Study on the influence of time-varying characteristics of mud cake on the safe density window of drilling fluid

Abstract

During overbalanced drilling, the presence of mud cake can effectively reduce the pressure of fluid in the well on the wellbore. However, the existing wellbore stability models mostly ignore the pressure loss caused by mud cake, which leads to the deviation of the safe density window of drilling fluid. Therefore, a time-dependent model of physical parameters of mud cake is introduced in this study. By coupling this model with seepage theory, the changes of pore pressure and water saturation near the wellbore during the dynamic generation of mud cake are analyzed, and then the wellbore instability model is reconstructed by considering the double influence of mud cake on the stress around the wellbore and the mechanical parameters of rock. The results show that with the continuous generation of mud cake, the effective pressure difference of shaft wall gradually decreases from the initial liquid column pressure and tends to be stable, and the deterioration degree of rock mechanical parameters decreases. After 30 h of drilling, the collapse pressure of the wellbore decreased from 1.42 g/cm³ to 1.33 g/cm³, while the fracture pressure increased from 1.71 g/cm³ to 1.87 g/cm³. The wellbore instability model considering mud cake proves that the drilling fluid system that is easy to form high-quality mud cake can expand the safe density window and achieve safe and efficient drilling with reduced density and increased drilling speed. This study provides a theoretical basis for determining the accurate safe density window of drilling fluid.

Introduction

The wellbore collapse pressure and formation fracture pressure are critical parameters for maintaining wellbore stability. Together, they define the lower and upper limits of the safe density window of drilling fluid. Reliable prediction of these pressures is essential in the design of field operations such as drilling and fracturing^1,2,3. In particular, in complex geological environments, the safe density window of drilling fluid is often very narrow^4,5; and accurate prediction of collapse and fracture pressures is directly related to effectively avoiding major engineering risks such as wellbore stability and lost circulation, as well as the associated economic losses.

Mud cake, as a key factor in maintaining wellbore stability, can effectively block pressure transmission within the wellbore and inhibit the invasion of solid particles and drilling fluid⁶. By adjusting the solid-phase composition and rheological properties of the drilling fluid, a high-quality dynamic filtration blocking layer can be constructed, significantly enhancing the long-term stability of the wellbore. To explore the dynamic evolution of mud cake properties, many scholars have conducted numerical simulation studies. Fisher et al.⁷ developed a 3D finite element dynamic mesh method to analyze the distribution characteristics of mud cake thickness in an eccentric annular space, finding that increased eccentricity leads to uneven mud cake thickness distribution. Sepehrnoori et al.⁸ demonstrated, based on Darcy’s law, that in high-permeability formations, mud cake thickness is primarily controlled by drilling fluid properties, while in low-permeability formations, it is more influenced by formation characteristics. Salehi et al.⁹ simulated particle migration and deposition using computational fluid dynamics (CFD) methods and verified the reliability of the results through permeability blocking devices. Ma et al.¹⁰ based on the Fisher model, introduced probability density functions and particle deposition probabilities to systematically study the impact of particle size distribution on mud cake thickness.

Although a series of equations have been established to describe the characteristics of mud cakes, it is still difficult to quantitatively evaluate the mechanical effects of mud cakes if the collapse pressure and fracture pressure are not combined for joint analysis¹¹. Therefore, some scholars have conducted numerical simulations based on wellbore stability models to reveal the underlying mechanical mechanisms. Tran et al.¹² proposed a mechanical model that characterizes dynamic mud cake thickness as a function of wellbore pressure drop. They highlighted the importance of a time-dependent mud cake model by comparing the collapse and fracture pressures under static and dynamic mud cake conditions. Rudyak and Seryakov¹³ simulated the effect of heterogeneous mud cake on wellbore pore pressure and stress field distribution, finding significant differences compared to the assumption of homogeneous mud cake. Feng et al.¹⁴ considered the thickness and strength parameters of static mud cake, and pointed out that the influence of mud cake on fracture initiation behavior mainly stems from its constraint on pore pressure, rather than its own mechanical strength. Li et al.¹⁵ further developed a mud cake model for analyzing wellbore fracture morphology, which simultaneously accounts for the effects of plastic strain softening and damage-induced permeability changes.According to previous research, the seepage of drilling fluid and its filtrate into the formation induces an additional stress field around the wellbore, consequently elevating the collapse pressure. Meanwhile, other investigations have highlighted that in high-permeability formations, the larger invasion radius and higher filtration flux result in more pronounced radial variations in water saturation near the wellbore compared to low-permeability formations, thereby accelerating rock degradation and further increasing collapse pressure.

Existing simulations of mud cake mechanisms primarily rely on experimental data or static mud cake models. In addition, the formation of mud cake leads to a reduction in rock water saturation in the near-wellbore zone, which can provide protection to formations affected by hydration. Therefore, the mathematical equation of the dynamic characteristics of the mud cake is established, and the dynamic permeability and thickness of the mud cake are transformed into the equivalent pore pressure boundary condition based on Darcy’s law. Combined with the porous elasticity theory, a wellbore stability analysis model suitable for the dynamic plugging condition of the mud cake is constructed. On this basis, the changes of mud cake performance, water content, rock deterioration process, pore pressure, collapse pressure and fracture pressure were systematically analyzed, aiming to reveal the influence mechanism of mud cake time-varying behavior on the safe density window of drilling fluid.

