtures of different compositions, the reduction averaged from the minimum and maximum values is considered here (marked with crosses). These points were used to approximate the dashed line shown in Fig. 15a. Wittke-Gattermann and Wittke (2004) state after Kiehl (1990) that the swelling can be modeled in each principal stress direction as dependent on the compressive stress in that direction. The following graph (Fig. 15b) presents only the averaged series. It differs from the previous one in that the strain values, originally measured in conditions of constrained horizontal strain, were recalculated to free swelling strain. It was done by subtracting the fraction of vertical strain resulting from compression acting in horizontal directions (with σ max stress). The isotropic nature of the material (and its swelling) was assumed, along with the previously mentioned values of E (150 MPa) and ν (0.35).
The presented model, based on test results and an extrapolation thereof, predicts that in the absence of stress, the final linear swelling strain of the sample is 0.0214 in each of the three orthogonal directions. Under compressive stress no greater than 8 kPa (σ min ), there is no reduction in swelling strain, and the relationship is logarithmic in the range of 8-444 kPa. Then, above 444 kPa (σ max ), further swelling stops. It applies independently to each of the three orthogonal directions (denoted by i in the formula). The value of the swelling-stopping stress σ max , both input and the result of the model, was calculated by successive approximations to be equal to 444 kPa -the x-intercept of the line of best fit (Fig. 15b). The transition from linear (low loads) to logarithmic relation (loads > σ min ) is a simplification -the model is not intended to represent such low loads accurately; it is more focused on higher load values (which are more important in a rock medium). Despite that, the initial linear part seems to be a good enough fit for the obtained results.
The load-swelling relationship, shown in Fig. 15b, can be described by Eq. (1): where: ε i,swell -final swelling strain in the direction of the principal stress i (−), ε 0 -final swelling strain under no load (here: 0.0214) (−), k -a constant describing the load-swelling relation (here: −0.249) (-), σ i -normal (compressive) stress in the i direction, constant during the swelling process (kPa), σ min -minimum stress value that causes a reduction in swelling (here: 8 kPa), σ max -normal stress value causing a complete reduction in swelling (here: 444 kPa).
As already mentioned, the swelling reduction ratio is almost constant over time. Thus, the load-swelling relationship described by Eq. (1) and in Fig. 15b can be directly transformed into a load-swelling rate relationship. To do so, the value of the final swelling strain (ε) should be replaced by the value of the swelling rate at a given instant t − (ε(t)), which is equal to the derivative of the swelling curve as in Figs. 8, 12, and 13. In that case, it is not enough to provide the final strain value

Fig. 15
Final swelling strain (after 140 days of testing) depending on the load (compressive stress), semi-logarithmic scale. a CBR test specimen, strain reduction envelope based in Fig. 14; b free swell-ing strain, linear-logarithmic model of the relationship according to Wittke-Gattermann and Wittke (2004) ε 0 ; knowledge of the swelling strain as a function of time for the zero-load sample is needed instead (ε 0 (t)). Despite this inconvenience, such an interpretation is more practical, as it allows for modeling the swelling of the material under dynamic conditions, such as time-varying stress, which is almost certain to occur when the material's volumetric expansion takes place. The application of such a model in the Abaqus/ CAE 2019 software is presented in the dissertation (Krysiak 2022) or of a similar one (though on a different material) in the Plaxis software, e.g., in Djellali et al. (2017). Although only tentatively drafted, the discussed approach appears to be a potential improvement over simulations that impose uniform volumetric swelling (Kwiecień et al. 2017), e.g., by enforcing it through the manipulation of the temperature and thermal expansion coefficient. Such simple models allow retrospective analysis but cannot a priori predict deformations in modeled structures and might lead to an overestimation of normal stress under deformation constraints.
