The Oil and Gas industry works based on the safe and efficient extraction of hydrocarbons from underground reservoirs, lost circulation is a common problem in the construction of oil and gas wells. The lost circulation control may hinder production operations in oil and gas wells, which can significantly increase nonproduction costs and time. (Amadi-Echendu and Yakubu, 2014; Albattat and Hoteit, 2019; Zhang et al., 2019; Li et al., 2022; Li et al., 2022; Wan et al., 2023). Fractures, whether natural or hydraulic, can be reopened when the wellbore pressure is higher than the initial fracture pressure, leading to significant mud loss. During lost circulation, with the increase of wellbore pressure, the stress on the natural fracture or induced fracture tip gradually increases. When the fracture tip stress intensity factor (SIF) reaches a critical value (K_IC), the fracture will reopen and continue to propagate. The critical value, known as the toughness coefficient, is one of the properties of the formation material, indicating the resistance of the formation to fracture propagation, the toughness coefficient of different strata is not the same. The SIF magnitude is dependent on the fracture surface width, size, and stress distribution. (Mirabbasi et al., 2020).

In general, the lost circulation materials (LCMs) used are a key factor in determining the lost circulation control, injection of commonly available materials into the fracture to reduce pressure or seal fracture (Ashoori et al., 2022). There are many ways to strengthen wellbore, among which there are three main types of fracture reinforcement. a) Fracture propagation resistance: Yili et al. (2014) believe that under the induction of the formation, the LCMs react in the fracture, forming a high-strength structure, isolation wellbore pressure and increasing bearing capacity of formation pressure. (Fuh et al., 1992; van Oort et al., 2011); b) Stress-cage theory: Alberty and Mclean (2004) believe that to balance the wellbore pressure and formation stress field, the stress field at the fracture tip or tangential stress field around the wellbore can be adjusted to effectively control drilling fluid loss. (Aston et al., 2004); c) Fracture closure stress: Loloi et al. (2010) consider the fracture tip to be isolated with LCMs to prevent drilling fluid pressure from reaching the fracture tip, thereby resistance fracture propagation (Dupriest and Koederitz, 2005). Although these techniques and theories have different approaches, they all use LCMs in fracture areas to plug natural and induced fractures. Consequently, it is possible to reduce the fracture pressure because of the isolating action of the LCMs on the fracture. Theoretically, an increase in fracture reopening pressure (FROP) can be achieved as a result of the lower fracture pressure being transferred to the fracture tip to induce fracture propagation. (Feng et al., 2016a; Feng and Gray, 2017). Therefore, it is essential to quickly calculate the fracture geometry and SIF in determining the Fracture Reopening Pressure (FROP) and LCM design. (Qin et al., 2023).

Many scholars have established analytical and numerical models to understand the mechanics of responses before and after fracture plugging. The numerical modeling, such the Cohesive zone model by Kostov et al. (2015), the PKN model by Wang et al. (2018), the finite element model by Feng and Gray. (2016b) and the boundary element model by Liu et al. (2020). Compared to numerical modeling, the analytical/semi-analytical model is more accurate in predicting the fracture aperture and stress intensity factor at different conditions and more computationally efficient. The semi-analytical method has been widely applied to the calculation of the stress intensity factor and fracture width in fracture models (Warren, 1982; Shahri et al., 2014; Shahri et al., 2015; Xu et al., 2017; Xu et al., 2020). Mehrabian and Abousleiman. (2015) and Mehrabian and Abousleiman (2018) presented a linear elastic fracture mechanics based model to calculate the extended drilling margin and also considered the case of multiple fractures. However, these studies do not take into account the effects of time and fluid flow to cause pressure changes in the fracture during fracture propagation. Zhong et al. (2017), (Zhong et al., 2018) considered the influence of drilling time on fracture propagation but believed that fluid pressure drop in fractures was too small to be ignored. Pressure exchange between fractures and formation also greatly affected fracture propagation. Wan et al. (2023) analyzed the influence of fluid flow on fracture propagation in triple porosity medium: LCMs, formation, and mud cake. However, for tight formations (formations permeability ≤ 1 mD), the pressure transfer from wellbore or fracture to formation is slow, and the fracture pressure leakage to formation efficiency is low, as Figure 1. With the decrease of formation permeability, the difference of stress intensity factor (SIF) between the two models becomes small, especially in tight formation, where the difference is negligible. So the effect of mud cake on inhibiting fracture growth is negligible, and considering triple porosity media increases the calculation cost in tight formation too.

