#### Abstract

Drilling fluid loss in depleted reservoir has been an import issue faced by further tapping the potential of old oil fields. Accurate evaluation of the fracture pressure is the foundation to avoid mud loss. Traditional views suggest that tensile failure is the only fracture mode and the fracture pressure should be determined by a tensile failure criterion, which are not suitable for wells drilled in the depleted reservoir. In this paper, the analysis focuses on the fracture mode and fracture pressure in depleted reservoir, and case studies show that three fracture modes may first occur, and the fracture mode will be changed with reservoir depletion which highly depends on reservoir depletion degree, well azimuth and deviation angle, and the in situ stress state; different failure criteria at different stages of reservoir depletion should be selected to accurately evaluate the fracture pressure. For the vertical well, fracture pressure is no longer a single linear reduction with reservoir depletion; instead, a three-step and two-step reduction may appear, and for the directional well, the fracture pressure is not always decreased; the other patterns such as increase and first increase then decrease may also appear for the wells drilled in reverse and strike fault stress regimes.

#### 1. Introduction

At present, more than 70% hydrocarbon resources are obtained from secondary and tertiary recovery [1]. Further tapping the potential of old oil fields is one of the main directions to improve oil recovery rate in the future. But due to the long-term hydrocarbon production, reservoir pressure often decreases which extends with time from wellbore vicinity to the far field of the reservoir over the lifetime of the oil field unless sufficient pressure support is supplemented. The in situ stress also changes due to reservoir compaction deformation along with pressure depletion [2]. Therefore, the fracture mode and fracture pressure are both changed when wellbore is drilled through highly depleted reservoirs to make new horizontal wells as producers or injectors, or develop new untapped reservoirs in deeper horizons. Field evidences have indicated that the fracture pressure was decreased with reservoir depletion and more drilling fluid losses occurred which have serious impact on drilling safety and costs [3–10].

Extensive related studies have been conducted in order to avoid drilling fluid loss in depleted reservoir, but most of the previous research focused on the fracture pressure [11–14] and only few on fracture mode in depleted reservoir, which result in the wrong fracture pressure calculation model selection and inaccurate fracture pressure calculation results. For example, the previous research have concluded that when the drilling fluid pressure inside the wellbore is high enough, tensile fracture first occurs at the wellbore along the direction of minimum principal stress, and fracture pressure should be determined by a tensile failure criterion [15]. On the basis of those facts, we can reach the conclusion that the fracture pressure of vertical well decreases linearly with reservoir depletion.

In fact, with reservoir depletion, the original tensile failure mode may be transformed into shear failure mode, and shear fracture may first occur at the wellbore when the drilling fluid pressure inside the wellbore is high enough. In this case, fracture pressure should not be determined by a tensile failure criterion, and the reduction of fracture pressure of vertical well will present a more complex law. Furthermore, the problem becomes more complicated in directional wells due to changes in well inclination and azimuth. Aiming at this problem, the analytical method for wellbore fracture mode and fracture pressure in depleted reservoir is established in this paper, and the fracture mode and fracture pressure in different reservoir and wellbore conditions are analyzed in depth.

#### 2. Changes of In Situ Stresses with Reservoir Depletion

For a developed reservoir, the long-term recovery of oil and gas resources will lead to the decrease of pore pressure, unless the energy is fully replenished by water injection or the other operations. The reservoir with an overall reduction in pore pressure is defined as depleted reservoir. Using reservoir numerical simulation method, by setting the number and location of producing and injection wells, reservoir characteristic parameters (permeability, porosity, and saturation), production parameters (production rate, injection rate, and production time), etc. can simulate and determine the pressure depletion degree at different stages.

Reservoir depletion will induce its compaction deformation. As a result, the magnitudes of in situ stresses will be changed [2]. The change of in situ stresses depends on a number of factors, such as pore pressure depletion, reservoir properties, and boundary conditions. But fortunately, a simple linear relationship usually exists between the change of horizontal stress and the reservoir depletion for all different scenarios, which can be expressed as follows: where and are the changes of the horizontal maximum and minimum stress due to reservoir depletion, respectively; is the pore pressure depletion; is a correlation coefficient which is related to rock mechanical properties and reservoir boundary conditions.

When the hydrocarbon reservoir is considered to be horizontal and infinite in lateral extent, while its thickness is finite, the correlation coefficient can be written as follows: where is the Poisson ratio and is the Biot coefficient.

