CN109063334B - Fluid-solid coupling model construction method of low-permeability porous medium material - Google Patents

Fluid-solid coupling model construction method of low-permeability porous medium material Download PDF

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CN109063334B
CN109063334B CN201810874858.9A CN201810874858A CN109063334B CN 109063334 B CN109063334 B CN 109063334B CN 201810874858 A CN201810874858 A CN 201810874858A CN 109063334 B CN109063334 B CN 109063334B
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王环玲
巢志明
徐卫亚
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Abstract

The invention discloses a fluid-solid coupling model construction method of a low-permeability porous medium material, which comprises the following steps: establishing a mathematical model of the influence of a low-permeability rock seepage field on a stress field; establishing a relation between the porosity of the low-permeability rock and the absolute permeability, and establishing a mathematical model of the influence of the stress field of the low-permeability rock on the seepage field on the basis of the relation; and (3) combining the mathematical model of the influence of the seepage field of the low-permeability rock on the stress field and the mathematical model of the influence of the stress field on the seepage field, and establishing the fluid-solid coupling relationship of the low-permeability rock. When a mathematical model of the influence of a stress field on the seepage field is researched, the absolute permeability of the low-permeability rock is measured by using gas as a seepage medium, and the influence of the gas slippage effect of the low-permeability rock is eliminated by adopting function fitting, so that the established mathematical model of the influence of the stress field on the seepage field is relatively accurate, and the finally established low-permeability rock fluid-solid coupling relationship is accurate and reliable; in addition, the method does not need to carry out in-situ test, thereby saving material resources and financial resources.

