Fractal characteristics of pore networks and sealing capacity of Ordovician carbonate cap rocks: a case study based on outcrop analogs from the Tarim Basin, China

Understanding and predicting the main controls on the sealing capacity of carbonate cap rocks is of great significance for ultra-deep carbonate reservoir exploration and production. This study focuses on revealing the pore networks and sealing capacity of the Ordovician carbonate cap rocks in the Tarim Basin, by analyzing samples from outcrop analogs using optical and scanning electron microscopy and a combination of mercury intrusion capillary pressure and nitrogen gas adsorption. Three classes of cap rocks are defined here according to their pore throat structure, fractal dimension and sealing capacity. These carbonate cap rocks are dominated by limestones and dolomitic limestones. Four pore types are identified: microfracture, intragranular pore, intercrystalline pore and intracrystalline pore. Six pore structure types show multiscale variability from macropores to micropores. The pore structures present multiple fractal behaviors, with fractal dimensions showing an increasing trend as the pore diameter decreases. The cover coefficient, a parameter that allows characterization of the cap rock sealing performance, shows an increasing trend along with increasing the fractal dimension of pore structure. The average cover coefficients of six pore 3 structure types not only show good correlations (either exponential or linear) with certain fractal dimensions, but they also demonstrate a strong positive correlation with the average fractal dimension. These results suggest that the sealing capacity of the studied rocks increases with increasing fractal dimension. The sealing performance of cap rocks significantly decreases with increasing the amount of macropores. This work provides a relevant case study for further understanding of pore structures and sealing capacity of carbonate cap rocks.


Introduction
In recent decades, carbonate reservoirs have attracted increasing attention owing to the abundant oil and gas reserves they store worldwide (Garing et al., 2014;Gundogar et al., 2016;Burberry and Peppers, 2017;Volatili et al., 2019). The Tarim Basin in Northwest China is the largest sedimentary basin in the country with an area of 56×10 4 km 2 (21.6×10 4 mi 2 ) (Fig. 1A, B) (Lü, et al., 2017;Pang et al., 2018). This complex basin has become one of the key petroleum exploration and production locations due to the abundant hydrocarbon reserves stored within the Ordovician carbonate strata (Kang, 2005). For example, there are major hydrocarbon breakthroughs in ultra-deep carbonate reservoirs (at depths ranging from 6,000 to 8,000 m [19,685 to 26,247 ft]) from the Tahe oilfield in the Tabei Uplift. This field has an 4 annual production of 730×10 4 tons (53.509 mbl) of crude oil and 15×10 8 m 3 (52.972 bcf) of natural gas (Lü et al., 2014).
Pervasively distributed carbonate cap rocks acting as effective sealing layers play a significant role in the preservation of hydrocarbons within carbonate reservoir strata (Lü et al., 2014;Zhang et al., 2015;Rahmani et al., 2018). Accordingly, these cap rocks have received increasing attention, especially the carbonates of the Ordovician Yingshan Formation, because they are considered one of the key sealing rocks in the Tarim Basin (Jin, 2014;Lü et al., 2017;Wu et al., 2019a).
Despite recent studies focusing on the identification and characterization of carbonate cap rocks that have been carried out in the Tazhong uplift (Lü, et al., 2017) and the Tahe oilfield (Wu et al., 2018a(Wu et al., , b, 2019a of the Tarim Basin, only a few researchers have focused on analyzing the pore structures of these cap rocks due to their heterogeneity and complex diagenetic history. The morphology and connectivity of pore networks in such rocks are key factors that determine their sealing potential (Norbisrath et al., 2015;Lohr and Hackley, 2018).
Several methods can be utilized to characterize the heterogeneous pore structure of carbonate rocks, including their pore geometry, connectivity and size distribution.
These techniques include optical microscopy, scanning electron microscopy (SEM), field emission scanning electron microscopy (FE-SEM), mercury intrusion capillary pressure (MICP), nitrogen gas adsorption (N2GA), nano X-ray computed tomography 5 facility (CT), nuclear magnetic resonance (NMR), and digital image analysis (Rezaee et al., 2012;Anovitz et al., 2013;Clarkson et al., 2013;Norbisrath et al., 2015;Gundogar et al., 2016;Cao et al., 2018;Njiekak et al., 2018;Xiao et al., 2018;El Sharawy and Gaafar, 2019;Górniak, 2019). The combined use of multiple pore structure characterization methods (such as optical microscope, SEM, and the combination of MICP and N2GA tests) can reveal the wide range of pore sizes and pore size distributions (PSDs) from the nanometer to the micrometer scale (Ross and Bustin, 2009;Schmitt et al., 2013;Cao et al., 2015;Gao et al., 2019), their full aperture and the overall pore throat structure of tight carbonate cap rocks (Wu et al., 2019a). The capillary leakage limits the capacity for retention of natural hydrocarbon accumulations (Vavra et al., 1992;Ingram et al., 1997;Wilkinson et al., 2014). The sealing capacity of carbonate cap rocks is controlled by the capillary pressure (McPhee et al., 2015;Wu et al., 2018a) and can be estimated by measuring the displacement pressure at 10% mercury saturation during MICP analysis (El Sharawy and Gaafar, 2019). This technique has recently been adopted by Lohr and Hackley (2018) to evaluate the sealing capacity of marine shales from the Tuscaloosa Group in Mississippi (USA).The fractal geometry theory can be used to study the features of pore networks of sedimentary rocks and to characterize reservoirs (Pfeifer and Avnir, 1983). The fractal dimensions calculated through different models, such as the Brunauer-Emmett-Teller (BET) (Brunauer et al., 1938), the Barret-Joyner-Halenda (BJH) (Joyner et al., 1951) and the Frenkel-Halsey-Hill (FHH) models (Avnir and Jaroniec, 1989) quantitatively reflect the 6 heterogeneity of rock pore structures. The fractal characteristics of carbonate rocks have historically been less studied than those of tight sandstone and shale reservoirs (Clarkson et al., 2013;Cao et al., 2015;Zhang et al., 2018;El Sharawy and Gaafar, 2019). This is partly because carbonate rocks undergo complex diagenesis Adelinet et al., 2019), resulting in heterogeneous pore connectivity, and therefore pore structures that are challenging to characterize quantitatively (Krohn, 1988;Lesniak and Such, 2006;Gundogar et al., 2016;Wu et al., 2019a).
Despite the huge economic importance of carbonate cap rocks, there are very few systematic studies relating pore structures to sealing capacity (e.g., Kaldi and Atkinson, 1997;Górniak, 2019). Furthermore, the complex relations between pore networks and fractal dimensions are poorly understood (Anovitz et al., 2013). The present study contributes to filling this knowledge gap by systematically characterizing the pore structures of the Ordovician Yingshan Fm. carbonate cap rocks in the Tarim Basin, as well as characterizing the fractal dimensions of the rocks, and using that information to evaluate their sealing capacity.
Previous attempts to systematically characterize pore size distributions and qualitatively investigate the relationships between a series of parameters and sealing capacity of subsurface carbonate cap rocks of the Tahe oilfield have been carried out using samples from a core from a burial depth of >5,500 m (18,045 ft) (Wu et al., 2019a).
However, statistical calculations of the dominant lithological types of carbonate cap 7 rocks in the Tahe oilfield are likely to be inaccurate due to the limited volume that the core represents. Thus, a systematic characterization of the spatial distribution of these cap rocks cannot be fully achieved because of their deep burial. To overcome this limitation, four outcrop sections with well-exposed carbonate cap rocks of the Ordovician Yingshan Fm. in the Keping and Bachu counties, along the cliff of the Keping uplift in the Tarim Basin have been chosen in this study (Fig. 1C) (Zhu and Ma, 1991). These outcrops expose continuous carbonate successions that can contribute to estimates of the relative importance of lithological types, and at the same time allow a full description of lateral and vertical variability. In order to establish a link between outcrops and wells by the same authors , the terms used to categorize the carbonate rocks studied at outcrops are also same to those in the Tahe oilfield. These outcrops are analogs to the subsurface Ordovician Yingshan Fm. carbonate reservoir fracture-vug-cavity system, because they present the same rock fabric, were deposited in the same settings, present a very similar diagenetic history (Kang, 1989), and were affected by the same tectonic events resulting in a similar deformation history Zhou et al., 2019).
The aims of this study are to: (1) analyze the pore structures of the Yingshan Fm.
carbonate cap rocks and their fractal dimensions, (2) unravel the links between fractal variations and rock sealing capacity, and (3) classify the carbonate cap rocks based on their pore structures and sealing capacity. This study provides useful insights into the sealing properties of ultra-deep carbonate cap rocks and can also be relevant for the 8 exploration and production of deep carbonate reservoirs in other basins worldwide.

