#### Abstract

As a fluid flows through a porous media, a drag force, called seepage force in the paper, will be formed on the matrix of the media in the fluid flowing direction. However, the seepage force is normally ignored in the analysis of wellbore fracturing during hydraulic fracturing operation. In this paper, an analytical model for seepage force around a vertical wellbore is presented based on linear elasticity theory, and the effect of the seepage force on wellbore breakdown has been analyzed. Also studied are the effects of the two horizontal principal stresses and the reservoir permeability on the action of seepage force. The paper proves that seepage force lowers formation breakdown pressure of a vertical wellbores; the deeper a formation is, the greater action of the seepage force; seepage force contributes more to breakdown formation with small difference of the two horizontal stresses such as unconventional reservoirs; seepage force increases as rock permeability decreases, and it should not be ignored in hydraulic fracturing analysis, especially for low-permeability formation.

#### 1. Introduction

The evaluation of the stress fields around wellbores in porous media attracts plenty of interests due to its relevance in oil and gas production, with special emphasis on hydraulic fracturing. The distributions of the stress and pore pressure fields in wellbore rock are essential to studying the initiation of hydraulic fractures.

Haimson and Fairhurst [1–4] systematically studied the stress field around a wellbore during the initiation of fractures and proposed that when fluid was injected into a wellbore, three stress fields acted together to induce the breakdown of permeable reservoirs. These three stress fields include in situ stress, wellbore pressure, and poroelastic stress caused by pore pressure variations in wellbore rock after wellbore fluid flows into a reservoir.

Hubbert and Willis [5] and Medlin and Masse [6] investigated the mechanics of hydraulic fracture initiation by comparing laboratory experiments with theoretical predictions based on poroelasticity. In the recent 20 years, quite a few scholars [7–11] have analyzed the influencing factors of formation breakdown pressure based on Haimson’s theory combined with experimental results.

Based on the model of Hubbert and Hamison, Ito [12, 13] proposed a newly constructed fracture criterion that can explain the effects of wellbore diameter and pressurization rate on the breakdown pressure. Jin et al. [14] presented a weight function method to predict the breakdown pressure of two general symmetrical radial fractures emanating from a wellbore. Fatahi et al. [15] presented a simulation model based on a distinct element method to study the breakdown pressure during hydraulic fracturing tests. Xiao et al. [16] proposed a fracture initiation model for carbon dioxide fracturing under various bottom hole pressure and temperature conditions.

However, flowing into reservoir rock of wellbore fluid not only increases in situ pore pressure and thus creates poroelastic stress but also results in a pore pressure gradient along the fluid flowing direction in wellbore rock. Under the action of fluid dragging, the flowing fluid exerts a force, called seepage force, on rock skeleton affecting stress fields around the wellbore. Mourgues et al. [17], Cobbold et al. [18], and Zanella and Cobbold [19] verified the existence of the seepage force during fluid flowing in a porous media based on sandbox experiments and evaluated the influence of seepage force on media structure. Rozhko et al. [20, 21] investigated the influence of seepage force on the failure of a porous elastic media and calculated the stress field created by the seepage force. Zhou et al. [22] analyzed the behavior of stabilizing piles for landslides in the Three Gorges Reservoir under the effect of seepage force. Zou et al. [23] derived theoretical solutions for a circular opening in an elastic-brittle-plastic rock mass incorporating the out-of-plane stress and seepage force. AlKhafaji et al. [24] studied the bearing capacity problem of shallow rigid foundations on rock matrix subjected to horizontal seepage force.

Although the seepage force has been proved and discussed in soil engineering, and applied in dam stability analysis for decades, it has been ignored in oil and gas production. Few literatures in hydraulic fracturing analysis discuss the effect of the seepage force on hydraulic fracturing process.

In this work, seepage force will be introduced into the analysis of a vertical well during hydraulic fracturing operation. A model analyzing the stress field around the wellbore by seepage force will be presented. To study the effect of seepage force, traditional stress field analyses around the wellbore during hydraulic fracturing process will be discussed firstly. Seepage force contribution to the wellbore breakdown will be compared by the results from cases with and without considering the seepage force. Also explored are the effects of formation confining pressure and permeability on wellbore fracture pressure in the case of considering the seepage force.