Description

Fluid-solid coupling model construction method of low-permeability porous medium material
Technical Field
The invention relates to a fluid-solid coupling model construction method, in particular to a fluid-solid coupling model construction method of a low-permeability porous medium material.
Background
Low permeability rocks are the primary media for many civil engineering projects, such as coal mining, CO2 storage, petroleum fuel recovery, and nuclear waste disposal. The mechanical properties of low-permeability rocks have a significant influence on engineering safety, and a lot of researches on the mechanical properties of the low-permeability rocks have been made by many researchers, and the researches generally do not consider the influence of liquid seepage on the mechanical properties of the low-permeability rocks; however, the mechanical response of low permeability rocks considering fluid-solid coupling is much different from that of low permeability rocks not considering seepage. Therefore, the research on the mechanical behavior of the low-permeability rock under the condition of fluid-solid coupling is urgent.
The method for constructing the low-permeability rock analysis model is a method for effectively researching the mechanical response of the low-permeability rock, but the method for constructing the fluid-solid coupling analysis model of the low-permeability rock is difficult. The liquid permeability (absolute permeability) of the low-permeability rock is difficult to measure by using liquid as a medium, and the absolute permeability of the low-permeability rock measured by gas causes a larger measurement result due to the slip effect. Therefore, how to select a suitable method to construct a fluid-solid coupling analytical model of low permeability rocks requires further research.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems in the prior art, the invention provides a fluid-solid coupling model construction method of a low-permeability porous medium material.
The technical scheme is as follows: the invention relates to a fluid-solid coupling model construction method of a low-permeability porous medium material, which comprises the following steps:
step 1, establishing a mathematical model of the influence of a low-permeability rock seepage field on a stress field;
step 2, establishing a relation between the porosity of the low-permeability rock and the absolute permeability, and establishing a mathematical model of the influence of the stress field of the low-permeability rock on the seepage field on the basis of the relation;
and 3, combining the mathematical model of the influence of the seepage field of the low-permeability rock on the stress field with the mathematical model of the influence of the stress field on the seepage field, and establishing the fluid-solid coupling relationship of the low-permeability rock.
In the step 1, the seepage field of the low-permeability rock influences the distribution of the stress field by changing the seepage volume force of the low-permeability rock, so that a mathematical model of the influence of the seepage field of the low-permeability rock on the stress field is established by the following steps:
in a two-dimensional seepage environment, the seepage volume force of low-permeability rock is in direct proportion to the hydraulic gradient, and the relationship between the seepage volume force and the hydraulic gradient is shown as the following formula (1):
Figure GDA0002622260790000021
in the formula (1), f is the seepage volume force, rwIs the specific gravity of water, H is the head, fxAnd fyIs the component of the osmotic volumetric force in the x and y directions, JxAnd JyAre the components of the hydraulic gradient in the x and y directions;
and secondly, converting the seepage volume force of the unit into equivalent node load, which is expressed by the following formulas (2) and (3):
Figure GDA0002622260790000022
Figure GDA0002622260790000023
in the formula (2), { FsIs the equivalent nodal force generated by the seepage volume force, [ N ]]Ω is a boundary condition of known forces as a function of cell shape; in formula (3), { Δ FsThe equivalent node force generated by seepage volume force increment is multiplied;
on the basis, a mathematical model of the influence of the low-permeability rock seepage field on the stress field is obtained, and the mathematical model is as follows (4):
Figure GDA0002622260790000024
in the formula, kliqAbsolute permeability of low permeability rock, { Δ F } node load increment, { } displacement, face0Is the initial displacement, { Delta } is the displacement increment, σijIs stress, njIs an equivalent node, SuFor a known displacement boundary condition, SaIs a defined stress boundary condition.
The influence of the stress field of the low-permeability rock on the seepage field is mainly realized by influencing the porosity of the low-permeability rock so as to influence the permeability of the low-permeability rock. Therefore, in step 2, the mathematical model of the seepage field considering the influence of the stress field is as shown in formula (5):
Figure GDA0002622260790000031
in the formula (5), phi is the porosity, kliq(phi) is the function relationship between the absolute permeability and porosity of the low permeability rock, H is the sample height, H is the head, q is1Is the flow rate;1in order to have a boundary condition that determines the head,2for a defined flow boundary condition, n2Is composed of2A normal to the boundary condition boundary;3as a mixed boundary condition, i.e.
Figure GDA0002622260790000032
α1And β1Is a constant number, n3Is composed of3Normal to the boundary condition boundary.
Therefore, the key to determining the effect of the stress field on the seepage field is to determine the relationship between the permeability and porosity of low permeability rocks. In step 2, establishing the relationship between the porosity and the absolute permeability of the low-permeability rock comprises the following steps:
step 21, measuring the permeability of the low-permeability rock under different confining pressures, and determining the corresponding absolute permeability;
step 22, measuring the porosity of the low-permeability rock under the corresponding confining pressure;
step 23, respectively determining a functional relation between the absolute permeability of the low-permeability rock and the confining pressure and a functional relation between the porosity of the low-permeability rock and the confining pressure;
and 24, deducing and establishing the relationship between the porosity of the low-permeability rock and the absolute permeability through a formula.
Specifically, in step 21, fitting the gas permeability and the reciprocal pore pressure of the low-permeability rock under different confining pressures by using the following formula (6) to obtain the absolute permeability of the low-permeability rock under different confining pressures for eliminating the slippage effect:
Figure GDA0002622260790000033
in the formula (6), kliqIs absolute permeability, k is gas permeability, b is slip factor, a is secondary slip factor, pcIs the pore pressure.
Thus, in step 23, the absolute permeability of the low permeability rock as a function of the confining pressure is given by the following equation (7):
Figure GDA0002622260790000034
in the formula (7), d1And h1Is constant and p is confining pressure.
In step 23, the porosity of the low permeability rock as a function of confining pressure is given by the following equation (8):
φ＝φ0e-ip(8)；
in the formula (8), phi0I is a constant for the initial porosity.
In step 24, combining the formula (7) and the formula (8), determining the relationship between the porosity of the low-permeability rock and the absolute permeability, as shown in the following formula (10):
