A method for calculating grouting pressure and stability of borehole considering seepage softening effect
By constructing a calculation method for borehole grouting pressure and stability based on the seepage softening effect, the problem of coupling between grout seepage and soil softening was solved, thereby improving the accuracy and safety of grouting pressure design.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SHAANXI PROVINCIAL NATURAL GAS
- Filing Date
- 2025-12-13
- Publication Date
- 2026-06-09
Smart Images
Figure CN122174420A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of borehole stability analysis technology in geotechnical engineering, and particularly to a method for calculating borehole grouting pressure and stability applicable to horizontal directional drilling (HDD) projects. This method achieves accurate prediction of borehole stability and critical pressure during grouting by coupling the grout seepage effect with the soil softening effect upon contact with water. Background Technology
[0002] During the grouting process in geotechnical engineering, the grout seeps into the soil around the borehole under pressure, triggering two key physical processes: First, the grout seepage changes the pore water pressure distribution in the soil, generating additional seepage volume forces and disrupting the original stress balance of the soil; Second, the water in the grout invades the soil skeleton, weakening the cementation between soil particles and significantly reducing its shear strength parameters (cohesion c and internal friction angle φ), i.e., a "softening" effect occurs.
[0003] Currently, classical theories analyzing borehole stability, such as the circular borehole expansion theory, typically treat the soil as an ideal elasto-plastic medium, failing to adequately account for the fluid-solid coupling effects during the aforementioned construction process. Although some studies have attempted to consider seepage forces or soil softening separately, no mature theoretical model has yet been established to effectively couple these two interrelated effects within a unified mechanical framework. This theoretical simplification leads to biases in the prediction of grouting pressure: underestimating the pressure may cause borehole collapse, while excessively high pressure may lead to formation fracturing and grout loss, thereby threatening construction safety and increasing costs.
[0004] Therefore, there is an urgent need in this field for a theoretical model that can accurately consider the dual effects of seepage pressure and soil softening, so as to provide more reliable theoretical guidance for the design and construction of grouting projects. Summary of the Invention
[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for calculating borehole grouting pressure and stability that can simultaneously couple the seepage effect and the soil softening effect, so as to solve the problem that the prediction results of traditional models do not match the actual working conditions and improve the accuracy and reliability of grouting pressure design.
[0006] To achieve the above objectives, the present invention adopts the following technical solution: a method for calculating borehole grouting pressure and stability considering the seepage softening effect, the specific steps of which are as follows:
[0007] S1. Parameter Acquisition and Problem Definition;
[0008] Obtain the initial parameters required for engineering calculations, including: initial borehole radius r0, initial far-field in-situ stress P0, soil elastic modulus E, Poisson's ratio ν, cohesion c in unsoftened soil, internal friction angle φ, and shear dilatation angle. Softening coefficient, which characterizes the degree of strength reduction after contact with water. 0 < ≤ 1, and the permeability coefficient of the soil, etc.
[0009] S2. Establish the governing equations that take into account the seepage effect;
[0010] (1) Construct the equilibrium differential equation;
[0011] Based on the axisymmetric assumption, considering the seepage volume force caused by the pore water pressure gradient generated by grout seepage, the radial equilibrium differential equation of the soil surrounding the borehole is established, and its general form is:
[0012] (1);
[0013] in P represents radial and circumferential stresses, respectively. wat Let the fluid pressure satisfy the following equation:
[0014] (2);
[0015] (3);
[0016] In the above formula, P w Where P is the osmotic pressure, k is the grouting pressure, and P is the grouting pressure. w r is the grouting pressure coefficient. p r is the plastic radius. i This is the inner diameter after the hole is enlarged.
[0017] (2) Constructing a softened constitutive relation;
[0018] Introducing softening coefficient The Mohr-Coulomb yield criterion is modified to describe the strength degradation of soil upon exposure to water. The modified yield criterion expression is as follows:
[0019] (4);
[0020] In the above formula, C1 is the internal friction angle of the soil after softening (the internal friction angle is not assumed to remain constant in this invention), and C2 is the cohesion after softening, which satisfies the following formula:
[0021] (5);
[0022] Where C1 is the initial cohesion.
