According to the fault displacement and the relative thickness of the faulted sand and mudstone, the lateral sealing of fault can be divided into two types, namely the juxtaposition sealing and the fault rock sealing. When the fault displacement is less than the thickness of the faulted sand and mudstone, the structure of the fault zone is not developed, or the porosity and permeability of the fault zone is better than that of surrounding rock, the target reservoir rock is juxtaposed to the opposite impermeable stratum, so the fault is lateral sealed, this type of sealing is juxtaposition sealing. When the fault displacement is greater than the thickness of the faulted single sand layer, and the strata are completely discon-nected, whether the oil and gas migrating in the target reservoir rock can be blocked near the fault depends on the sealing ability of fault rock. This sealing type is fault rock sealing^[1-2].

There are three methods to quantitatively analyze the lateral sealing ability of extensional faults with fault rock sealing. The first method is the clay content calculation method, mainly including the method of “ SSF (Shale Smear Factor)”^[3], “ CSP(Clay Smear Potential)”^[4] and “ CCR (Clay Content Ratio)”^[5]. All of them evaluate fault sealing ability indirectly and semi-quantitatively by calculating the clay smearing degree in the fault fracture, and are mostly suitable for specific geological environments. The second method is the SGR (Smear Gouge Ratio) method^[6]. By calculating the clay content of fault rock and combining with the pressure statistics of known fault trap reservoirs, this method establishes the mathematical statistical relationship between clay content of fault rock and hydrocarbon column height, to achieve the purpose of quantitative evaluation of fault lateral sealing state and ability^[7⇓-9]. Although this method is widely used by geologists to evaluate the lateral sealing ability of faults, it still has the following three shortcomings: (1) It is not based on the sealing mechanism, so the results of this method only have statistical significance. The fault sealing ability is not always proportional to the SGR value. (2) This method doesn’t consider fault dip angle and consider diagenetic degree (buried depth of breakpoint) roughly. (3) The data are all from existing oil and gas reservoirs, and the established evaluation equation is only applicable to the prediction of the height of the sealed hydrocarbon column in the effective fault trap, while will misjudge invalid traps inevitably. The third method is the fault-reservoir entry pressure differential method, which analyze the lateral sealing state and sealing ability of fault by calculating the entry pressure of fault rock and reservoir rock, then comparing the difference between the two^[10-11]. Among the above three extensional fault sealing evaluation methods, only the third fault-reservoir entry pressure differential method is a quantitative evaluation method established based on the fault sealing mechanism. Besides, when verifying the above methods with the heights of hydrocarbon column sealed by known fault traps, the evaluation results of the fault-reservoir entry pressure differential method are the closest to the actual situation^[11]. Therefore, based on the current research status of fault sealing at home and abroad, it is both meaningful and necessary to further study and improve the quantitative evaluation of lateral sealing of extensional faults by the fault-reservoir entry pressure differential method.

The fault-reservoir entry pressure differential method evaluating the extensional fault sealing was originally based on calculating the normal pressure and clay content of the target fault rock, since the rocks with the same clay content and same static pressure have the same diagenetic degree. Therefore, under the assumption of the same entry pressure, the entry pressure of sedimentary rocks as fault rocks with the same clay content and the same static pressure is calculated. This entry pressure is equal to that of the target fault rock. Then the value is subtracted from that of the juxtaposed reservoir rock, if the difference is negative, it means the fault is open; while if it is positive, it means the fault is sealed, and the larger the difference, the stronger the fault sealing ability is^[10]. As the research progresses, it is found that although the fault rock and sedimentary rock currently have the same clay content and diagenetic pressure, they are different in diagenetic degree due to their different diagenetic time, so their entry pressures are also different. Simple analogy between the fault rock and reservoir rock causes large errors in the evaluation of the sealing ability of actual extensional faults, so the evaluation method of sealing ability of extensional fault considering diagenetic time is proposed^[11]. Although this method enhances the accuracy of fault sealing evaluation, it considers the overlying sedimentary strata after the fault ceases activity as a whole, and does not consider the difference in deposition rate of strata of different burial depths and the pressure on the fault rock is a value that gradually accumulates with the increase of diagenetic time, the evaluation result of the extensional fault seal ability has significant difference from the actual situation inevitably, and the earlier the fault become inactive, the greater the difference between the evaluation results and the actual situation. In addition, this method also ignores the effect of in-situ stress during diagenetic process. Accordingly, based on the study of the fault-reservoir entry pressure differential method, this article fully considers the formation and evolution process of fault sealing, and establishes an integral mathematical-geological model, so as to evaluate the lateral sealing of extensional faults more accurately and quantitatively.