Mathematical model of drilling fluid invasion into the formation under dynamic mud cake formation

Drilling fluid is composed of solid particles and liquid phase. Under a differential pressure, the liquid phase invades the formation, while the solid particles are subjected to gravitational force, buoyancy, shear force, lateral lift force, and seepage drag force. Figure 1 schematically illustrates the mechanism of mud cake formation during drilling fluid invasion into the formation. When the forces acting on the particles reach a moment equilibrium, the particles adhere to the wellbore wall and form a mud cake¹⁶. With continued particle accumulation, the mud cake thickness increases and its permeability decreases, thereby inhibiting further filtrate invasion. After long-time invasion, an invaded zone and an uninvaded zone are formed around the wellbore.

Fig. 1

Schematic diagram of mud cake formation process and drilling fluid invasion in drilling process.

After the mud cake reaches a stable state, the pressure tends to remain constant. Therefore, to investigate the dynamic evolution of pressure, it is first necessary to examine the time-dependent evolution of mud cake physical properties. In this study, the drilling mud cake prediction equation proposed by Zhao Jingyuan¹⁷ is adopted, whose applicability is derived from the physical mechanism of crossflow filtration theory on which the model is based. The validity of this equation strongly depends on the composition and stability of the drilling fluid system. Its core assumptions include a predictable particle size distribution and solid concentration in the drilling fluid, as well as stability under shear conditions, thereby ensuring that particle deposition and compaction remain mechanically dominated processes throughout the filtration process.Based on reasonable simplifications of the actual physical processes, the coupled model is established on the following fundamental assumptions: (1) Only oil and water two-phase fluids exist in the reservoir; (2) The multiphase seepage process in the formation around the wellbore conforms to Darcy’s law; (3) The influence of capillary pressure is neglected.”

Mud Cake Thickness Model:

$${L_c}(t)=\sqrt {{{\left( {k\frac{{\Delta {x_m}}}{{{k_m}}}} \right)}^2}+2\frac{{{\varepsilon _{so}}}}{{{\varepsilon _s} - {\varepsilon _{so}}}}\frac{k}{\mu }\Delta P\gamma t} - k\frac{{\Delta {x_m}}}{{{k_m}}}$$

(1)

where $k$ is the mud cake permeability, m²; $\varDelta{x}_{\text{m}}$ is the thickness of the filtration medium, m;${k}_{\text{m}}$ is the permeability of the filtration medium, m²; ${\epsilon}_{\text{s}\text{o}}$ is the solid content of the drilling fluid; ${\epsilon}_{\text{s}}$ is the solid content of the mud cake;$\mu$ is the liquid viscosity, Pa·s; $\varDelta P$ is the filtration pressure difference, Pa; $\gamma$ is the probability of spherical particles depositing on the surface of the mud cake; $t$ is the time, s.

Dynamic Permeability Model:

$$k(t)=(0.197\Delta P - 1.09){L_{c(t)}} - 1.5v+3.9$$

(2)

Where ${L}_{\text{c}\left(\text{t}\right)}$ is the mud cake thickness, mm;$v$ is the drilling fluid flow rate, m/s;

In the process of drilling fluid invasion into the formation, the mass transport process in the porous medium follows Darcy’s law. The Darcy flow equation is given by:

$${V_1}= - \frac{K}{\mu }\nabla P$$

(3)

Where ${V}_{1}$is the Darcy velocity, m/s; $K$is the formation permeability, mD, and $\nabla P$is the driving force, MPa.

During drilling fluid invasion, in the absence of sources and sinks, the continuity equations for the fluid and solid phases are expressed as:

$$\frac{\partial }{{\partial t}}\left( {{\varphi _f}{\rho _1}} \right)+\nabla \cdot \left( {{\rho _{}}{V_1}} \right)=0$$

(4)

Where $t$ is time, s; ${\varphi}_{f}$ is the formation porosity (dimensionless); $\rho$ is the fluid density, kg/m³.

Substituting the motion equations into the continuity equation and simplifying, the resulting seepage differential equation is obtained.

$$\frac{\partial }{{\partial t}}\left( {{\varphi _f}{\rho _{}}} \right)=\nabla \cdot \frac{{{\rho _{}}K}}{\mu }\left( {\nabla P} \right)$$

(5)

For controlling fluid dynamic viscosity, there are differences between the two-phase Darcy law and the porous medium multiphase flow module. The equation for fluid viscosity control in the two-phase Darcy law is:

$$\bar {\mu }=\frac{{\bar {\rho }}}{{\frac{{{k_{r1}}{\rho _l}}}{{{\mu _1}}}+\frac{{+{k_{r2}}{\rho _2}}}{\mu }}}$$

(6)

The control equation for multiphase flow in porous media is:

$$\frac{1}{{\bar {\mu }}}={S_l}\frac{{{k_{rl}}}}{{{\mu _1}}}+{S_w}\frac{{{k_{r2}}}}{{{\mu _{}}}}$$

(7)

Where ${\mu}_{\text{eff}}$ is the equivalent dynamic viscosity of the two-phase mixture, Pa·s; ${\mu}_{1}$ is the viscosity of the oil phase; Pa·s; ${k}_{r1}$ is the relative permeability of the oil phase; ${k}_{r2}$ is the relative permeability of the water phase; ${\rho}_{1}$ is the drilling fluid density, g/cm³; ${\rho}_{2}$ is the density of the formation oil, g/cm³.

