CN111832110A - Calculation method for karst area circular tunnel seepage field analytic solution - Google Patents

Calculation method for karst area circular tunnel seepage field analytic solution Download PDF

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CN111832110A
CN111832110A CN202010691203.5A CN202010691203A CN111832110A CN 111832110 A CN111832110 A CN 111832110A CN 202010691203 A CN202010691203 A CN 202010691203A CN 111832110 A CN111832110 A CN 111832110A
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tunnel
plane
karst
circumference
radius
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蒋冲
韩海霞
谢寒松
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Central South University
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Abstract

The invention discloses a calculation method of an analytic solution of a karst area circular tunnel seepage field, which comprises the steps of obtaining geological parameters of the karst area circular tunnel seepage field; mapping the tunnel and the karst cave in the infinite plane to a w plane through conformal mapping; calculating a functional relation between a total pressure head and the radius on the circumference of the tunnel in the w plane; and obtaining the final seepage flow of the circular tunnel in the karst area. The calculation method of the karst area circular tunnel seepage field analytical solution provided by the invention provides a theoretical method for the stability of karst water flowing into the circular tunnel in the karst development area; the method can calculate the seepage flow of the karst water flowing into the tunnel, and can judge the stability of the tunnel; in addition, the tunnel seepage flow analytic formula provided by the invention is suitable for conditions of different tunnel radiuses, different karst cave positions and the like, and has a wider application range.