Once a fault is formed and active, a fracture will come up between the hanging wall and the footwall^[12]. The dislocation of the two blocks of the fault will scrape the rock on both sides of the fault, resulting in debris falling into the fault fracture. The fracture formed during the fault activity is also fluid cavity. Pore water in permeable formations on both sides of the fault would enter into the fracture fluid cavity under the action of the formation pressure difference, and mix with the debris in the fault fracture to form fracture filling material which is not compacted and contains a lot of pore water. When the fault isn’t active anymore, the filling material of the fault gradually discharges the pore water under the effect of vertical stress and horizontal in-situ stress to transform into fault rock^[13]. Similar to normal sedimentary strata, the greater the diagenetic pressure the fault filling withstands, the longer the diagenetic time, the higher the diagenetic degree of the fault rock and the greater the entry pressure of the fault rock under the same clay content will be. When the entry pressure of the fault rock reaches that of the target reservoir rock to be sealed, the fault begins to have lateral sealing ability^[14-15]. As the fault rock is buried deeper and deeper, it withstands increasingly high diagenetic pressure and experiences longer diagenetic time, it enhances in diagenetic degree, entry pressure, and sealing ability constantly.

The main factors affecting the entry pressure of the extensional fault rock are clay content and diagenetic degree of the fault rock. While the diagenetic degree of fault rock depends on the vertical stress, horizontal in-situ stress and diagenetic time, the vertical stress is controlled by the depth and dip angle of the fault. The smaller the fault dip and the greater the depth of the fault, the greater the vertical stress will be. The horizontal in-situ stress is controlled by buried depth, dip angle of the fault and the angle between the stress direction and fault strike^{[9,16 -17]}.

In previous quantitative studies on the lateral sealing state and ability of the fault rock, when considering the influence of diagenetic time on the entry pressure of fault rock, the diagenetic time and pressure were often considered separately^[11], and the internal relationship between diagenetic pressure and diagenetic time was not recognized. In fact, with the increase of diagenetic time, the buried depth of fault rock at the breakpoint increases gradually, and the diagenetic pressure also increases, that is, the diagenetic pressure is a variable that increases with the increase of diagenetic time. In the long-term diagenetic process, the diagenesis of fault rock is actually a process of pressure accumulation. Therefore, taking the extension fault in the sand-mudstone stratum as the research objective, we calculated the diagenetic degree of the fault rock by integrating diagenetic pressure to diagenetic time in the geological period, in the hope to evaluate the lateral sealing ability of the fault rock more accurately.

Fault rock is the product of diagenesis of fault fillings, while the fillings are a mixture of formation water and fragments formed by scraping the sandstone and mudstone strata on both sides of the fault due to the fault activity entering into the fault crack. The argillaceous component in the fault rock comes from the sand and mudstone strata on both sides of the fault, its content is mainly related to the number and thickness of the sand and mudstone strata broken by the fault as well as the size of the fault displacement. The SGR value can be calculated by the equation proposed by Yielding^[6]:

According to previous studies, there is an obvious positive correlation between the entry pressure of the rock and the product of its clay content and buried depth^[16]. The higher the clay content and the deeper the burial depth of the rock, the greater its entry pressure will be. Therefore, when calculating the entry pressure of rock, a series of typical rock samples in the study area can be selected, and the fitting statistical equation of the rock entry pressure with its clay content and buried depth can be established based on measured data.

When an extensional fault is active, its hanging wall and footwall grind with each other, and the surrounding rock debris cut by sliding falls into the fault zone. At this time, the fault fillings have not undergone diagenesis yet. When the fault activity ceases, the fault fillings will gradually discharge pore water and slowly undergo diagenesis under the effect of overlying sedimentary loading and horizontal in-situ stress. That is, the diagenesis of the fault rock actually begin after the fault ceases activity, which corresponding to the beginning of the deposition of the stratum ③ in Fig. 1.

