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Volume 32 Issue 4
Aug.  2021
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You Zhou, Songtao Wu, Zhiping Li, Rukai Zhu, Shuyun Xie, Xiufen Zhai, Lei Lei. Investigation of Microscopic Pore Structure and Permeability Prediction in Sand-Conglomerate Reservoirs. Journal of Earth Science, 2021, 32(4): 818-827. doi: 10.1007/s12583-020-1082-7
Citation: You Zhou, Songtao Wu, Zhiping Li, Rukai Zhu, Shuyun Xie, Xiufen Zhai, Lei Lei. Investigation of Microscopic Pore Structure and Permeability Prediction in Sand-Conglomerate Reservoirs. Journal of Earth Science, 2021, 32(4): 818-827. doi: 10.1007/s12583-020-1082-7

Investigation of Microscopic Pore Structure and Permeability Prediction in Sand-Conglomerate Reservoirs

doi: 10.1007/s12583-020-1082-7
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  • The microscopic pore structure of sand-conglomerate rocks plays a decisive role in its exploration and development of such reservoirs. Due to complex gravels-cements configurations and resultant high heterogeneity in sand-conglomerate rocks, the conventional fractal dimensions are inadequate to fully characterize the pore space. Based on the Pia Intermingled Fractal Units (IFU) model, this paper presents a new variable-ratio factor IFU model, which takes tortuosity and boundary layer thickness into consideration, to characterize the Triassic Karamay Formation conglomerate reservoirs in the Mahu region of the Junggar Basin, Northwest China. The modified model has a more powerful and flexible ability to simulate pore structures of porous media, and the simulation results are closer to the real conditions of pore space in low-porosity and low-permeability reservoirs than the conventional Pia IFU model. The geometric construction of the model is simplified to allow for an easing of computation. Porosity and spectral distribution of pore diameter, constructed using the modified model, are generally consistent with actual core data. Also, the model-computed permeability correlates well with experimental results, with a relative error of less than 15%. The modified IFU model performs well in quantitatively characterizing the heterogeneity of sand-conglomerate pore structures, and provides a methodology for the study of other similar types of heterogeneous reservoirs.
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    Gao, S. S., 2012. Research on Seepage Theory and Use of Petroleum Reservoir Engineering of Sang-Conglomerate Reservoir Formation in Mobei Oilfield, Xinjiang: [Dissertation]. China University of Geosciences, Beijing (in Chinese with English Abstract)
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    Li, C. X., Lin, M., Ji, L. L., et al., 2017. Investigation of Intermingled Fractal Model for Organic-Rich Shale. Energy & Fuels, 31(9): 8896-8909. https://doi.org/10.1021/acs.energyfuels.7b00834 doi:  10.1021/acs.energyfuels.7b00834
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    Liu, Z. X., Yan, D. T., Niu, X., 2020. Insights into Pore Structure and Fractal Characteristics of the Lower Cambrian Niutitang Formation Shale on the Yangtze Platform, South China. Journal of Earth Science, 31(1): 169-180. https://doi.org/10.1007/s12583-020-1259-0 doi:  10.1007/s12583-020-1259-0
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Investigation of Microscopic Pore Structure and Permeability Prediction in Sand-Conglomerate Reservoirs

doi: 10.1007/s12583-020-1082-7

Abstract: The microscopic pore structure of sand-conglomerate rocks plays a decisive role in its exploration and development of such reservoirs. Due to complex gravels-cements configurations and resultant high heterogeneity in sand-conglomerate rocks, the conventional fractal dimensions are inadequate to fully characterize the pore space. Based on the Pia Intermingled Fractal Units (IFU) model, this paper presents a new variable-ratio factor IFU model, which takes tortuosity and boundary layer thickness into consideration, to characterize the Triassic Karamay Formation conglomerate reservoirs in the Mahu region of the Junggar Basin, Northwest China. The modified model has a more powerful and flexible ability to simulate pore structures of porous media, and the simulation results are closer to the real conditions of pore space in low-porosity and low-permeability reservoirs than the conventional Pia IFU model. The geometric construction of the model is simplified to allow for an easing of computation. Porosity and spectral distribution of pore diameter, constructed using the modified model, are generally consistent with actual core data. Also, the model-computed permeability correlates well with experimental results, with a relative error of less than 15%. The modified IFU model performs well in quantitatively characterizing the heterogeneity of sand-conglomerate pore structures, and provides a methodology for the study of other similar types of heterogeneous reservoirs.