In the porous media multiphase flow module, there is a lack of relevant control conditions for the fluid volume fraction inside the porous medium. For simulating the migration and diffusion of drilling fluid invasion in porous media, the two-phase Darcy law module, due to the control of related conditions, provides a more ideal and uniform simulation of drilling fluid migration and diffusion results.

$${S_{\text{l}}}+{S_2}=1$$

(8)

Where ${S}_{1}$ is the drilling fluid volume fraction and ${S}_{w}$ is the formation oil volume fraction.

Based on the above theory, the overall governing equation for the two-phase Darcy law simulating drilling fluid invasion is derived.

Expanding the single-phase mass conservation Eq. (5) to the two-phase mass equation:

$$\frac{\partial }{{\partial t}}\left[ {{\varphi _f}\left( {{S_1}{\rho _1}+{S_2}{\rho _2}} \right)} \right]=\nabla \cdot \left[ {{\rho _1}\frac{{K{k_{r1}}}}{{{\mu _1}}}\nabla P+{\rho _2}\frac{{K{k_{r2}}}}{{{\mu _{}}}}\nabla P} \right]$$

(9)

If ${\rho}_{1}={\rho}_{2}=\rho$(constant), then:

$$\rho \frac{\partial }{{\partial t}}\left( {{\varphi _f}\left( {{S_1}+{S_2}} \right)} \right)=\rho \nabla \cdot \left[ {K\left( {\frac{{{k_{r1}}}}{{{\mu _1}}}+\frac{{{k_{r2}}}}{{{\mu _{}}}}} \right)\nabla P} \right]$$

(10)

According to the above formula :

$$\nabla \cdot \left[ {K{\lambda _t}\nabla P} \right]=0$$

(11)

From the above equation, for the saturation ${S}_{2}={C}_{g}$, the volume conservation equation is:

$$\frac{\partial }{{\partial t}}\left( {{\varphi _f}{c_g}} \right)+\nabla \cdot \left[ { - \frac{{K{k_{r2}}({c_g})}}{{{\mu _{}}}}\nabla P} \right]=0$$

(12)

The overall governing equation for the two-phase Darcy law simulating drilling fluid invasion is as follows:

$$\frac{\partial }{{\partial t}}\left( {{\varphi _f}{c_g}} \right)+\nabla \left( {{c_g}\left( { - \frac{K}{{{\mu _1}}}\nabla P} \right)} \right)=0$$

(13)

Analysis of variation law of wellbore stress and rock water saturation under the time-dependent effects of mud cake

Model verification and solution

The drilling fluid invasion model for the filter cake-formation coupling system integrates fluid dynamics and seepage theory analysis. The dynamic filter cake generation model is utilized. Table 1 presents the basic parameters used in the calculations.

Table 1 Basic parameters of numerical simulation.

Fig. 2

Dynamic evolution law of mud cakes.

Figure 2 presents the dynamic variation of filter cake properties and the pressure at the wellbore interface under different time conditions. As shown in Fig. 2a, the filter cake thickness increases non-linearly over time. With the increase in drilling fluid invasion time, the filter cake thickness grows, but the permeability of the filter cake decreases, and this reduction in permeability occurs at a rapid rate. During the initial phase (0–18 h), the filter cake thickness increases rapidly, while the permeability significantly decreases. After 18 h, the properties of the filter cake tend to stabilize, indicating that the filter cake formation process is nearly complete.

Due to the transient changes in the pressure exerted by the drilling fluid on the borehole wall caused by the dynamic evolution of the mud cake, and the involvement of multi-parameter coupling effects in the model, it is relatively complex to analyze the drilling fluid invasion process using analytical methods. To address this issue, this study employs the finite element method to numerically simulate the seepage process between the mud cake and the formation. The computational schematic is shown in Fig. 3, which clearly defines the boundary conditions for the mud cake and drilling fluid invasion.As time progresses, the thickness of the mud cake gradually increases, leading to a corresponding reduction in the annular flow region. In the seepage simulation, quadrilateral elements are used to discretize the computational domain, with local mesh refinement implemented near the wellbore to effectively capture the pressure and seepage gradient variations in this region, thereby ensuring the accuracy of the fully coupled calculation. The geometric model of the mud cake and formation, along with the meshing results, are presented in Fig. 3.

Fig. 3

Geometric modeling and meshing of drilling fluid intrusion into formation.

To evaluate the impact of grid density on the numerical simulation results, a grid independence analysis was conducted in this study. As shown in Fig. 4, by comparing the drilling fluid invasion depths simulated with six different grid densities (21,018, 59,597, 24,240, 15,364, 12,708, and 6,722), it was found that when the grid density is below 24,240, there are significant differences in the simulated invasion depth. However, when the grid density exceeds this value, the results tend to stabilize. Simultaneously, the computation time increases significantly with the grid density, and beyond 24,240, the computational cost rises sharply. Considering both accuracy and efficiency, a grid density of 24,240 was ultimately selected for all subsequent numerical simulations in this study.

Fig. 4

Results and time of calculation at different grid numbers.

To verify the accuracy of the proposed model, the dynamic mud cake calculation results presented in this paper were compared with those from Fan Yiren et al.¹⁸ to validate the accuracy of drilling fluid invasion distance in the reservoir.