Firstly, it is assumed that the material composition and structure in the fault zone are the same as those in the surrounding rock, so the entry pressure of the fault rock can be studied indirectly by that of the surrounding rock. This is out of the following two considerations: one is safety factor, well blowout or bit jumping may occur when drilling in fault zone; the other is that fault zones are limited, there are relatively few wells drilled in fault zones, and the fault rock is fragile, so it is difficult to obtain samples of fault rock for experiment, and fault rock is not representative of the whole area due to the influence of well location distribution. In Fig. 1, in the study of the entry pressure at fault point A, if there is a point K in the surrounding rock, its clay content and diagenetic degree are the same as those of the fault rock at point A (equations (3) and (4)), then the entry pressure at point K is equal to that at point A. The diagenetic degree is a function of diagenetic pressure and diagenetic time, which is affected by both vertical stress and horizontal in-situ stress. In this article, the integral value of the normal pressure of fault surface and horizontal in-situ stress of the fault rock in the corresponding diagenetic time is used to express the diagenetic degree.

As shown in Fig. 1, the clastic fillings at breakpoint A in the fault zone begins diagenesis at t₃. According to the buried depth of point A and the average deposition rate v₁, v₂, v₃ of the overlying strata ①, ②, ③, the deposition thickness of the overlying stratum is determined, that is, the relationship between the buried depth of point A and the diagenetic time it has experienced:

The normal pressure of the fault surface at breakpoint A has the following relationship with its buried depth:

The relationship between the normal pressure of the fault surface and the diagenetic time can be obtained by combining equation (5) and equation (6). The integral value between the two parameters can indicate the contribution of vertical stress to the diagenetic degree of fault rock (Fig. 2, equation (7)). If the strata in the study area have obvious uplift and denudation, equation (7) can be further modified into the form expressed by equation (8).

During the diagenesis process, besides the vertical stress of the overlying sedimentary loading, the fault rock is also affected by the horizontal in-situ stress. In 1978, Brown and Hoek summarized the variation of vertical in-situ stress with buried depth in different regions of the world (equation (9))^[18]. Combining with the relationship between the average horizontal in-situ stress and vertical in-situ stress of sedimentary rocks in China (equation (10))^[19], the horizontal in-situ stress of target rock can be calculated. Meanwhile, comprehensively considering the magnitude and direction characteristics of the in-situ stress in different geological periods, an integral mathematical- geological model of the horizontal in-situ stress diagenetic degree of fault rock, equation (11), is established.

By accumulating the contributions of the vertical stress and horizontal in-situ stress to the diagenetic degree of fault rock, the diagenetic degree of fault rock can be determined.

As shown in Fig. 1, according to the average deposition rate of each sedimentary stratum in the surrounding rocks, the quantitative relationship between the overburden thickness, i.e. the burial depth of point K, and the deposition time can be calculated when the different surrounding rock stratum (① or ② or ③ or ④) is taken as the initial sedimentary stratum and the deposition period continues to t₀ (present). The following equations (12)-(15) respectively correspond to the cases with the point K located in the stratum ① to ④:

The geostatic pressure of overlying strata has the following relationship with the buried depth:

By substituting equations (12)-(15) into equation (16), the following relationship can be obtained:

Combining the above equations, the relationship between the geostatic pressure of a stratum and the deposition time can be obtained (Fig. 3). The diagenetic degree of a surrounding rock layer under the effect of vertical stress can be calculated by integrating the geostatic pressure of the overlying strata to the deposition time (equations (21)-(24)):

Based on the above analysis of the contribution of horizontal in-situ stress to diagenetic degree of fault rock, the diagenetic degrees of different surrounding rock layers under the effect of horizontal in-situ stress are established:

By accumulating the contributions of vertical stress and horizontal in-situ stress to the diagenetic degree of different surrounding rock layers, the diagenetic degrees of surrounding rock layers can be determined when any stratum is taken as the initial sedimentary stratum.

Based on equations (8) and (11), the diagenetic degree of the fault rock at point A can be calculated. There is a point K in the surrounding rock with diagenetic degree equal to that of the fault rock at point A (equation (3)).

