CN110717295B - Method for tracking streamline distribution of tight sandstone reservoir by using finite element method - Google Patents

Method for tracking streamline distribution of tight sandstone reservoir by using finite element method Download PDF

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CN110717295B
CN110717295B CN201910954959.1A CN201910954959A CN110717295B CN 110717295 B CN110717295 B CN 110717295B CN 201910954959 A CN201910954959 A CN 201910954959A CN 110717295 B CN110717295 B CN 110717295B
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张烈辉
罗毅
赵玉龙
冯胤
刘启国
张东旭
刘沙
梁海鹏
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Southwest Petroleum University
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Abstract

The invention relates to a method for tracking streamline distribution of a compact sandstone reservoir in a two-dimensional unstructured grid by using a finite element method, which comprises the following steps of: establishing a numerical simulation model of the tight sandstone reservoir; given a real space initial point P0Coordinates, obtaining P0The mesh number, the three vertex coordinates and the three vertex speeds of the triangular mesh in which the triangular mesh is positioned; converting the speed in the Master Element by using the three vertex speeds of the Jacobian inverse matrix; calculating an initial point P by Shape Function0The speed of (d); calculating an exit coordinate; converting exit coordinates to real space coordinates Pe(ii) a Judging whether to leave the system; if leaving the grid system, starting to track the next streamline, otherwise PeAssign to P0And repeating the steps until the flow line is traced. The method can track the streamline in the triangular mesh system by using a finite element method, and accurately track the streamline distribution condition in the dense oil reservoir.

Description

Method for tracking streamline distribution of tight sandstone reservoir by using finite element method
Technical Field
The invention relates to a method for tracking the streamline distribution of a compact sandstone reservoir by using a finite element method, belonging to the technical field of unconventional oil and gas exploration and development.
Background
In the 50 s of the 20 th century, with the development of numerical simulation methods such as a flow pipe, an overlay theory, a well mirror image method and the like, a streamline simulation method starts to be developed in the petroleum field, and the method can effectively track a fluid path and visually research fluid velocity distribution in the calculation field. In the 90 s of the 20 th century, manyScholars began to propose modern streamline simulation concepts and began to study modern streamline simulation methods which put forward the term "time of flight" (TOF, denoted by the symbol τ) to describe the time of motion of particles in the motion domain. According to this definition, a coordinate transfer function is proposed
Figure BDA0002226970980000011
The conversion function can convert a multidimensional operator ▽ into a one-dimensional TOF coordinate system, which generates a key principle, and a modern streamline simulation method can simplify a two-dimensional or three-dimensional problem into a series of one-dimensional subproblems along a streamline, wherein the multidimensional problem is a problem under a space (x, y) coordinate system, and the one-dimensional subproblem is a problem under a time coordinate (TOF, tau) system.
According to most of the streamline simulation methods, the classical principle of streamline simulation mainly comprises the following steps:
(1) tracking a streamline by using the concept of 'flight time', and constructing a one-dimensional coordinate system related to the streamline;
(2) describing transport equations (e.g., mass or energy transfer equations) with TOF;
(3) periodically updating the streamlines to account for the evolution of the velocity field;
(4) numerically solving or analytically solving a one-dimensional expression of the transport equation along each streamline, and expressing the one-dimensional expression by TOF;
(5) computing the transverse flux (or other cross-streamline behavior) uses an operator splitting technique.
According to the above principle, the most important step is to trace the streamline based on the velocity field obtained by solving the continuity equation and the material equation in the porous medium by the finite element method (Darcy's law in the porous medium). Once the TOF grid system along each streamline is established, one-dimensional mass transfer equations for TOF can be solved. A multi-dimensional grid system formed by finite element methods can be used to solve the flow problem and trace streamlines, and each streamline can be discretized into a series of TOF-based computing nodes.
