Multiscale and Multiphysical Approaches to Fluids Flow in Unconventional Reservoirs
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Lele Liu, Nengyou Wu, Changling Liu, Qingguo Meng, Haitao Tian, Yizhao Wan, Jianye Sun, "Maximum Sizes of FluidOccupied Pores within HydrateBearing Porous Media Composed of Different Host Particles", Geofluids, vol. 2020, Article ID 8880286, 14 pages, 2020. https://doi.org/10.1155/2020/8880286
Maximum Sizes of FluidOccupied Pores within HydrateBearing Porous Media Composed of Different Host Particles
Abstract
Hydraulic properties of hydratebearing sediments are largely affected by the maximum size of pores occupied by fluids. However, effects of host particle properties on the maximum size of fluidoccupied pores within hydratebearing sediments remain elusive, and differences in the maximum equivalent, incircle, and hydraulic diameters of fluidoccupied pores evolving with hydrate saturation have not been well understood. In this study, numerical simulations of graincoating and porefilling hydrate nucleation and growth within different artificial porous media are performed to quantify the maximum equivalent, incircle, and hydraulic diameters of fluidoccupied pores during hydrate formation, and how maximum diameters of fluidoccupied pores change with hydrate saturation is analyzed. Then, theoretical models of geometry factors for incircle and hydraulic diameters are proposed based on fractal theory, and variations of fluidoccupied pore shapes during hydrate formation are discussed. Results show that host particle properties have obvious effects on the intrinsic maximum diameters of fluidoccupied pores and introduce discrepancies in evolutions of the maximum pore diameters during hydrate formation. Porefilling hydrates reduce the maximum incircle and hydraulic diameters of fluidoccupied pores much more significantly than graincoating hydrates; however, hydrate pore habits have minor effects on the maximum equivalent diameter reduction. Shapes of fluidoccupied pores change little due to the presence of graincoating hydrates, but porefilling hydrates lead to much fibrous shapes of fluidoccupied pores.
1. Introduction
Natural gas hydrates vastly stored in marine sediments along the continental margins have a great potential to become one of global unconventional hydrocarbon energy resources [1–3]. Currently, the exploitation of this potential energy resource is still not economically feasible and requires innovative production methods and techniques [4, 5]. New methods and techniques should be well evaluated by numerical simulators prior to field applications, and results of these numerical evaluations largely depend on proper characterizations of coupled processes and appropriate quantifications of physical properties of hydratebearing sediments [6–9]. Hydraulic properties of hydratebearing sediments are quite crucial [10–12], and they still represent an ongoing research issue although great effort has been made [13–16].
Hydraulic properties (e.g., saturated water permeability, water retention curve, and relative permeability to water and gas) of hydratebearing sediments are inherently governed by porescale structures of the solid matrix and pores [17–20]. The pore space occupied by water and/or gas within hydratebearing sediments shrinks due to solid hydrate formation, and structures of fluidoccupied pores can be highly diverse due to different hydrate pore habits (e.g., graincoating and porefilling) even though hydrate saturations are identical. These diverse pore structures are experimentally observed, and how they change during hydrate formation or dissociation has been quantified by using varieties of parameters with clear physical significances. Examples include porosity, shape factor, Euler characteristic of individual hydrate cluster, fractal dimensions, pore surface, and pore volume and size [21–24]. Various pore sizes (e.g., the critical, mean, and maximum pore sizes) have been correlated to the hydraulic permeability of porous media, and larger pore sizes generally lead to higher values of the hydraulic permeability [25–27]. Hydraulic properties of porous media are significantly affected by the maximum pore size [27–29], and the maximum pore size is experimentally and theoretically correlated with porosity, permeability, and particle size [30–33]. Grain sizes of marine sediments hosting natural gas hydrates are of a wide range from clays and silts to coarsegrained sands, and sand particle shapes are generally different [34–36]. Effects of host sediments properties (e.g., porosity, grain size, and shape) on the maximum size of fluidoccupied pores within hydratebearing sediments remain elusive, although papers have been published to clarify how the maximum size of fluidoccupied pores evolve with hydrate saturation during hydrate formation and dissociation [23, 37].