By accumulating the diagenetic degree of the vertical stress and horizontal in-situ stress of the corresponding stratum in equations (16)-(28), the diagenetic degree of the surrounding rock at any point can be obtained. Assuming that the diagenetic degree of point A is between those of the bottom interface of surrounding strata ③ and ④, that is Q_r3<Q_fA<Q_r4. It can be sure that point K, which has the same diagenetic degree as point A of the fault rock, is located in the stratum ④. According to equations (24) and (28), the following relation can be obtained:

On the basis of determining the deposition time to present (t_K) of point K, its burial depth can be obtained according to the average deposition rate of stratum ④. This burial depth is expressed as the compacted diagenetic burial depth of the target fault rock (point A), which is of diagenetic significance after considering the effect of diagenetic time on diagenetic pressure. Since the clay content of point K is the same as that of point A, and the burial depth of point K is known, so the entry pressure of point K can be calculated according to equation (2), that is, the entry pressure of fault rock at point A.

The above is the process of calculating the entry pressure of fault rock by using the integral mathematic-geological model. It should be specially pointed out that when considering the integration process of vertical diagenetic pressure to diagenetic time by equations (3)-(29), although the diagenetic pressure is characterized by the compaction diagenetic pressure of rock, the diagenetic time does not simply represent the diagenetic time experienced by the fault rock or reservoir rock, but the synthesis of all the diagenetic factors experienced by the rock during the whole diagenesis period. Besides, when establishing the fitting relationship of rock entry pressure shown in equation (2), the typical rock samples in the study area have actually experienced the comprehensive effects of mechanical compaction, cementation, and dissolution etc. Considering the above two aspects, this model accords with the actual geological conditions that the underground rocks experienced a variety of diagenetic processes during the geological history, rather than only considering mechanical compaction. Only in the actual analysis, the influence of all diagenetic processes on fault rock or reservoir rock is reflected in its diagenetic burial depth, which corresponds to the burial depth of surrounding rock K with the same diagenetic degree as fault rock A in the previous study. The geological significance of this value (diagenetic burial depth) includes, but is by no means limited to, the compaction characteristic.

According to the gamma ray curves of typical wells near extensional faults to be evaluated, the variation law of clay content in reservoir rock can be obtained^[17]. Combined with the buried depth of the reservoir rock in the target block, its entry pressure can be calculated by substituting the buried depth into the functional relationship between the rock entry pressure, clay content and buried depth of the study area (equation (2)).

By comparing the entry pressure of fault rock determined above with that of reservoir rock, if the former is lower than the latter, the fault does not have lateral sealing ability. If the former is greater than or equal to the latter, the fault is laterally sealed, and the greater the difference between the two, the stronger the lateral sealing ability of the fault is. The sealing ability of the fault can be expressed by the height of the hydrocarbon column^[10]:

It is worth noting that the establishment of the above integral mathematical-geological model is based on the assumption that the material composition and structure in the fault zone are the same as those in the surrounding rock. But in general, the fault rock is more susceptible to cementation than the surrounding rock. If the degree of cementation in the study area is relatively weak, the result of this method is closer to the actual situation when evaluating the lateral sealing of extensional faults. If the degree of cementation in the study area is extremely strong, the evaluation result represents the minimum sealing ability of the extensional fault, and this value is the safest value in actual oilfield development. But this method is applicable to the evaluation of lateral sealing of extensional faults regardless of the geological conditions.

Taking fault traps F₁, F₂ and F₃ in the 1^st structural zone of Nanpu Sag in Bohai Bay Basin as examples, the lateral sealing state and ability of faults were evaluated by the integral mathematical-geological method and the fault-reservoir entry pressure differential methods without considering the diagenetic time and simply considering the diagenetic time respectively. The evaluation results obtained by these method were compared to analyze the rationality of the method proposed in this paper.

The 1^st structure of Nanpu is one of the major oil and gas enrichment areas in the Nanpu Sag of the Bohai Bay Basin located in the southwestern slope belt of the Nanpu Sag. Affected by the NW-SE (before 40 Ma) and nearly NS (after 40 Ma) extensional tectonics of Cenozoic rift basin in eastern China^[20], it appears as a drape anticline structure in NE strike with a large number of extensional faults developed under the background of buried hill^[21-23](Fig. 4). The sedimentary strata in this area include Shahejie Formation (Es) and Dongying Formation (Ed) of Paleogene, Guantao Formation (Ng) and Minghuazhen Formation (Nm) of Neogene, Quaternary (Q) from bottom to top. Among them, the second and third members of Shahejie Formation (shortened as Es₂ and Es₃ Formation) and the first member of Shahejie Formation (Es₁ Formation for short)-the third member of Dongying Formation (Ed₃ Formation for short) are the main oil and gas generation strata in Nanpu Sag, and the mudstone stratum of the second member of Dongying Formation (Ed₂ for short) is the main regional caprock^[24]. Core and field measurement and other means have revealed that the 1^st structure of Nanpu Sag has undergone two times of in-situ stress state variation during the geological history period, and the current maximum principal stress direction is near NS-EW^[25]. According to the homogenization temperature of fluid inclusions and well burial history, the oil and gas charging in this area happened in 24.6-25.5 Ma and 6.0-9.5 Ma, respectively, corresponding to the late depositional stage of Dongying Formation and the middle depositional stage of Minghuazhen Formation^[26].