Geological setting
The Keping uplift is located in the northwestern region of the Tarim Basin ( Fig.   1B, C), with an area of 18×10 4 km 2 (6.95×10 4 mi 2 ) . The studied outcrops in the Keping uplift experienced deformation during multiple tectonic phases, including the late Caledonian and the early Hercynian orogenies .
The Penglaiba Fm. is characterized by fine crystalline and medium crystalline dolomites that correspond to a carbonate platform ramp in which there was evaporation . In addition, the Yingshan carbonate formation in this study is about 250-400 m (~820-1,312 ft) thick at the outcrop sections and can be further subdivided into two parts according to lithological differences. The lower part primarily comprises lime-rich dolomite and peloidal-bearing dolomitic limestone, and has been interpreted as resulting from sedimentation in a restricted carbonate platform (lagoon and tidal flat) 9 (Ji et al., 2013). The upper part is composed of intraclastic grainstone, mudstone and packstone, and was deposited in open platform conditions (intraplatform shoal and intershoal sea) (Wu et al., 2018b;Wang et al., 2019). The dense intervals of the Yinghshan Fm. act as local cap rocks for karst reservoirs with fracture-vug-cavity systems . The Yijianfang Fm. is composed of bioclastic grainstone, framestone, and oolitic grainstone. This association presents reef and shoal complexes along the carbonate platform margin . This variation of the depositional environment was probably caused by relative sea-level fluctuation (Chen et al., 1999;Gao et al., 2016).
The burial history for the studied Yingshan Fm. rocks in carbonate successions directly entered the progressive burial stage after deposition, within a short period of time (Fig. 3). The maximum burial depth reached 4,000 meters (13,123 ft) (Du et al., 2017). Most importantly, this succession underwent significant uplift during the late Hercynian orogeny during the Late Permian, resulting in the whole strata exposed to the surface (Kang, 1989(Kang, , 2005 (Dong et al., 2013). Six main diagenetic processes affected these rocks, including micritization, cementation, dolomitization, dissolution, and mechanical and chemical compaction (Du et al., 2016(Du et al., , 2017. The timing of each diagenetic process can be seen in Fig. 3 Dunham (1962). Because the Ordovician Yingshan Fm. carbonate rocks are exposed to the surface and thus they are susceptible to weathering and erosion. Two steps were taken during sample collection: (i) removal of weathered carbonate layers at the outcrop surface to expose fresh rock surface; and (ii) the rocks were then drilled to a depth of 40~50 cm (1.31~1.64 ft) using a Shaw single backpack drill rig, to obtain intact cylindrical rock samples. Coring was carried out to minimize damage as much as possible. The samples were measured using multiple analytical techniques, including thin-section optical and electron microscope petrography, a combination of MICP and N2GA analysis, porosity and permeability tests, and fractal dimension calculation.

Optical and electronic microscope petrography
To study the petrographic properties of the Ordovician Yingshan Fm. carbonate cap rocks of this study, 263 samples from outcrops were selected for thin section 11 preparation. Three criteria for the sample selection were used: (1) sampling was carried out in the upper and lower parts of the Yingshan Fm.; (2) the sampling locations were chosen to gather samples from strata with varying thicknesses (including thin-bedded, medium-bedded, thick-bedded and massive rocks); (3) almost the same number of samples were collected from each of the four outcrops, of which 66 samples were from the Penglaiba, Kepingshuinichang and Dabantage outcrops and 65 samples from the Yangjikan outcrop. Thin sections of the samples were studied under a standard polarized microscope at the China University of Geosciences (Beijing). 126 of these sections were half-stained with Alizarin Red S to differentiate calcite and dolomite, and 108 of them were impregnated with epoxy resin dyed with methylene blue to highlight the pore systems and their distributions. The dolomite content was quantitatively determined from more than ten photomicrographs from each thin section through measuring volumetric percentages with image analysis software.
Scanning electron microscopy (SEM) was used to characterize the size of pore space and its location and cement morphology in the cap rock intervals. Fractured surfaces of seven samples coated with gold were examined using a FEI Quanta FEG450 SEM with a working current set at an accelerating voltage of 20 kV at the experimental research center of unconventional technology research institute of China National Offshore Oil Corporation (CNOOC).