In this paper, based on Wan et al. (2023), a dual porosity media model combined with a fracture-mechanics model based on dislocation is established in tight formation. The influence of fluid flow and filtration on fracture width is discussed. The fracture reopening pressure (FROP) is calculated and its influencing factors are analyzed. The model simplifies the influence of other fractures around the wellbore, accurately reproduces the interaction between the plugging zone and the formation and the pressure transfer from well to fracture tip. Compared with other models, the accuracy and efficiency of SIF and fracture width calculation results are improved, and the calculation process of FROP is supplemented for Wan et al. (2023). The model can reasonably design LCMs and calculate FROP, which has important guiding significance for wellbore reinforcement technology that prevents continuous fracture propagation, collapse and leakage around the well.

In the process of using LCMs to plug induced and naturally occurring fractures to prevent further fracture propagation and leakage of a large amount of drilling damage formation, the dual porosity medium for plugging zone (Part of the LCMs plugged in the fracture) and formation matrix in the fracture area can be considered as a system in which fluid flow follows Darcy’s law (Wang et al., 2018), as shown in Figure 2.

Assuming that the system of wellbore, plugging zone and formation matrix is quasi-stable, fluid flows into the plugging zone from the wellbore and the wellbore pressure will transfer to the formation matrix, thus forming the pressure difference inside and outside the fracture. We assume that the pressure difference is linear, and establish the following fluid flow model in the dual porosity medium, (See the detailed derivation Wan et al., 2023), ϕ m K w + 1 − ϕ m K m ∂ p m ∂ t + ∇ ⋅ − k m μ ∇ p m = ω p f − p m (1) C b + ϕ f K w ∂ p f ∂ t + ∇ ⋅ − k f μ ∇ p m = − ω p f − p m (2) where p is the fluid pressure, ϕ is porosity, C _b is bulk compressibility coefficient, k is permeability, μ is the viscosity of fluid and ω is the transfer coefficient. The subscripts m and f represent matrix and fracture.

According to Wan et al. (2023), the final stress intensity factor and fracture width distribution are W t k = π b − R 1 − v 2 2 m ∑ k = 1 N f t k (3) K I = 2 π b − R 2 m E ′ 8 ∑ k = 1 m − 1 k − 1 f t k 1 − t k 1 + t k 1 2 (4)

The distribution function f (t _k ) can be calculated by a semi-analytical model (Warren, 1982). The model can be Split into two subproblems, shown in Figure 3. Subproblem I without fractures, the tangential stress distribution is given by Kirsch's equation, as Eq. 5: σ θ θ r = − σ H + σ h 2 1 + R 2 r 2 + σ H − σ h 2 1 + 3 R 4 r 4 cos ⁡ 2 θ + p w R 2 r 2 (5)

Subproblem II with fractures, the stress distribution of the fracture surface can be expressed according to the potential of the complex function, as Eq. 6 (Muskhelishvili, 1953; Warren, 1982) σ θ θ f x = 1 π ∫ R b f ξ 1 x − ξ + K x , ξ + 1 − x − ξ + K − x , ξ d ξ (6)

The superimposed gravity generated by the above two subproblems extending the fracture surface is equal to the pressure inside the fracture, and the following equation can be established: E ′ 4 σ θ θ f x + σ θ θ x = − p f x (7)

Where G is the shear modulus, v is the Poisson’s ratio and E ′ = 2 G / 1 − ν . (See Wan et al. (2023) for the solution of Eq. 7).

When the fracture tip stress intensity factor (K _I ) reaches the fracture toughness (K _IC ) as well as the wellbore pressure (p _w ) is adjusted, this wellbore pressure is the fracture reopening pressure (FROP). The steps of solving the FROP coupling model are shown in Figure 4. Solved step by step as follows: a) The finite element software COMSOL was used to solve Eqs 1, 2, the fracture pressure of plugging zone was obtained; b) Using Matlab to solve Eqs 3, 4, the fracture width and fracture tip stress intensity factor (SIF) can be obtained; c) Finally, the fracture reopening pressure (FROP) was determined by comparing the K _I to K _IC .

In this paper, considering that the effect of mud cake on the tight formation is negligible, the three-pore medium model was simplified to be dual porosity model, combined with the semi-analytical fracture mechanics model to calculate the fracture width and fracture reopening pressure (FROP) in tight formations. The model parameterization studies before and after the use of LCMs were performed, and the following conclusions were drawn:

(1) In the formation with large horizontal principal stress anisotropy, low Young’s modulus and Poisson’s ratio, the fracture width is larger. Wellbore radius and pressure is another factor influence the fracture width, and the increase of fracture width occurs in conditions with higher wellbore radius and pressure.