Previous results have revealed that the strata in the Nanpu Sag buried less than 3100 m and greater than 3500 m deep are mainly subjected to mechanical compaction, while the strata with a buried depth of 3100-3500 m mainly suffer cementation by calcite^[27]. At present, the first member of Dongying Formation (referred to as Ed₁) to Quaternary in the 1^st structure of Nanpu Sag are less than 3100 m in burial depth, indicating that the rocks in these strata have low probability of cementation. In addition, the log interpretation results in the study area also revealed that typical wells in this area have porosity changing exponentially with depth, and no obvious increase or decrease sections of porosity^[28], which also indicate that the rocks in this area are mainly affected by mechanical compaction and have low probability of cementation. Therefore, the integral mathematic-geological model established above can be used to quantitatively evaluate the lateral sealing state and ability of target extensional faults.

The F₁ fault is an important sealing fault in the 1^st structure of Nanpu Sag, with an extension length of about 2.3 km, a fault displacement of 20-200 m, and a dip angle of 40-60° (Fig. 5), which constitutes a fault trap with inclined strata. This fault trap has good configuration of hydrocarbon accumulation factors such as oil source supply and reservoir physical properties^[28], so whether the oil and gas can accumulate in the trap mainly depends on whether the fault is laterally sealed. Moreover, oil and gas have already been found in shallow formations such as Nm Formation and Ng Formation of this trap. With shallow burial depth, the fault rock is mainly subjected to mechanical compaction, which meets the assumed conditions of the above integral mathematics-geological model. Therefore, the F₁ fault in the 1^st structure of Nanpu Sag was taken as a typical extensional fault to verify the rationality and accuracy of the integral mathematical-geological model established in this article in quantitatively evaluating the lateral sealing state and ability of extensional fault.

(1) Determination of the clay content of the fault rock. Based on the results of 3D seismic interpretation in the study area, a fault-stratigraphic structural geological model was established, combined with the logging curve data of typical wells near the F₁ fault, the variation law of clay content of the surrounding rocks with the buried depth was worked out. And then the clay content of fault rock at any point in the 3D space was calculated by equation (1) (Fig. 6). The results show that the clay content of F₁ fault is relatively high from the Nm Formation to the lower part of the Ed₁. The SGR value of the Ng Formation is more than 25%, and even as high as 70% in some part of the F₁ fault, except in some parts of the central and eastern fault (less than 20%).

(2) Determination of the entry pressure of rock samples. 34 typical rock samples of 2.5 cm×1 cm (diameter×height) were drilled from different strata and depths in the study area. Based on the buried depth and clay content (obtained from logging curve or measured by X-ray diffraction method) of the rock samples, the functional relationship between the entry pressure and the product of the burial depth and clay content can be established (Fig. 7). There is a good exponential relationship between the three parameters, and the corresponding fitted relation is shown as equation (31). Therefore, with the clay content and burial depth of rock given, the entry pressure of the corresponding rock can be calculated by the following equation:

where $q=\frac{{{V}_{\text{sh}}}Z}{100}$

(3) Determination of the diagenetic degree of fault rock. According to the calculation steps of the integral mathematical-geological model proposed above, based on the time when the F₁ fault ceased activity (the 3D seismic interpretation results indicated that the fault ceased activity at the late depositional stage of the Nm Formation, so the diagenetic time experienced by the fault rock was the period from the end of Nm Formation to the present, about 2.58 Ma), the buried depth of point A, the matrix density (Fig. 8), the deposition rates of different formations obtained from data such as deposition time and deposition thickness (among which, the deposition rates of Q, Nm, Ng, Ed, Es₁, Es₂, Es₃ Formation are 791 m/Ma, 23 m/Ma, 53 m/Ma, 194 m/Ma, 798 m/Ma and 600 m/Ma, respectively), and the clarification of the subsidence history of the 1^st structure in Nanpu Sag, the diagenetic degrees of vertical stress and horizontal in-situ stress of the fault rock at point A were worked out by equations (5)-(8) and (9)-(11).