Combination of MICP and N 2 GA analyses
A total of 19 cylindrical plugs (Fig. 4

Pore size distributions obtained from the mercury intrusion capillary pressure (MICP) tests
An AUTOPORE IV9520 Micropore Structure Analyzer was utilized to perform the mercury intrusion capillary pressure (MICP) tests following the Chinese standards of GB/T 21650.1-2008. All cylindrical plugs were dried in a vacuum oven at 105ºC [221℉, ~378 K] for 24 hours. Subsequently, each plug was placed in the permeameter, then the sealed permeameter was inserted into the low-pressure port of the device and the air was evacuated by injecting mercury. Next, the mercury-intruded permeameter was removed from the low-pressure port and weighted. After that, the permeameter was placed in the high-pressure port and was loaded with mercury under a stepwise applied pressure (Hao et al., 2017;Guo et al., 2020).
The pore volume of each rock sample (Vp) can be calculated with: where Vp is the pore volume of the rock sample(in mL), φ is the porosity (in %), MHg 13 is the rock sample mass (in g), ρ is the rock density (in g/cm 3 ).
To calculate mercury saturation, the total number of test points (n) and the increments of mercury saturation at the first test data point (ΔSHg)1 can be expressed as: where VHg1 is the intrusion volume of mercury at the first point (in mL) and Vb1 is the intrusion volume of mercury in the blank test (in mL).
Similarly, the increments of mercury saturation at other test points ((ΔSHg)i, in %) can be calculated: where i is the serial number of test points in the range of 1-n, VHgi and VHg(i-1) are the cumulative intrusion volume of mercury corresponding to each test pressure (in mL), Vbi and Vb(i-1) are the cumulative intrusion volume of mercury at each test point in the blank tests (in mL), VHgi-VHg(i-1) is the intrusion volume of mercury caused by the pressure increase from Pi-1 to Pi (in mL) and Vbi-Vb(i-1) is the intrusion volume of mercury caused by the pressure increase from Pi-1 to Pi in the blank tests (in mL). In The cumulative intrusion volume of mercury corresponding to each test point (SHg)i can be calculated using: Therefore, the pore space not occupied by mercury (φi) can be expressed as:

Pore size distributions obtained from the nitrogen gas adsorption (N2GA) tests
Nitrogen gas adsorption (N2GA) tests were conducted using a JWBK-22 Surface Area Analyzer to determine the full-aperture and pore size distributions of the study rocks following the Chinese standard of GB/T 21650.2-2008. About 0.5 grams powder (<100 mesh) from each sample was heated to degas and remove adsorbed moisture at 150ºC [302℉, ~423K] for at least two hours. N2 adsorption volumes at -195.85ºC [-320.5℉, 77.3K] liquid nitrogen condition were obtained under a wide range of relative pressures (P/Po) ranging from 0.001 to 0.990. In general, one milliliter of nitrogen gas can be condensed into 1.547×10 -3 milliliters of liquid nitrogen under standard conditions (Gregg et al., 1967). Hence, the total pore volume was converted from the maximum N2 adsorption. The volume of pore size in the range of 2-200 nm was calculated using the BJH model (Joyner et al., 1951).
The increments of saturation from the first test point to the second one where V0 is the adsorption volume at the first test point (in mL), V1 is the total adsorption volume at the second test point and is also the first point connecting with the MICP data (in mL), ρ is the apparent rock sample density (in g/cm 3 ) and Ma is the rock mass (in g).
Thus, the increments of adsorption saturation at each test point ( where j is the serial number of test points in the range of 1-n', Vj-Vj+1 is pore volume with pore radii ranging from rj to rj+1 (in mL).

Linking the results of nercury intrusion capillary pressure and nitrogen gas adsorption analyses
The connection between the results of the MICP and N2GA analyses can be divided into three steps: Step (1): Converting the measured laboratory capillary curves into the capillary pressure under the gas-water phase condition and the corresponding capillary radii.
The pore throat sizes were quantified from the pressure versus intrusion volume data according to the Washburn equation (Washburn, 1921): where Pc is the capillary pressure (in MPa), σ is the interfacial tension (in N/m), θ is the wetting angle (in °) and rc is the pore radius that is intruded by mercury (in μm).
Step ( Step (3): Linking test datum of the MICP and N2GA analyses 18 As mentioned before, although MICP tests can theoretically quantify porosity defined by pores with radii ranging from 6 to 10,000 nm , it is convenient to also measure large-scale PSDs because this procedure may create new microcracks under high-pressure conditions during mercury intrusion. The N2GA technique is mainly used to characterize the distribution of pore sizes with pore radii in the range of 1-10 nm. On the basis of conversion in the first two steps, taking the logarithm of the capillary pressure under the gas-water phase condition as the ordinate and the cumulative saturation as the abscissa, a full-pore capillary pressure curve is obtained from the combination of MICP and N2GA analyses (Cheng et al., 2006).
Accordingly, nine parameters comprising the measured porosity, permeability, breakthrough pressure (Pb), breakthrough radius (Rb), median pressure (Pm), median radius (Rm), height of gas column (HGC), cover coefficient (CC) and specific surface area (SSA) can be obtained to quantitatively characterize the pore structure and sealing capacity of cap rocks (Cheng et al., 2006). The critical accumulation height of oil and/or gas is defined as the height of the cap rock gas column (HGC) (Berg, 1975;Wilkinson et al., 2014;Shu et al., 2017;Lohr and Hackley, 2018). The HGC trapped in a reservoir is potentially controlled by the breakthrough pressure (Pb) of cap rocks (Ingram et al., 1997). The sealing capacity of cap rocks for each hydrocarbon field is expressed by the cover coefficient (CC) in a specific trap (Cheng et al., 2006). Accordingly, two important parameters (HGC and CC) can be calculated using the following equations (14) and (15). The ratio of the surface in contact with fluid and the total volume can be 19 expressed by the specific surface area (SSA) (Zambrano et al., 2018).
where Pb is the breakthrough pressure (in MPa), and ℎ are the density of water and hydrocarbon (in g/cm 3 ), respectively, g is the acceleration of gravity (9.8m/s 2 ), CC is the cover coefficient (in %) and Z is the closure of the structure (in m). Porosities and permeabilities were analyzed using a QK-98 Helium Gas Porosimeter (YQ2-98-01) and a GDS-90F Helium Gas Permeameter (YQ2-12-01) following