Pores and fluid channels within most porous media in nature are generally nonspherical and noncircular [38–40]. For these irregularly shaped pores, several definitions of pore diameter are applied to quantify pore sizes. Examples include the equivalent diameter [41], the incircle diameter, and the hydraulic diameter [42]. The equivalent diameter is a diameter of a circle having an area equal to the pore area. The incircle diameter is determined by using the maximum ball method [43], and it is widely applied to pore network extractions from porous media [16, 24, 44]. The hydraulic diameter is defined as , where is the crosssectional area and is the wetted perimeter of the crosssection. The hydraulic diameter is commonly used to simplify fluid flow in noncircular tubes and channels as round pipe flow [17, 45, 46]. For fluidoccupied pores within hydratebearing sediments, the maximum pore size is expected to decrease with increasing hydrate saturation. However, similarities and differences in the maximum equivalent, incircle, and hydraulic diameters of fluidoccupied pores evolving with hydrate saturation have not been well understood.
This study is aimed at clarifying the effects of host particle properties on the maximum size of fluidoccupied pores within hydratebearing porous media and further the understanding of different maximum pore diameters evolving with hydrate saturation. Graincoating and porefilling hydrates are randomly nucleated and grew within different artificial porous media to quantify the maximum equivalent, incircle, and hydraulic diameters of fluidoccupied pores at selected hydrate saturations, followed by analyses of maximum fluidoccupied pore diameters changing due to the presence of gas hydrates. Then, theoretical models for incircle and hydraulic diameterrelated geometry factors are proposed based on the fractal theory, and these proposed models are further applied to provide insights into the hydrate saturation and morphologydependent pore shape changes during hydrate formation.
2. Methods
Ten square images of artificial porous media shown in Figure 1(a) are generated by using the method of [47] for further numerical simulations of hydrate nucleation and growth in this study. Each of these porous media is constructed by randomly placing black particles with unrestricted overlap into a white square image until the desired porosity has been reached. The porosity is calculated as the ratio of white over total pixel numbers in a square image, and the square image has a side length of 300 pixels. The shape of a solid particle is characterized by using a concept of sphericity which is defined as , where is the area and is the perimeter of the solid particle [48]. Particle sphericity values are normally no bigger than 1, and represents a circular solid particle. In addition, stands for an elliptical solid particle with a major over minor diameter ratio of 2, and for a diameter ratio of 3. Sphericity and size values of solid particles used for porous media constructions are summarized in Figure 1(b), and Figure 1(c) shows the intrinsic porosity and median particle diameter map for those artificial porous media.
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Random nucleation and growth of graincoating and porefilling hydrates within artificial porous media is numerically simulated by using the method of [37]. These numerical simulations continually calculate minimum distances of fluidoccupied pore pixels to solid particle and hydrate pixels, and fluidoccupied pore pixels are selectively turned into hydrate pixels until the desired hydrate saturation has been reached. Different nucleation and growth preferences represent different hydrate pore habits. Graincoating hydrate nucleation and growth is simulated by stochastically changing candidate pore pixels with into hydrate pixels. Porefilling hydrate nucleation and growth is modeled by preferentially seeding hydrate in candidate pore pixels with the highest , followed by randomly changing hydratetouched pore pixels into hydrate pixels, and the value of the growth parameter is set to be 0.7 in this study. Each case of numerical simulations is 100 times performed to obtain probabilistically acceptable results. For more details, please refer to our previous paper [37].
The maximum equivalent , incircle , and hydraulic diameters of fluidoccupied pores within hydratebearing porous media are quantified from to at intervals of . When a preselected hydrate saturation is reached, fluidoccupied pores are extracted from hydratebearing porous media by using the function named “bwlabel” with (i.e., 4 connected pixels) in MATLAB 2016Ra, followed by calculations of the area and perimeter for all the fluidoccupied pores. Calculated values of the area and perimeter are further used to determine values of equivalent and hydraulic diameters, and the incircle diameter is quantified by using minimum distance values. Assuming that there is a rectangular fluidoccupied pore with side lengths of 5 px and 6 px in hydratebearing porous media (Figure 2(a)), the equivalent diameter is calculated to be , the incircle diameter , and the hydraulic diameter according to their definitions (Figure 2(b)). Values of the maximum equivalent, incircle, and hydraulic diameters of pores within hydratefree porous media are determined and summarized in Figure 2(c) as intrinsic maximum pore diameters.
