CN111274731A - Fractured stratum fracturing fracture extension track prediction method - Google Patents
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Abstract
The invention provides a method for predicting a fracture extension track of fractured stratum, which comprises the following steps of: (1) collecting formation parameters; (2) collecting construction parameters; (3) establishing a strong form of a fracture extension control equation set in a porous medium; (4) establishing a weak form of a fracture extension control equation set in the porous medium; (5) controlling the finite element dispersion of an equation set; (6) controlling the linearization of an equation set and establishing an iterative format; (7) and (3) inputting the parameters obtained in the steps (1) and (2) into the iterative calculation equation set established in the step (6), and predicting the fracture extension track under the parameters. The invention does not need to re-divide the grids after the fracture extends, and does not need to introduce additional criteria to track the fracture track.
Description
Technical Field
The invention relates to the field of yield increase transformation of oil and gas fields, in particular to a method for predicting a fracture extension track of fractured stratum.
Background
Unconventional reservoirs such as a Chinese shale reservoir, a coal bed, a tight sandstone reservoir and the like contain abundant natural gas resources, but because the permeability of the reservoirs is low, artificial fractures need to be built in the formations through a hydraulic fracturing technology, and the natural gas resources in the formations can be exploited. And a fracture simulation model is adopted to predict fracture tracks before fracturing construction, which is beneficial to optimizing a construction scheme. And after fracturing construction, the fracture morphology is predicted by adopting a fracturing fracture simulation model, so that evaluation after fracturing is facilitated. Therefore, the fracture extension track prediction technology becomes a core technology of hydraulic fracture optimization design. However, reservoirs such as shale, coal rock, tight sandstone and the like contain a large number of natural fractures, and the existence of the natural fractures can influence the hydraulic fracture extension track. Therefore, the method for predicting the fracture extension track in the fractured stratum has great significance for the fracturing optimization design of the fractured stratum.
Disclosure of Invention
The invention establishes a method for predicting the fracture extension track of fractured stratum by comprehensively applying multidisciplinary knowledge such as seepage mechanics, rock mechanics, finite element theory, phase field method theory and the like.
A fractured stratum fracturing fracture extension track prediction method comprises the following steps:
(1) collecting formation parameters;
(2) collecting construction parameters;
(3) establishing a strong form of a fracture extension control equation set in a porous medium;
(4) establishing a weak form of a fracture extension control equation set in the porous medium;
(5) controlling the finite element dispersion of an equation set;
(6) controlling the linearization of an equation set and establishing an iterative format;
(7) and (3) inputting the parameters obtained in the steps (1) and (2) into the iterative calculation equation set established in the step (6), and predicting the fracture extension track under the parameters.
Further, the formation parameters collected in step (1) include: the ground stress parameters, the rock elastic modulus and poisson's ratio, the rock body critical cohesion, the natural fracture length, the natural fracture azimuth, the natural fracture permeability, the natural fracture critical cohesion.
The construction parameters collected in the step (2) include: injection displacement, injection time and fracturing fluid viscosity.
The step (3) of establishing a strong form of a fracture extension control equation set in the porous medium comprises the following steps:
(3.1) establishment of stress equilibrium equation of porous medium
The stress balance equation of the porous elastic rock is as follows:
in the formula: σ is the stress tensor, MPa.