[0023] (3) Reconstruction and general solution of equilibrium differential equations;
[0024] Combining (5) and substituting (2) and (4) into (1), we can obtain the following equilibrium differential equation considering the dual effects of seepage pressure and soil softening:
[0025] (6);
[0026] In the above formula, A and B are two parameters characterizing the mechanical properties of soil, and their expressions are as follows:
[0027] (7);
[0028] Solving equation (6) yields the general solution as follows:
[0029] (8);
[0030] In the above formula, C is the integration constant.
[0031] S3. Solving for displacement and stress field in the elastic region;
[0032] When the grouting pressure exceeds the plastic limit pressure, the soil around the borehole is in an elastic-plastic state. Therefore, the boundary conditions at the interface between the plastic zone and the outer elastic zone are as follows:
[0033] (9);
[0034] Elastic stress and strain satisfy Hooke's law and geometric equations, from which the displacement equation and stress equation for the elastic region can be derived:
[0035] (10);
[0036] (11);
[0037] Combining the continuous boundary condition (9) at the interface of the elastoplastic region, the critical plastic stress P can be calculated. rp for:
[0038] (12);
[0039] S4. Solving for displacement and stress field in the plastic zone;
[0040] (1) Determine the stress field in the plastic zone;
[0041] To determine the particular solution of the stress function in the plastic zone, it is also necessary to consider the boundary conditions. The stress continuity boundary condition must be satisfied at the interface between the plastic and elastic zones, i.e., equation (9). Furthermore, for the inner diameter of the cavity, the following stress boundary conditions must be satisfied.
[0042] (13);
[0043] In the above equation, P is the grouting pressure inside the borehole wall. Substituting the boundary conditions (9) and (13) into (8), the integration constant C can be determined as follows:
[0044] (14);
[0045] In the above formula
[0046] (15);
[0047] (16);
[0048] in, The ratio of the radius of the plastic zone to the inner diameter after reaming reflects the relative development degree of the plastic zone. Combining this with equation (14), we can obtain the value of... The transcendental equations are as follows:
[0049] (17);
[0050] The relative development of the plastic zone can be accurately estimated by using the Newton iteration method. Substituting (14) into (8) and combining it with (4) will determine the plastic radial and circumferential stress fields.
[0051] (18);
[0052] (19);
[0053] (2) Determine the displacement field in the plastic region;
[0054] To determine the displacement field in the plastic region, geometric equations need to be applied, and the elastic and plastic strains in the plastic region need to be determined. The geometric equations are as follows:
[0055] (20);
[0056] in, Radial strain; For circumferential strain; This represents radial displacement.
[0057] Based on the generalized Hooke's law, the elastic strain in the plastic region can be obtained as follows:
[0058] (twenty one);
[0059] (twenty two);
[0060] in:
[0061] ,
[0062] ,
[0063] ,
[0064]
[0065] For soil and rock materials, the non-associated flow rule better suits their frictional characteristics; therefore, the non-associated flow rule is selected for plastic strain in the plastic region.
[0066] (twenty three);
[0067] Where: m is the soil dilatation parameter. .
[0068] Substituting equations (21) and (22) into (20) and combining them with (23), we can obtain the differential equation for the radial displacement as follows:
[0069] (twenty four);
[0070] Solving the above equations, and combining them with the continuous boundary conditions for the displacement at the interface between the elastic and plastic regions, the definite solution for the displacement in the plastic region can be determined:
[0071] (25);
[0072] in: , , .
[0073] Furthermore, the displacement generated under initial stress can be expressed as:
[0074] (26);
[0075] The expression for the relative radial displacement in the plastic zone is as follows:
[0076] (27);
[0077] S5. Determination of borehole inner diameter, plastic radius, and ultimate grouting pressure;
[0078] During the grouting and borehole enlargement process, the distance the grout-soil interface moves is the displacement at the borehole enlargement radius r1. Substituting r1 into (27), we can obtain:
[0079] (28);
[0080] in, Solving the above equation yields the inner diameter r1 of the expanded hole.