Under the same conditions, the vertical principal stress component does not change with reservoir depletion, i.e.,

#### 3. Fracture Mode and Fracture Pressure of Vertical Well in Depleted Reservoir

When the drilling fluid pressure inside the wellbore is high enough to fracture the rock surrounding the borehole, the critical pressure is defined as fracture pressure. To calculate the fracture pressure, the first is to get the effective stress state around vertical wellbore wall under the effect of in situ stress and drilling fluid pressure inside the borehole, which has been presented by many published literatures as follows [16, 17]: where , , and are the effective radical, tangential, and axial stresses in the borehole coordinate system, respectively; is the drilling fluid pressure inside the borehole; is the angle from the direction of to the radius vector of the borehole.

When reservoir pressure is depleted by , the effective stress state around vertical wellbore can be derived by substituting Equation (1) into Equation (4):

The previous research have concluded that when the drilling fluid pressure inside the wellbore is high enough, tensile fracture first occurs at the wellbore along the direction of minimum principal stress, and fracture pressure should be determined by a tensile failure criterion [15].

But a new phenomenon has also been observed after reservoir depletion. The change of the effective stress state around vertical wellbore with reservoir depletion is calculated according to the above equation, and the results (Figure 1) show that tensile fracture occurred first at the location of when the drilling fluid pressure is high enough, but when the reservoir pressure is depleted (from 20 MPa to 10 MPa), the minimum effective tangential stress is changed very little, and the maximum stress differences of and at the location of are both getting wider and wider; hence, shear fracture may first occur with reservoir depletion according Mohr-Coulomb criterion. As a result, the wellbore fracture mode in the depleted reservoir can be divided into three types which depend on the reservoir depletion and so on: (1) Mode 1: tensile fracture, (2) Mode 2: shear fracture and , and (3) Mode 3: shear fracture and .

According to the above analysis, fracture always first occurred at the location of for any of the three modes due to its minimum effective tangential stress and the maximum stress differences around the borehole. So, if we want to get the fracture pressure, the effective stress state at the location of should be first derived based on Equation (5):

For Mode 1, fracture pressure should be determined by a tensile failure criterion: where is the tensile strength.

Then, we can get the fracture pressure in this mode:

For Mode 2, fracture pressure should be determined by Mohr-Coulomb shear failure criterion when : where is the cohesive force; is the internal friction angle.

Then, we can get the fracture pressure in this mode by taking Equation (6) into Equation (9): where .

For Mode 3, fracture pressure should be determined by Mohr-Coulomb shear failure criterion when :

Then, we can get the fracture pressure in this mode by taking Equation (6) into Equation (11):

Through the comparison of the fracture pressures in the three modes, the fracture pressure of the vertical well in depleted reservoir can be determined as follows:

The fracture mode and corresponding conditions can be determined by the following principles: (1)If , tensile fracture first occurs and the fracture mode is Mode 1, and the occurrence conditions is and (2)If , shear fracture first occurs and the fracture mode is Mode 2, and the occurrence conditions is and (3)If , shear fracture first occurs and the fracture mode is Mode 3, and the occurrence conditions is and

#### 4. Fracture Mode and Fracture Pressure of Directional Well in Depleted Reservoir

In order to further tap the potential of old oil fields, more directional wells including horizontal wells are drilled. But due to unequal change of the vertical and horizontal principal stress, fracture mode and fracture pressure of directional well in depleted reservoir are more complicated. For example, the fracture pressure in Mode 1 of the horizontal wells drilled in the direction of maximum horizontal in situ stress should be described in two types, and the influences on the fracture pressure of reservoir depletion for the two types are both different from the vertical well: (1)In normal and strike fault stress regime, the in situ stress state is and , respectively [18]. Hence, the fracture pressure is as follows, and the influence coefficient of reservoir depletion is(2)In reverse fault stress regime, the in situ stress state in reverse fault regimes is . Hence, the fracture pressure is as follows, and the influence coefficient of reservoir depletion is

For an arbitrarily deviated well, the stress components of in situ ground stresses in borehole coordinate system can be derived by coordinate system transformation [17]: where is the well deviation angle and is the well azimuth angle relative to the maximum horizontal principal stress orientation.

