A method for studying gas invasion laws in fractured formations

Abstract

The present invention relates to the technical field of fracture gas invasion research, and in particular discloses a method for studying the laws of gas invasion in fractured formations. By establishing a reservoir deformation model, a fluid flow model, and a fluid-solid cross-coupling model, the numerical form of the mathematical model is derived using the finite element method, and discrete fractures are constructed using MATLAB. Finally, the method is solved using commercial software to obtain the laws of gas invasion in fractured formations. By understanding the characteristics and influencing factors of gas invasion, corresponding measures can be formulated to reduce gas invasion and mitigate the adverse effects on the drilling process and oil and gas field development. By studying gas invasion caused by negative pressure differential drilling, the design and optimization of drilling fluid can be improved to reduce the risk of gas invasion into the wellbore. The discussion of the research results has great significance for actual oilfield development.

Description

Technical Field

[0001] The present invention relates to the technical field of fracture gas invasion research, and in particular to a method for studying the law of gas invasion in fractured formations. Background Art

[0002] Over 40% of the world's oil and gas reservoirs are naturally fractured, and major oil distribution regions such as the Middle East, North America, South America, and Southeast Asia all harbor naturally fractured reservoirs, making their extraction process complex. In most oil and gas reservoirs, the presence of natural fractures increases the permeability and storage capacity of the reservoir, making them the primary pathways for the flow and transport of formation fluids. Because most natural fractures are randomly oriented and have varying pore sizes, discrete fractures exhibit a higher degree of heterogeneity and diversity compared to traditional porous media.

[0003] The drilling safety window for fractured reservoirs is extremely small, especially for carbonate reservoirs, where the safety window may not even exist. Therefore, gas invasion or drilling fluid loss is very likely to occur during the drilling process. Commonly used methods for analyzing fractured reservoirs include empirical methods, analytical methods, semi-analytical methods, and numerical simulation methods. The empirical method generally uses relationship regression from experimental data or field data to obtain models such as linear relationships and exponential relationships, and then analyzes them based on simple mathematical models. Although these methods provide engineers with "rules of thumb" guidance, the analytical accuracy of this method is low. Although some established predictive models can characterize the flow state of reservoir fluids to a certain extent,

[32] However, it cannot accurately describe the geomechanical properties of porous media and the impact of rock deformation on gas invasion. Summary of the Invention

[0004] In response to the above problems, this research method mainly focuses on the problem of gas invasion in fractured reservoirs, comprehensively considers the influence of matrix deformation and fracture aperture changes on fluid flow, and establishes a fluid-solid coupling mathematical model of gas invasion in fractured reservoirs.

[0005] This method comprises the following steps:

[0006] Before constructing the mathematical model, the following basic assumptions are made: the reservoir is a homogeneous, isotropic and elastic continuum; the fluid flow in the reservoir is single-phase flow and conforms to Darcy's law; the reservoir rock is slightly compressible and has a constant compression coefficient; the influence of gravity is ignored; and the fluid flow in the reservoir is isothermal.

[0007] This method comprises the following steps:

[0008] S1. Establish reservoir deformation model and fluid flow model;

[0009] For the reservoir deformation model, the Navier equation of the mechanical field during gas invasion, which is derived from the equilibrium equation of porous media deformation, geometric equation, and stress-strain relationship, is as follows:

[0010]

[0011] Where G = E / 2(1+v) is the shear modulus of the porous elastic material; E and v are the Young's modulus and Poisson's ratio of the porous elastic material, respectively; u ij is the displacement component; a is the Biot coefficient; p is the fluid pressure; f i is the body force component.

[0012] The process of establishing the fluid flow model is as follows:

[0013] The continuity equation in the matrix pores is:

[0014]

[0015] Where ρ g is the gas density under reservoir conditions; is the porosity of the porous medium; v m is the fluid velocity in the matrix pores; q m is the mass of the pore fluid.

[0016] Using Darcy's law to describe the flow rate of fluid in porous media and the continuity equation of the matrix storage behavior into the matrix pores, the governing equation for the flow of fluid in the matrix pores can be obtained as follows:

[0017]

[0018] Where k m is the permeability of the matrix pores; μ is the viscosity of the fluid; p is the pressure in the pores; S m is the storage coefficient.

[0019] The continuity equation inside the crack is:

[0020]

[0021] Where d f is the aperture size of the crack, is the porosity of the crack; T is the tangential gradient; v f is the flow velocity of the fluid in the fracture; q f is the mass source of the fluid in the fracture.