(3.2) fluid pressure calculation equation establishment
According to Darcy's percolation theorem and Biot pore elasticity theory, a fluid flowing pressure calculation equation in the porous medium can be obtained:
wherein M is the Biot modulus, MPa, and can be calculated by formula (3), α is the Biot coefficient, and can be calculated by formula (4), and εiiIs the volume strain; k is the anisotropic permeability tensor, m2Can be obtained by calculation according to the formula (5); mu.sfIs the fluid viscosity, pa.s;
in the formula: kfIs the bulk modulus of the fluid, MPa; phi is the porosity of the porous medium; k is the bulk modulus of the porous medium, MPa; ksIs the bulk modulus, MPa, of the rock skeleton particles;
in the formula: k is a radical of0Is the initial permeability of the rock, m2(ii) a k is the permeability of the rock after being influenced by external force, m2Establishing a calculation mode shown as a formula (6); θ is the angle of direction of the maximum principal strain, as shown in equation (7):
in the formula: b1，b2And b3The parameters to be determined are obtained by fitting experimental data; epsilon1Is the maximum principal strain; epsilonyIs strain in the Y direction; gamma rayxyFor shear strain:
(3.3) establishment of fracture extension control equation based on phase field method
During hydraulic fracturing, the total free energy per unit volume of fully saturated porous media can be broken down into three fractions, namely:
in the formula: ΨeffIs the elastic strain energy density, MPa, stored in the rock skeleton; ΨfluidFluid energy density, MPa; ΨfracFracture energy density, MPa;
rock can cause the tensile elasticity strain energy density of rock skeleton to reduce at the damage in-process, and compression elasticity strain energy density remains unchanged, then splits rock skeleton elasticity strain energy density into impaired tensile part and compression part in equation (8), promptly:
in the formula: g (c) is a decay function, and the decay function is defined as shown in the formula (10); psi+ effAnd psi- effTensile and compressive elastic strain energy densities, respectively, which can be calculated by equation (11), MPa:
g(c)＝(1-c)2(10)
in the formula: λ and G are Lame constants, MPa; epsiloni(i ═ 1, 2, 3) as the main strain; function(s)<x>+＝(|x|+x)/2，<x>-＝(|x|-x)/2；
The fluid energy density and the fracture energy density in equation (8) can be calculated by equations (12) and (13), respectively:
in the formula: l0Is a length scale parameter for controlling the diffusion crack region width, m; gcThe critical energy release rate, mpa.m, can be calculated by formula (14);
in the formula: sigmacCritical cohesion, MPa;
the crack phase field evolution is described by a micro-force balance equation; the micro-force balance equation of the porous medium is that without considering the influence of micro-inertia and external micro-force:
in the formula: hmMicro traction force, MPa.m; kmInternal micro force, MPa;
the internal energy balance equation of the porous medium can be written as:
in the formula: rho is the density of the porous medium, kg/m3(ii) a e is entropy, J/(mol.K); ζ is the increment of the volume content of the fluid, as shown in formula (17):
the fracturing process can be considered an isothermal process, and the total mechanical dissipation in this process is non-negative, i.e. the process can be described using the Clausius-Duhem inequality:
the derivative of the total free energy density over time in the above equation can be expressed as:
wherein:
substituting equations (16), and (19) to (22) into equation (18) yields:
all thermodynamic processes must satisfy the above inequality, that is to say
And
taking any value, the above inequality holds, then:
taking equation (24) into equation (1), the stress balance equation can be rewritten as:
in the formula: i is the unit tensor, in the two-dimensional case [ 110]T；σ+ effEffective tensile stress, MPa; sigma- effEffective compressive stress, MPa;
substituting equations (26) and (27) into equation (15) yields the equation for the evolution of the fracture phase field during fracturing:
wherein:
(3.4) boundary conditions corresponding to the governing equation
The boundary conditions of the stress balance equation, the fluid pressure calculation equation and the fracture phase field evolution equation are respectively shown in formulas (31) to (33);
in the equations (31) to (33),
and Γ are Dirichlet boundaries for displacement, pressure and fracture phase fields respectively,
and
the Neumann boundaries of the displacement field, pressure field and fracture phase field, respectively.
The step (4) of establishing a weak form of a fracture extension control equation set in the porous medium comprises the following steps:
partial differential equations (28), (2) and (29) are multiplied by a heuristic function w, respectivelyu、wpAnd wcAnd is integrated in the calculation domain in such a way that,then, by adopting the divergence theorem and combining the boundary conditions (31) to (33), a weak form of the control equation can be obtained:
the step (5) of controlling the finite element dispersion of the equation set comprises the following steps:
finite element discretization is carried out on the calculation area by adopting 4-node quadrilateral units, and for each calculation unit, the interpolation forms of displacement, pressure, fracture phase field and corresponding weight function and corresponding gradient are respectively shown as a formula (37) and a formula (38):
in the formula: u. ofh、phAnd chRespectively calculating displacement, pressure and fracture phase field values at the nodes; w is ah u、wh pAnd wh cRespectively calculating displacement, pressure and fracture phase field trial function values at nodes; n is a radical ofu、NPAnd NcRespectively are interpolation shape functions of a displacement field, a pressure field and a fracture phase field; b isuAnd Bu volRespectively a strain matrix and a volume strain matrix; b ispAnd BcRespectively, a derivative matrix of an interpolation shape function of the pressure and the fracture phase field;
substituting equations (37) and (38) into equations (34) to (36), respectively, yields:
the subscript n in equation (40) represents the value of the nth time step; Δ t is the time step, s;
the step (6) of controlling the linearization of the equation system and the establishment of the iterative format comprises the following steps:
the invention adopts a Newton-Raphson (NR) iterative method to solve seepage-stress coupling equation sets (39) and (40), and the Newton-Raphson (NR) iterative format of the ith iterative step is as follows:
wherein:
the displacement increment delta u of the ith iteration step can be obtained by equation (42)hAnd pressure increase δ phThen, the heuristic solution of the (i + 1) th iteration step displacement and pressure can be obtained, namely:
after a tentative solution of the displacement and pressure of the (i + 1) th iteration step is obtained, then the fracture phase field value can be obtained by solving equation (47), and the displacement and pressure values are fixed in the process:
wherein
When the displacement, the pressure and the fracture phase field all meet the convergence condition shown in the formula (48), ending the iteration, and entering the calculation of the next time step, otherwise, continuing the iteration;
||Ru||≤||Ru0||，||Rp||≤||Rp0||，||ci+1-ci||≤||Rc0|| (50)
in the above formula, Ru0、Rp0And Rc0Respectively displacement field, pressure field, and crack phase field convergence tolerance.