After reservoir depletion, the stress components will be changed due to in situ stress change which can be derived by substituting Equation (1) into Equation (16):

Assuming the plane strain is normal to the borehole axis, concentrated stress component solutions surrounding an arbitrarily deviated well wall in terms of cylindrical polar coordinates can be written as follows:

Because, is one of the three principal stresses, and the other two principal stresses can be derived as follows:

Based on the analysis of vertical well, failure criterion in different fracture modes should be described as the following principles: (1)For Mode 1, fracture pressure should be determined by a tensile failure criterion(2)For Mode 2, fracture pressure should be determined by Mohr-Coulomb shear failure criterion(3)For Mode 3, fracture pressure should also be determined by Mohr-Coulomb shear failure criterion

The fracture pressure in the three fracture modes can be solved by substituting Equation (18) and Equation (19) into Equation (20) to Equation (22), respectively. But because there is no analytical solution to the fracture pressure of directional wells in depleted reservoir, the fracture pressures (, , and ) in the three fracture modes should be solved numerically by iteration. The final fracture pressure and fracture mode can be determined by using Equation (13), and fracture mode discrimination principles are the same with that of the vertical well.

#### 5. Case Study and Discussion

Using the theory described above, case studies are performed in this section to illustrate the evolution of fracture mode and fracture pressure with reservoir depletion. The following parameters are chosen for the base case: (i)Reservoir mechanical parameters: , , , , and .(ii)(iii)The direction of maximum horizontal in situ (iv)Three types of initial in situ stress states are considered: (a)Normal fault stress regime: (b)Reverse fault stress regime: (c)Strike fault stress regime:

##### 5.1. Vertical Well

For the vertical well drilled in different fault stress regimes, the changes of fracture mode and fracture pressure with reservoir depletion are illustrated in Figures 2–4, respectively. In the figures, the blue line represents the fracture pressure in Mode 1, the red line represents the fracture pressure in Mode 2, and the green line represents the fracture pressure in Mode 3, and the final fracture pressure should be the minimum of the three. Furthermore, the pale blue areas represent Mode 1 first occurs when the well is fractured, the pale red areas represent Mode 2 first occurs, and the pale green areas represent Mode 3 first occurs.

The results show several differences from conventional views: (1)Tension failure is not the only fracture mode; two types of shear fracture modes also occur; for example, in reverse fault stress regime (Figure 3), Mode 3 always first occurs when the vertical well is fractured with reservoir depletion, and the fracture pressure should always be determined by the Mohr-Coulomb shear failure criterion, and the tensile failure criterion used in previous calculations may make a great overestimate (about 13 MPa) for the fracture pressure(2)Fracture mode will be changed with reservoir depletion, so different failure criteria at different stages of reservoir depletion should be selected; for example, in normal fault stress regime (Figure 2), Mode 1 first occurs in initial stage of reservoir depletion (), but Mode 3 first occurs in the medium-term stage (), and in the later period (), Mode 2 first occurs(3)Fracture pressure is no longer a single linear reduction with reservoir depletion; instead, three-step and two-step reduction may appear as shown in normal and strike fault stress regimes, respectively (Figures 2 and 4)

##### 5.2. Directional Well

The changes of fracture mode with reservoir depletion of directional wellbore drilled in different fault stress regimes are illustrated in Table 1. In the table, the polar angle of the pies represents the well azimuth angle , ranging from 0 to 360°, and the polar radius represents the well deviation angle , ranging from 0 to 90°. Moreover, the blue areas represent Mode 1 first occurs when the well is fractured, the red areas represent Mode 2 first occurs, and the green areas represent Mode 3 first occurs.

The results show that fracture mode highly depends on reservoir depletion degree, well azimuth and deviation angle, and the in situ stress state; the specific relationships are as follows: (1)In initial stage of reservoir depletion (), Mode 3 is the only fracture mode for the arbitrary directional well drilled in the reverse and strike fault stress regimes, but for the normal fault stress regime, Mode 1 also appears in the wells drilled towards the direction of maximum horizontal in situ stress(2)In the medium-term stage of reservoir depletion (), the fracture modes remain unchanged for the arbitrary directional well drilled in the reverse and strike fault stress regimes. But for the normal fault stress regime, the fracture modes are changed; all of the three fracture modes come out. The fracture modes for some lowly deviated wells drilled towards the direction of maximum and minimum horizontal in situ stress are changed into Mode 3 and Mode 2, respectively(3)In the later period of reservoir depletion (), only for the arbitrary directional well drilled in the reverse fault stress regimes, the fracture modes remain unchanged. For strike fault stress regimes, the fracture modes for some lowly deviated wells drilled towards the direction of minimum horizontal in situ stress are changed into Mode 2. And for normal fault stress regimes, the fracture modes for the lowly deviated wells towards arbitrary direction are all changed into Mode 2