Description

Calculation method for karst area circular tunnel seepage field analytic solution
Technical Field
The invention belongs to the technical field of civil engineering, and particularly relates to a calculation method for an analytic solution of a karst area circular tunnel seepage field.
Background
With the development of economic technology and the improvement of living standard of people, road construction has been widely spread in China, and brings endless convenience to production and life of people.
In the course of road construction, geological conditions are an extremely critical influencing factor. Geological conditions in karst development areas are complex, and specific geological conditions built around tunnels are generally difficult to find before the tunnels are built. Tunnels under such conditions present a certain risk: on one hand, the seepage of karst water is easy to cause tunnel water burst, and the safety, economic benefit and the safety and stability of later-stage operation of tunnel construction are seriously influenced; on the other hand, the phenomenon that the existence of karst causes collapse, rockfall and changes of the physical and mechanical properties of surrounding rocks during tunnel excavation. Therefore, the research on the seepage field of the circular tunnel in the karst area has important engineering significance.
At present, a lot of researches are carried out on a seepage field of underground water which stably flows into a circular tunnel, a method of conformal mapping in a semi-infinite plane is mainly adopted, the researches on karst water of karst caves in the infinite plane which flows into the circular tunnel are less, and especially, the researches on the influence of the karst caves in different directions on the seepage field of the tunnel are extremely less.
Disclosure of Invention
The invention aims to provide a calculation method for the analytic solution of the seepage field of the circular tunnel in the karst area, which can be used for solving the analytic solution of the seepage field of the circular tunnel in the karst area and has high reliability and good practicability.
The invention provides a method for calculating a seepage field analytical solution of a circular tunnel in a karst area, which comprises the following steps of:
s1, acquiring geological parameters of a seepage field of a circular tunnel in a karst area;
s2, mapping the tunnels and the karst caves in the infinite plane to a w plane through conformal mapping;
s3, calculating a functional relation between a total pressure water head on the circumference of the tunnel in the w plane and the radius;
and S4, calculating to obtain the final seepage quantity of the circular tunnel in the karst area.
Mapping the tunnel and the karst cave in the infinite plane to the w plane through conformal mapping in the step S2, specifically, mapping by using the following steps:
A. the following formula is adopted as a complex variable function formula of the tunnel and the karst cave in the infinite plane, which are mapped to the w plane through conformal mapping:
Figure BDA0002589438370000021
Figure BDA0002589438370000022
Figure BDA0002589438370000023
wherein z is any point in the z plane; w is any point in the w plane; a is a group consisting of d and rwAnd a first constant determined by r, and d-rw＜a＜d+rw(ii) a r is the tunnel radius; d is the central distance between the tunnel and the karst cave; r iswIs the karst cave radius; r0Is composed of d and rwAnd a second constant determined by R, and 0 < R0＜1；
B. The following formula is adopted as a mapping function formula of the karst cave in any direction:
Figure BDA0002589438370000024
w1＝-eiαz
Figure BDA0002589438370000025
wherein z is any point in the z plane; w is a1Is w1At any point in the plane, and w1The plane is a transition plane; w is a2Is w2At any point in the plane, and w2The plane is a mapping plane; r is the tunnel radius; d is the central distance between the tunnel and the karst cave; r iswIs the karst cave radius; a. the1Is composed of d and rwAnd a third constant determined by r(ii) a Alpha is an included angle between a central connecting line of the tunnel cave and a transverse axis in the z plane, and the transverse axis is a u axis.
Step S3, calculating a functional relationship between the total pressure head and the radius on the circumference of the tunnel in the w plane, specifically, calculating by using the following steps:
a. the following formula is adopted as a functional relation between the total pressure head and the radius on the circumference of the tunnel in the w plane:
Figure BDA0002589438370000031
in the formula C1、C2、C3、C4、C5And C6The constants are all constants determined by boundary conditions of the mapped tunnel circumference and karst cave circumference; ρ is the radius of any point in the w plane and R0Rho is not less than 1; is as follows; phi is the total pressure head of any point in the w plane;
Figure BDA0002589438370000032
is the included angle between the inner radius of the w plane and the transverse axis;
b. the following formula is adopted as a calculation relation of the karst cave boundary condition:
Figure BDA0002589438370000033
c. determining the boundary condition of the tunnel circumference by adopting the following rules:
the first condition is as follows: if the hydrostatic pressure of the circumference of the tunnel is 0, the total pressure head of the circumference of the tunnel is the position head of the circumference of the z-plane tunnel; expressed as the following equation:
Figure BDA0002589438370000034
the relationship between the total pressure head and the radius at any point in the plane w is given by:
Figure BDA0002589438370000041
case two: the total pressure head of the circumference of the tunnel is constant as Ht(ii) a Expressed as the following equation:
Figure BDA0002589438370000042
the relationship between the total pressure head and the radius at any point in the plane w is given by:
Figure BDA0002589438370000043
step S4, calculating to obtain a final karst area circular tunnel seepage amount, specifically calculating by using the following formula:
Figure BDA0002589438370000044
Figure BDA0002589438370000045