You Zhou, Songtao Wu, Zhiping Li, Rukai Zhu, Shuyun Xie, Xiufen Zhai, Lei Lei. Investigation of Microscopic Pore Structure and Permeability Prediction in Sand-Conglomerate Reservoirs. Journal of Earth Science, 2021, 32(4): 818-827. doi: 10.1007/s12583-020-1082-7
Citation: You Zhou, Songtao Wu, Zhiping Li, Rukai Zhu, Shuyun Xie, Xiufen Zhai, Lei Lei. Investigation of Microscopic Pore Structure and Permeability Prediction in Sand-Conglomerate Reservoirs. Journal of Earth Science, 2021, 32(4): 818-827. doi: 10.1007/s12583-020-1082-7
  • Compared with conventional sandstone reservoirs, conglomerate reservoirs are highly heterogeneous due to their complex detrital composition and texture, which makes pore structure characterization difficult (Xu et al., 2019; Yang et al., 2017). Moreover, conglomerate reservoirs often exhibit high content of clay matrix and friable composition, making it difficult to measure some physical properties such as porosity and permeability using conventional methods (Xu et al., 2020; Chen et al., 2015; Wu et al., 2011). Given the complexity of pore-throat structures of conglomerate reservoirs, pore structures have been classified into unimodal, bimodal, and complex modals depending on the size and relative content of grains (Qian et al., 2016). However, the concept of 'modal' was proposed based on an ideal grain model, which may not represent the reality of complex pore space. Additionally, the integration of Micro-CT and MAPS analyses indicates that conventional statistical method is inadequate for the quantitative characterization of the spatial distribution of pores in conglomerate reservoirs (Zhou et al., 2018; Kuang et al., 2017).

    In recent years, fractal theory has been widely applied in quantitative characterization of pore structures in complex reservoirs. He and Hua (1998) pointed out that rock pore space exhibits fractal characteristics within a certain scale range. Therefore, it is possible to construct an equivalent fractal porous medium based on self-similarity of rock pore structure to simulate real pore space (Liu et al., 2020; Wan et al., 2020; Erol et al., 2017; Ju et al., 2014). The fractal dimension is a highly significant parameter in fractal theory. However, with the development of fractal theory, some scholars have successively proposed that using the single-scale fractal dimension will inevitably miss part of effective information (Lyu et al., 2017). In other words, the single-scale fractal dimension is inadequate to accurately describe the characteristics of complex fractal systems. This is because fractal objects with different geometric features may have the same, or similar fractal dimension (Zhang, 2011).

    Given the limitations of the single-scale fractal dimension, an intermingled fractal units (IFU) model was introduced to simulate rock pore space by altering the iteration rule of the Sierpinski carpet (Pia and Casnedi, 2017; Pia, 2016; Pia et al., 2016a, b, 2015; Pia and Sanna, 2014a, 2013). This model has proven useful in simulating porous media in terms of thermal conductivity, elastic deformation, and permeability. Intuitively, the IFU model, considered to be several superimposed fractals, can present a detailed characterization of diverse rock space. Pia and Sanna (2014b) simulated real porous media using the IFU model, and then derived a formula for computing permeability. In this model, the ratio factor (multiple of equal division of side length in each iteration) is set to be three. However, this is a simple iteration method, because the side lengths of pores generated in each iteration have a rigid variation (Atzeni et al., 2008). Concretely, the values of the pore diameters generated after each iteration are always one-third of those before the iteration, resulting in that once the maximum value of pore diameter is given, the values of all other pore diameters are then fixed. Consequently, the fixed-ratio factor in the iteration process will inevitably lead to the limited ability of the model to simulate the pore size distribution spectrum of porous media. Therefore, to enhance the simulation capability of the model, this limitation must be removed to permit the generated pore diameters to take more values. Additionally, previous studies have shown that conglomerate reservoirs in the study area are classified as typical dense reservoirs with low porosity and permeability (Shan et al., 2016), thus their seepage characteristics are quite different from those of reservoirs with medium and high permeability. Some scholars have found that the percolation characteristics of conglomerate reservoirs exhibit distinct non-Darcy features (Gao, 2012), and this phenomenon is largely due to the effect of the boundary layer (Lou et al., 2014; Shi et al., 2011). Therefore, the thickness of the boundary layer cannot be neglected in the calculation of permeability of conglomerate reservoirs. However, the pia IFU model assumes that pore space is composed of simple parallel capillary bundles, an assumption that tends to produce large errors in calculating the permeability of such reservoirs.