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3. Results
The maximum equivalent , incircle , and hydraulic diameters of fluidoccupied pores within ten images of porous media containing gas hydrates are shown in Figure 3, and all the pore diameters decrease with increasing hydrate saturation . The average value of intrinsic maximum equivalent diameters of fluidoccupied pores within ten porous media (Figure 1(a)) is with a confidence interval of 95.4% (the same below), and it decreases to (19.2% of the intrinsic value) and (3.6% of the intrinsic value) when hydrate saturation is 0.8 for graincoating (Figure 3(a)) and porefilling (Figure 3(b)) hydrates, respectively. The average value of intrinsic maximum incircle diameters of fluidoccupied pores within ten porous media is , and it decreases to (58.6% of the intrinsic value) for graincoating hydrates (Figure 3(c)) and (2.1% of the intrinsic value) for porefilling hydrates (Figure 3(d)) when . The average value of intrinsic maximum hydraulic diameters of fluidoccupied pores within ten porous media is , and it decreases to (54.3% of the intrinsic value) and (23.9% of the intrinsic value) when for graincoating (Figure 3(e)) and porefilling (Figure 3(f)) hydrates, respectively. It is obvious that porefilling hydrates reduce values of the maximum equivalent, incircle, and hydraulic diameters more efficiently than graincoating hydrates when , and the maximum hydraulic diameter is the least sensitive to hydrate saturation.
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Host particle properties (e.g., intrinsic porosity, particle size, and sphericity) obviously affect the intrinsic maximum equivalent , incircle , and hydraulic diameters of fluidoccupied pores (Figure 2(c)) and introduce diversities (41.1 px for , 23.3 px for , and 9.1 px for ) in values of the intrinsic maximum pore diameter. All the intrinsic diversities decrease due to the presence of gas hydrates, and this implies that effects of host particle properties decrease with increasing hydrate saturation. In addition, porefilling hydrates (Figures 3(b), 3(d), and 3(f)) have more significant effects on the intrinsic diversity reductions than grainfilling hydrates (Figures 3(a), 3(c), and 3(e)) when .
In order to obtain further understandings of hydrate saturation and morphologydependent maximum equivalent, incircle, and hydraulic diameters, Figure 4 shows normalized maximum equivalent , incircle , and hydraulic diameters of fluidoccupied pores within hydratebearing porous media. Normalized maximum pore diameters are defined as ratios of maximum pore diameters within hydratebearing over hydratefree porous media. All the values of normalized maximum equivalent diameters of fluidoccupied pores within porous media containing graincoating (Figure 4(a)) and porefilling (Figure 4(b)) hydrates generally fall into the region between upper and lower curves, with values partially lower than when hydrate saturation is lower (i.e., ) for graincoating hydrates and higher (i.e., ) for porefilling hydrates. These two models are derived based on the assumption that gas hydrates uniformly grow into all pores with different sizes, and for their derivations, refer to [29, 37]. Porefilling hydrates reduce normalized maximum incircle (Figure 4(d)) and hydraulic (Figure 4(f)) diameters of fluidoccupied pores more significantly than graincoating hydrates (Figures 4(c) and 4(e)). Normalized maximum incircle and hydraulic diameters of fluidoccupied pores decreasing due to the presence of graincoating hydrates (Figures 4(c) and 4(e)) can be generally described by using . Normalized maximum incircle diameter of fluidoccupied pores within porous media containing porefilling hydrates decreases with increasing hydrate saturation as when in a general trend (the blue dot curve in Figure 4(d)), and normalized maximum hydraulic diameter can be generally depicted by using (Figure 4(f)).
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An empirical model [23, 37] is applied to fit values of different normalized maximum pore diameters, and values of empirical parameters are summarized in Figure 4(g). The empirical model is an alternative form for the weighted average of theoretical models for porefilling hydrates and for graincoating hydrates [23]. Values of empirical parameters and are largely controlled by hydrate pore habits. For graincoating hydrates, the normalized maximum equivalent diameter of fluidoccupied pores can be described by setting and , normalized maximum incircle diameter by setting and , and normalized maximum hydraulic diameter by setting and . For porefilling hydrates, the normalized maximum equivalent diameter of fluidoccupied pores can be described by setting and , normalized maximum incircle diameter by setting and , and normalized maximum hydraulic diameter by setting and .