In the aspect of calculating the streamline, a Runge-Kutta type solver is generally used for numerical tracking, but because the difficulty of generating the TOF grid by the Runge-Kutta type solver is high, and the numerical tracking cannot be performed by a longge-coutta solver in streamline simulation, a semi-analytic method such as a Pollock algorithm is widely used. These methods are referred to as semi-analytic methods because the analytic solution is derived from the control body, not the entire domain. Pollock proposes a piecewise analytical solution, derived by TOF, when assuming that the velocity of each direction in the velocity field varies linearly, and is independent of the velocities of other directions. Given the start point coordinates and velocity for any hypothetical fluid particle, the algorithm can determine the exit coordinates from the grid and the time it takes to pass through the grid (this is TOF and is also the residence time of the particle within a grid). Although Pollock can efficiently generate TOF grids while tracing streamlines and can reasonably assume patterns for streamlines in each computational domain, it is only applicable to structured grids, and it cannot trace streamlines when there are sources or sinks (represented by the flow in the grid) within the grid. Although streamline simulation in unstructured grids has been studied by many scholars, these algorithms suffer from at least one of the following disadvantages:
(1) these formulas rely on the concept of a control volume, requiring flow to be applied to each face of the mesh, and a certain flow in each mesh;
(2) in this process, ordinary differential equation numerical solution is involved, and TOF grid cannot be calculated and obtained efficiently.
Therefore, the streamline obtained by the current method can only be used for visualizing the velocity field, and the one-dimensional mass transfer equation cannot be mapped on the streamline.
Disclosure of Invention
Aiming at the problems, the invention mainly overcomes the defects in the prior art and provides a method for tracking the streamline distribution of a compact sandstone reservoir by using a finite element method.
The technical scheme provided by the invention for solving the technical problems is as follows: a method for tracking the streamline distribution of a tight sandstone reservoir by using a finite element method comprises the following steps:
s1, establishing a numerical simulation model of the compact sandstone reservoir of the two-dimensional unstructured grid by using a finite element method, and acquiring a pressure field and a velocity field of the compact sandstone reservoir according to parameters of the compact sandstone reservoir;
step S2, setting the injection as the real space initial point P0(x0,y0) (ii) a Obtaining P0The number of the located grid is CellID, and the coordinates (x) of three vertexes of the grid are obtained1,y1)，(x2,y2)，(x3,y3) And vertex velocity (v)x1,vy1)，(vx2,vy2)，(vx3,vy3)；
Step S3, converting the real space initial point P by the coordinate conversion formula0(x0,y0) Conversion to Master Element (η)0,ξ0)；
Step S4, converting the three vertex speeds of the grid into (v) by using a Jacobian inverse matrixξ1,vη1)，(vξ2,vη2)，(vξ3,vη3) (ii) a The velocity (v) of the initial point is calculated by Shape Functionξ0,vη0)；
Step S5, solving a track equation in the MasterElement, obtaining the time of the particles to reach three sides of the triangular unit through a Newton-Raphson iterative algorithm, taking the minimum positive value as the flight time delta tau in the unit, and calculating an exit coordinate Pe *(ηe,ξe)；
Step S6, utilizing Shape Function to convert Pe *(ηe,ξe) Switch to trueReal space coordinate Pe(xe,ye) And accumulating the flight time delta tau;
step S7, judgment Pe(xe,ye) Whether to leave the grid system;
step S8, if leaving the grid system, starting to track the next streamline; otherwise P will bee(xe,ye) Assign to P0(x0,y0) And repeating the steps S2 to S7 until the flow line is traced.
Further technical solution is that, in the step S2, the real space initial point P is obtained under the FEniCS system through programming0(x0,y0) Three vertex coordinates (x) of the located grid1,y1)，(x2,y2)，(x3,y3) And vertex velocity (v)x1,vy1)，(vx2,vy2)，(vx3,vy3)。
The further technical solution is that in step S3, the expression of η and ξ is solved by the following formula:
x＝x1(1-ξ-η)+x2ξ+x3η
y＝y1(1-ξ-η)+y2ξ+y3η
Figure BDA0002226970980000041
Figure BDA0002226970980000042
wherein x is the x coordinate of real space, y is the y coordinate of real space, ξ is the transverse coordinate of Master Element (as shown in FIG. 3) corresponding to the x coordinate, η is the longitudinal coordinate of Master Element (as shown in FIG. 3) corresponding to the y coordinate, x is the x coordinate of real space, y is the y coordinate of real space, and1，y1)，(x2，y2)，(x3，y3) The coordinates of three vertexes of the triangle in the real space are respectively converted into (0, 0), (1, 0), (0,1) in the MasterElement.