#### 2. Stresses around a Vertical Wellbore

##### 2.1. Tradition Model

Conventionally, the stress analysis of a vertical wellbore at an interested depth during hydraulic fracturing is simplified as a plane problem [25, 26]. As shown in Figure 1, a well with a radius of is drilled in a formation with in situ horizontal principal stresses and (). In a cylindrical coordinate system , the stresses at any point are the radial stress and circumferential stress , and tensile stresses were assumed to be negative in this paper. The circumferential stress of the hollow square can be expressed as follows (all symbols in this paper are shown in Table 1):

When the wellbore is pressurized by the injected fracturing fluid during hydraulic fracturing, two circumferential stresses and will be yielded at any point. The is caused by the fluid pressure at the borehole wall, which can be viewed as an internal pressure acting on a hollow thick cylinder.

The third stress is introduced by the pore pressure variation in the formation when the fracturing fluid penetrates into the formation and flows through its pores. When the formation pore pressure changes, the rock skeleton undergoes an uneven elastic deformation, causing so-called stress under the mutual constraint of the skeleton elements [27–30]: where is the pore pressure distribution around the wellbore (Pa); is the initial pore pressure (Pa); is Poisson’s ratio, dimensionless; and and are the frame and matrix bulk moduli of the rock (Pa).

If the wellbore pressurization rate is relatively small and the fluid is noncompressible, the pore pressure distribution around the wellbore can be regarded as steady-state. This quasistatic pressure field in a domain with constant permeability is governed by the Laplace equation:

Once wellbore pressure and pore pressure at outer boundary were determined, Equation (5) was solved by polar coordinates, and pore pressure distribution around the wellbore during steady-state flow was obtained as follows [2, 31, 32]:

By taking the pore pressure distribution around the wellbore in Equation (6) into Equation (4), circumferential stress due to pore pressure variation at any point under steady-state flow was obtained as ()

##### 2.2. Introduction of Seepage Force

###### 2.2.1. Mechanism of Seepage Force

As a viscous fluid flows through the pores of a porous media, the fluid imparts a friction force and normal thrust to the solid element of the matrix. The force is normally called seepage force in soil mechanics. Seepage force has long been considered in geotechnical engineering to assess the stability of a slope or the risk of sand liquefaction of dams [22, 33, 34] although its definition is still arguing in the engineering. In the paper, seepage force is defined as follows:
where denotes pressure head gradient, dimensionless, and is the unit weight of fluid (N/m^{3}).

For the radial flow in Figure 1, the seepage force is simplified as follows:

The direction of the seepage force is in the opposite direction of the pressure gradient.

During hydraulic fracturing, high pressure wellbore fluid flows into rock and the in situ pore pressure changes. Pore fluid flowing zone around the wellbore with certain pore pressure gradient will be formed. As shown in Figure 2, rock matrix around a vertical wellbore satisfies the total stress equilibrium equation under stable state. where is the radial volume force per unit volume and is the circumferential volume force per unit volume. When the fluid flows radially, , . Under axisymmetric conditions, both the circumferential stress and the radial stress are only functions of , and the shear stress is 0. Therefore, Equation (10) is reduced to

As shown in Figure 2, when the fluid was under hydrostatic pressure state, pore pressure gradients in radius direction is equal to 0, and there is no seepage force. However, when fracturing fluid flows in the radial direction during hydraulic fracturing, there are pore pressure gradients , and there is a seepage force . The direction of is consistent with the direction of fluid flowing.

###### 2.2.2. Calculation of Seepage Force Stress Field

To study the effect of seepage force, solve the seepage force equation (Equation (11)). The internal and external boundary stresses are zero as the effects of wellbore pressure and pore pressure at outer boundary are already taken into account in Equation (11).

Combining Equations (6), (11), and (12), the circumferential stress formed by the seepage force under the plane strain condition during the steady-state flowing of wellbore fluid into a vertical wellbore:

It is well known that rock failure is controlled by the Terazaghi’s effective stress [35]. The theory assumes when the effective circumferential stress reaches the tensile strength of a rock (), at borehole wall, tensile fracture at the wellbore occurs [2, 5, 12].