Porosity and permeability tests
Chinese standard industry methods (GB/T 29172-2012) . Through the difference between the measured skeletal rock density and bulk density, the He-porosity was obtained for each sample using the helium expansion method .
The detailed description of the experimental setup can be found in Wang and Yu (2017)., Permeability was tested under a constant pressure gradient using a bubble flowmeter to allow helium pass through the core sample until the permeability value obtained was stable. Subsequently, permeability was calculated using Darcy's law. 20

Fractal dimension calculation
The fractal theory is widely used to characterize the irregularity and roughness of the pore space in sedimentary rocks (Krohn, 1988). Many fractal models have been utilized to obtain the fractal dimension of the rock's pore structure (e.g., Brunauer et al., 1938;Joyner et al., 1951;Avnir and Jaroniec, 1989). The computed values of fractal dimension based on these models differ from each other. Their relationship with the rock heterogeneity is always similar in a way that the greater the fractal dimension, the stronger the heterogeneity of the rock's pore structure (Li, 2010;Li et al., 2017;Zhang et al., 2017;Song et al., 2018;Xiao et al., 2018;Zhang et al., 2018;Sharawy and Gaafar, 2019). This is because these models are derived theoretically and interrelated on the basis of fractal modeling of porous media (Li, 2004;Li et al., 2010;Song et al., 2018).
Accordingly, here the He-Hua (1998) model was adopted to calculate the fractal dimension of carbonate rocks. There are two main reasons that support the choice of this model over the others: (1) The carbonate cap rocks of the Lower-Middle Ordovician Yingshan Fm. are typically characterized by highly heterogeneous pore structures due to the influence of depositional setting variations, diagenetic overprints and the effects of tectonic events (Garing et al., 2014;Gundogar et al., 2016;Lü et al., 2017;Wu et al., 2018b;. The He-Hua model utilized here to calculate the fractal dimension of the rock's pore structures has been widely used to quantitatively characterize pore structures in tight sandstones, shales and also carbonate rocks (e.g., Hollis et al., 2010; 21 Rezaee et al., 2012;Clarkson et al., 2013;Cao et al., 2015;Brunauer and Emmett, 2017;Hao et al., 2017;Li et al., 2017;Zhang et al., 2017;Zhang et al., 2018;Guo et al., 2020).
Moreover, the cap rocks analyzed here present very low porosity and permeability, making this method appropriate (Lü et al., 2017;Wu et al., 2018a;Zhou et al., 2019).
Plus, previous analyses of cap rocks in the Tahe oilfield proved the validity of the He-Hua model for these rock types (Wu et al., 2019a).
(2) Previously published studies showed that the He-Hua model can comprehensively reflect the rock's pore structure over a wide range of scales and also allows the evaluation of the influence of the remaining pores that are not intruded by mercury, as well as those isolated or enclosed Guo et al., 2020).
This not only helps to achieve an accurate description of the pore structures with a fullaperture, but also contributes to the most appropriate evaluation of the carbonate cap rocks sealing capacity (Jin, 2014). Quantitatively assessing the pore structures of carbonate cap rocks, including pore irregularities and their complex connectivity, has become one of the key tasks for the cap rock classification (Lü et al., 2017).
Subsequently, the derivation process of the He-Hua model in this paper is as follows: The number of pores with a radius larger than r is first counted N(>r).
According to the fractal geometry theory, N(>r) will exhibit a good relationship with the pore radius (r) if the pore size distribution presents a fractal structure Wu et al., 2019b): where rmax and f(r) are the maximum pore radius and the distribution density of pore radius, respectively; a and D are a proportionality coefficient associated with the pore shapes and the fractal dimension of the pores, respectively (Hao et al., 2017).
The distribution density of pore radius f(r) can be expressed by the derivative of the pore radius (r), Eq. (17) can be computed from Eq. (16).
The volume of pores with radius less than r, V(<r) can be calculated by integrating r, as expressed in Eq. (18): where β is a constant related to the geometric shape of the pore and rmin is the minimum pore radius in the cap rocks.
Consequently, the total pore volume (V) of the sample can be obtained: Combined with Eqs. (18) and (19), the cumulative volume fraction S(<r) of pores with a radius less than r in terms of the total pore volume can be transformed as follows: Considering rmin<<rmax, r, Eq. (20) can be re-arranged as: According to the relationship between capillary pressure (Pc) and rock pore radius (Berg, 1975;Zhang et al., 2017), and assuming that the pore size does not influence the wetting capillary pressure (Mcphee et al., 2015), the value rmax corresponds to the minimum capillary pressure (Pmin): where σ is the interfacial tension, N/m and θ is the wetting angle, °.
Therefore, by combing Eqs. (21), (22) and (23) the cumulative volume percentage S(<r) with pore radius less than r can be expressed as: Taking the logarithm of both sides in Eq. (24), Eq. (25) can be written as: where S(<r) is the wetting phase saturation corresponding to Pc. The wetting phase in this paper is air and the non-wetting phase is mercury.
Due to the non-wetting properties of mercury in carbonates, it cannot penetrate the pores until an external pressure is applied (Mcphee et al., 2015;Gao et al., 2019). The 24 pores with larger sizes are more likely to be filled with mercury. The cumulative mercury volume intruded into pores at a certain pressure can be recorded as VHg, which is equal to the pore volume with pore radius larger than r .
In addition, the mercury saturation (SHg) defined as the ratio of cumulative mercury volume and the total volume can be obtained: According to the method adopted by Zhu et al. (2018) and Wu et al. (2019b), the cumulative volume percentage with a pore radius less than r can be expressed as: The following formula can be derived by combing Eqs. (25) and (28): The derivation of Eq. (29) is consistent with the conclusion drawn by Yang et al. (2015), who proposed another way for getting the Brooks-Corey type capillary pressure model. Consequently, a linear trend in the cross-plot between lg(1-SHg) and lgPc can be found when the pore-throat features match the fractal theory . The slope of the straight line (B) in the log-log plot can be obtained and the fractal dimension of pore structures in carbonate cap rocks will be calculated using the Eq. (30) as follows: According to Eq. (29), the double-logarithm coordination (lg(1-SHg)-lgPc) shows overall linearity when the structures from large to small pore throats are almost similar.
This means that their fractal dimensions are very close and can be classified as an integral (i.e., single) fractal. Otherwise, the plot of lg(1-SHg) versus lgPc shows a curved trend and has various inflection points, which can be divided into several segments to calculate the corresponding fractal dimension of distinct pore throats (Hao et al., 2017;Zhang et al., 2017;Song et al., 2018). According to the method reported in several publications Zhang et al., 2017;Wu et al., 2019a), the multi-fractal dimension characterizing the whole pore-throat sizes in combination with the weighted average porosity was used to calculate the total fractal dimension (Dtotal).
where ϕi and Di are the porosity and fractal dimension of the corresponding pore-throat size.