[0081] (29);
[0082] The relative development of the plastic region obtained by combining S4 solution This allows us to determine the radius of the plastic zone.
[0083] For equation (28), when the expanded radius r1 is infinitely large relative to the initial orifice diameter r0, i.e. r1 / r0→0, the limit value of the grouting pressure P can be obtained. u transcendental equations:
[0084] (30);
[0085] The grouting pressure limit value P can be accurately estimated using the Newton-Raphson iteration method. u .
[0086] Compared with the prior art, the present invention has the following significant advantages:
[0087] 1. Highly original: For the first time, seepage force and soil softening effect are fully coupled in a unified analytical model, which more realistically reflects the complex mechanical mechanism of the grouting process.
[0088] 2. Rigorous Theory: Starting from basic mechanical principles, the derivation process is rigorous, providing a complete semi-analytical solution from stress to displacement, with clear physical meaning.
[0089] 3. High practical value: The model ultimately boils down to solving well-defined equations, is easy to program and can be quickly used for engineering calculations, parameter sensitivity analysis and safety assessment.
[0090] 4. Wide applicability: By adjusting parameters (such as setting...) = 0 (the softening effect can be ignored, or the seepage force can be set to zero), the model can be flexibly degenerated into a specific case that only considers a single effect, which is convenient for mechanism research and comparative analysis, and has strong universality. Attached Figure Description
[0091] Figure 1 Comparative analysis of the effects of permeation and softening on the stress distribution of surrounding rock: (a) circumferential stress; (b) radial stress.
[0092] Figure 2 The effect of the permeability pressure coefficient on the ultimate grouting pressure and plastic radius, (a) plastic radius; (b) ultimate grouting pressure.
[0093] Figure 3 The effect of softening coefficient on ultimate grouting pressure and plastic radius, (a) plastic radius; (b) ultimate grouting pressure. Detailed Implementation
[0094] The invention will now be described in detail with reference to the accompanying drawings and a specific horizontal directional drilling (HDD) engineering embodiment. The invention adopts the following technical solution, and its core process is as follows: Figure 1 As shown, the specific steps are as follows:
[0095] S1. Parameter Acquisition and Problem Definition;
[0096] Obtain the initial parameters required for engineering calculations, including: initial borehole radius r0, initial far-field in-situ stress P0, soil elastic modulus E, Poisson's ratio ν, cohesion c in unsoftened soil, internal friction angle φ, and shear dilatation angle. Softening coefficient, which characterizes the degree of strength reduction after contact with water. And the permeability coefficient of the soil, etc.
[0097] S2. Establish the governing equations that take into account the seepage effect;
[0098] (1) Construct the equilibrium differential equation;
[0099] Based on the axisymmetric assumption, considering the seepage volume force caused by the pore water pressure gradient generated by grout seepage, the radial equilibrium differential equation of the soil surrounding the borehole is established, and its general form is:
[0100] (1);
[0101] in P represents radial and circumferential stresses, respectively. wat Let the fluid pressure satisfy the following equation:
[0102] (2);
[0103] (3);
[0104] In the above formula, P w Where P is the osmotic pressure, k is the grouting pressure, and P is the grouting pressure. w r is the grouting pressure coefficient. p r is the plastic radius. i This is the inner diameter after the hole is enlarged.
[0105] (2) Constructing a softened constitutive relation;
[0106] Introducing softening coefficient The Mohr-Coulomb yield criterion is modified to describe the strength degradation of soil upon exposure to water. The modified yield criterion expression is as follows:
[0107] (4);
[0108] In the above formula, C1 is the internal friction angle of the soil after softening (the internal friction angle is not assumed to remain constant in this invention), and C2 is the cohesion after softening, which satisfies the following formula:
[0109] (5);
[0110] Where C1 is the initial cohesion.
[0111] (3) Reconstruction and general solution of equilibrium differential equations;
[0112] Combining (5) and substituting (2) and (4) into (1), we can obtain the following equilibrium differential equation considering the dual effects of seepage pressure and soil softening:
[0113] (6);
[0114] In the above formula, A and B are two parameters characterizing the mechanical properties of soil, and their expressions are as follows:
[0115] (7);
[0116] Solving equation (6) yields the general solution as follows:
[0117] (8);
[0118] In the above formula, C is the integration constant.