The further technical scheme is that the specific process of the step S4 is as follows:
converting the three vertex velocities of the grid into (v) in the MasterElement by using a coordinate conversion formulaξ1,vη1)，(vξ2,vη2)，(vξ3,vη3) (ii) a Calculating an initial point P by Shape Function0Velocity (v) ofξ0,vη0)。
Wherein the Jacobian inverse matrix is:
Figure BDA0002226970980000043
Figure BDA0002226970980000051
wherein
Figure BDA0002226970980000052
Figure BDA0002226970980000053
In the formula: (x)1，y1)，(x2，y2)，(x3，y3) Coordinates of three vertexes of a triangle in a real space are respectively; upsilon isxAnd upsilonyVelocity in real space x and y, respectively, upsilonξAnd upsilonηThe velocities in the ξ and η directions in MasterElement.
The further technical scheme is that the specific process of the step S5 is as follows:
step S51, setting particle slave P0Starting from the triangle, P can be calculated separately from any side of the triangle0The time from the side 1, the side 2, and the side 3 is Δ τ1、Δτ2、Δτ3；
Step S52, three edge time delta tau is taken1、Δτ2、Δτ3Is the time of flight Δ τ within the cell;
step S53, determining departure coordinate P in MasterElemente(ξe,ηe) Wherein ξ is obtained by substituting the time of flight Δ τ obtained in the second step into the following equatione,ηe；
Figure BDA0002226970980000054
Figure BDA0002226970980000055
Figure BDA0002226970980000056
Figure BDA0002226970980000057
Wherein
Figure BDA0002226970980000058
In the formula: (v)ξ1,υη1)，(υξ2,υη2)，(υξ3,υη3) The speeds of ξ and η directions of three vertexes of a triangle in the MasterElement are respectively, and lambda is1，2Is a characteristic equation
Figure BDA0002226970980000061
Two characteristic values of
Figure BDA0002226970980000062
e is Euler number about 2.7183, Δ τ is time of flight ξe,ηeThe exit coordinates of the particles in the Master Element from the triangle.
Step S54, determining departure coordinate P under real coordinate systeme *(ηe,ξe) ξ in the third stepe,ηeConversion to real coordinates P by Shape Functione *(ηe,ξe)。
The invention has the following beneficial effects:
(1) similar to Pollock algorithm, the exit coordinates can be determined from the entry coordinates in the triangular unit mesh;
(2) the calculation only needs the data (coordinates and flow) of the vertex of the triangular mesh, and does not need to consider the flow on the boundary of the triangular unit;
(3) the invention provides an analytic solution for describing the streamline locus in the triangular unit, so that the TOF can be accurately and efficiently determined.
Drawings
Obtaining a grid vertex coordinate code of an arbitrary point in the FENICS system in FIG. 1;
FIG. 2 is a graph showing a graph vertex velocity code of a mesh vertex where an arbitrary point is located in the FENICS system;
FIG. 3 is a triangular solid graph and coordinate distribution in Master element;
FIG. 4 is a solution process for streamline tracing in unstructured mesh triangle cells generated using finite elements;
FIG. 5 validation of results of streamline tracing in triangle unit in MasterElement;
FIG. 6 is a technical route diagram of a tight sandstone reservoir utilizing a finite element tracing streamline method according to an embodiment of the invention;
FIG. 7 is a computational grid diagram of a tight sandstone reservoir in an embodiment of the present invention;
figure 8 is a pressure field diagram of a tight sandstone reservoir in an embodiment of the present invention;
FIG. 9 is a velocity field plot for a tight sandstone reservoir in an embodiment of the present invention;
figure 10 is a tight sandstone reservoir flowline profile according to an embodiment of the present invention.
Detailed Description
The present invention will be further described with reference to the following examples and the accompanying drawings.