[0022] Considering the natural fracture as the flow between two plates, the seepage velocity and fracture storage, combined with the continuity equation within the fracture, yield the governing equation for the fluid flow in the natural fracture:

[0023]

[0024] Where S f is the total compression coefficient of saturated cracks.

[0025] S2. Establishing a fluid-solid cross-coupling model

[0026] The dynamic fluid-solid cross-coupling model mainly analyzes the impact of changes in rock permeability and porosity.

[0027] The permeability k of the matrix pores in step S1 is m The calculation formula of matrix permeability can be obtained from the dynamic permeability of the matrix and the specific surface area at different times as shown below:

[0028]

[0029] Where ε v is the volume strain, is the porosity of the reservoir matrix

[0030] The crack aperture d in step 1) f The formula for calculating the total length of the crack during deformation is as follows:

[0031] d f =a n +a s +a res

[0032] S3. Use the finite element method to derive the numerical form of the mathematical model

[0033] In order to solve the unknown variables, it is necessary to integrate and simplify the control equations, and the finite element formula is obtained as follows:

[0034] S4. Discrete cracks were constructed using MATLAB and solved using the commercial software COMSOL Multiphysics.

[0035] The beneficial effects of the present invention are as follows: the discussion of the research results has great significance for actual oil field development.

[0036] 1. Gas invasion estimation: By understanding the characteristics and influencing factors of gas invasion, corresponding measures can be formulated to reduce gas invasion and minimize its adverse effects on the drilling process and oil and gas field development.

[0037] 2. Drilling fluid design and optimization: By studying the gas invasion caused by negative pressure differential drilling, the design and optimization of drilling fluid can be improved to reduce the risk of gas intrusion into the wellbore. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The present invention will be further described below with reference to the accompanying drawings.

[0039] Figure 1 The embodiment is a schematic diagram of a single crack;

[0040] Figure 2 Pressure distribution at different times;

[0041] Figure 3 Test points at different distances from the wellbore;

[0042] Figure 4 Pressure variations in fractures at different distances from the wellbore;

[0043] Figure 5 The degree of permeability reduction of fractures at different locations from the wellbore. DETAILED DESCRIPTION

[0044] The following is a geometric model of a single crack (such as Figure 1 The embodiments of the present invention are fully described in conjunction with the accompanying drawings, taking the embodiments of the present invention as an example. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. All other embodiments derived by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0045] The specific implementation steps are as follows:

[0046] S1. Establish reservoir deformation model and fluid flow model

[0047] The mathematical model of porous media deformation includes equilibrium equations, geometric equations, and constitutive equations. Based on the poroelasticity theory, the equilibrium equation of reservoir deformation can be defined as:

[0048] σ ij，i +f i =0 (1)

[0049] Where, σ ij is the sum of stress components; f i is the body force component;

[0050] Under the assumption of small deformation, the geometric equation (strain-displacement relationship) can be defined as:

[0051]

[0052] Where, ε ij is the strain component; u ij is the displacement component;

[0053] The stress-strain relationship of rock materials obeys the linear poroelastic law. According to the Biot effective stress, it can be obtained:

[0054] σ ij =σ ij -αpδ ij (3)

[0055] Where σ` ij is the effective stress; σ ij is the sum of stress components; a is the Biot coefficient; p is the fluid pressure; δ ij is the Kronecker function;

[0056] Substituting equations (2) and (3) into (1), we can obtain the Navier equation of the mechanical field during the gas invasion process:

[0057]

[0058] Where G = E / 2(1+v), G is the shear modulus of the porous elastic material, E and v are the Young's modulus and Poisson's ratio of the porous elastic material, respectively; u ij is the displacement component; a is the Biot coefficient; p, i is the pore pressure; f i is the body force component;

[0059] For the fluid flow model, the continuity equation in the matrix pores is:

[0060]

[0061] Using Darcy's law to describe the flow rate of fluid in porous media and the matrix storage behavior can be defined as follows:

[0062]

[0063] Substituting formulas (6) and (7) in the steps into (5), the governing equations for matrix pore fluid flow are as follows:

[0064]

[0065] The continuity equation inside the crack is:

[0066]