Petrography
Two main lithological types of the Ordovician Yingshan Fm. carbonate cap rocks in the Tarim Basin are recognized according to the field and petrographic observations: limestones and dolomitic limestones. (2) Well-cemented peloidal grainstone: the Dabantage, Penglaiba and Kepingshuinichang outcrops have light grey medium-to thick-bedded dense peloidal grainstones alternating with thin-bedded mudstones. However, the Yangjikan outcrop 27 has dark grey and thick-bedded peloidal grainstones that exhibit no dissolution features.
Thin sections reveal that intraclasts in this lithology have a wide variation, ranging from 0.18 to 2.4 mm in size (Fig. 7C, D, F, 8A, C). The amount of cement ranges between 9.4% and 47.4%. Elongate or bladed calcite crystals around intraclasts mostly occur as first generation calcite cements (Fig. 7C). Three generations of calcite cements include elongate blocky, drusy and blocky cements along microfractures, and another two generations of calcite cements, namely isopachous and blocky cements formed around micrite envelops of intraclasts (Fig. 8A). Drusy and/or blocky cements occlude the pore space between intraclasts with various geometries and sizes (Fig. 7D, F). These peloidal grainstones experienced intense compaction due to the overburden of overlying strata during burial (Dong et al., 2013). The intraclasts appear tightly packed, and there are some diffuse stylolites of the sharp peak and simple wave-like types distributed within the crystal mosaic. Broken bioclastic fragments are rarely present and appear surrounded by drusy calcite crystals (Figs. 6C, 7B, 8C).

Dolomitic limestones
The dolomitic limestone cap rocks of this study are mainly characterized by light grey medium-to thick-bedded and grey thin-to medium-bedded rocks that appear alternated with bioclastic limestones at the four outcrops (Fig. 6A, C). The dolomitic limestones are principally composed of scattered subhedral to anhedral dolomite rhomb crystals (Fig. 8B), varying from 18.6 to 210 μm in size. These crystals float locally in the peloidal grainstone, and sometimes display crystals that appear clustered along 28 stylolites (Fig. 5H). Fine-to medium-crystalline dolomite crystals account for around 10%-52% of the total volume of dolomitic limestones. Dolomitic crystals with intermediate size (100-250 μm) were commonly subjected to intense recrystallization and mechanical compaction (Fig. 8D) (Ji et al., 2013). Accordingly, these crystals do not retain the rhomb shapes, and show an anhedral and/or subhedral crystal appearance (Fig. 6C).

Pore types
Thin section optical petrography and SEM analysis indicate that primary intergranular pores formed during deposition were rarely preserved as a result of complex diagenetic alteration that started just after deposition and continued through the burial stage (Dong et al., 2013;Wang et al., 2019). Secondary pores are mainly microfractures, and intragranular, intercrystalline and intracrystalline pores, accounting for the vast majority of porosity of these carbonate cap rocks.

Microfractures
Microfractures are abundant in the mudstones and well-cemented peloidal grainstones of this study, and can be observed in the outcrops and thin section Residual microfractures are always short in length and have a small aperture (Fig. 9B,   D). Pervasively recrystallized calcite crystals fill microfractures, and dolomite crystals are locally distributed at stylolite peaks (Fig. 6F). Additionally, completely-filled microfractures are present within the algal-bearing zone (Fig. 6E).

Intragranular pores
Intragranular pores observed in this study are characterized by irregular and sharpedged pore structures (Fig. 10E). Irregular pores appear pervasively occluded by calcite crystals with minor pyrite, illite and/or pyrobitumen. Some intragranular pores are partly filled with blocky calcite cements (Fig. 10D). Calcite cements are commonly present in vuggy and fracture porosity (Fig. 9B). Two to three different generations of calcite cements (marine, meteoric and burial) can be recognized .
Additionally, SEM analysis shows that there are abundant calcite crystals with various geometries (that vary from 20 to 300 μm in size) filling intragranular pores and microfractures (Fig. 10B). Minor amounts of illite are found locally also filling intragranular pores. Intragranular pores are rarely well preserved due to continued mechanical and chemical compaction. The residual pore space is partially filled by late quartz or calcite crystals from the margin to the center. In rare cases, illite is the least 30 common fracture-and intragranular pore-filling product in the carbonate cap rock (Fig.   10H).

Intercrystalline pores
Intercrystalline pores observed in this study are generally occluded by coarse calcite crystals (Fig. 5H). Initial elongate blocky calcite cements occlude a major volume of the pore space. Alternatively, some intercrystalline pores were partially or completely filled with anhedral calcite and/or dolomite crystals (Figs. 5C, 9A, C), and also by other infills (e.g., pyrobitumen, pyrite and clay minerals) (Fig. 6C, D, H).