For the experiment, a fine sandstone formation module was selected with a porosity of 14.21% and permeability of 10 mD. Other experimental parameters were set as follows: brine drilling fluid system was adopted, formation water salinity was 1 g/L, drilling fluid pressure differential was 1 MPa, and the filtration process was dynamic filtration. Mud cake samples were collected at 0.82, 1.67, 2.63, 4.68, 7.75, 12.78, and 44.00 h. The experiment was conducted at 25 °C. The drilling fluid formulation was as follows: sodium carbonate 2.1 kg/m³, bentonite 42.3 kg/m³, polyacrylamide 2.1 kg/m³, and sodium chloride 10.6 kg/m³.

Fig. 5

Comparison results of numerical simulation and experimental data.

Figure 5a presents the simulated cloud map of drilling fluid invasion after 40 h, while Fig. 5b shows the comparison between experimental results and simulation results at different time points.

The results indicate that the average agreement between our and those from previous studies reaches 96.45%, with a relative error of 3.55%. All error indicators are controlled within 10%, demonstrating that the proposed model possesses high computational accuracy.

Analysis of wellbore stress distribution

Figure 6a, b show the distribution of formation pressure under the condition of considering and not considering the time-varying characteristics of mud cake respectively. It can be seen from the pressure cloud diagram that when considering the time-varying influence of the mud cake, the thickness and permeability of the mud cake continue to change with time, and the drilling fluid needs to pass through the mud cake to act on the wellbore, resulting in the pressure at the wellbore gradually decreasing with the dynamic formation process of the mud cake. The formation of mud cake is a dynamic process: the initial thickness is small, the permeability is high, and gradually becomes dense and stable with time, and the thickness and permeability continue to change. The formation of mud cake in high permeability reservoir is lagging and denser. The effective pressure of drilling fluid on the wellbore is the total pressure of the liquid column minus the pressure loss of the mud cake. The variation characteristics are as follows: the formation pressure rises rapidly to the pressure value of drilling fluid column in the initial stage; after the mud cake is formed, the pressure value gradually decreases, but the pressure propagation range gradually expands with time. Figure 6b shows the change of formation pressure without considering the influence of mud cake. The results of the cloud map show that the pressure in the near-wellbore zone remains stable and does not change with time, but its influence range gradually expands with time.

Fig. 6

Pressure distribution cloud maps with and without mud cake. (a) Simulation cloud map considering the time-varying effects of mud cakes, (b) Simulation cloud map not considering the time-varying effects of mud cakes.

Fig. 7

Analysis of wellbore stress under different conditions.

Figure 7a, b show the variation curves of formation pressure under different conditions. It can be seen from Fig. 6a that when considering the influence of mud cake, when the time is 0.1 h, the liquid column pressure at the wellbore is 19.49 MPa; with the extension of time to 30 h, the pressure gradually decreased to 17.4 MPa, with a cumulative decrease of 2.6 MPa. This pressure decay phenomenon can be attributed to the dynamic formation process of mud cake: with the increase of mud cake thickness and compactness, it hinders the direct pressure transfer of drilling fluid to the formation, thus effectively protecting the wellbore. As a comparison, Fig. 7b shows the simulation results without considering the influence of mud cake. Under this condition, the pressure at the wellbore is always maintained at 20 MPa and does not change with time, indicating that the wellbore pressure acts completely on the wellbore without any pressure dissipation behavior when there is no mud cake formation mechanism.

Analysis of wellbore water saturation variation

Figure 8a, b show the distribution of formation water content under the condition of considering and not considering the time-varying characteristics of mud cake. It can be seen from the diagram that regardless of whether the influence of mud cake is considered or not, the formation water content gradually increases with time, and the water content increases rapidly in the early stage of invasion, and the growth trend tends to be gentle in the later stage. Figure 8a, b show the spatial distribution and temporal evolution of formation water content under the two conditions of considering the time-varying characteristics of mud cake and not considering the characteristics respectively. It can be clearly observed from the figure that under the two simulation conditions, the formation water content shows a general trend of gradual increase with the extension of drilling fluid invasion time. In the initial stage of invasion, the water content of the formation rises rapidly due to the infiltration of a large amount of fluid. With the continuation of the intrusion process, the growth rate of water content slowed down significantly, and the curve shape gradually tended to be gentle, reflecting that the seepage process was gradually close to the dynamic equilibrium state.

Fig. 8

Analysis of wellbore water content under different conditions.

The results of further comparative analysis show that under the condition of considering the time-varying characteristics of mud cake, the water content of the formation near the wellbore increases from 0.46 to 0.66 between 0.1 h and 30 h. When the effect of mud cake was not considered, the water content increased from 0.46 to 0.80 during the same period. The significant difference between the two sets of data shows that the presence of mud cake not only effectively blocks the direct transmission of drilling fluid pressure in the wellbore to the formation, but also greatly inhibits the invasion of filtrate to the near-wellbore area, so that the growth rate of water saturation in this area is significantly reduced. This mechanism significantly slows down the hydration of drilling fluid on the formation, especially for mudstone, shale and other weak rock formations that are prone to hydration and expansion. The formation and evolution of mud cakes play a key role in maintaining wellbore stability.