[0067] In formulas (5) to (9), ρ g is the gas density under reservoir conditions; is the porosity of the porous medium; v m is the fluid velocity in the matrix pores; q m is the mass of pore fluid; k m is the permeability of the matrix pores; μ is the viscosity of the fluid; p is the fluid pressure; ▽ p is the pressure gradient; Sm is the storage coefficient; d f is the crack aperture, is the porosity of the crack; T is the tangential gradient; v f is the flow velocity of the fluid in the fracture; q f is the mass source of the fluid in the fracture. Considering the natural fracture as the flow between two flat plates, its seepage velocity and fracture storage behavior can be defined as:

[0068]

[0069] Substituting formula (10) (11) into formula (9) we can obtain the governing equations for fluid flow in natural fractures:

[0070]

[0071] In formulas (10) to (12), v f is the flow velocity of the fluid in the crack; k f is the fracture permeability; μ is the viscosity of the fluid; d f is the crack aperture; p is the fluid pressure; ▽ p is the pressure gradient; ρ g is the gas density under reservoir conditions; is the porosity of the fracture; S f is the total compression coefficient of saturated cracks; T is the tangential gradient; q f is the mass source of the fluid in the fracture.

[0072] S2. Establishing a fluid-solid cross-coupling model

[0073] Assuming the surface area of ​​the rock particles remains constant, the total volume change of the rock is equal to the change in pore volume. The specific surface area at different times can be expressed as:

[0074]

[0075] In formulas (13) and (14), V b is the volume of the rock; V s is the change in rock volume, ε v is the volume strain;

[0076] The calculation formula of matrix permeability can be obtained by combining equations (13) and (14):

[0077]

[0078] Where k m is the permeability of matrix pores; k m0 is the initial permeability; ε v is the volume strain; is the initial porosity;

[0079] The total pore size when the crack is deformed is:

[0080]

[0081] Where a n 、a s is the normal deformation, shear deformation, and a of the crack res The size of the residual aperture, is the effective expansion angle.

[0082] S3. Use the finite element method to derive the numerical form of the mathematical model

[0083] In order to solve the unknown variables, it is necessary to integrate and simplify the control equations, and the finite element formula is obtained as follows:

[0084]

[0085] S4. Discrete cracks were constructed using MATLAB and solved using the commercial software COMSOL Multiphysics.

[0086] Example 1:

[0087] Assume that the geometry of a fracture is known. The fracture is parallel to the x-axis and intersects the wellbore. The geometric model is as follows: Figure 1 Some of the parameters used in the simulation are shown in Table 1.

[0088] Table 1

[0089] Parameter name Numerical Reservoir area 8m×8m Reservoir thickness 8m Initial formation pressure 24MPa bottomhole pressure 20MPa Fluid viscosity <![CDATA[4.74×10 -5 Step]]> Fluid compressibility <![CDATA[4×10 -4 MPa -1 ]]> Pore ​​compressibility <![CDATA[7.5×10 -4 MPa -1 <!-- 5 -->]]> Crack compression coefficient <![CDATA[7.5×10 -3 MPa -1 ]]> Matrix porosity 0.07 Fracture porosity 0.0001 Poisson's ratio of rock 0.25 Initial crack width 0.5mm Biot coefficient 0.6

[0090] The outer boundary is set as a non-flowing boundary, and the inner boundary wellbore is set to a fixed pressure. Some of the formulas used are as follows:

[0091]

[0092]

[0093] The commercial software COMSOL Multiphysics was used to solve the problem. Figure 2 The pressure distribution at different times is shown, and it can be seen that the pressure inside the crack decreases rapidly along the length of the crack. Figure 3 There are 8 test points at different distances from the wellbore.

[0094] After drilling into the cracks, Figure 4It can be seen that the pressure in the crack near the wellbore will drop instantaneously and approach the pressure at the bottom of the well, while the pressure in the crack away from the wellbore will drop but change more slowly, and the pressure difference between the crack and the bottom of the well will also increase. Based on the effective stress and the cubic law combined with the change in pressure, the change law of permeability in the crack can be inferred, that is, the closer to the wellbore, the greater the decrease in permeability of the crack. The obtained law is also consistent with the results obtained by simulation. Figure 5 consistent.

[0095] This paper establishes a mathematical model to simulate and explore the characteristics of fluid flow and rock deformation in fractures when drilling encounters fractures. The pressure in the fracture will drop rapidly and approach the bottomhole pressure. The decrease in formation pressure increases the effective stress, which leads to a decrease in the permeability of the fracture.