Example 1
The invention discloses a method for tracking streamline distribution of a compact sandstone reservoir by using a finite element method, which comprises the following steps of:
step S1, firstly, establishing a compact sandstone reservoir numerical simulation model of a two-dimensional unstructured grid (triangular grid) by using a finite element method;
wherein the size of the compact sandstone reservoir is 500m multiplied by 500m, the reservoir permeability is 5mD, 1 injection well and 4 production wells are arranged, the bottom pressure of the injection well is 14MPa, the bottom pressure of the production well is 12.5MPa, and a pressure field (shown in figure 8) and a velocity field (shown in figure 9) of the compact sandstone reservoir are obtained;
step S2, injection well is given as real space initial point P0(x0,y0) Obtaining P0The grid number CellID is located, and the coordinates (x) of three vertexes of the grid are read1,y1)，(x2,y2)，(x3,y3) And vertex velocity (v)x1,vy1)，(vx2,vy2)，(vx3,vy3)；
The vertex coordinates of the mesh where the point is located are obtained under the FEniCS system through programming, and the code is as follows (as shown in fig. 1, programmed by using Python language):
the vertex speed of the mesh where the point is located is obtained under the FEniCS system through programming, and the code is as follows (as shown in fig. 2, using Python language for programming):
s3, converting the real space initial point P through a coordinate conversion formula0(x0,y0) Conversion to MasterElement (η)0,ξ0)；
Wherein by solving for the expression of η and ξ:
x＝x1(1-ξ-η)+x2ξ+x3η
y＝y1(1-ξ-η)+y2ξ+y3η
Figure BDA0002226970980000081
Figure BDA0002226970980000082
that is, the initial point P in real space can be obtained0(x0,y0) Conversion to MasterElement (η)0,ξ0)。
S4, converting the three vertex velocities (v) of the grid by using a Jacobian inverse matrixξ1,vη1)，(vξ2,vη2)，(vξ3,vη3) (ii) a The velocity (v) of the initial point is calculated by Shape Functionξ0,vη0)；
The MasterElement rates were:
Figure BDA0002226970980000083
wherein
Figure BDA0002226970980000084
Figure BDA0002226970980000085
Figure BDA0002226970980000086
Figure BDA0002226970980000087
S5, solving a track equation in Master Element, obtaining the time of the particles to reach three edges of a triangular unit through a Newton-Raphson iterative algorithm, taking the minimum positive value as the flight time delta tau in the unit, and calculating an exit coordinate Pe *(ηe,ξe)；
Master Element is a right-angled isosceles triangle with three vertexes as P1(0,0)，P2(1,0)，P3(0, 1); the three sides are respectively: edge 1 (P)1-P2) Edge 2 (P)2-P3) Edge 3 (P)3-P1) As shown in fig. 3.
P in one particle from Master Element Unit0From a point P on one side of the triangleeThe time taken for departure is time of flight (TOF),denoted by Δ τ.
The trajectory equation is solved as follows:
and solving a trajectory equation of any point P in a standard triangle in the Master element according to the first-order Lagrange Shape Function.
Three vertex coordinates P of triangle1(0,0),P2(1,0),P2(0,1), where the coordinate of any point P is (ξ), the initial condition, t is 0, ξ is ξ0,ξ'＝υξ0,η＝η0,η'＝υη0。
Arbitrary point P (ξ) velocity upsilonξIs composed of
Figure BDA0002226970980000091
Wherein
Figure BDA0002226970980000092
Since the velocity distribution in a velocity field is arbitrary, the coefficients of the equation have uncertainty, and it is necessary to discuss the equation coefficients to solve the equation:
(1) when b is2Is equal to 0, and c1When 0, the differential equation can be converted to a directly separable set of variable differential equations whose solution is:
Figure BDA0002226970980000093
Figure BDA0002226970980000094
Figure BDA0002226970980000101
Figure BDA0002226970980000102
(2) when b is2When not equal to 0, the differential equation is a non-homogeneous linear squareThe set of programs is a set of programs,
when Δ ═ b1-c2)2+4b2c1＞0
Figure BDA0002226970980000103
Specific solutions of two distinct solid roots
Figure BDA0002226970980000104
Wherein
Figure BDA0002226970980000105
Figure BDA0002226970980000106
(3) When Δ ═ b1-c2)2+4b2c1＝0，
Figure BDA0002226970980000107
Special solution of two equal root
Figure BDA0002226970980000108
Wherein:
Figure BDA0002226970980000109
Figure BDA00022269709800001010
(4) when Δ ═ b1-c2)2+4b2c1＜0
Figure BDA0002226970980000111
Special solution of two conjugate complex roots
Figure BDA0002226970980000112
Wherein
Figure BDA0002226970980000113
Figure BDA0002226970980000114
Specifically, according to the master element trajectory equation, the coordinate PeAnd time of flight Δ τ can be calculated by the following four steps:
the first step is as follows: suppose that this particle is from P0Starting from the triangle, P can be calculated separately from any side of the triangle0The time from the side 1, the side 2, and the side 3 is Δ τ1、Δτ2、Δτ3。
Given a position, given equation (4) for example, the time for the particle to reach this position can be determined, and thus given η equal to 0, Δ τ can be determined1The determination of Δ τ can be made when ξ + η ═ 12ξ is 0, Δ τ can be determined3。
Based on these principles, the following equation (7) can be summarized, where Δ τ1、Δτ2、Δτ3Obtained by the Newton-Raphson algorithm.