    This paper, based on Pia and Sanna's study (2014b), attempts to modify the Pia IFU model to construct a more generic model with a variable-ratio factor. The value of the scale factor can be determined according to the actual conditions, making the model more flexible in simulating pore size distribution in the porous medium. Also, taking tortuosity and the thickness of boundary layer into consideration, a new expression for permeability is derived to investigate the permeation performance of conglomerate rocks.

  • The reservoir in the study area has large burial depths (1 000-4 500 m) and diverse types of storage spaces. Also, it has undergone multiple stages of diagenetic superimposition and transformation, making the lithological characteristics of the reservoir extremely complex. According to statistics, rock types in the study area can be roughly divided into three categories. Among them, about 35% are sandstones, 15% are pebbly sandstones and 50% are conglomerates, and among the conglomerates, fine-grained conglomerates are dominant. The porosity of the tested samples ranges from 7.4% to 13.5% with an average value of 10.5%, and the permeability ranges from 0.005 4 to 18.6 mD with an average value of 2.14 mD, showing typical characteristics of low-porosity and low-permeability.

  • In recent years, many researchers have simulated the pore space of real porous media based on two classical fractal structures: the Sierpinski carpet, and the Menger sponge (the 3D form of the Sierpinski carpet) (Vita et al., 2012; Sergeyev, 2009). The Sierpinski carpet is an exactly self-similar fractal, which can be generated by applying certain iteration rules (Fig. 1).

    Figure 1.  Generation of the Sierpinski carpet.

    A Sierpinski carpet is derived from a square which is considered to be rock matrix. In the first iteration, the side length is divided equally into three sections. Specifically, the parent square is divided into nine child squares. The child square situated in the center is removed, in the representation of a pore. The procedure is repeated with the other eight child squares, and so on to infinity. In Fig. 1, the pores and matrix are sketched in white and black, respectively. In general, the more types of fractal units, the higher the accuracy of simulating porous media. However, the increase in the number of fractal units will lead to massive calculations. Therefore, in practice, three or four types of fractal units are generally selected to satisfy the accuracy requirements (Li et al., 2017; Pia and Sanna, 2014b). Next, construction rules of the IFU model are presented in detail.

    Figure 2 shows the iteration process of three fractal units with different iteration parameters. Unit A is a standard Sierpinski carpet fractal unit, while Unit B and C are variant carpet fractal units. In Unit B, two child squares are removed from the parent square in the first iteration. Besides, the grey sections in Unit C are excluded from iteration and retained as matrix with no pore formation. Np is the number of the child squares removed from parent squares in each iteration (i.e., the number of the pores formed in each iteration). Thus, we have NAp=1 and NBp=NCp=2. Nsolid is the number of child squares that do not take part in the iteration. Thus, we have NAsolid=NBsolid=0 and NCsolid=1. The value i is the iteration times. According to fractal geometric theory, the fractal dimension is dependent on the number of squares that participate in the next iteration (Niteration) and the ratio factor (b).

    Figure 2.  Schematic diagram showing the iteration process of different fractal units.

    Thus, we have DfA=log8/log3=1.89; DfB=log7/log3=1.77; and DfC=log6/log3=1.63. Without loss of generality, Unit C is employed to discuss the calculations of some other iteration parameters.

    As shown in Fig. 3a, let dmax be the maximum pore side length, the side length of the fractal unit will then be 3dmax. With an increase in the iteration times, the side lengths of pores generated exhibit a regular variation of dmax/3, dmax/9, dmax/27, and so on. In general, let b be the ratio factor. The side length of the new pores formed in the i-th iteration is then given by

    Figure 3.  Schematic diagram of fixed and variable ratio factor IFU.

    The number of new pores formed in the i-th iteration (Ni) is expressed as follows (Li et al., 2018).

    Afterwards, the pore volume produced in the iteration process and the porosity of the fractal units are easy to calculate (Pia et al., 2014a), making it possible to reproduce the pore size distribution spectra with several different fractal units. Finally, these various fractal units with different parameters can be superimposed to form an IFU model.