Study on the impact of time-varying mud cakes on the safety density window of drilling fluids

The boundary range of the safe density window of drilling fluid is mainly controlled by the stress state around the well and the rock mechanics parameters. After the drilling fluid invades the formation, it will lead to the increase of pore pressure and the redistribution of near-well stress. At the same time, it will cause the change of original water saturation, and then induce the gradual deterioration of rock mechanical properties. Finally, it is manifested as the decrease of rock strength modulus and the deviation of safe density window boundary. In order to reveal the mechanism of mud cake time-varying behavior on the safety window, a quantitative correlation between the evolution of stress field around the well and the mechanical parameter responses due to rock hydration damage during drilling fluid invasion is stablished in this study. The dynamic evolution of collapse pressure and fracture pressure under different operating conditions is systematically revealed.

Safety density window of drilling fluid under different conditions

In order to facilitate mathematical analysis, the local wellbore coordinate system is usually used to study the stress distribution around the wellbore. The stress tensor can be transformed from the global in-situ coordinate system to the local wellbore coordinate system by the following equations :

$${R_1}=\left[ {\begin{array}{*{20}{c}} {\cos {\alpha _1}\cos {\beta _1}}&{\sin {\alpha _1}\cos {\beta _1}}&{ - \sin {\beta _1}} \\ { - \sin {\alpha _1}}&{\cos {\alpha _1}}&0 \\ {\cos {\alpha _1}\sin {\beta _1}}&{\sin {\alpha _1}\sin {\beta _1}}&{\cos {\beta _1}} \end{array}} \right]$$

(14)

The drilling in the formation rock mass will cause the stress concentration effect around the well. Considering that the rock mass is a homogeneous, linear elastic and isotropic medium, Bradley ( 1979 )¹⁸ gave the general solution of the stress equation around the well based on the Kirsch solution. The analytical solution is actually obtained by the superposition of the solutions under a single stress, and its expression is as follows:

$$\left\{ \begin{gathered} {\sigma _r}=\frac{{{\sigma _x}+{\sigma _y}}}{2}\left( {1 - \frac{{r_{i}^{2}}}{{{r^2}}}} \right)+\frac{{{\sigma _x} - {\sigma _y}}}{2}\left( {1+\frac{{3r_{i}^{4}}}{{{r^4}}} - \frac{{4r_{i}^{2}}}{{{r^2}}}} \right)\cos 2\theta +{\tau _{xy}}\left( {1+\frac{{3r_{i}^{4}}}{{{r^4}}} - \frac{{4r_{i}^{2}}}{{{r^2}}}} \right)\sin 2\theta +\frac{{r_{i}^{2}}}{{{r^2}}}{p_i} - \alpha {p_p} \hfill \\ {\sigma _\theta }=\frac{{{\sigma _x}+{\sigma _y}}}{2}\left( {1+\frac{{r_{i}^{2}}}{{{r^2}}}} \right) - \frac{{{\sigma _x} - {\sigma _y}}}{2}\left( {1+\frac{{3r_{i}^{4}}}{{{r^4}}}} \right)\cos 2\theta - {\tau _{xy}}\left( {1+\frac{{3r_{i}^{4}}}{{{r^4}}}} \right)\sin 2\theta - \frac{{r_{i}^{2}}}{{{r^2}}}{p_i} - \alpha {p_p} \hfill \\ {\sigma _z}={\sigma _{{z_b}}} - 2\nu \left( {{\sigma _x} - {\sigma _y}} \right)\frac{{r_{i}^{2}}}{{{r^2}}}\cos 2\theta - 4\nu {\tau _{xy}}\frac{{r_{i}^{2}}}{{{r^2}}}\sin 2\theta - \alpha {p_p} \hfill \\ {\tau _r}_{\theta }=\left[ {\frac{{{\sigma _x} - {\sigma _y}}}{2}\sin 2\theta +{\tau _{xy}}\cos 2\theta } \right]\left( {1 - \frac{{3r_{i}^{4}}}{{{r^4}}}+\frac{{2r_{i}^{2}}}{{{r^2}}}} \right) \hfill \\ {\tau _r}_{z}=\left[ {{\tau _{x{z_b}}}\cos \theta +{\tau _{y{z_b}}}\sin \theta } \right]\left( {1 - \frac{{r_{i}^{2}}}{{{r^2}}}} \right) \hfill \\ {\tau _\theta }_{z}=\left[ { - {\tau _{x{z_b}}}\sin \theta +{\tau _{y{z_b}}}\cos \theta } \right]\left( {1+\frac{{r_{i}^{2}}}{{{r^2}}}} \right) \hfill \\ \end{gathered} \right.$$

(15)

Where σ_ij is each stress component of σ_B; θ is the Angle between the radial direction of a certain point around the wellbore and the direction of the maximum horizontal principal stress; r_i is the radius of the wellbore; r is the distance from the center of the wellbore; a is the Biot coefficient, usually taken as 1; ν is the Poisson’s ratio.

When considering the introduction of the mud cake condition, the Pp is modified to:

$${P_p}={P_m} - {P_s}$$

(16)

Wellbore instability model

In the process of drilling fluid invasion, for water-sensitive formations, due to the presence of expansive clay minerals in the formation, drilling fluid water molecules enter the clay mineral crystal layer structure, which will expand the interlayer distance and replace the lattice, resulting in clay mineral volume expansion. The underground rock is in a constrained state, which will inevitably produce expansion stress in the stratum. The superposition of this stress and the original stress of the stratum may increase the trend and possibility of shear failure of the stratum. Domestic scholars have analyzed different samples and found that the strength characteristics of rocks are related to the water content of the formation. Domestic scholars have also carried out water absorption tests on sandstone, and found that the cohesion and internal friction angle of rock samples after hydration are regular with water content.