Figure BDA0002226970980000115
The second step is that: from Δ τ1、Δτ2、Δτ3Finding the true time of flight Δ τ. The calculated TOF value may be positive or negative and the true time of flight Δ τ is the smallest positive value, denoted Δ τ -min (Δ τ)1,Δτ2,Δτ3)。
The third step: determining departure coordinates P in a master elemente(ηe,ξe) The time of flight Δ τ obtained in the second step is substituted into equation (8) to be calculated ξe,ηe
Figure BDA0002226970980000121
The fourth step: determining departure coordinates P in a real coordinate systeme *(ηe,ξe) I.e. ξ in the third stepe,ηeConversion to real coordinates P by Shape Functione *(ηe,ξe)；
S6, using Shape Function to convert Pe *(ηe,ξe) Conversion to real space coordinates Pe(xe,ye) And accumulating the flight time delta tau;
converting any point P (eta, xi) in the MasterElement into a formula in the Actual Element:
x＝x1(1-ξ-η)+x2ξ+x3η
y＝y1(1-ξ-η)+y2ξ+y3η
s7, judgment Pe(xe,ye) Whether to leave the system;
s8, if leaving the grid system, starting to track the next streamline, otherwise, Pe(xe,ye) Assign to P0(x0,y0) Steps S2 to S7 are repeated, with 200 streamlines traced starting from the injection well, with the results shown in fig. 10.
When initial conditions are given, the motion trajectory in a triangle unit is tracked by using the above formula and theory, as shown in fig. 4, fig. 4 shows the solving process of streamline tracing in an unstructured mesh triangle unit generated by using finite elements when Δ >0, fig. 4(a) shows the coordinates and the velocity of the vertex and the starting point in Actual Element, then the information in fig. 4(a) is converted from Actual Element to Master Element, and the motion trajectory of the particle is tracked, and as a result, as shown in fig. 4(b), the result needs to be converted into Actual Element after the Master Element tracing is finished, as shown in fig. 4 (c). The same applies when Δ is 0 and Δ < 0.
The results of the patent of the invention are verified through 4-order Runge-Kutta numerical algorithm results, as shown in FIG. 5, the lines in the graph represent the patent results of the invention, the black dots represent the 4-order Runge-Kutta numerical algorithm results, the errors of the two results are very small, and the accuracy of the patent method of the invention is verified.
Although the present invention has been described with reference to the above embodiments, it should be understood that the present invention is not limited to the above embodiments, and those skilled in the art can make various changes and modifications without departing from the scope of the present invention.

Claims (5)

1. A method for tracking the streamline distribution of a tight sandstone reservoir by using a finite element method is characterized by comprising the following steps:
s1, establishing a numerical simulation model of the compact sandstone reservoir of the two-dimensional unstructured grid by using a finite element method, and acquiring a pressure field and a velocity field of the compact sandstone reservoir according to parameters of the compact sandstone reservoir;
step S2, setting the position of the injection well as the real space initial point P0(x0,y0) (ii) a Obtaining P0The number Cell ID of the mesh where the grid is located, and the coordinates (x) of three vertexes of the mesh are obtained1,y1)，(x2,y2)，(x3,y3) And vertex velocity (v)x1,vy1)，(vx2,vy2)，(vx3,vy3)；
Step S3, converting the real space initial point P by the coordinate conversion formula0(x0,y0) Is converted into Master Element space coordinates (η)0,ξ0)；
Step S4, converting the three vertex speeds of the grid into (v) in Master Element by using Jacobian inverse matrixξ1,vη1)，(vξ2,vη2)，(vξ3,vη3) (ii) a Calculating an initial point P by Shape Function0Velocity (v) ofξ0,vη0)；
Step S5, solving a track equation in the Master Element, obtaining the time of the particles to reach three edges of the triangular unit in the Master Element through a Newton-Raphson iterative algorithm, taking the minimum positive value as the flight time delta tau in the unit, and calculating an outlet coordinate Pe *(ηe,ξe)；
Step S6, utilizing Shape Function to convert Pe *(ηe,ξe) Conversion to real space coordinates Pe(xe,ye) And accumulating the flight time delta tau;
step S7, judgment Pe(xe,ye) Whether to leave the grid system;
step S8, if leaving the grid system, starting to track the next streamline; otherwise P will bee(xe,ye) Assign to P0(x0,y0) And repeating the steps S2 to S7 until the flow line is traced.