  • As previously mentioned, making the ratio factor variable allows the IFU model to produce more differential values of pore diameters, which makes the model possess more powerful and flexible simulation capabilities. Figure 3b shows some fractal units with variable-ratio factor, and different values of the ratio factor can be selected depending on the actual conditions. Therefore, combining different fractal units with variable-ratio factors together, with the consideration of tortuosity, we can obtain a modified IFU model as shown in Fig. 4.

    Figure 4.  Schematic diagram showing the modified IFU model.

    Also, the thickness of boundary layer and tortuosity are two key parameters in the modeling process. The boundary layer is considered to be a layer of immobile wet phase fluid attached to the inner wall of the pore, causing the effective flow diameter (de) to be smaller than the real pore diameter (d) (Fig. 5). Therefore, the effective flow diameter (de) is

    Figure 5.  Schematic diagram of influence of boundary layer on fluid flow in tight reservoir.

    where h is the thickness of boundary layer, cm. Tian et al. (2014) and Meng et al. (2017) proposed a quantitative expression for the thickness of boundary layer by reorganizing the micro-tube experimental data.

    where r is the pore radius, μm; μ is the viscosity of fluids, MPa·s; $ \nabla p $ is the pressure gradient, MPa/m. Since the pressure gradient in the experiment was considerably greater than 1 MPa/m, the formula below was chosen for calculation.

    Clearly, considering the tortuosity, according to the Posenille Law, the flow of a single capillary tube with diameter d is given by

    where q is the flow, cm3/s; Δp is the pressure difference between the ends of the capillary tube, atm; L0 is the characteristic length of the capillary tube, cm; and τ is the tortuosity (dimensionless parameter), which is the ratio of the capillary tube's real length (Lt) to the characteristic length (L0). Then

    To facilitate theoretical calculations, it was assumed that the percolation section of each square pore is its maximum incircle. Besides, based on the average tortuosity-porosity analytical model for porous media consisting of square particles (Yu and Li, 2004), a more favorable expression was proposed to calculate the average tortuosity for porous media consisting of circular particles (Yun et al., 2008).

    where ϕ is the porosity, %. Obviously, the tortuosity of a fractal unit is 1 if its porosity is 100%, which means that the capillary tube's real length Lt is equal to its characteristic length L0, showing a straight flowline. While the porosity of the fractal unit decreases to 0, then its tortuosity tends towards infinity. This is consistent with previous theoretical studies.

    Combining Eqs. (2) and (6), we obtain the flow (q(d)) of a single pore produced after the i-th iteration

    The total flow (Qi) of new pores produced after the i-th iteration is obtained by combining Eqs. (3) and (9).

    Thus, the cumulative flow (Q) of the pores generated from the first to the k-th iteration is given by

    According to the equivalent percolation principle, the cumulative flow of the assumed rocks model is equivalent to that of the real rocks.

    where K is the permeability of a particular fractal unit, μm2; A is the section surface, cm2. Finally, the expression for calculating K is derived as follows.

    Clearly, the expression is a function of some iteration parameters and reflects the characteristics of the microscopic pore structure of porous media. The empirical constant is not required in this expression, and each item has specific physical significance. Similarly, in the case of an IFU model containing several fractal units, Eqs. (4), (8) and (11) can be adopted to compute their effective flow diameters, tortuosity and cumulative flow, respectively. Then, the respective flows are accumulated and the total permeability of the IFU model is easily obtained using the equivalent percolation principle.

    In particular, if the IFU model contains only one fractal unit, then A is given by

    Thus, the permeability of the model is derived as follows.

  • In simulating real rock pore systems using the IFU model, it is essential to ensure that the pore size distribution spectra and porosity given by the model are consistent with the experimental data. In this paper, the modeling process is summarized as "fit first, then predict". The specific steps for permeability computation are given as follows (Fig. 6).

    Figure 6.  Calculation process of model parameter (redrawn and modified based on the image from Pia and Sanna, 2014b).

    (1) Measure the porosity, permeability and pore size distribution of the rock samples. A variety of methods are available for measurement of pore size distribution, such as mercury injection capillary pressure (MICP), nuclear magnetic resonance (NMR), SEM and CT scanning. Then, pore size distribution is converted into cumulative curve of pore size distribution to simultaneously display the pore size distribution and porosity, which is convenient for subsequent fitting and comparison.