(4) Determination of entry pressure of the fault rock. First of all, based on the average deposition rate, deposition time, and in-situ stress characteristics of each stratum, the diagenetic degrees of vertical stress and horizontal in-situ stress corresponding to any stratum as initial sedimentary stratum were calculated by using equations (12)-(24) and equations (25)-(28) respectively, and they were added to get the diagenetic degree of the surrounding rock. Then, by comparing the relationship between the diagenetic degree of the fault rock and different surrounding rocks, the position and buried depth of point K in the surrounding rocks with the same clay content and diagenetic degree as the target fault rock (point A) were determined by using equations (3) and (29), which is the compacted diagenetic buried depth of point A. Among them, the compacted diagenetic burial depths of the fault rocks opposite to the reservoir rocks ①-⑤ in well NP103X2 are 958-1 399 m. Finally, combined with the above determined SGR value of the fault rock at point A, the entry pressure of the surrounding rock at point K was calculated by equation (31). Since the entry pressure at point K is equal to that at point A, so this value is the entry pressure of point A, which is 0.61-0.84 MPa (Table 1).

(5) Calculation of the entry pressure of reservoir rocks. According to the buried depth and clay content of the reservoir rock juxtaposing to point A, the entry pressure fitting relationship (equation (31)) established by the actual data in the Nanpu Sag was used to calculate the entry pressure of the reservoir rock.

(6) Quantitative evaluation of lateral sealing of F₁ fault. The minimum entry pressure difference between the fault rock and the reservoir rock at each point is regarded as the sealing ability of the fault. The height of the oil and gas column that can be sealed at each point of the fault was calculated by equation (30) at 70-165 m (Table 1). The results in Table 1 show that the entry pressures of the reservoir rocks in the target block are 0.51-0.63 MPa, while the entry pressures of the fault rocks at different buried depths of the F₁ fault calculated by the above integral mathematical-geological model are 0.61-0.84 MPa. The entry pressure difference between the fault rock and the reservoir rock is 0.09-0.21 MPa, and all the evaluation values are greater than 0, which indicates that these fault rocks have a certain sealing ability and can seal hydrocarbons to form reservoir. Combined with the density of crude oil in the study area, the height of the oil column predicted by the integral mathematical-geological model established by this article is very close to the actual height of the oil column, with an error of only 0-3 m (Table 1). Similarly, the above integral quantitative evaluation process of lateral sealing of extensional fault was repeated to analyze the F₂ and F₃ faults in the 1^st structure of Nanpu Sag (Fig. 9). The evaluation results obtained are consistent with the actual oil distribution of Well NP1 sealed by F₂ fault and Well NP1-90 sealed by F₃ fault. The prediction error of F₂ fault is 0-3.8 m, while the prediction error of F₃ fault is 0.5-8.2 m, which further confirms the reliability of the integral mathematical-geological method in quantitatively evaluating the lateral sealing of extensional fault.