Intracrystalline pores
Intracrystalline pores in the cap rocks are small in number, and show angular to subangular shapes with sizes ranging from 0.43 μm to 0.96 mm (Fig. 9B). In rare cases, intracrystalline pores appear not connected and scattered in the peloidal grainstone and mudstone (Figs. 9F, 10A, D, F). Both pyrite and clay minerals plug intracrystalline pores in highly-fractured intervals (Fig. 5F). In addition, some goethite within intracrystalline pores is distributed along or perpendicular to microfractures and presents different shapes (e.g., beam-shaped and lumpy) (Fig. 10E).

Types of pore structures
The pore size distributions (PSDs) of carbonate cap rocks derived from the 31 combination of MICP and N2GA data are plotted in Figs. 11, 12 and 13. Six types of pore throat structures in the carbonate cap rocks are identified based on the cross-plot morphology of capillary vs. mercury saturation, median pressure and dominant pore throat diameters. The detailed descriptions can be seen in Table 2.
(1) The dominant PSDs for the type A pore structure are larger than 2,500 nm, and account for 58.41% of the total pore volume. The pore throat volume with diameters of 25-50 nm, 100-250 nm, 250-500 nm, 500-1,000 nm represents less than 13% (Fig. 11A, C, E).
(2) The pore throat diameters of the type B pore structure are distributed in a wide size range from 100 to 8,000 nm. 32.56% of the pore volume corresponds to pore throat diameters greater than 2,500 nm, followed by 18.38% of the total volume with pore diameters in the range of 100-250 nm (Fig. 11B, D, F).
(3) The volume proportion of different pore throat distributions in the type C pore structure significantly increases from 1.18% to 41.51%, with increasing pore throat diameter (Fig. 12A, C, E).
(4) Proportionately, the pore throats with diameters greater than 2,500 nm in the type D pore structure are dominant, and account for 51.7% of the total pore volume ( Fig. 12B, D, F).
(5) For the type E pore structure, the pore throat volume with pore diameters greater than 2,500 nm and in the range of 500-1,000 nm make up 39.51% and 18.12% 32 of the total pore volume, respectively (Fig. 13A, C, E).
(6) The highest peak of the unimodal PSD in the type F pore structure skews towards smaller pore sizes compared to the other five types. There is 45.78% of the total pore volume that corresponds to a PSD in the range of 2-5 nm, followed by 21.53% with pore throats greater than 2,500 nm in diameter (Fig. 13B, D, F).

Porosity and permeability
The studied cap rocks of the Yingshan Fm. in the Tarim basin have ultra-low porosity and permeability. The porosity within the carbonate cap rocks has a narrow range between 0.22% and 1.27%, with an average of 0.55%. The permeability of these rocks is in the range of 0.00428×10 -3 μm 2 -0.036×10 -3 μm 2 with an average of 0.011×10 -3 μm 2 . The cross-plot of both parameters shows that there is a poor relationship between porosity and permeability, with a very low correlation coefficient for the 19 carbonate cap rocks analyzed (Fig. 14). For instance, sample S-7 presents the largest permeability of 0.036×10 -3 μm 2 but has a porosity of 0.67%. However, this is significantly lower than that of sample S-14 (1.27%), which has a lower permeability of 0.00858×10 -3 μm 2 (Table 1).

Fractal dimension characteristics
Based on the cross-plots of lg(Pc) and lg(1-SHg) of carbonate cap rock samples at the Tarim Basin outcrops, the fractal characteristics of pore structures can be derived from the combination of MICP and N2GA analyses (Fig. 15). According to various logarithmic capillary pressure inflection points for the six pore structure types (Zhu et al., 2018), the pore structures of Tarim Basin carbonate cap rocks with various pore size ranges have different fractal characteristics (Yang et al., 2016).
As mentioned in Section 4.4, and following the classification scheme defined in a 34 previous study (Wu et al., 2019b), the pore size range of carbonate cap rocks can be divided into four groups (Table 3). The first group corresponds to the pore size range between 4,422 and 12,720 nm, while the second, third and fourth groups correspond to 2,770 to 5,525 nm, 180 to 3145 nm, and 2 to 325 nm, respectively. The detailed pore size ranges of all the cap rock samples in this study are listed in Table 3. In addition, the fractal dimensions (D1, D2, D3 and D4) of each of these four groups of pore size ranges can be obtained.
In terms of the type D pore structure of cap rocks in this study, and taking sample S-25 as an example, the relation of lg(1-SHg) and lgPc shows an overall linear trend, which indicates that the fractal dimensions of D1, D2, D3 and D4 (from large pores to small ones) are very close and can be described by an integral (i.e., single) fractal (Yang et al., 2016;Wu et al., 2019b). This implies that all the fractal dimensions from D1 to D4 are equal to the integral fractal dimension. Therefore, the method proposed by Wu et al., (2019b) can be referred and used when comparing fractal dimensions within a certain pore size range. Otherwise, the plots of lg(1-SHg) versus lgPc for the other five types of pore structures show a curved trend with various inflection points (Hao et al., 2017). They can be divided into several segments to calculate the corresponding fractal dimension of distinct pore throats . In summary, the fractal dimensions (D) can be divided into one-segment (type D), two-segment (types A, B, C and F) and three-segment (type E) patterns. Each segment is associated with different pore throat diameter distributions (Zhu et al., 2018). This suggests that the pore fractal 35 characteristics of carbonate cap rock samples from the Ordovician Yingshan Fm. show multiple fractal structures with various pore size distributions .
The values of fractal dimensions from D1 to D4 that correspond to different pore diameters (from macropores to micropores) for the cap rock samples were calculated using Eqs. (29) and (30) and the results are listed in Fig. 16 and Table 3. The correlation coefficients of the double logarithmic cross plot between lg(Pc) and lg(1-SHg) are greater than 0.94. This implies that carbonate cap rocks within a certain range of pore size distributions from the Tarim Basin present a fractal behavior (Lesniak and Such, 2006).
However, some fractal dimension values are lower than 2.0 and therefore they do not conform to the fractal theory for rock heterogeneity (Pfeifer and Avnir, 1983;Zhu et al., 2018). According to Nooruddin et al. (2014) and He et al. (2016), this result may be due to detection errors of experimental instruments during micropore measuring.
Therefore, these meaningless fractal dimension values were excluded in this study (Zhu et al., 2018;Wu et al., 2019a (Fig. 16, Table 3). Although each pore structure type has various fractal dimensions, their fractal behaviors can be compared among these six types because each fractal dimension corresponding to almost the same pore size range was defined to characterize the heterogeneity of pore structure in this studied cap rocks, as proposed by Hao et al. (2017) and Zhu et al. (2018). 36 The fractal dimensions from D1 to D4 for the six pore throat types show an increasing trend as the pore throat diameter decreases. This result suggests that small pores of the cap rocks present more complex and rougher surfaces with higher fractal dimension values than large pores with lower fractal dimension values (Krohn, 1988;Yang et al., 2016;Zhu et al., 2018).
Consequently, the slope of the straight line (B) in the log-log plot can be read and the fractal dimension of pore structures in the study cap rocks can be calculated using Eq. (30). The total fractal dimensions of types A, B and F pore structure can be calculated using Eq. (32) to reveal the multi-fractal dimension characterizing the whole pore-throat sizes in combination with the weighted average porosity Wu et al., 2019b). Meanwhile, by combing with Eqs. (33), (34) and (35), the total fractal dimensions corresponding to the types C, D and E pore structure were also calculated in