    (2) Select fractal units with different variable-ratio factors to replicate the pore size distribution and the porosity of the samples. Note that the size of the model (e.g., height and section size) is consistent with the real rock samples. Meanwhile, programming is used to adjust the iteration parameters to reduce the gap between the data obtained by the model and the experimental ones. During the adjustment process, the ratio factor (b), the number of fractal units (N), and the number of new pores produced in each iteration (Np) have the greatest influence on the fitting effect, thus they are the key parameters that need to be adjusted (see Section 2.2). Normally, fitting runs from the maximum pore size (which is determined by the experimental data) to the minimum pore size.

    (3) Fractal expression of permeability is employed to calculate the permeability of the model.

  • In this section, MICP measurement data are utilized to build the IFU model, and a satisfying fitting result is obtained (Fig. 7). Additionally, compared with the results in Pia and Sanna (2014b), it can be seen that by introducing variable-ratio factors into the modeling process, the modified model fits the experimental data better, showing a more powerful simulation ability. This provides a solid foundation for subsequent permeability forecasting.

    Figure 7.  Comparison of cumulative curve of pore size between the IFU model and experimental data for typical samples.

    Table 1 shows the parameters of the IFU models of some typical samples. Using these input parameters, the effective flow diameter, tortuosity and cumulative flow of each IFU model can be calculated, and then, the total permeability of the sample can be determined.

    Sample ID Number of fractal units Number of pores generated Iteration times Ratio factor Maximum pore diameter (μm) Number of units excluded from iteration
    FN16-19 Unit A 50 1 2 8 80 1
    Unit B 1×104 3 4 4 10 1
    Unit C 1.4×108 3 3 4 0.18 0
    Unit D 1.6×109 8 1 3 0.03 0
    M20-04 Unit A 4.2×105 1 2 5 1.26 1
    Unit B 2.5×107 5 3 3 0.26 0
    Unit C 2.3×109 3 2 3 0.037 0
    Unit D 1.3×1010 1 1 3 0.02 0
    B64-33 Unit A 6×103 8 2 3 5 0
    Unit B 2.5×106 2 3 3 1.7 0
    Unit C 2.2×109 3 3 2 0.05 0
    Unit D 0 0 0 0 0 0
    B65-8 Unit A 7×104 1 2 3 3.2 0
    Unit B 2×106 2 3 4 1.1 0
    Unit C 5×108 5 2 3 0.077 0
    Unit D 1.6×109 2 2 4 0.04 0
    B64-3 Unit A 3.5×104 3 2 3 2.1 0
    Unit B 9×106 1 3 2 0.72 0
    Unit C 1.2×108 2 3 2 0.19 0
    Unit D 1.5×109 3 2 4 0.05 0
    B64-38 Unit A 1×103 1 2 3 3.2 0
    Unit B 1.4×107 1 6 2 1.1 0
    Unit C 3.2×109 1 2 3 0.04 0
    Unit D 0 0 0 0 0 0

    Table 1.  Parameters of the IFU models of some typical samples

    Comparison between permeability results by experimental tests and model calculations are presented in Table 2 and Fig. 8, illustrating a good agreement between the predictions of the modified model and the experimental data. Additionally, the calculation accuracy of the improved model is higher than that of the Pia model, demonstrating its validity and reliability in simulating pore space of real porous media, with a relative error of less than 15%. Furthermore, it is found that lower permeability of samples leads to larger deviations in the Pia model. This is because the Pia model does not consider the effect of the boundary layer in tight reservoirs, causing the calculation error to be amplified in permeability prediction of dense rock samples.

    Sample ID Experimental permeability(10-15 m2) Model-computed permeability and relative error
    Modified model Pia model
    Permeability(10-15 m2) Relative error (%) Permeability(10-15 m2) Relative error (%)
    FN16-19 18.6 18.698 0.5 21.617 16.2
    M20-04 0.041 4 0.040 2 2.9 0.058 2 40.6
    B64-33 3.056 3.051 0.2 3.695 20.9
    B65-8 0.327 0.347 6.1 0.398 21.7
    B64-3 0.183 0.202 10.4 0.244 33.3
    B64-38 0.551 0.602 9.3 0.639 16.0
    B64-42 0.368 0.387 5.2 0.455 23.6
    B64-29 0.033 0.038 15.2 0.044 33.3
    M5-1 0.403 0.411 2.0 0.496 23.1
    M101-2-2 0.034 0.037 8.8 0.045 32.4
    M5-6 0.005 4 0.006 11.1 0.008 48.1

    Table 2.  Comparison between model-computed permeability and experimental ones

    Figure 8.  Comparison between permeability results by experimental test and model calculations. Solid line represents the perfect agreement between the experimental data and model predictions.