The fitting formula is:

$$C^{\prime}=0.0325{\omega ^3} - 0.4481{\omega ^2}+0.8950\omega +C$$

(17)

$$\varphi ^{\prime}=0.1806{\omega ^3} - 2.4524{\omega ^2}+5.9385\omega +\varphi$$

(18)

Where C is the cohesion of the rock under dry conditions, MPa; $\phi$ It is the internal friction Angle of the rock under dry conditions, °; ${\phi}^{\prime}$ is the friction Angle on the shear failure surface after degradation, °; ${C}^{\prime}$is the cohesion of the degraded rock, MPa. $\omega$ is rock moisture content.

The rock water saturation has a significant weakening effect on the tensile strength. With the increase of saturation, water invades the rock particle gap and weakens its cementation force, and reduces the cohesive energy of the mineral surface through physical and chemical action, resulting in a sharp decrease in the tensile strength of the rock. Through a large number of experiments, domestic scholars have deduced the relationship between rock tensile strength under different water content as follows :

$${\sigma _t}={\sigma _{t0}} \cdot {e^{ - b \cdot W}}$$

(19)

Where ${\sigma _t}$ is the tensile strength of the rock, MPa; W is water saturation, dimensionless; b is an empirical parameter, 0.5 for sandstone formations.

It can be seen from the parameters provided on site that the cohesion of the rock in a certain block is 10 MPa and the internal friction angle is 30 ° under dry conditions. It should be noted that this fitting formula is derived from water absorption test data of shallow high-permeability sandstone in a specific block, where the high-permeability sandstone is primarily composed of montmorillonite and illite and has a shallow burial depth. Therefore, this fitting formula is only applicable to high-permeability sandstone formations with similar mineral composition and burial depth. For other sandstone formations, the hydration effects may differ significantly, and it is recommended that calibration be performed based on regional geological characteristics when applying the formula.The initial value is brought into the fitting equation to obtain the variation law of rock mechanical parameters under different water contents. Figure 9 shows the variation law of rock mechanical parameters with different water content. It can be found from the fitting results that when the water content increases in the early stage, the cohesion and internal friction angle of the rock decrease rapidly, and when the water content is greater than 0.8, the magnitude of the decrease decreases.

Fig. 9

Variation of mechanical parameters of rocks under different water contents.

According to the Mohr-Coloumb criterion, the theory holds that the shear force of the same material is equal to the sum of the cohesion independent of the normal stress and the frictional resistance generated by the normal stress on the shear surface¹⁹, that is:

$$\tau =C+\sigma ^{\prime}\tan \varphi$$

(20)

In terms of principal stress, the Mohr-Coloumb criterion is:

The Mohr-Coloumb criterion expressed by principal stress is :

$${\sigma ^{\prime}_1}=\frac{{1+\sin \varphi }}{{1 - \sin \varphi }}{\sigma ^{\prime}_3}+\frac{{2C\cos \varphi }}{{1 - \sin \varphi }}$$

(21)

Where σ₁’ is the maximum effective principal stress; σ₃’ is the minimum effective principal stress; C is the cohesive strength of the rock, MPa; σ’ is the normal effective stress on the shear failure plane; φ is the friction angle on the shear failure plane, in degrees, °.

The stress distribution on the vertical wellbore is as follows: at or, when $\theta =\frac{\pi }{2}$ or $\frac{{3\pi }}{2}$ the circumferential and radial stress differences of the wellbore are the greatest and the wellbore is most prone to collapse, the stress distribution on the wellbore is:

$$\left\{ \begin{gathered} {\sigma _r}={P_w} - {P_s} \hfill \\ {\sigma _\theta }=3{\sigma _H} - {\sigma _h} - {P_w} \hfill \\ {\tau _{r\theta }}=0 \hfill \\ \end{gathered} \right.$$

(22)

When substituting it into the Mohr-Coloumb criterion can be expressed as:${\sigma _1}={\sigma _\theta }$${\sigma _3}={\sigma _r}$

$${\sigma _\theta } - \alpha ({P_p} - {P_s})=\frac{{1+\sin \varphi }}{{1 - \sin \varphi }} \cdot ({\sigma _r} - \alpha {P_p})+\frac{{2{C_0}\cos \varphi }}{{1 - \sin \varphi }}$$

(23)

According to the theoretical collapse pressure calculation formula, it is denoted as:

According to the theoretical collapse pressure calculation formula, it is recorded as :

$${P_b}=\frac{1}{2}(3{\sigma _H} - {\sigma _h})(1 - \sin \varphi )+\alpha ({P_p} - {P_s})\sin \varphi - C\cos \varphi$$

(24)

Where:${\sigma}_{H}$ is the maximum horizontal ground stress, MPa;${\sigma}_{h}$ Is the minimum horizontal ground stress, MPa; ${P}_{p}$ Is formation pressure, MPa;

The formula for collapse pressure after degradation is:

$${P_b}=\frac{1}{2}(3{\sigma _H} - {\sigma _h})(1 - \sin \varphi ^{\prime})+\alpha ({P_p} - {P_s})\sin \varphi ^{\prime} - C^{\prime}\cos \varphi ^{\prime}$$

(25)

For the fracture pressure, according to the maximum tensile stress theory, the formation fracture is caused by the fact that the circumferential stress of the borehole wall rock exceeds the tensile strength of the rock due to the excessive density of the drilling fluid in the well. Most people think that the beginning of tensile yield can be determined by whether the minimum effective stress on the borehole wall is less than the tensile strength of the formation (assuming that the compressive stress is positive). The tensile yield criterion can be expressed as:

$${\sigma ^{\prime}_3} \leqslant - \left| {{\sigma _t}} \right|$$

(26)

Where: ${\sigma ^{\prime}_3}$ is the minimum effective stress.