To better assess the practical significance of the loadswelling relationship, one can imagine a 10-m-high embankment built of the spoils-ash mixture. At its bottom, the vertical compressive stress induced by the dead load is about 165 kPa. This suggests that the stresses encountered in typical earth structures can lead to a partial, but not a complete, reduction in the swelling strain of the material. At the same time, if deformation constraints are present in the structure, such as retaining walls, additional stress can occur due to swelling pressure, even of the order of σ max . In the article (Kwiecień et al. 2017), one example of the potential consequences was presented, in the form of the closure of expansion joints of an overpass, necessitating their repair. It is likely not the only danger associated with the embankment material's swelling; one should also consider potential rippling of the road surface, displacement of structures and installations founded in the embankment (e.g., lamp posts), displacement of retaining walls, etc.
The model presented here is preliminary and does not consider, for example, the creep of the material under longterm loading or the variation of E and ν values over time (as setting progresses).

Conclusions
The presented swelling tests on mixtures of colliery spoils and fluidized bed bottom ash allow to characterize the potential for volumetric expansion of the material and to determine the effect of composition and loading (compressive stress) on the phenomenon. The conclusions are as follows: 1. The mixtures undergo monotonic and gradually fading volume growth over up to 150 days; the range of measured linear strain of individual samples is 0.35-3.20% (excluding the first 24 h). Significant swelling of the material occurs even when cured under isolation from moisture access and over several months after the mixture is mixed and compacted; 2. The primary cause of swelling is the presence of fluidized bed ash, which reacts with water resulting in binding and volumetric expansion of the mixtures. The phenomena occurring in the hydrated grains of colliery spoils contribute little to the swelling observed in the experiment. However, its combination with fluidized bed ash creates exceptionally favorable conditions for swelling compared to other aggregates of similar grain size; 3. The ash content relative to the total mass of the mixture is a good predictor of swelling -a positive correlation is observed; 4. No additional water above the absorption capacity of the colliery spoils (16.5%) is needed for the swelling (elongation of up to 1.77%). With the water content of the spoils reduced to 10.5%, the swelling of the mixture is relatively short-term and notably smaller (0.35%). At the same time, excessive water (above the saturation of the spoils) amplifies the phenomenon up to 3.20% linear strain; 5. All the test mixtures (except the one with the lowest water content) developed a hardened form after 150 days of curing, resembling a low-strength concrete (UCS up to 7.3 MPa). Strength positively correlates with swelling strain, indicating that the same chemical reactions lead to bonding and expansion. It does not seem feasible to create a mixture with strength sufficiently high to resist the cause of the volumetric expansion of the material effectively; 6. Compressive stress in the range of about 0-9 kPa does not affect the strain values noticeably under the conditions of the experiment. However, loads of 25, 50, and 100 kPa cause a marked reduction in the final swelling strain of the material. This reduction reaches about 20-40%, 40-50%, and 50-60%, respectively, compared to mixtures without loading and depends on the mix composition; 7. The relationship between the swelling strain and the opposing compressive stress in a given direction can be approximated by the logarithmic model ε = k · ln (σ/σ max ), the one described i.a. in Wittke-Gattermann and Wittke (2004). This relationship is valid in the stress range from σ min = 8 kPa to σ max = 444 kPa. Note that the estimation of σ max is based on extrapolation beyond the maximum load value assumed in the experiment, which is 100 kPa; 8. The reduction of strain under constant load is almost constant over time. Thus, the presented functional relationship can similarly describe the final swelling strain or the instantaneous swelling rate (compared to the swelling of an unloaded sample at the same instant).
The discussed relationships and model are not universal and cannot be directly (quantitatively) transferred to other, even apparently similar spoils-ash mixtures. The inherent variability of the properties of the two components, differences in how the mixture can be prepared and compacted, etc., do not allow to draw such general conclusions.
Funding This research was partially funded by a grant awarded in 2021 by the Dean of the Faculty of Building Services, Hydro and Environmental Engineering at Warsaw University of Technology.
Data availability All collected and processed data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.