    Also, it is observed that the calculated values of the improved model are generally slightly larger than the experimental ones. This may be due to the development of microcracks in the rock samples caused by the excessive pressure in the MICP measurements, which in turn leads to larger calculations.

    Besides, combined with the model parameters in Table 1, it can be seen that the IFU model with more large-sized fractal units will have higher permeability value, basically regardless of the number of small-sized units. This reveals that although a large number of micropores have large reservoir capacity, their contribution to permeability is quite limited, while the developmental degree of macropores with a relatively small volume fraction is an important factor in improving permeability. This is consistent with previous theoretical and experimental studies, demonstrating that the model can provide a better understanding of seepage characteristics and rules of reservoir rocks.

  • Iteration parameters control the microstructure and permeability of pore systems. An analysis of a specific fractal unit was performed to further illustrate the effects of these parameters on permeability. The primary iteration parameters are iteration times (i), the number of pores produced in the iteration (Np), the number of child squares excluded from iteration (Nsolid), and the ratio factor (b). The permeability of a single fractal unit is determined by Eq. (15).

  • Let Np=1, Nsolid=0 and b=3, the maximum pore diameter of the model varies between 0.001 to 100 μm. Figure 9 reveals that many small pores are generated in the model with the increase of iteration times, resulting in increases in porosity and permeability, and then, permeability tends to be a constant. This is because the newly produced micropores contribute little to permeability. Namely, it has a diminishing effect on permeability as the number of iteration times increases. Therefore, an excessive iterative process is unnecessary, which can also lead to a significant increase in the calculation amount of the model. In general, iteration times of 2 to 4 may be enough.

    Figure 9.  Change of permeability with i.

  • Let i=5, Nsolid=0 and b=3. Np has a direct influence on the porosity, and the number of pores in the model. Figure 10 shows that permeability increases significantly with the increase of Np. This is because greater values of Np allow the presence of more large pores, and more pores are generated in each iteration, leading to a higher permeability. Equation (15) indicates that a power function relationship exists between permeability and Np when the other parameters are fixed. Therefore, Np has a significant influence on permeability and should be given priority in the process of parameter adjustments. In other words, when we are going to reproduce the pore size distribution spectrum of some samples with high porosity, a vital issue we should consider is to adjust the value of Np to be large enough, which will facilitate the adjustment of other parameters.

    Figure 10.  Change of permeability with Np.

  • Let i=5, Np=1 and b=3. The permeability tends to decrease slightly as Nsolid increases. This is because an increase in Nsolid means that a portion of the model will be excluded from iteration and retained permanently as matrix, resulting in a decrease in porosity and permeability. Equation (15) exhibits a linear relationship, rather than a power function relationship, between permeability and Nsolid when the other parameters are fixed. As a result, the adjustment of this parameter does not bring significant changes to the permeability (Fig. 11). In some cases, we can even ignore the effect of this parameter. Generally, a value of 0 or 1 is sufficient.

    Figure 11.  Change of permeability with Nsolid.

  • Let i=5, Np=1 and Nsolid=0. The ratio factor has a distinct influence on pore structure in the initial stage and permits a significant decrease in permeability. However, as the ratio factor continues to increase, the decrease in permeability becomes progressively smaller. This is because the larger the ratio factor leads to the smaller generated pores in the model with the same iteration times, and these microcapillaries have limited contribution to the permeability (Fig. 12). Although, the later stage of parameter adjustment for b have little effect on permeability, the effect in the initial stage (i.e., ranges from 2 to 6) is non-negligible and even decisive. The ratio factor allows for fine control of the microscopic pore structure, which theoretically explains the reason why the models with variable-ratio factor has more flexible simulation capabilities than the fixed-ratio factor ones.

    Figure 12.  Change of permeability with b.

  • (1) A modified variable-ratio factor IFU model is proposed, which has a relatively simple geometric structure and can be regarded as an important porous medium equivalent. Compared with the fixed-ratio factor IFU model, the modified model allows better fitting with experimental data and has more flexibility in simulating pore size distribution spectra.

    (2) A new expression for the permeability calculation, taking tortuosity and thickness of boundary layer into consideration, is derived in the modified model. The empirical constant is not required for this expression, and each item has specific physical significance. The effectiveness and accuracy of the model have been proven by correlating computed values with experimental data.

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