The results calculated by the fault-reservoir entry pressure differential method without considering the diagenetic time^[10] and roughly considering the diagenetic time^[11] and the results from the integral mathematical-geological method established in this article are listed in Table 1, and the predicted results are compared with the actual height of oil column to find out which results are closer to the actual data. According to the evaluation results shown in Table 1, the sealing ability of F₁ fault is greatly underestimated when evaluating by the fault-reservoir entry pressure differential method without considering the diagenetic time, only one of the five reservoirs was sealed and the predicted oil column height (20 m) was far lower than the actual sealed oil column height (162 m). The underestimation of sealing ability is mainly due to the neglect of diagenetic time, which also reveals that the diagenetic time is an important factor affecting the diagenesis of fault rocks. The evaluation results by the fault-reservoir entry pressure differential method roughly considering the diagenetic time also have a large deviation from the actual situation (Table 1). This is mainly due to the fact that the total deposition time of overlying strata is simply taken as the diagenetic time of fault rocks and sedimentary rocks when calculating the entry pressure. However, in fact, the deposition time of each overlying stratum is different. The deposition time of one thin stratum may be longer than that of a thick stratum, and the pressure exerted by the thin stratum on underlying strata or fault rocks is different from the thick stratum. If two strata differ greatly in thickness and deposition time, although the total deposition time of the two sets of strata is certain, the two strata make quite different contribution to the diagenetic degree of the underlying strata, which results in variation of diagenetic degree of the underlying strata with the deposition rate of the overlying strata in a certain period of time. Therefore, the calculated entry pressure of fault rock and reservoir rock will not be accurate, and the evaluation result of extensional fault sealing state is bound to deviate from the actual situation. Moreover, the larger the difference of the deposition rates of overlying strata, the greater the deviation between the evaluation result and the actual one. In fact, as time goes on and overlying strata get thicker, the pressure of overlying strata increases constantly. The diagenetic degree of rock is not an instantaneous quantity, but a variable that accumulates continuously with the growth of diagenetic time and formation thickening. Therefore, reproducing the diagenetic process experienced by fault rock by integrating diagenetic time with diagenetic pressure considers the influence of diagenetic time on diagenetic rock reasonably, and the calculated sealing ability is quite close to the actual situation. In summary, the integral mathematics-geological model proposed in this article can evaluate the fault sealing state and ability more accurately.

Since formed, a fault would be affected by a variety of diagenetic processes. In the process of forming lateral sealing ability, the diagenetic pressure of target fault rock is a variable that accumulates continuously with the increase of diagenetic time and thickness of sedimentary strata, that is, the longer the diagenetic time, the greater the diagenetic pressure and the greater the entry pressure of the fault rock, and the more likely it is for the extensional fault to form lateral sealing. By establishing the integral mathematical-geological model of the diagenetic pressure of the fault rock and surrounding rock with the diagenetic time to characterize its diagenetic degree, the entry pressure of fault rock can be worked out by the fitting relationship of the study area based on the burial depth and clay content of the surrounding rock which has the same diagenetic degree with the target fault rock, then the entry pressure of reservoir rock is compared with it to quantitatively evaluate the sealing ability of the extensional fault. The evaluation results of typical faults (F₁, F₂ and F₃) in the 1^st structure of Nanpu Sag show that the predicted hydrocarbon column heights are consistent with the actual drilling results, with an error of only 0-8 m, and the evaluation results are in good agreement with the actual underground conditions.

The integral mathematical-geological model characterizing the lateral sealing of extensional fault by the integration of diagenetic time to diagenetic pressure is established under the assumption that the material composition and structure in the fault zone are the same as the surrounding rocks. Since the fault rock is more susceptible to cementation than surrounding rock in general, the applicable conditions of the integral mathematical-geological model to evaluate the lateral sealing of extensional fault is that the strata in the area to be evaluated have no cementation, in this case, the quantitative evaluation results of lateral sealing of extensional fault obtained by this model are more in line with the underground reality. But this does not mean that the integral mathematical-geological model can’t be used to evaluate the lateral sealing ability of extensional faults in zones with cementation, but the evaluation results obtained in this kind of cases are the minimum sealing ability of the faults. The occurrence of cementation will inevitably increase the sealing ability of fault, and the value calculated by this model is the safest value for maintaining sealing of the extensional fault in actual oilfield development.

It is difficult to obtain enough representative fault rock samples of a whole fault zone or even the whole area to directly analyze the lateral sealing of extensional faults, so only samples of surrounding rocks can be used to indirectly evaluate the lateral sealing of extensional faults. But there are still many difficult problems to be solved in this evaluation method, such as the heterogeneity of fault rock (the difference in the development degree of dissolution and cementation), and how to combine the measured data with the integral mathematical-geological model etc. Therefore, the quantitative evaluation of lateral sealing of extensional faults needs to be studied further.