4. Discussion: Pore Shape Changes due to Hydrate Formation
Fractal theory is widely used to characterize porescale structures and investigate various physical (e.g., hydraulic and electrical) properties of porous media [49, 50]. In these investigations, pores within porous media are treated as circles with equivalent areas in the twodimensional space, and the maximum equivalent diameter can be calculated as [50] where is the total area of porous media, is porosity, and is poresize fractal dimension [28] which can be determined by using the boxcounting method [51].
For the case that size of a pore within porous media is quantified by using incircle diameter (Figure 5(a)), area of the pore can be calculated as and is a geometry factor for incircle diameter. For references, the geometry factor for incircle diameter for squareshaped pores, for regular triangleshaped pores, and for circular pores. Then, the maximum incircle diameter can be easily calculated as
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If hydraulic diameter is used to quantify sizes of pores within porous media (Figure 5(a)), area of an individual pore can be calculated as and is a geometry factor for hydraulic diameter. For references, the geometry factor for hydraulic diameter for squareshaped pores, for regular triangleshaped pores, and for circular pores. Then, the maximum hydraulic diameter can be easily calculated as
If graincoating hydrates uniformly grow in pores with different sizes and do not alter the shape of fluidoccupied pores (Figure 5(b)), area , incircle , and hydraulic diameters of these pores decrease with increasing hydrate saturation as , , and , respectively. In these equations, subscript 0 represents the intrinsic (i.e., hydratefree) condition and subscript c stands for graincoating hydrates. Then, it is easy to obtain according to Equation (2), and based on Equation (4). Normalized geometry factor for incircle diameter in porous media containing graincoating hydrates can be defined as according to Equation (6), and normalized geometry factor for hydraulic diameter as according to Equation (7).
If porefilling hydrates uniformly grow in pores with different sizes and the hydrate shape is identical with the intrinsic pore shape (Figure 5(c)), area , incircle , and hydraulic diameters of these pores decrease with increasing hydrate saturation as , , and , respectively. In these equations, subscript f represents porefilling hydrates. Then, it is easy to obtain according to Equation (2), and based on Equation (4). Equation (10) is applicable for , and when . Normalized geometry factor for incircle diameter in porous media containing porefilling hydrates can be defined as according to Equation (10), and normalized geometry factor for hydraulic diameter as according to Equation (11).
Values of the poresize fractal dimension for fluidoccupied pores within porous media containing graincoating and porefilling hydrates are summarized in Tables 1 and 2, respectively. The pore diameter ratio defined as the ratio of the minimum over maximum pore diameters can be calculated by [52] and values of the pore diameter ratio within porous media containing graincoating and porefilling hydrates are shown in Figure 6. It is obvious that all the values are generally smaller than , and the fractal theory can be used to analyze properties of porous media containing gas hydrates in this study [52].


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Values of geometry factor for incircle diameter within hydratebearing porous media can be calculated by using Equations (3), where . Geometry factors for incircle diameter changing due to the presence of graincoating and porefilling hydrates are shown in Figures 7(a) and 7(b), respectively. It is obvious that hydrate saturation has little effects on values compared with values. The normalized geometry factor for incircle diameter changing due to the presence of graincoating hydrates is shown in Figure 7(c) and porefilling hydrates in Figure 7(d). It is shown that values generally stay close to the horizontal red line (Figure 7(c)), and values go through an evolutionary process from below to above the red curve (Figure 7(d)). These discrepancies between numerical simulated data and corresponding theoretical models (i.e., Equations (8) and (12)) are mainly due to differences in hydrate pore habits since graincoating and porefilling hydrate growths can hardly follow the uniform and selfsimilar way (Figure 5) strictly. Based on simulated data, an empirical model is proposed as to depict how values evolve with hydrate saturation. Equation (15) with can capture the essential physics of pore shape changes in hydratebearing porous media during hydrate formation, while and set the lower and upper bounds.
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Geometry factors and normalized geometry factors for hydraulic diameter evolving with hydrate saturation are shown in Figure 8. It is obvious that hydrate saturation and morphology effects on and values are similar with those on and values. Normalized geometry factor for hydraulic diameter in porous media containing graincoating hydrates changes much more mildly compared with normalized geometry factor . The black curve in Figure 8(d) drawn by using Equation (13) agrees well with the general trend of values increasing due to the presence of porefilling hydrates. Based on simulated data, another empirical model is proposed for the normalized geometry factor prediction, which is with describing the general trend, setting the lower bound, and setting the upper bound.
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