According to the method for predicting the fracture extension track of the fractured stratum provided by the invention, the mesh does not need to be re-divided after the fracture is extended, and the fracture track does not need to be tracked by introducing an additional criterion.
Drawings
FIG. 1 is a schematic diagram of the calculation region and boundary conditions in embodiment 1;
FIG. 2 is a graph of fracture trajectories after 8s of injection in example 1;
FIG. 3 plot of fracture trajectories after 16s of injection in example 1;
figure 4 fracture tracings after 24s of injection in example 1.
Detailed Description
The invention is described in further detail below with reference to certain well parameters, but without limiting the invention in any way, wherein the basic parameters used in the calculations are shown in table 1.
Table 1 table of basic parameters used for the calculation of example 1
The first step is as follows: establishing a computational physical model, and setting boundary conditions of a computational area, as shown in fig. 1, in this embodiment, the computational area is a square area of 20m × 20m, a fracture with an initial length of 2m is located in the center of the computational area, and fracturing fluid is injected from the center of the initial fracture. And a natural crack which forms an angle of 30 degrees with the y direction and is 5m long is arranged at a position 3m above the injection point. The minimum horizontal ground stress in the x direction is 20MPa at the right boundary of the calculation region, the maximum horizontal ground stress in the y direction is 25MPa at the upper boundary of the calculation region, the displacement of the left boundary of the calculation region in the x direction is fixed to 0, and the displacement of the lower boundary of the calculation region in the y direction is fixed to 0.
The second step is that: and (4) mesh subdivision, namely dividing a calculation area into 80 multiplied by 80 square units with four nodes uniformly.
The third step: the parameters in table 1 were substituted into the set of equations established in the present invention for simulation. The simulation results are shown in fig. 2 to 4. It can be seen from fig. 2 that the hydraulic fracture propagates along a straight line before it intersects the natural fracture, and from fig. 3, it can be seen that in this embodiment the hydraulic fracture will propagate along the natural fracture after it intersects the natural fracture. As can be seen in fig. 4, after the hydraulic fracture propagates along the natural fracture to the natural fracture tip, it will turn to propagate in the direction of maximum horizontal stress.
Claims (7)
1. A fractured stratum fracturing fracture extension track prediction method is characterized by comprising the following steps:
(1) collecting formation parameters;
(2) collecting construction parameters;
(3) establishing a strong form of a fracture extension control equation set in a porous medium;
(4) establishing a weak form of a fracture extension control equation set in the porous medium;
(5) controlling the finite element dispersion of an equation set;
(6) controlling the linearization of an equation set and establishing an iterative format;
(7) and (3) inputting the parameters obtained in the steps (1) and (2) into the iterative calculation equation set established in the step (6), and predicting the fracture extension track under the parameters.
2. A method for predicting a fracture propagation trajectory of a fractured formation of a fractured reservoir as claimed in claim 1, wherein the formation parameters collected in the step (1) comprise: the method comprises the following steps of crustal stress parameters, rock elastic modulus and Poisson ratio, rock body critical cohesion, natural fracture length, natural fracture direction angle, natural fracture permeability and natural fracture critical cohesion.
3. A method for predicting a fracture propagation trajectory of a fractured stratum according to claim 1, wherein the construction parameters collected in the step (2) comprise: injection displacement, injection time and fracturing fluid viscosity.