F_fab—normal pressure of the fault surface acting on point A before denudation of the eroded section ab, Pa;

F_fi—normal pressure of the fault surface acting on point A by the overlying strata with the thickness of Z_Ai, Pa;

F_rK—geostatic pressure on point K of surrounding rock, Pa;

F_r—geostatic pressure of overlying rock, Pa;

g—gravity acceleration, m/s²;

h₁—distance from the bottom interface of stratum ① to the earth's surface, m;

h₂—distance from the bottom interface of stratum ② to the earth's surface, m;

h₃— distance from the bottom interface of stratum ③ to the earth's surface, m;

h₄—distance from the bottom interface of stratum ④ to the earth's surface, m;

h_K—distance from K to the earth's surface, m;

H_h—height of hydrocarbon column sealed by fault rock, m;

H—vertical displacement of fault, m;

k—sample number;

m₁—stages of in-situ stress experienced during the depositional period of stratum ①;

m₂—stages of in-situ stress experienced during the depositional period of strata ①-②;

m₃—stages of in-situ stress experienced during the depositional period of strata ①-③;

m₄—stages of in-situ stress experienced during the depositional period of strata ①-④;

m—stages of in-situ stress experienced from the beginning of rock diagenesis to the present (can be divided according to the size and direction of in-situ stress);

n—number of surrounding strata sliding through the breakpoint of the fault;

p_df—entry pressure of fault rock, MPa;

p_dr—entry pressure of reservoir rock, MPa;

p_d—entry pressure of rock sample, MPa;

Q_fA—diagenetic degree of the fault rock at point A, Pa·Ma;

Q_fvA—contribution of vertical stress to diagenetic degree of the fault rock at point A, Pa·Ma;

Q_fσA—contribution of horizontal in-situ stress to diagenetic degree of the fault rock at point A, Pa·Ma;

Q_rK—diagenetic degree of the surrounding rock at point K, Pa·Ma;

Q_rv1—diagenetic degree of stratum ① under vertical stress, Ma;

Q_rv2—diagenetic degree of strata ①-② under vertical stress, Ma;

Q_rv3—diagenetic degree of strata ①-③ under vertical stress, Ma;

Q_rv4—diagenetic degree of strata ①-④ under vertical stress, Ma;

Q_rvK—contribution of vertical stress to diagenetic degree of the surrounding rock at point K, Pa·Ma;

Q_rσ1—diagenetic degree of stratum ① under horizontal in-situ stress, Ma;

Q_rσ2—diagenetic degree of strata ①-② under horizontal in-situ stress, Ma;

Q_rσ3—diagenetic degree of strata ①-③ under horizontal in-situ stress, Ma;

Q_rσ4—diagenetic degree of strata ①-④ under horizontal in-situ stress, Ma;

Q_rσK—contribution of horizontal in-situ stress to diagenetic degree of the surrounding rock at point K, Pa·Ma;

SGR—clay content of fault rock, %;

t₀—deposition time to present, Ma;

t₁—initial deposition time of stratum ①, Ma;

t₂—initial deposition time of stratum ②, Ma;

t₃—initial deposition time of stratum ③, Ma;

t_a—initial time of stratum uplift and erosion in sectionab, Ma;

t_b—end time of stratum uplift and erosion in sectionab, Ma;

t_js—initial time of the j^th phase in-situ stress experienced by the stratum, Ma;

t_jf—end time of the j^th phase in-situ stress experienced by the stratum, Ma;

t_K—time from the deposition of point K to the present, Ma;

t—deposition time of strata, Ma;

v₁—average deposition rate of stratum ①, m/Ma;

v₂—average deposition rate of stratum ②, m/Ma;

v₃—average deposition rate of stratum ③, m/Ma;

V_shi—clay content of thei^th stratum slipping through the breakpoint of fault rock, %;

V_sh—clay content of rock sample, %;

Z₀—vertical distance between point A and the top interface of stratum ④, m;

Z₁—vertical distance between point A and the top interface of stratum ③, m;

Z₂—vertical distance between point A and the top interface of stratum ②, m;

Z₃—vertical distance between point A and the top interface of stratum ①, m;

Z_A—burial depth of point A, m;

Z_Ai—thickness of the overlying strata on fault rock point A, m;

Z—burial depth of rock, m;

γ—angle between the horizontal in-situ stress direction and the fault strike, (°);

ΔZ_i—thickness of thei^th stratum slipping through the breakpoint of fault rock, m;

θ—dip angle of fault, (°);

ρ_o—density of hydrocarbon, kg/m³;

ρ_r—matrix density of overlying sedimentary rock, kg/m³;

ρ_w—density of formation water, kg/m³;

σ_hj—horizontal in-situ stress experienced by the strata in the j^th phase, Pa;

σ_hmax—maximum horizontal in-situ stress, Pa;

σ_hmin—minimum horizontal in-situ stress, Pa;

σ_h—horizontal in-situ stress, Pa;

σ_v—vertical in-situ stress, Pa.