When the minimum effective stress on the rock ${\sigma ^{\prime}_3}$ is less than the tensile strength of the rock $- \left| {{\sigma _t}} \right|$, the rock does not fail. The minimum effective stress ${\sigma ^{\prime}_3}$ is equal to the minimum principal stress minus the pore pressure and mud cake pressure difference loss, that is:

$${\sigma ^{\prime}_3}={\sigma _3} - \alpha ({P_p} - {P_s})$$

(27)

Rock fracture occurs at the minimum point, that is, at 180°, at which point:${\sigma _\theta }$$\theta =0^\circ$

$${\sigma _\theta }_{{\hbox{min} }}=3{\sigma _h} - {\sigma _H} - {P_w} - {P_s}$$

(28)

According to the tensile failure criterion, the safety conditions for keeping the wellbore from fracturing can be obtained:

$$P_{w} \le 3\sigma _{h} - \sigma _{H} - \alpha (P_{p} - P_{s} ) + \left| {\sigma _{t} } \right|$$

(29)

Based on this theory, the calculation formula of fracture pressure is obtained:

$$P_{f} = 3\sigma _{h} - \sigma _{H} - \alpha (P_{p} - P_{s} ) + \left| {\sigma _{t} } \right|$$

(30)

The variation law of safe density window of drilling fluid under different conditions

In this section, the change law of fracture pressure and collapse pressure under the condition of wellbore with mud cake and without mud cake will be analyzed. The basic parameters used in the calculation are listed in Table 2.

Table 2 Basic parameters for calculating collapse pressure and rupture pressure.

Figure 10 shows the variation of wellbore collapse pressure and fracture pressure with time and well angle without considering the influence of mud cake. The analysis shows that both pressures increase with the increase of the well angle. At the same time, over time, the safe drilling fluid density window continues to narrow. The fundamental reason is that the drilling fluid filtrate invades the formation, resulting in the deterioration of rock mechanical parameters such as cohesion, internal friction angle and tensile strength, which in turn causes the collapse pressure to rise and the fracture pressure to decrease. Specifically, when t = 0.1 h, equivalent density is 1.36 g/cm³, the peak value of wellbore collapse pressure appears at 90 ° well angle. When t = 30 h, the equivalent density of collapse pressure is further increased to 1.42 g/cm³ due to the continuous deterioration of rock parameters.

Fig. 10

Safety density window of drilling fluid without considering mud cake action.

Figure 11 shows the variation of wellbore collapse pressure and fracture pressure with time and well angle when considering the effect of mud cake. The analysis shows that both pressures increase with the increase of the well angle, which is consistent with the law when the mud cake is not considered. The collapse pressure gradually decreases with time. For example, at t = 0.1 h, the maximum equivalent density of collapse pressure decreases from 1.36 g/cm³ to 1.31 g/cm³ without considering the mud cake. This change is mainly due to two reasons : on the one hand, the barrier effect of the mud cake reduces the fluid pressure acting on the wellbore and reduces the pore pressure of the formation, resulting in an increase in the stress around the well ; on the other hand, the mud cake inhibits the increase of water saturation in the wellbore and delays the deterioration of rock mechanical parameters. The mud cake used in this paper has the characteristics of low permeability and large thickness, which is high-quality mud cake. Its barrier effect is significantly greater than the rock deterioration effect caused by water intrusion, so the collapse pressure decreases with time. It can be seen that high-quality mud cake can effectively reduce wellbore collapse pressure, and its change with time depends on the synergistic effect between rock degradation characteristics and mud cake barrier effect.

When t = 0.1 h, the minimum equivalent density of fracture pressure increased from 1.73 g/cm³ to 1.87 g/cm³ without considering the mud cake. Since the tensile strength of the rock is less affected by the water content (according to the empirical formula), the barrier effect of the mud cake directly reduces the stress around the well, so that the fracture pressure gradually increases with time.

Fig. 11

Consider the safe density window of the drilling fluid under mud cake action.

As shown in Fig. 12, for the single well calculation of drilling fluid safety density window. If the traditional stress model without considering the mud cake is used, the collapse pressure is 1.42 g/cm³ and the fracture pressure is 1.71 g/cm³ after 30 h of drilling. When considering the formation of mud cake, the collapse pressure is reduced to 1.33 g/cm³, the fracture pressure is increased to 1.87 g/cm³, and the safe density window of drilling fluid is expanded. The main reason is that mud cake, as a physical barrier, inhibits the invasion of drilling fluid filtrate and slows down the deterioration of rock strength. On the other hand, its mechanical support improves the stress distribution around the well, thus significantly improving the anti-collapse and anti-cracking ability of the wellbore. Therefore, the construction of high-quality mud cake is an important engineering measure to ensure wellbore stability.

Fig. 12

Changes in the safe density window of drilling fluid.