4. A method for predicting a fracture propagation trajectory of a fractured stratum according to claim 1, wherein the step (3) establishes a strong form of a fracture propagation control equation system in a porous medium, and comprises the following steps:
(3.1) establishment of stress equilibrium equation of porous medium
The stress balance equation of the porous elastic rock is as follows:
▽·σ＝0 (1)
in the formula: sigma is stress tensor, MPa;
(3.2) fluid pressure calculation equation establishment
According to Darcy's percolation theorem and Biot pore elasticity theory, a fluid flowing pressure calculation equation in the porous medium can be obtained:
wherein M is the Biot modulus, MPa, and can be calculated by formula (3), α is the Biot coefficient, and can be calculated by formula (4), and εiiIs the volume strain; k is the anisotropic permeability tensor, m2Can be obtained by calculation according to the formula (5); mu.sfIs the fluid viscosity, pa.s;
in the formula: kfIs the bulk modulus of the fluid, MPa; phi is the porosity of the porous medium; k is the bulk modulus of the porous medium, MPa; ksIs the bulk modulus, MPa, of the rock skeleton particles;
in the formula: k is a radical of0Is the initial permeability of the rock, m2(ii) a k is the permeability of the rock after being influenced by external force, m2Establishing a calculation mode shown as a formula (6); θ is the angle of direction of the maximum principal strain, as shown in equation (7):
in the formula: b1，b2And b3The parameters to be determined are obtained by fitting experimental data; epsilon1Is the maximum principal strain; epsilonyIs strain in the Y direction; gamma rayxyFor shear strain:
(3.3) establishment of fracture extension control equation based on phase field method
During hydraulic fracturing, the total free energy per unit volume of fully saturated porous media can be broken down into three fractions, namely:
ψ(ε，ζ,c,▽c)＝ψeff(ε)+ψfluid(ε，ζ)+ψfrac(c,▽c) (8)
in the formula: ΨeffIs the elastic strain energy density, MPa, stored in the rock skeleton; ΨfluidFluid energy density, MPa; ΨfracFracture energy density, MPa;
rock can cause the tensile elasticity strain energy density of rock skeleton to reduce at the damage in-process, and compression elasticity strain energy density remains unchanged, then splits rock skeleton elasticity strain energy density into impaired tensile part and compression part in equation (8), promptly:
in the formula: g (c) is a decay function, and the decay function is defined as shown in the formula (10); psi+ effAnd psi- effTensile and compressive elastic strain energy densities, respectively, which can be calculated by equation (11), MPa:
g(c)＝(1-c)2(10)
in the formula: λ and G are Lame constants, MPa; epsiloni(i＝1, 2, 3) is main strain; function(s)<x>+＝(|x|+x)/2，<x>-＝(|x|-x)/2；
The fluid energy density and the fracture energy density in equation (8) can be calculated by equations (12) and (13), respectively:
in the formula: l0Is a length scale parameter for controlling the diffusion crack region width, m; gcThe critical energy release rate, mpa.m, can be calculated by formula (14);
in the formula: zetacCritical cohesion, MPa;
the crack phase field evolution is described by a micro-force balance equation; the micro-force balance equation of the porous medium is that without considering the influence of micro-inertia and external micro-force:
▽·Hm-Km＝0 (15)
in the formula: hmMicro traction force, MPa.m; kmInternal micro force, MPa;
the internal energy balance equation of the porous medium can be written as:
in the formula: rho is the density of the porous medium, kg/m3(ii) a e is entropy, J/(mol.K); ζ is the increment of the volume content of the fluid, as shown in formula (17):
the fracturing process can be considered an isothermal process, and the total mechanical dissipation in this process is non-negative, i.e. the process can be described using the Clausius-Duhem inequality:
the derivative of the total free energy density over time in the above equation can be expressed as:
wherein:
substituting equations (16), and (19) to (22) into equation (18) yields:
all thermodynamic processes must satisfy the above inequality, that is to say
And
take any value of aboveIf the equations are true, then:
taking equation (24) into equation (1), the stress balance equation can be rewritten as:
in the formula: i is the unit tensor, in the two-dimensional case [ 110]T；σ+ effEffective tensile stress, MPa; sigma- effEffective compressive stress, MPa;
substituting equations (26) and (27) into equation (15) yields the equation for the evolution of the fracture phase field during fracturing:
wherein:
(3.4) boundary conditions corresponding to the governing equation
The boundary conditions of the stress balance equation, the fluid pressure calculation equation and the fracture phase field evolution equation are respectively shown in formulas (31) to (33);
in the equations (31) to (33),
and Γ are Dirichlet boundaries for displacement, pressure and fracture phase fields respectively,
and
the Neumann boundaries of the displacement field, pressure field and fracture phase field, respectively.