[0119] S3. Solving for displacement and stress field in the elastic region;
[0120] When the grouting pressure exceeds the plastic limit pressure, the soil around the borehole is in an elastic-plastic state. Therefore, the boundary conditions at the interface between the plastic zone and the outer elastic zone are as follows:
[0121] (9);
[0122] Elastic stress and strain satisfy Hooke's law and geometric equations, from which the displacement equation and stress equation for the elastic region can be derived:
[0123] (10);
[0124] (11);
[0125] Combining the continuous boundary condition (9) at the interface of the elastoplastic region, the critical plastic stress P can be calculated. rp for:
[0126] (12);
[0127] S4. Solving for displacement and stress field in the plastic zone;
[0128] (1) Determine the stress field in the plastic zone;
[0129] To determine the particular solution of the stress function in the plastic zone, it is also necessary to consider the boundary conditions. The stress continuity boundary condition must be satisfied at the interface between the plastic and elastic zones, i.e., equation (9). Furthermore, for the inner diameter of the cavity, the following stress boundary conditions must be satisfied.
[0130] (13);
[0131] In the above equation, P is the grouting pressure inside the borehole wall. Substituting the boundary conditions (9) and (13) into (8), the integration constant C can be determined as follows:
[0132] (14);
[0133] In the above formula
[0134] (15);
[0135] (16);
[0136] in, The ratio of the radius of the plastic zone to the inner diameter after reaming reflects the relative development degree of the plastic zone. Combining this with equation (14), we can obtain the value of... The transcendental equations are as follows:
[0137] (17);
[0138] The relative development of the plastic zone can be accurately estimated using the Newton-Raphson iteration method. Substituting (14) into (8) and combining it with (4) allows us to determine the radial and circumferential stress fields of the plastic zone.
[0139] (18);
[0140] (19);
[0141] (2) Determine the displacement field in the plastic region;
[0142] To determine the displacement field in the plastic region, geometric equations need to be applied, and the elastic and plastic strains in the plastic region need to be determined. The geometric equations are as follows:
[0143] (20);
[0144] in, Radial strain; For circumferential strain; u r This represents radial displacement.
[0145] Based on the generalized Hooke's law, the elastic strain in the plastic region can be obtained as follows:
[0146] (twenty one);
[0147] (twenty two);
[0148] in:
[0149] ,
[0150] ,
[0151] ,
[0152]
[0153] For soil and rock materials, the non-associated flow rule better suits their frictional characteristics; therefore, the non-associated flow rule is selected for plastic strain in the plastic region.
[0154] (twenty three);
[0155] Where: m is the soil dilatation parameter. .
[0156] Substituting equations (21) and (22) into (20) and combining them with (23), we can obtain the differential equation for the radial displacement as follows:
[0157] (twenty four);
[0158] Solving the above equations, and combining them with the continuous boundary conditions for the displacement at the interface between the elastic and plastic regions, the definite solution for the displacement in the plastic region can be determined:
[0159] (25);
[0160] in: , , .
[0161] Furthermore, the displacement generated under initial stress can be expressed as:
[0162] (26);
[0163] The expression for the relative radial displacement in the plastic zone is as follows:
[0164] (27);
[0165] S5. Determination of borehole inner diameter, plastic radius, and ultimate grouting pressure;
[0166] During the grouting and borehole enlargement process, the distance the grout-soil interface moves is the displacement at the borehole enlargement radius r1. Substituting r1 into (27), we can obtain:
[0167] (28);
[0168] in, Solving the above equation yields the inner diameter r1 of the expanded hole.
[0169] (29);
[0170] The relative development of the plastic region obtained by combining S4 solution This allows us to determine the radius of the plastic zone.
[0171] For equation (28), when the expanded radius r1 is infinitely large relative to the initial orifice diameter r0, i.e. r1 / r0→0, the limit value of the grouting pressure P can be obtained. u transcendental equations:
[0172] (30);
[0173] The grouting pressure limit value P can be accurately estimated using the Newton-Raphson iteration method. u .