Field example well verification

This study is based on the historical drilling data of Well G. As shown in Fig. 13, the actual drilling fluid density used in the 6000 m to 6500 m well section is significantly lower than the design density recommended by the traditional model. In order to further explore the mechanism of this difference, the measured formation parameters (including porosity, permeability and formation pressure, etc.) and drilling fluid density data of the target layer are substituted into the classical wellbore stability calculation model and the improved model constructed in this paper for comparison and verification. The results of comparative verification show that the main causes of the actual drilling fluid density lower than the recommended value of the traditional model can be summarized as the following two aspects : First, the well section encountered oil and gas reservoirs with high porosity and good permeability, and the stability of the formation itself is relatively good ; second, more importantly, a drilling fluid system with high solid content and high viscosity characteristics was applied in the field. Under the action of downhole high pressure difference, the system can quickly form a layer of mud cake with high quality, high strength and excellent compactness on the surface of the wellbore. This high-quality mud cake not only effectively plugs the micro-cracks and pores of the formation, but also blocks the invasion of drilling fluid filtrate into the formation, and significantly enhances the mechanical support and chemical wall protection around the wellbore. Its comprehensive plugging efficiency and wall protection stability are far beyond the preset scope of the traditional theoretical model.

Fig. 13

Usage of drilling fluid density and drilling fluid parameters for G Well.

The comparison results show that the main reason for the above deviation is that the oil layer with good porosity and permeability is drilled in this well section, and the drilling fluid system with high solid phase and high viscosity is used in the field. This drilling fluid forms a high-quality and high-strength dense mud cake on the surface of the wellbore, and its plugging efficiency and wall protection effect are far beyond the expected range of the traditional model. Therefore, in the actual drilling process, the wellbore stability is significantly enhanced, so that the actual lower limit of the safe density of drilling fluid can be reduced, so as to achieve safe and efficient drilling under the condition of lower than the traditional recommended density.

Fig. 14

Comparison between the traditional model and the model of the present invention, and on-site parameters.

According to the comparison results, as shown in Fig. 14, under the formation conditions involved in this study, the drilling fluid formed a continuous and dense mud cake on the surface of the wellbore. Therefore, for the traditional theoretical model, the calculated collapse pressure at 6400 m is 1.624 g/cm³.After introducing the calculation model of the safe density window of drilling fluid under the condition of mud cake, the calculated lower limit is 1.602 g/cm³, which is closer to the actual value used in the field. It explains why the density of drilling fluid can be reduced for safe drilling after the density of drilling fluid is increased. Therefore, under such geological and engineering conditions, the actual lower limit value of the drilling fluid safety density window is lower than the traditional theoretical prediction. Based on this understanding, the drilling fluid density can be appropriately reduced in the field operation, thereby reducing the pressure holding effect of the liquid column pressure on the bottom hole rock, which is conducive to improving the penetration rate and achieving rapid drilling under the premise of safety.

Conclusion

In this study, the time-varying model of the physical parameters of the mud cake is established. Combined with the seepage theory, the changes of pore pressure and water saturation near the wellbore during the dynamic evolution of the mud cake are analyzed. Then, the double influence of the mud cake on the stress state and the mechanical properties of the rock is reconstructed. The wellbore instability analysis model, and then the following conclusions are obtained :

• (1) With the accumulation of mud cakes in the wellbore, the effective pressure of drilling fluid on the wellbore gradually decreases. When the liquid column pressure is 20 MPa, after the mud cake is completely formed, the pressure on the wellbore is reduced to 17.2 MPa, which is reduced to 84% of the drilling fluid column pressure.

• (2) The formation of mud cake effectively inhibits the increase of water content of shaft wall rock. Without considering the influence of mud cake, the water content of borehole wall rock increased to 0.8 after 30 h. After considering the impermeability of mud cake, the water content of rock is only 0.64, indicating that mud cake can significantly slow down the deterioration of rock.

• (3) Based on the comprehensive analysis of the stress mechanism of the wellbore and the water content of the formation, this study further considers the influence of the dynamic formation of the mud cake on the basis of the traditional model, and reconstructs the calculation model of wellbore instability. Based on this, a method for determining the safe density window of drilling fluid based on the mud cake formation process is proposed. The calculation results show that using the traditional wellbore instability model, under the simulation conditions after 30 h of drilling, the equivalent density of wellbore collapse pressure is 1.42 g/cm³, and the equivalent density of fracture pressure is 1.71 g/cm³. After introducing the mud cake formation effect, the calculation results based on the model built in this paper show that the equivalent density of collapse pressure decreases to 1.33 g/cm³, and the equivalent density of fracture pressure increases to 1.87 g/cm³. The safety density window of drilling fluid is significantly widened, which provides a theoretical basis for improving drilling safety and optimizing drilling fluid design.

• (4) The example of Well G shows that the equivalent density of collapse pressure predicted by the traditional model at 6400 m is 1.624 g/cm³, while the lower limit of calculation of the model with mud cake generation is 1.602 g/cm³, which is closer to the actual situation. This confirms that the mud cake can improve the mechanical stability of the borehole wall, and the actual lower limit of the safety density window is about 0.022 g/cm3 lower than the theoretical value. Based on this, the drilling fluid density can be reduced in the field to reduce the pressure holding effect, improve the penetration rate while ensuring the safety of the wellbore to achieve safe and efficient drilling.For drilling operations in high-permeability formations, it is recommended to employ a drilling fluid system that promotes mud cake development, thereby enhancing operational safety.”