Figure GDA0002622260790000041
in the final step 3, the mathematical models obtained in the steps 1 and 2 are linked, and the fluid-solid coupling relation of the low permeability rock is established as the following formula (13):
Figure GDA0002622260790000042
in the formula (13), { f } is a seepage field head distribution function.
Has the advantages that: compared with the prior art, the invention has the remarkable advantages that: (1) the fluid-solid coupling relation of the low-permeability rocks is obtained by combining a mathematical model of the influence of a seepage field of the low-permeability rocks on a stress field and a mathematical model of the influence of a stress field on the seepage field, wherein when the mathematical model of the influence of the stress field on the seepage field is researched, the absolute permeability of the low-permeability rocks is measured by adopting gas as a seepage medium, and the influence of the gas slip effect of the low-permeability rocks is eliminated by adopting function fitting, so that the established mathematical model of the influence of the stress field on the seepage field is relatively accurate, and the finally established fluid-solid coupling relation of the low-permeability rocks is accurate and reliable; (2) the method determines the fluid-solid coupling relationship of the low-permeability rock through indoor tests and mathematical formula deduction, does not need to perform in-situ tests, is simple and easy to implement, and saves a large amount of material resources and financial resources.
Detailed Description
The technical solution of the present invention is further explained below.
The method for constructing the fluid-solid coupling model of the low-permeability porous medium material is explained by taking the determination of the fluid-solid coupling relationship of the low-permeability rock of a certain underground water-sealed cave depot as an example.
Step 1, establishing a mathematical model of the influence of a low-permeability rock seepage field on a stress field
The seepage field of the low-permeability rock influences the distribution of the stress field by changing the seepage volume force of the low-permeability rock, and in a two-dimensional seepage environment, the seepage volume force of the low-permeability rock is in direct proportion to the hydraulic gradient, as shown in a formula (1):
Figure GDA0002622260790000051
wherein f is the seepage volume force, rwIs the specific gravity of water, H is the head, fxAnd fyIs the component of the osmotic volumetric force in the x and y directions, JxAnd JyAre the components of the hydraulic gradient in the x and y directions;
the following equation can convert the cell's seepage volumetric force to an equivalent nodal load, as shown in equations (2) and (3):
Figure GDA0002622260790000052
wherein, { FsIs the equivalent nodal force generated by the seepage volume force, [ N ]]Is a cell shape function and Ω is a boundary condition for a known force.
Figure GDA0002622260790000053
Wherein, { Δ F [ ]sIs the equivalent node force produced by the seepage volume force increment.
A mathematical model of the stress field taking into account the influence of the seepage field is thus obtained, as shown in equation (4).
Figure GDA0002622260790000054
In the formula, kliqAbsolute permeability of low permeability rock, { Δ F } node load increment, { } displacement, face0Is the initial displacement, { Delta } is the displacement increment, σijIs stress, njIs an equivalent node, SuFor a known displacement boundary condition, SaIs a defined stress boundary condition.
Step 2: the method comprises the following steps of establishing a mathematical model of the influence of a stress field of low-permeability rock on a seepage field:
the influence of the stress field on the seepage field of the low-permeability rock is mainly realized by influencing the porosity of the low-permeability rock so as to influence the permeability of the low-permeability rock. Therefore, a two-dimensional model of the seepage field considering the influence of the stress field is shown in formula (5):
Figure GDA0002622260790000061
wherein phi is the porosity, kliq(phi) is the function relationship between the absolute permeability and porosity of the low permeability rock, H is the sample height, H is the head, q is1Is the flow rate;1in order to have a boundary condition that determines the head,2for a defined flow boundary condition, n2Is composed of2A normal to the boundary condition boundary;3as a mixed boundary condition, i.e.
Figure GDA0002622260790000062
α1And β1Is a constant number, n3Is composed of3Normal to the boundary condition boundary.
Therefore, the key to determining the effect of the stress field on the seepage field is to determine the relationship between the permeability and porosity of low permeability rocks.
And step 3: determining the functional relationship between the permeability of low-permeability rock and confining pressure
And (3) fitting the gas permeability and the reciprocal pore pressure of the low-permeability rock under different confining pressures by adopting a formula (6) to obtain the absolute permeability of the low-permeability rock under different confining pressures for eliminating the slippage effect.
Figure GDA0002622260790000063
In the formula, kliqIs absolute permeability, k is gas permeability, b is slip factor, a is secondary slip factor, pcIs the pore pressure.
The absolute permeability of the low permeability rock obtained by fitting is shown in table 1.
TABLE 1 Absolute Permeability of Low permeable rock
Figure GDA0002622260790000064
And (3) obtaining the exponential function relation between the absolute permeability of the low-permeability rock and the confining pressure as formula (7) through function fitting, wherein the fitting parameters are shown in table 2.
Figure GDA0002622260790000065
In the formula d1And h1Is constant and p is confining pressure.
TABLE 2 fitting parameters
Figure GDA0002622260790000071
And 4, step 4: determining the functional relationship between the porosity of low-permeability rock and confining pressure
Fitting the porosity of the low-permeability rock and confining pressure by a function to form an exponential function relation, wherein the equation is (8); the fitting parameter i is 0.01.
φ＝φ0e-ip(8)
In the formula, phi0Is the initial porosity and i is a constant.
And 5: and deducing and establishing the relationship between the porosity and the permeability of the low-permeability rock through a formula.
And combining the formula (7) and the formula (8) to obtain a formula (9).
Figure GDA0002622260790000072
After simplification, the absolute permeability of the low-permeability rock is plotted against the porosity as shown in formula (10).
Figure GDA0002622260790000073
Therefore, the absolute permeability of the low-permeability rock is in a power function relationship with the porosity, and the values of the parameters are obtained by substituting the formula (10) and the formula (11).
kliq＝0.313φ2(11)
Step 6: and (3) combining the mathematical model of the influence of the seepage field of the low-permeability rock on the stress field and the mathematical model of the influence of the stress field on the seepage field, and establishing the fluid-solid coupling relationship of the low-permeability rock.
Equation (11) is substituted into equation (5), and equation (12) is obtained.
Figure GDA0002622260790000074
Therefore, the relationship of fluid-solid coupling of low permeability rocks is obtained by combining equation (12) and equation (4) to obtain equation (13).
Figure GDA0002622260790000081
Where { f } is the seepage field head distribution function.