To compare with traditional methods, three cases are considered: rock is impermeable, pressured wellbore fluid flows into rock without considering seepage force effect, and pressured wellbore fluid flows into rock with seepage force effect.

When considering the rock is impermeable, the total effective circumferential stress is expressed as follows:

When considering the fact that pressured wellbore fluid flows into the rock without considering seepage force effect,

When considering the effect of seepage force (), the total effective circumferential stress is

Note, in Equation (17), is not included as the effect of wellbore pressure (pa) is already considered in .

#### 3. Analyses of Fracture Initiation under Seepage Force

##### 3.1. Comparison of Wellbore Breakdown Process for Three Cases

In Section 2, circumferential stress fields around a wellbore wall and corresponding formation breakdown conditions are derived for steady-state fluid flow condition. During hydraulic fracturing operation, wellbore pressurization rates are relatively slow. The operation can be modeled using quasistatic conditions [2].

The calculation conditions are given as follows. Considering a vertical circular wellbore borehole drilled in an isotropic geologic medium, a 2D isotropic plane is used in the analysis (Figure 1). The maximum and minimum horizontal stresses are MPa and MPa. The initial wellbore pressure and pore pressure is assumed as *5* MPa. The tensile strength of the rock is taken as *0* MPa. The Poisson’s ratio in the isotropic plane is 0.25. The inner and outer diameters of the wellbore are supposed as 1 dm and 10 dm, respectively. It is assumed that starting from s, the wellbore pressure starts to increase at a rate of 1 MPa/s until the wellbore wall is broken. That is, the effective circumferential stress reaches the tensile strength of the rock . The pore pressure at the outer boundary of wellbore during the entire pressurization process is maintained at ().

Equations (15)–(17) represent the three cases: rock is impermeable, pressured wellbore fluid flows into rock without considering seepage force effect, and pressured wellbore fluid flows into rock with seepage force effect. Using the given conditions, the wellbore breakdown processes for the three cases are calculated under quasistatic conditions.

The calculated results for the three cases are shown in Figures 3–5 (color ruler scale corresponds to the effective circumferential stress value ).

From Figures 3–5, at time s, the wellbore has not been pressurized, and there is apparent stress concentration at the wellbore wall. As the wellbore is pressurized, the effective circumferential compressive stress around the wellbore gradually decreases and the stress concentration gradually disappears. The breakdown points at the wellbore wall for the three cases are in the directions of maximum horizontal principal stress ( and ). When fluid flows into the formation (Figure 4), the stress field varies more intensely comparing to Case 1 (Figure 3). This proves that fluid flowing into wellbore process should be considered during the analysis of hydraulic fracturing. Under the action of seepage force, as shown in Figure 5, the wellbore wall will be broken very quick, taking only 7.9 s. The effect of seepage force is markedly conducive to wellbore breakdown.

Figures 6–8 show the variations of the effective circumferential stresses along the direction of the maximum horizontal principal stress for the three cases. From Figures 6–8, the effective circumferential stress from the wellbore wall decreases the fastest, and the decreasing speed becomes slower gradually as increases. For impermeable rock (Case 1), trends to no variation anymore from the distance of (Figure 6). When fluid flows in the wellbore, due to the variation of the pore pressure , varies in the whole modeling area. For the effect of seepage force (Case 3), the effective circumferential compressive stress around the outer boundary even exceeds the initial stress state, indicating that the seepage force gives a compression effect at the outer boundary at the case condition (Figure 8).

##### 3.2. Wellbore Breakdown Pressure under Different Outer Boundary Conditions

To study the effect of formation in situ stresses on wellbore breakdown for the three cases, two situations of varying formation maximum and minimum horizontal principal stresses are given. One situation is and increasing uniformly while keeping the difference between and unchanged. The other situation is that the difference between and is different.