The changes of fracture pressure with reservoir depletion of directional wellbore drilled in normal, reverse, and strike fault stress regimes are illustrated in Tables 2–4, respectively. In the table, the polar angle and radius of the pies represent the same meaning as in Table 1. The color represents specific value of the fracture pressure. The results show that the fracture pressure is not always decreased with reservoir depletion which depends on the well azimuth and deviation angle and the in situ stress state; the specific change rules are as follows: (1)For normal fault stress regimes, all of the four fracture pressures in different modes show basically the same change rule with reservoir depletion; the fracture pressures are all decreased for the arbitrary directional wells(2)For reverse fault stress regimes, all of the four fracture pressures in different modes also show basically the same change rule with reservoir depletion; the fracture pressures for the some lowly deviated wells drilled towards the direction of maximum horizontal in situ stress are decreased, but the fracture pressure of the other wells showed a pattern of increase(3)For reverse fault stress regimes, all of the four fracture pressures in different modes also show basically the same change rule with reservoir depletion; the fracture pressures for the some highly deviated wells drilled towards the direction of minimum horizontal in situ stress are first increased and then decreased, but the fracture pressure of the other wells showed a pattern of decrease

#### 6. Model Validation

The field data are collected from M well drilled in the South China Sea, and the maximum horizontal principal stress orientation is N165°. The reservoir depth is 1300~1400 m, and initial pore pressure is 13.5~14.5 MPa. The gas field has now been in production for nearly 20 years, and pore pressure has been depleted into 6.5~7.0 MPa. In order to stabilize the yield of the gas field, the extended reach horizontal well M was drilled with 88°~94° deviation angle towards N250°E direction. According to the Addis and Li and Gray’s model [11, 15], the fracture pressure after reservoir depletion is basically over 21 MPa (Figure 5), and the fluid pressure inside the wellbore was set below the fracture pressure to ensure the borehole stability. But several mud losses were induced during the drilling process.

Based on the analysis by using the model of this study, it is found that the fracture mode belongs to Mode 3, and the pressure of this well is basically below 21 MPa at present. The pressure at the location of mud loss reaches the fracture pressure resulting in mud loss. The model of this study is more consistent with the actual drilling condition, which verifies the effectiveness of the model.

#### 7. Conclusion

Hydrocarbon extraction usually leads to pore pressure decline within a reservoir, which further changes the wellbore fracture mode and fracture pressure drilled in the depleted reservoir. Accurate evaluation of the fracture pressure on the basis of correct judgment of the fracture mode is the foundation of safe and efficient drilling. The expression is somewhat unclear. It is suggested to change it to:Traditional studies have considered that tensile failure is the fracture mode and the fracture pressure should be determined by a tensile failure criterion, which are not suitable for the depleted reservoir.

In this paper, the studies focus on the change of the fracture mode and fracture pressure with reservoir depletion; the analysis results show the following: (1)Tension failure is not the only fracture mode; any of the three fracture modes including shear fracture when the drilling fluid pressure inside the borehole is high enough may first occur in the well drilled in depleted reservoir(2)Fracture mode will be changed with reservoir depletion, and the fracture mode highly depends on reservoir depletion degree, well azimuth and deviation angle, and the in situ stress state, so different failure criteria at different stages of reservoir depletion should be selected to accurately evaluate the fracture pressure(3)For the vertical well, the fracture pressure is decreased with reservoir depletion, but the difference from conventional views is that fracture pressure is no longer a single linear reduction with reservoir depletion; instead, three-step and two-step reduction may appear due to the change of fracture mode(4)For the directional well, the fracture pressure is not always decreased with reservoir depletion; the other patterns such as increase and first increase then decrease may also appear for the wells drilled in reverse and strike fault stress regimes

#### Data Availability

The data used to support the findings and results of this study are available from the corresponding authors upon request.

#### Conflicts of Interest

The authors declare that they have no conflict of interest.

#### Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant No. 52074224 and Grant No. 52074221), the Research Foundation of Key Laboratory of Deep Geodrilling Technology, Ministry of Natural Resources, China University of Geosciences (Beijing) (Grant No. KF201902), and the Key Research and Development Program of Shandong Province (2019GGX103025).