[0174] This embodiment is intended to help understand the essence of the present invention, rather than to limit the scope of protection.
[0175] Example: Grouting pressure assessment during the borehole enlargement stage of HDD engineering;
[0176] S1. Initial parameters;
[0177] The initial borehole radius r0 = 0.2 m, and the initial ground stress P0 = 100 kPa. Soil parameters: unsoftened cohesion C1 = 20 kPa, internal friction angle φ1 = 18°, and shear dilatation angle. The elastic modulus E = 30.8 MPa, and Poisson's ratio ν = 0.25. Based on geotechnical tests, the softening coefficient is set... This is the osmotic pressure coefficient.
[0178] S2. Solving for relevant parameters;
[0179] Given a grouting pressure P and a permeability coefficient, substituting the initial parameters into the equation and solving it allows us to calculate the ratio of the plastic zone radius to the inner diameter after reaming, i.e., the relative development degree r of the plastic zone. Substituting the obtained r into the equation allows us to solve for the inner diameter r1 after reaming, and further obtain the plastic zone radius r. p Based on this, the ultimate grouting pressure value can be calculated.
[0180] S3. Stress distribution around the hole;
[0181] Combining the parameters obtained from S2, in the polar coordinate system, given a radius r, we compare r with the radius r of the plastic region. p The size is used to determine the region: if r is located in the elastic region (i.e., r > r) p If r is located in the plastic region (i.e., r ≤ r), then substitute it into formula (1.10) to calculate the circumferential and radial stress distribution in the elastic region; p If the values are not specified, then substitute them into formulas (1.17) and (1.18) to calculate the corresponding circumferential and radial stress distributions in the plastic zone. The calculation results are as follows: Figure 1 As shown.
[0182] S4. Parametric analysis to investigate the effects of softening and seepage forces;
[0183] Under the given conditions of four grouting pressures P (250, 300, 350, 400) kPa, the radius r of the plastic zone is systematically investigated. p With osmotic pressure coefficient k w The variation pattern of the softening coefficient β (0 ~ 0.3) is further analyzed. Furthermore, given four softening coefficients β (0.0, 0.3, 0.6, 0.9), the ultimate grouting pressure P is further analyzed. u With osmotic pressure coefficient k w Response characteristics for changes in (0 ~ 0.3). Results are as follows. Figure 2 As shown.
[0184] Under the given conditions of four grouting pressures P (250, 300, 350, 400) kPa, the radius r of the plastic zone is systematically investigated. p The variation of softening coefficient β (0 ~ 0.9) is shown. Furthermore, four osmotic pressure coefficients k are given. w (0.0, 0.1, 0.2, 0.3), further analysis of the ultimate grouting pressure P u The response characteristics vary with the softening coefficient β (0 ~ 0.9). The results are as follows: Figure 3 As shown.