2. The method for tracking streamline distribution of tight sandstone reservoir by using finite element method as claimed in claim 1, wherein said step S2 is implemented by obtaining real space initial point P under FENICS system0(x0,y0) Three vertex coordinates (x) of the located grid1,y1)，(x2,y2)，(x3,y3) And vertex velocity (v)x1,vy1)，(vx2,vy2)，(vx3,vy3)。
3. The method for tracking streamline distribution of tight sandstone reservoir according to claim 1, wherein the expression of η and ξ is solved in step S3 by the following formula:
x＝x1(1-ξ-η)+x2ξ+x3η
y＝y1(1-ξ-η)+y2ξ+y3η
Figure FDA0002587952920000021
Figure FDA0002587952920000022
wherein x is the x coordinate of real space, y is the y coordinate of real space, ξ is the transverse coordinate of Master Element corresponding to the y coordinate, η is the longitudinal coordinate of Master Element corresponding to the x coordinate, (x)1，y1)，(x2，y2)，(x3，y3) The coordinates of three vertexes of the triangle in the real space are respectively converted into (0, 0), (1, 0), (0,1) in the Master Element.
4. The method for tracking streamline distribution of tight sandstone reservoir by using finite Element method as claimed in claim 1, wherein in step S4, the three vertex velocities of the grid are converted into (v) in Master Element by using Jacobian inverse matrixξ1,vη1)，(vξ2,vη2)，(vξ3,vη3) (ii) a Calculating an initial point P by Shape Function0Velocity (v) ofξ0,vη0)；
The Jacobian inverse matrix is:
Figure FDA0002587952920000023
wherein
Figure FDA0002587952920000024
In the formula: (x)1，y1)，(x2，y2)，(x3，y3) Coordinates of three vertexes of a triangle in a real space are respectively; upsilon isxAnd upsilonyVelocity in real space x and y, respectively, upsilonξAnd upsilonηThe speeds in the directions of ξ and η in the Master Element.
5. The method for tracking the streamline distribution of the tight sandstone reservoir by using a finite element method as claimed in claim 1, wherein the specific process of the step S5 is as follows:
step S51, setting particle slave P0Starting from the triangle, P can be calculated separately from any side of the triangle0The time from the side 1, the side 2, and the side 3 is Δ τ1、Δτ2、Δτ3；
Step S52, taking the time delta tau of three edges of the particle1、Δτ2、Δτ3Is the time of flight Δ τ within the cell;
step S53, determining an outlet coordinate P in Master Elemente *(ηe,ξe) Wherein ξ is obtained by substituting the time of flight Δ τ obtained in the second step into the following equatione,ηe；
Figure FDA0002587952920000031
Figure FDA0002587952920000032
Wherein
Figure FDA0002587952920000033
Figure FDA0002587952920000034
In the formula: (v)ξ1,υη1)，(υξ2,υη2)，(υξ3,υη3) Are respectively asξ, η directional speeds of three vertexes of a triangle in MasterElement, lambda1，2Is a characteristic equation
Figure FDA0002587952920000035
Two characteristic values of
Figure FDA0002587952920000036
e is Euler number about 2.7183, Δ τ is time of flight ξe,ηeThe coordinates of the outlet of the particles in the Master Element from the triangle;
step S54, determining outlet coordinates P under a real coordinate systeme *(ηe,ξe) ξ in the third stepe,ηeConversion to real space coordinates P by Shapefunctione(xe,ye)。
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