Claims (7)

1. A method for constructing a fluid-solid coupling model of a low-permeability porous medium material is characterized by comprising the following steps of:
step 1, establishing a mathematical model of the influence of a low-permeability rock seepage field on a stress field;
first, the seepage volume force of low permeability rock is determined as follows (1):
Figure FDA0002622260780000011
in the formula (1), f is the seepage volume force, rwIs the specific gravity of water, H is the head, fxAnd fyIs the component of the osmotic volumetric force in the x and y directions, JxAnd JyAre the components of the hydraulic gradient in the x and y directions;
and secondly, converting the seepage volume force of the unit into equivalent node load, which is expressed by the following formulas (2) and (3):
Figure FDA0002622260780000012
Figure FDA0002622260780000013
in the formula (2), { FsIs the equivalent nodal force generated by the seepage volume force, [ N ]]Ω is a boundary condition of known forces as a function of cell shape; in formula (3), { Δ FsThe equivalent node force generated by seepage volume force increment is multiplied;
and finally, establishing a mathematical model of the influence of the low-permeability rock seepage field on the stress field, wherein the mathematical model is as follows (4):
Figure FDA0002622260780000014
in the formula, kliqAbsolute permeability of low permeability rock, { Δ F } node load increment, { } displacement, face0Is the initial displacement, { Delta } is the displacement increment, σijIs stress, njIs an equivalent node, SuFor a known displacement boundary condition, SaIs a determined stress boundary condition;
step 2, establishing a relation between the porosity of the low-permeability rock and the absolute permeability, and establishing a mathematical model of the influence of the stress field of the low-permeability rock on the seepage field on the basis of the relation; the mathematical model of the influence of the stress field of the low-permeability rock on the seepage field is as follows (5):
Figure FDA0002622260780000021
in the formula (5), phi is the porosity, kliq(phi) is the function relationship between the absolute permeability and porosity of the low permeability rock, H is the sample height, H is the head, q is1Is the flow rate;1in order to have a boundary condition that determines the head,2for a defined flow boundary condition, n2Is composed of2A normal to the boundary condition boundary;3as a mixed boundary condition, i.e.
Figure FDA0002622260780000022
α1And β1Is a constant number, n3Is composed of3Of boundary conditionsA normal line;
and 3, combining the mathematical model of the influence of the seepage field of the low-permeability rock on the stress field with the mathematical model of the influence of the stress field on the seepage field, and establishing the fluid-solid coupling relationship of the low-permeability rock.
2. The method for constructing a fluid-solid coupling model of low-permeability porous medium material according to claim 1, wherein in the step 2, the establishing the relationship between the porosity and the absolute permeability of the low-permeability rock comprises the following steps:
step 21, measuring the permeability of the low-permeability rock under different confining pressures, and determining the corresponding absolute permeability;
step 22, measuring the porosity of the low-permeability rock under the corresponding confining pressure;
step 23, respectively determining a functional relation between the absolute permeability of the low-permeability rock and the confining pressure and a functional relation between the porosity of the low-permeability rock and the confining pressure;
and 24, deducing and establishing the relationship between the porosity of the low-permeability rock and the absolute permeability through a formula.
3. The method for constructing the fluid-solid coupling model of the low-permeability porous medium material of claim 2, wherein in step 21, the gas permeability and the reciprocal pore pressure of the low-permeability rock under different confining pressures are fitted by using the following formula (6) to obtain the absolute permeability of the low-permeability rock under different confining pressures for eliminating the slip effect:
Figure FDA0002622260780000023
in the formula (6), kliqIs absolute permeability, k is gas permeability, b is slip factor, a is secondary slip factor, pcIs the pore pressure.
4. The method for constructing a fluid-solid coupling model of low-permeability porous medium material according to claim 3, wherein in step 23, the absolute permeability of the low-permeability rock is as a function of confining pressure as shown in the following formula (7):
Figure FDA0002622260780000031
in the formula (7), d1And h1Is constant and p is confining pressure.
5. The method for constructing a fluid-solid coupling model of low-permeability porous medium material according to claim 4, wherein in step 23, the porosity of the low-permeability rock is in function of confining pressure as shown in the following formula (8):
φ＝φ0e-ip(8)；
in the formula (8), phi0I is a constant for the initial porosity.
6. The method for constructing a fluid-solid coupling model of low-permeability porous medium material according to claim 5, wherein in step 24, the relationship between the porosity of the low-permeability rock and the absolute permeability is determined by combining formula (7) and formula (8), and the relationship is represented by the following formula (10):
Figure FDA0002622260780000032
7. the method for constructing a fluid-solid coupling model of low-permeability porous medium material according to claim 1, wherein in step 3, the fluid-solid coupling relationship of the low-permeability rock is finally established as shown in the following formula (13):
Figure FDA0002622260780000033
in the formula (13), { f } is a seepage field head distribution function.
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