Claims
1. A method for calculating borehole grouting pressure and stability considering the seepage softening effect, characterized in that, The specific steps are as follows: S1. Parameter Acquisition and Problem Definition; Obtain the initial parameters required for engineering calculations, including: initial borehole radius r0, initial far-field in-situ stress P0, soil elastic modulus E, Poisson's ratio ν, cohesion c in unsoftened soil, internal friction angle φ, and shear dilatation angle. The softening coefficient β, which characterizes the degree of strength reduction after contact with water, 0 < β ≤ 1, and the permeability coefficient of the soil; S2. Establish the governing equations that take into account the seepage effect; (1) Construct the equilibrium differential equation; Based on the axisymmetric assumption, considering the seepage volume force caused by the pore water pressure gradient generated by grout seepage, the radial equilibrium differential equation of the soil surrounding the borehole is established, and its general form is: （1）； in , P represents radial and circumferential stresses, respectively. wat Let the fluid pressure satisfy the following equation: （2）； （3）； In the above formula, P w Where P is the osmotic pressure, k is the grouting pressure, and P is the grouting pressure. w r is the grouting pressure coefficient. p r is the plastic radius. i The inner diameter after reaming; (2) Constructing a softened constitutive relation; A softening coefficient β is introduced to modify the Mohr-Coulomb yield criterion to describe the strength degradation of soil after exposure to water; the modified yield criterion expression is as follows: （4）； In the above formula, φ2 is the internal friction angle of the soil after softening (in this invention, it is not assumed that the internal friction angle remains unchanged), and C2 is the cohesion after softening, which satisfies the following formula: （5）； Where C1 is the initial cohesion; (3) Reconstruction and general solution of equilibrium differential equations; Combining (5) and substituting (2) and (4) into (1), we obtain the following equilibrium differential equation considering the dual effects of seepage pressure and soil softening: （6）； In the above formula, A and B are two parameters characterizing the mechanical properties of soil, and their expressions are as follows: （7）； Solving equation (6) yields the general solution as follows: （8）； In the above formula, C is the integration constant; S3. Solving for displacement and stress field in the elastic region; When the grouting pressure exceeds the plastic limit pressure, the soil around the borehole is in an elastic-plastic state. Therefore, the boundary conditions at the interface between the plastic zone and the outer elastic zone are as follows: （9）； Elastic stress and strain satisfy Hooke's law and geometric equations, from which the displacement equation and stress equation for the elastic region can be derived: （10）； （11）； Combining the continuity boundary condition (9) at the interface of the elastoplastic region, the critical plastic stress P is calculated. rp for: （12）； S4. Solving for displacement and stress field in the plastic zone; (1) Determine the stress field in the plastic zone; The stress continuity boundary condition is satisfied at the interface between the plastic and elastic regions, i.e., equation (9). In addition, for the inner diameter of the cavity, the following stress boundary condition is satisfied. （13）； In the above formula, P is the grouting pressure inside the borehole wall; substituting the boundary conditions (9) and (13) into formula (8) respectively, the integration constant C is determined as follows: （14）； In the above formula （15）； （16）； in, The ratio of the radius of the plastic zone to the inner diameter after reaming reflects the relative development degree of the plastic zone. Combining this with equation (14), we can obtain the value of... The transcendental equations are as follows: （17）； The relative development degree of the plastic zone can be accurately estimated by using the Newton iteration method; by substituting (14) into (8) and combining it with (4), the plastic radial and circumferential stress fields can be determined; （18）； （19）； (2) Determine the displacement field in the plastic region; To determine the displacement field in the plastic region, geometric equations need to be applied, and the elastic and plastic strains in the plastic region need to be determined; the geometric equations are as follows: （20）； Where, ε r For radial strain; ε θ For circumferential strain; u r Radial displacement; Based on the generalized Hooke's law, the elastic strain in the plastic region is obtained as follows: （21）； （22）； in: ， ； ， ； ， ； ； For soil and rock materials, the non-associated flow rule better suits their frictional characteristics; therefore, the non-associated flow rule is selected for plastic strain in the plastic region. （23）； Where: m is the soil dilatation parameter. ; Substituting equations (21) and (22) into (20) and combining them with (23), we obtain the differential equation for the radial displacement as follows: （24）； Solve the above equations, and combine them with the continuous boundary conditions for the displacement at the interface between the elastic and plastic regions, to determine the boundary value for the displacement in the plastic region: （25）； in: , , ; Furthermore, the displacement generated under initial stress is expressed as: （26）； The expression for the relative radial displacement in the plastic zone is as follows: （27）； S5. Determination of borehole inner diameter, plastic radius, and ultimate grouting pressure; During the grouting and borehole enlargement process, the distance the grout-soil interface moves is the displacement at the borehole enlargement radius r1. Substituting r1 into (27), we can obtain: （28）； in, Solving the above equations yields the inner diameter r1 of the expanded hole. （29）； The relative development of the plastic region obtained by combining S4 solution Determine the radius of the plastic zone; For equation (28), when the expanded radius r1 is infinitely large relative to the initial orifice diameter r0, i.e. r1 / r0→0, the limit value of the grouting pressure P is obtained. u transcendental equations: （30）； The grouting pressure limit value P can be accurately estimated using the Newton-Raphson iteration method. u .