[0096] Throughout this specification, references to terms such as "one embodiment," "example," or "specific example" indicate that the specific features, structures, materials, or characteristics described in conjunction with that embodiment or example are included in at least one embodiment or example of the present invention. In this specification, schematic representations of these terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

[0097] The above contents are merely examples and explanations of the present invention. Those skilled in the art may make various modifications or additions to the described specific embodiments or replace them in similar ways. As long as they do not deviate from the invention or exceed the scope defined by the claims, they should all fall within the scope of protection of the present invention.

Claims

1. A method for studying the gas invasion law of fractured formations, characterized by: The research method includes the following steps: S1. Establish reservoir deformation model and fluid flow model The mathematical model of porous media deformation includes equilibrium equations, geometric equations, and constitutive equations. Based on the poroelasticity theory, the equilibrium equation of reservoir deformation can be defined as: （1） Where, is the sum of stress components; is the body force component; Under the assumption of small deformation, the geometric equation strain-displacement relationship can be defined as: （2） Where, is the strain component; is the displacement component; The stress-strain relationship of rock materials obeys the linear poroelastic law. According to the Biot effective stress, it can be obtained: （3） Where, is the effective stress; is the sum of stress components; a is the Biot coefficient; p is the fluid pressure; δ ij is the Kronecker function; Substituting equations (2) and (3) into (1), we can obtain the Navier equations of the mechanical field during the gas invasion process: （4） Where, G=E / 2(1+v) , G is the shear modulus of the porous elastic material, E and v are Young’s modulus and Poisson’s ratio of the porous elastic material, respectively; u ij is the displacement component; a is the Biot coefficient; f i is the body force component; For the fluid flow model, the continuity equation in the matrix pores is: （5） Using Darcy's law to describe the flow rate of fluid in porous media and the matrix storage behavior can be defined as follows: （6） （7） Substituting formulas (6) and (7) in the steps into (5), the governing equations for matrix pore fluid flow are as follows: （8） The continuity equation inside the crack is: （9） In formulas (5) to (9), ρ g is the gas density under reservoir conditions; φ m is the porosity of the porous medium; v m is the fluid velocity in the matrix pores; q m is the mass of the pore fluid; k m is the permeability of matrix pores; µ is the viscosity of the fluid; p is the fluid pressure; p is the pressure gradient; S m is the storage coefficient; d f is the crack aperture, φ f is the porosity of the crack; T is the tangential gradient; v f is the flow velocity of the fluid in the fracture; q f is the mass source of the fluid in the fracture. Considering the natural fracture as the flow between two flat plates, its seepage velocity and fracture storage behavior can be defined as: （10） （11） Substituting formula (10) (11) into formula (9) yields the governing equations for fluid flow in natural fractures: （12） In formulas (10) to (12), v f is the flow velocity of the fluid in the fracture; k f is the fracture permeability; µ is the viscosity of the fluid; d f is the crack aperture; p is the fluid pressure; p is the pressure gradient; ρ g is the gas density under reservoir conditions; φ f is the porosity of the fracture; S f is the total compression coefficient of saturated cracks; T is the tangential gradient; q f is the mass source of the fluid in the fracture; S2. Establishing a fluid-solid cross-coupling model Assuming that the surface area of ​​the rock particles remains constant and the total volume change of the rock is equal to the change in pore volume, the specific surface area at different times can be expressed as: （13） （14） In formulas (13) and (14), V b is the volume of the rock; V s is the change in rock volume, ɛ v is the volume strain; The calculation formula of matrix permeability can be obtained by combining equations (13) and (14): （15） Where, k m is the permeability of matrix pores; k m0 is the initial permeability; ɛ v is the volume strain; φ m0 is the initial porosity; The total pore size when the crack is deformed is: （16） In the formula a n 、 a s The normal deformation, shear deformation, a res The size of the residual aperture, φ dil is the effective expansion angle; S3. Use the finite element method to derive the numerical form of the mathematical model In order to solve the unknown variables, it is necessary to integrate and simplify the control equations, and the finite element formula is obtained as follows: （17） S4. Use MATLAB to construct discrete cracks and solve them using commercial software.

2. The method for studying gas invasion laws in fractured formations according to claim 1, characterized in that: Before constructing the reservoir deformation model and fluid flow model in step S1, the following basic assumptions are made: the reservoir is a uniform, isotropic and elastic continuum; the fluid flow in the reservoir is single-phase flow and conforms to Darcy's law; the reservoir rock is slightly compressible and has a constant compression coefficient; the influence of gravity is ignored; and the fluid flow in the reservoir is isothermal.

3. The method for studying gas invasion laws in fractured formations according to claim 1, characterized in that: The commercial software used in step S4 is COMSOL Multiphysics.

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