D type D type B D type C D type D D type E D type A
The type F pore structure has the greatest heterogeneity, and type A is the most homogeneous one.  (Cheng et al., 2006). The CC varies from 3.90% and 5.71%, with an average of 4.57%. The specific surface area (SSA) is in a range of 2.21-3.98 m 2 /g, with an average of 2.95 m 2 /g.

Relationship between fractal dimension and sealing capacity
As mentioned previously, better sealing capacity of carbonate cap rocks is revealed by a higher CC, while a poorer sealing capacity is represented by a low CC value (Cheng et al., 2006).

Relationships between different fractal dimensions and cover coefficients
Fig . 17 shows the relationship between the fractal dimension and the rock's sealing capacity. D2 has a poor positive correlation with the cover coefficient (R 2 =0.24), while D4 shows no apparent correlation with this parameter (R 2 =0.01). However, linear 38 relationships between cover coefficients and D1 and D3 arise, with R 2 =0.79 and 0.76, respectively (Fig. 17A, C). The cover coefficients of carbonate cap rocks show an increasing trend along with increasing fractal dimensions (e.g., D1 and D3). This suggests that, although D2 and D4 have little effect on the rock's sealing capacity, the fractal dimensions D1 and D3 have a significant impact on the sealing performance of the corresponding carbonate cap rocks (Fig. 17B, D). In addition, there is also a close relationship between the cover coefficient and Dtotal with a relatively higher correlation coefficient of R 2 =0.77 (Fig. 17E). This is probably disturbed by D2 and D4 together with extremely irregular pore structures, an observation supported by optical microscopy and SEM analyses (Figs. 9, 10). Pyrobitumen, pyrite and clay minerals occlude the pore throats and alter the pore morphology, resulting in poorly interconnected pore systems (Figs. 5, 6 and 10) . These pore-filling materials play a significant role in reducing the percolation properties of the rock (Ross and Bustin, 2009;Xiao et al., 2018), enhance the pore system heterogeneity and weaken the impact of D2 and D4 on the Dtotal.
Furthermore, the average cover coefficients in the six PSD types are strongly positively correlated with the average fractal dimension (Davg.) (R 2 =0.88) (Fig. 17F).
The close relationship between cover coefficients and certain fractal dimensions (D1, D3, Dtotal, and Davg.) reveals the significance of the fractal dimension in controlling the sealing capacity of the Tarim Basin carbonate cap rocks. This can be attributed to the fact that a higher fractal dimension suggests that the pore structures of these rocks are 39 characterized by highly heterogeneous pore networks and rough pore surfaces (Burberry and Peppers, 2017;Zhu et al., 2018). Accordingly, the sealing capacity of a carbonate cap rock increases with increasing the fractal dimension.

Relationships of average D1, D3 and average cover coefficients
The average cover coefficients of the six pore structures show good exponential relationships with the average D1, with a correlation coefficient of 0.83 (Fig. 18A).
Additionally, they also present a good linear correlation with the average D3, with R 2 =0.88 (Fig. 18B).
Eqs. (36) and (37) imply that not only the increase in average D1 but also the increase in average D3 result in a significant increase of average cover coefficients.

Relationships between pore size proportion and sealing capacity
Various types of pores with wide ranges of pore radii can be observed in the  (Wu et al., 2019a;Zhou et al., 2019). The poor relationship between porosity and permeability for all the carbonate cap rock analyzed samples (Fig. 14) reveals that the impact of porosity on permeability is quite limited for tight carbonate cap rocks . Another important controlling factor of permeability that cannot be ignored is the effect of the pore structure (e.g., pore radius), as proposed by Rezaee et al. (2012) and Lai et al. (2018).
The studied carbonate cap rocks present strong heterogeneity and complex relationship between porosity and permeability (Garing et al., 2014). The rather complex carbonate cap rock pore structures are probably a consequence of the abundant presence of nanometer-to micrometer-sized pores and throats (Chalmers et al., 2012;41 Cao et al., 2015). This results from the influence of the sedimentary environments in which the sediments forming these rocks were deposited together with the complex diagenetic alterations they underwent (Ross and Bustin, 2009;Anovitz et al., 2013;Lai and Wang, 2015;Liu et al., 2018;Rahmani et al., 2018).
The proportion of different pore sizes of the total pore volume reveals the heterogeneity of the pore throat structure (Sharawy and Gaafar, 2019), while the heterogeneity of these carbonate cap rocks can be reflected by their fractal dimensions (Volatili et al., 2019). The cross-plots of transitional pore (10-100 nm), mesopore (100-1,000 nm), macropore proportion (>1,000 nm and >2,500 nm) and average cover coefficient of the six pore structure types are summarized in Fig. 20.
From the regression analysis, the results show that the average cover coefficients have a very weak negative correlation with the transitional pore proportion (R 2 =0.21) (Fig. 20A), and almost no correlation with the proportion corresponding to mesopores (R 2 =0.18) (Fig. 20B). However, there is a very good negative logarithmic relationship between the average cover coefficients and macropore (>1,000 nm) proportion (Fig.   20C). Furthermore, the average of cover coefficients is strongly negatively correlated with the macropore proportion, with pore sizes larger than 2,500 nm and a correlation coefficient (R 2 ) of 0.91 in this case (Fig. 20D). This indicates that the cover coefficient utilized to indicate the sealing performance of carbonate cap rocks is significantly controlled by the amount of macropores. An increasing proportion of macropores 42 contributes to the decreasing sealing capacity of the carbonate cap rocks (Wu et al., 2019a). When the macopores and microfractures are connected by small pore throats and occupy the major part of the total pore volume, the flow of fluids along the pore space becomes easier (Wilkinson et al., 2014). Although this is obviously beneficial to improving the rock's reservoir quality (Lai et al., 2018), the reduction of the resistance force to migration is not appropriate for the rock's capacity for sealing the oil and gas in the underlying reservoirs (Lohr and Hackley, 2018;Wu et al., 2018b;Zhu et al., 2018).