wherein k is the permeability coefficient of the surrounding rock; q1The seepage flow of karst water flowing into the tunnel under the condition that the hydrostatic pressure of the circumference of the tunnel is 0 and the total pressure head is a position head is formed; q2The seepage flow of karst water flowing into the tunnel under the condition of constant water pressure at the circumference of the tunnel.
The calculation method of the karst area circular tunnel seepage field analytical solution provided by the invention provides a theoretical method for the stability of karst water flowing into the circular tunnel in the karst development area; the method can calculate the seepage flow of the karst water flowing into the tunnel, and can judge the stability of the tunnel; in addition, the tunnel seepage flow analytic formula provided by the invention is suitable for conditions of different tunnel radiuses, different karst cave positions and the like, and has a wider application range.
Drawings
FIG. 1 is a schematic process flow diagram of the process of the present invention.
FIG. 2 is a schematic view of a computational model of the method of the present invention.
FIG. 3 is a schematic diagram of a model for calculating karst caves in various directions according to the method of the present invention.
Detailed Description
FIG. 1 is a schematic diagram of a method flow of the method of the present invention, and FIG. 2 is a schematic diagram of a calculation model of the method of the present invention: the invention provides a method for calculating an analytic solution of a seepage field of a circular tunnel in a karst area, which comprises the following steps of:
s1, acquiring geological parameters of a seepage field of a circular tunnel in a karst area;
s2, mapping the tunnels and the karst caves in the infinite plane to a w plane through conformal mapping; specifically, the following steps are adopted for mapping:
A. the following formula is adopted as a complex variable function formula of the tunnel and the karst cave in the infinite plane, which are mapped to the w plane through conformal mapping:
Figure BDA0002589438370000051
Figure BDA0002589438370000052
Figure BDA0002589438370000053
wherein z is any point in the z plane; w is any point in the w plane; a is a group consisting of d and rwAnd a first constant determined by r, and d-rw＜a＜d+rw(ii) a r is the tunnel radius; d is the central distance between the tunnel and the karst cave; r iswIs the karst cave radius; r0Is composed of d and rwAnd a second constant determined by R, and 0 < R0＜1；
B. The following formula is adopted as a mapping function formula of the karst cave in any direction:
Figure BDA0002589438370000054
w1＝-eiαz
Figure BDA0002589438370000061
wherein z is any point in the z plane; w is a1Is w1At any point in the plane, and w1The plane is a transition plane; w is a2Is w2At any point in the plane, and w2The plane is a mapping plane; r is the tunnel radius; d is the central distance between the tunnel and the karst cave; r iswIs the karst cave radius; a. the1Is composed of d and rwAnd r is a third constant; alpha is an included angle between a central connecting line of the tunnel cave and a transverse axis in the z plane, and the transverse axis is a u axis; as shown in fig. 3;
s3, calculating a functional relation between a total pressure water head on the circumference of the tunnel in the w plane and the radius; specifically, the method comprises the following steps:
a. the following formula is adopted as a functional relation between the total pressure head and the radius on the circumference of the tunnel in the w plane:
Figure BDA0002589438370000062
in the formula C1、C2、C3、C4、C5And C6The constants are all constants determined by boundary conditions of the mapped tunnel circumference and karst cave circumference; ρ is the radius of any point in the w plane and R0Rho is not less than 1; is as follows; phi is the total pressure head of any point in the w plane;
Figure BDA0002589438370000063
is the included angle between the inner radius of the w plane and the transverse axis;
b. the following formula is adopted as a calculation relation of the karst cave boundary condition:
Figure BDA0002589438370000064
c. determining the boundary condition of the tunnel circumference by adopting the following rules:
the first condition is as follows: if the hydrostatic pressure of the circumference of the tunnel is 0, the total pressure head of the circumference of the tunnel is the position head of the circumference of the z-plane tunnel; expressed as the following equation:
Figure BDA0002589438370000071
the relationship between the total pressure head and the radius at any point in the plane w is given by:
Figure BDA0002589438370000072
case two: the total pressure head of the circumference of the tunnel is constant as Ht(ii) a Expressed as the following equation:
Figure BDA0002589438370000073
the relationship between the total pressure head and the radius at any point in the plane w is given by:
Figure BDA0002589438370000074
s4, calculating to obtain the final seepage quantity of the circular tunnel in the karst area; specifically, the method is calculated by adopting the following formula:
Figure BDA0002589438370000075
Figure BDA0002589438370000076
wherein k is the permeability coefficient of the surrounding rock; q1The seepage flow of karst water flowing into the tunnel under the conditions that the hydrostatic pressure of the circumference of the tunnel is 0 and the total pressure head is the position head is formed; q2The seepage flow of karst water flowing into the tunnel under the condition of constant water pressure at the circumference of the tunnel.
The process of the invention is further illustrated below with reference to a specific example:
(one) calculation conditions
The cross section of a certain tunnel is circular, the radius r of the tunnel is 6m, and the radius r of the karst cave isw5m, the central distance d between the tunnel and the karst cave is 16.5m, and the permeability coefficient k of the surrounding rock is 1.2 multiplied by 10-6cm/s, H when the tunnel circumference pressure water head is constanttIs 0m, a karst cave circumferential pressure water head Hw93.3 m;
the method of the invention is adopted to calculate, and the finally obtained tunnel seepage flow is 2.05 multiplied by 10-5m2/s。