From Equation (1), the circumferential compressive stress generated by and at the well wall in the direction of maximum stress ( or ) is . When and increase uniformly, indicates the change in the stress concentration of the well wall and the magnitude of the compressive stress value that needs to be overcome to breakdown the wellbore wall by the fluid pressure in the wellbore. Figure 9 shows the calculated breakdown pressure versus the values as and increase uniformly. From Figure 9, when and increase uniformly, the formation breakdown pressure increases linearly for the three cases. With increasing, the gap of the formation breakdown pressure values for the seepage force case (Case 3) is getting greater for other two cases. The effect of seepage force becomes more and more significant as the principal stresses of the formation increase uniformly.

Figure 10 gives the variation of formation breakdown pressure with the difference of the two in situ stresses (-). From Figure 10, for the three cases, with the increasing of -, formation breakdown pressure decreases linearly and the gap of the formation breakdown pressure values for seepage force case and the other cases is getting smaller. The higher the difference between and , the smaller the effect of the seepage force.

#### 4. Effect of Seepage Force for Different Permeability Rock

Steady-state fluid flow in porous media is controlled by Darcy’s law. For given fluid pressure in a wellbore and pore pressure, the smaller the rock permeability, the greater the pressure difference in wellbore radial direction. Therefore, the effect of seepage force on formation breakdown pressure will be affected by the rock permeability.

Using the same condition in Section 3.1, the formation breakdown pressures are analysed with various permeabilities for the three cases. Noting that in this section the pore pressure at the outer boundary is determined by the rock permeability ( cm^{3}/s; dm; mPa·s). During wellbore pressurization, different stable differential pressures are formed around the wellbore with different permeability ( mD, MPa; mD, MPa; mD, MPa).

The formation breakdown process of the three different permeability rocks under the acting of seepage force is shown in Figure 11.

As shown in Figure 11, low-permeability rock ( mD) is the first to initiate fracture under the action of seepage forces due to higher pressure differences and pore pressure gradients. As the pore pressure gradient is small and the effect of seepage force is not significant, the fracture initiation of high permeability rock ( mD) is slowest to get breakdown pressure. Although the pressure difference between the inner and outer boundaries of the high-permeability reservoir is small, the pore pressure at each point is relatively great. Therefore, the area where the effective circumferential stress field around the wellbore of the high-permeability reservoir is disturbed during which wellbore pressurization is more significant than that of the low-permeability rock.

In addition, the relationships between formation breakdown pressure and permeability for the three cases are shown in Figure 12. Figure 13 gives the ratio between the formation breakdown pressure of the case with seepage force and those from other two cases.

It can be seen from Figures 12 and 13 that the formation breakdown pressure for the case of fluid flowing into rock is smaller than that for impermeable rock. As the permeability increases, the effect of seepage force becomes less important, and the formation breakdown pressure is gradually approaching that without considering the effect of seepage force.

The effect of seepage force is more significant in low-permeability rock, where the formation breakdown pressure is only 20.65% of the breakdown pressure of ignoring seepage force. Therefore, the effect of seepage force should be applied during the analysis of hydraulic fracturing.

#### 5. Conclusions

(1)During hydraulic fracturing, wellbore fluid flows into wellbore and will yield a seepage force due to fluid drag forces. The seepage force acts on the rock matrix in the fluid flowing direction and could be measured by pore pressure gradient(2)For a vertical well, the seepage force is beneficial to breakdown the formation and thus lowers the wellbore breakdown pressure(3)The effect of seepage force on formation breakdown increases as the two horizontal principal stresses increase uniformly. Therefore, the deeper a formation, the greater action of the seepage force(4)The greater the difference of the two horizontal principal stresses, the lower effect of seepage force on formation breakdown. Therefore, seepage force gives more contribution to breakdown formation for isotropy formation or formations with small difference of the two horizontal stresses such as unconventional reservoirs(5)The greater the pore pressure gradient, the greater the seepage force. Seepage force increases as rock permeability decreases, and it should not be ignored in hydraulic fracturing analysis, especially for low-permeability formation

#### Data Availability

The [DATA TYPE] data used to support the findings of this study are available from the corresponding author upon request at [email protected].

#### Conflicts of Interest

The authors declare no conflict of interest.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51934005, 51874242, and 51904244); the National Science and Technology Major Project of China (Grant No. 2016ZX05050-009); the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2019JQ-364); and the Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 19JK0663).