Classification and evaluation of the rock's sealing capacity
The scattered distributions of fractal dimensions reveal the complexity and heterogeneity of the pore structure (Lesniak and Such, 2005). The capillary resistance force determines the sealing capacity of carbonate cap rocks (Berg, 1975;Kaldi and Atkinson, 1997;Wu et al., 2018a). According to the petrology, pore structures, their fractal dimensions and estimated sealing capacity, the studied carbonate cap rock samples are divided into three classes: I, II, and III (Fig. 22).
(1) Class I carbonate cap rocks are mainly dominated by well-cemented peloidal grainstones, include the types C and F pore structures, and present the best sealing capacity. Some residual dissolved intragranular pores can be detected due to the relatively low-image resolution of thin sections (Njiekak et al., 2018), while the SEM analysis reveals that there are quite minor amounts of pores connected by effective 43 microfractures due to the multiple generations of calcite cements (Fig. 10F). Very few partially residual intercrystalline pores and microfractures could potentially act as effective fluid flow pathway. Meanwhile, fluids within isolated intragranular pores and dead-end intracrystalline pores are considered immovable due to the precipitation of calcite cement (Fig. 10A) (Zambrano et al., 2018). The reduction of the pore space can be attributed to the complex cementation combined with compaction (Figs. 6G, 8A, C) (Anovitz et al., 2013;Wu et al., 2018b). The breakthrough pressures and median pressures are larger than those of Classes II and III. Fractal dimensions D1, D2, and D3 of this class are larger than 2.55, implying that the pore structures in Class I samples are complex and heterogeneous and result in poor pore connectivity (Gundogar et al., 2016;He et al., 2016;Zhu et al., 2018). In addition, the cover coefficient and sealing gas column height are the largest among the three classes proposed. The abundance of trimodal PSDs indicates that the heterogeneity of the pore throat system has a significant impact on the fractal dimension (Gundogar et al., 2016;Burberry and Peppers, 2017), as well as on the sealing capacity of cap rocks. Furthermore, this also indirectly provides qualitative evidence that the increase in the proportion of small but effectively connected pores and narrow strip-shaped throats does not only make a significant contribution to the enhancement of the pore structure heterogeneity and capillary force (Cai et al., 2019;Lai et al., 2019), but also favors the increase of the rock's sealing performance.
(2) Class II carbonate cap rocks are primarily composed of mudstones and 44 dolomitic limestones, cover the types B and E pore structures, and present a moderate sealing capacity. Some intragranular pores are rarely preserved because they have undergone extreme mechanical and chemical compaction (Fig. 10D, E) (Hollis et al., 2010;Dong et al., 2013). However, some intercrystalline pores induced by dissolution with local calcite cement fills can be observed in thin sections and through SEM analysis (Fig. 9A). The coexistence of almost completely filled intracrystalline and residual intragranular pores is characterized by bimodal PSDs obtained from the combination of MICP and N2GA analyses (Figs. 12B, 15B). The fractal dimensions of this class display a wide range from 2.392 to 2.655, implying that both types B and E pore structures have relatively strong intrinsic heterogeneity. Although the tortuous intragranular pores are rarely developed, the occurrence of large body pores but only interconnected by extremely narrow throats results in a moderate capillary force (Wilkinson et al., 2014). Hence, the Class II carbonate cap rocks also present moderate cover coefficient and sealing gas column height values.
(3) Class III carbonate cap rocks are predominately composed of well-cemented peloidal grainstones and mudstones, contain pore structures of the types A and D, and present a poor sealing capacity. The pore systems of this class are typically characterized by abundant microfractures and intercrystalline pores (Figs. 6C, E, 7A, 8A, 9B, D, F), and pores with large radii are always connected by smooth throats. Part of these completely filled microfractures have been subjected to surface-derived meteoric water leaching when the whole strata were uplifted (Dong et al., 2013). Vugs 45 formed as a consequence of dissolution during epigenetic diagenesis. Microfractures within highly-compacted zones are discontinuous and partially closed due to mechanical compaction during the burial stage. However, SEM analysis reveals that some of them were partially reopened during the exposure stage to form effective channels that get connected with intergranular pores (Fig. 10C).  (Ross and Bustin, 2009;Xiao et al., 2018). Additionally, stylolites provide pathways for fluid flow and hydrocarbon charging (Du et al., 2017). This finding is supported by Zhu et al. (2018), who proposed that smooth pore surfaces with small fractal dimensions are not beneficial for hydrocarbon preservation.

Conclusions
(1) Two main lithological types carbonate cap rocks of the Ordovician Yingshan 46 Fm. (Tarim Basin) can be defined by analyzing outcrop analogs. These include limestones and dolomitic limestones. Their pore types are dominated by microfractures and intragranular pores, followed by inter-and intracrystalline pores. Pore structures present a wide range and are divided into six types from A to F. There is no apparent correlation between porosity and permeability for these carbonate cap rocks.
(2) The carbonate cap rocks with a certain range (50-5,000 nm) of pore size distributions present multiple pore size distribution fractal behaviors. The values of the fractal dimensions from D1 to D4, corresponding to different pore diameters (from macropores to micropores), present an increasing trend as the pore throat diameters decrease. The average total fractal dimension of the type A pore structure is the highest among the six pore structure types defined in this study.
(3) The close relationships between the cover coefficients and fractal dimensions (D1,D3,Dtotal,and Davg.)                                               Bimodal shape: large pore throat diameter greater than 2,500 nm and small pore throat diameters lower than 2-5 nm.  Table 3