Claims (4)

1. A calculation method for an analytic solution of a karst area circular tunnel seepage field comprises the following steps:
s1, acquiring geological parameters of a seepage field of a circular tunnel in a karst area;
s2, mapping the tunnels and the karst caves in the infinite plane to a w plane through conformal mapping;
s3, calculating a functional relation between a total pressure water head on the circumference of the tunnel in the w plane and the radius;
and S4, calculating to obtain the final seepage quantity of the circular tunnel in the karst area.
2. The method according to claim 1, wherein the step S2 is performed by mapping the tunnels and caverns in the infinite plane to the w-plane through conformal mapping, specifically by performing mapping through the following steps:
A. the following formula is adopted as a complex variable function formula of the tunnel and the karst cave in the infinite plane, which are mapped to the w plane through conformal mapping:
Figure FDA0002589438360000011
Figure FDA0002589438360000012
Figure FDA0002589438360000013
wherein z is any point in the z plane; w is any point in the w plane; a is a group consisting of d and rwAnd a first constant determined by r, and d-rw＜a＜d+rw(ii) a r is the tunnel radius; d is the central distance between the tunnel and the karst cave; r iswIs the karst cave radius; r0Is composed of d and rwAnd a second constant determined by R, and 0 < R0＜1；
B. The following formula is adopted as a mapping function formula of the karst cave in any direction:
Figure FDA0002589438360000014
w1＝-eiαz
Figure FDA0002589438360000021
wherein z is any point in the z plane; w is a1Is w1At any point in the plane, and w1The plane is a transition plane; w is a2Is w2At any point in the plane, and w2The plane is a mapping plane; r is the tunnel radius; d is the central distance between the tunnel and the karst cave; r iswIs the karst cave radius; a. the1Is composed of d and rwAnd r is a third constant; alpha is an included angle between a central connecting line of the tunnel cave and a transverse axis in the z plane, and the transverse axis is a u axis.
3. The method for calculating the analytical solution of the seepage field of the circular tunnel in the karst area according to claim 2, wherein the step S3 is to calculate the functional relationship between the total pressure head and the radius of the tunnel circumference in the w plane, and specifically comprises the following steps:
a. the following equation is used as a functional relationship between the total pressure head and the radius in the w plane:
Figure FDA0002589438360000022
in the formula C1、C2、C3、C4、C5And C6The constants are all constants determined by boundary conditions of the mapped tunnel circumference and karst cave circumference; ρ is the radius of any point in the w plane and R0Rho is not less than 1; is as follows; phi is the total pressure head of any point in the w plane;
Figure FDA0002589438360000025
is the included angle between the inner radius of the w plane and the transverse axis;
b. the following formula is adopted as a calculation relation of the karst cave boundary condition:
Figure FDA0002589438360000023
→C1＝Hw，C3＝-C4，C5＝-C6
c. determining the boundary condition of the tunnel circumference by adopting the following rules:
the first condition is as follows: if the hydrostatic pressure of the circumference of the tunnel is 0, the total pressure head of the circumference of the tunnel is the position head of the circumference of the z-plane tunnel; expressed as the following equation:
Figure FDA0002589438360000024
Figure FDA0002589438360000031
C3＝0，
Figure FDA0002589438360000032
the relationship between the total pressure head and the radius at any point in the plane w is given by:
Figure FDA0002589438360000033
case two: the total pressure head of the circumference of the tunnel is constant as Ht(ii) a Expressed as the following equation:
Figure FDA0002589438360000034
Figure FDA0002589438360000035
C3＝C5＝0
the relationship between the total pressure head and the radius at any point in the plane w is given by:
Figure FDA0002589438360000036
4. the method for calculating the analytical solution of the karst region circular tunnel seepage field according to claim 3, wherein the final karst region circular tunnel seepage flow is obtained by the calculation in step S4, specifically by using the following formula:
Figure FDA0002589438360000037
Figure FDA0002589438360000038
wherein k is the permeability coefficient of the surrounding rock; q1The seepage flow of karst water flowing into the tunnel under the condition that the hydrostatic pressure of the circumference of the tunnel is 0 and the total pressure head is a position head is formed; q2The seepage flow of karst water flowing into the tunnel under the condition of constant water pressure at the circumference of the tunnel.
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US20160245541A1 (en) * 2013-07-12 2016-08-25 Best Technologies, Inc. Fluid control measuring and controlling device
CN104389635A (en) * 2014-09-29 2015-03-04 招商局重庆交通科研设计院有限公司 Method for determining reasonable discharging quantity of tunnel underground water
JP2016217897A (en) * 2015-05-21 2016-12-22 公益財団法人鉄道総合技術研究所 Analysis method of flow on roof of vehicle inside tunnel
CN107391778A (en) * 2016-05-17 2017-11-24 武汉大学 A kind of Analytic Calculation Method of circular tunnel seepage discharge
CN108920417A (en) * 2018-07-31 2018-11-30 中建隧道建设有限公司 The calculation method of Rock And Soil dehydration quicksand rate on the outside of a kind of shield subway tunnel section of jurisdiction

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