5. A method for predicting a fracture propagation trajectory of a fractured stratum according to claim 1, wherein the step (4) establishes a weak form of a fracture propagation control equation system in a porous medium, and comprises the following steps:
partial differential equations (28), (2) and (29) are multiplied by a heuristic function w, respectivelyu、wpAnd wcAnd integrating in a calculation domain, and then adopting a divergence theorem and combining boundary conditions (31) to (33) to obtain a weak form of a control equation:
6. a method for predicting a fracture propagation trajectory of a fractured formation of a fractured according to claim 1, wherein the step (5) controls the finite element dispersion of the equation system, and comprises the following steps:
finite element discretization is carried out on the calculation area by adopting 4-node quadrilateral units, and for each calculation unit, the interpolation forms of displacement, pressure, fracture phase field and corresponding weight function and corresponding gradient are respectively shown as a formula (37) and a formula (38):
in the formula: u. ofh、phAnd chRespectively calculating displacement, pressure and fracture phase field values at the nodes; w is ah u、wh pAnd wh cRespectively calculating displacement, pressure and fracture phase field trial function values at nodes; n is a radical ofu、NPAnd NcRespectively are interpolation shape functions of a displacement field, a pressure field and a fracture phase field; b isuAnd Bu volRespectively a strain matrix and a volume strain matrix; b ispAnd BcRespectively, a derivative matrix of an interpolation shape function of the pressure and the fracture phase field;
substituting equations (37) and (38) into equations (34) to (36), respectively, yields:
the subscript n in equation (40) represents the value of the nth time step; Δ t is the time step, s.
7. A method for predicting a fracture propagation trajectory of a fractured formation of a fractured according to claim 1, wherein the step (6) of controlling the linearization and iterative format establishment of the equation system comprises the following steps:
solving seepage-stress coupling equation systems (39) and (40) by using a Newton-Raphson (NR) iterative method, wherein the Newton-Raphson (NR) iterative format of the ith iteration step is as follows:
wherein:
the displacement increment delta u of the ith iteration step can be obtained by equation (42)hAnd pressure increase δ phThen, the heuristic solution of the (i + 1) th iteration step displacement and pressure can be obtained, namely:
after a tentative solution of the displacement and pressure of the (i + 1) th iteration step is obtained, then the fracture phase field value can be obtained by solving equation (47), and the displacement and pressure values are fixed in the process:
wherein
When the displacement, the pressure and the fracture phase field all meet the convergence condition shown in the formula (48), ending the iteration, and entering the calculation of the next time step, otherwise, continuing the iteration;
||Ru||≤||Ru0||，||Rp||≤||Rp0||，||ci+1-ci||≤||Rc0|| (50)
in the above formula, Ru0、Rp0And Rc0Respectively displacement field, pressure field, and crack phase field convergence tolerance.
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| CN112127882A (en) * | 2020-11-02 | 2020-12-25 | 西南石油大学 | Method for calculating dynamic fracture width of drilling fluid leakage of fractured formation |
| CN112487557A (en) * | 2020-12-03 | 2021-03-12 | 上海交通大学 | Method for predicting interface failure and microscopic crack propagation of composite material under hydraulic permeation load |
| CN112949119A (en) * | 2021-02-03 | 2021-06-11 | 中国石油天然气股份有限公司 | Crack form simulation method, device, equipment and medium |
| CN114548593A (en) * | 2022-03-02 | 2022-05-27 | 西南石油大学 | Conglomerate reservoir hydraulic fracturing complex fracture extension track prediction method |
| CN116110520A (en) * | 2023-01-16 | 2023-05-12 | 沈阳工业大学 | Near-field dynamics-based rock material constitutive model building and simulating method |
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| CN114548593A (en) * | 2022-03-02 | 2022-05-27 | 西南石油大学 | Conglomerate reservoir hydraulic fracturing complex fracture extension track prediction method |
| CN114548593B (en) * | 2022-03-02 | 2024-04-16 | 西南石油大学 | Method for predicting extension track of hydraulic fracturing complex cracks of conglomerate reservoir |
| CN116110520A (en) * | 2023-01-16 | 2023-05-12 | 沈阳工业大学 | Near-field dynamics-based rock material constitutive model building and simulating method |
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