CN110261282B - Shear seam shale core pulse attenuation permeability testing method - Google Patents
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Abstract
The invention discloses a method for testing pulse attenuation permeability of a shear joint shale core, which comprises the following steps: s1: establishing a shear joint shale core dual-medium physical model based on a multi-scale seepage model of gas in a shale shear joint core; s2: obtaining a finite volume discrete format corresponding to the physical model by using a finite volume method, solving a full implicit solving format of the physical model by combining a Newton-Laplacian iterative algorithm, and programming and solving the model; s3: performing a pulse attenuation experiment on the shear seam shale core by using a pulse attenuation method experiment device, and measuring pressure and time relation curves of an upstream chamber and a downstream chamber by using experiment data; s4: and (3) combining a shear fracture shale core dual-medium physical model, and obtaining the matrix permeability and the fracture permeability of the shear fracture shale core by adopting a curve fitting method. The permeability test method can be used for testing the permeability of the shale core with the shear seam.
Description
Technical Field
The invention relates to the technical field of shale gas development, in particular to a method for testing pulse attenuation permeability of a shear joint shale core.
Background
From the perspective of global energy development strategy, the development of shale gas is changing the global energy pattern and becoming the focus of global development. The total amount of shale gas reservoir resources worldwide is predicted to be about 456 x 1012m3The total resource amount of unconventional natural gas (coal bed gas, tight sandstone gas and shale gas) is about 50 percent. Meanwhile, exploration shows that the recoverable reserves of shale gas resources in China are very rich.
Commercial exploitation of shale gas in multiple basins has been successful in the united states, but many problems have been exposed in recent years of development practice, and many theoretical and technical bottlenecks have been encountered. Compared with North America, the shale gas reservoir in China has deeper buried depth, complex structure, large physical difference and the like, so that the problems encountered in shale development in China are more complex. The core permeability is the basis for knowing the seepage condition of the shale formation, and the shale core permeability test is particularly important to solve the problems.
The permeability test of the core is mainly divided into a steady-state method test and an unsteady-state method test. The steady state method is a method for obtaining permeability by using a Darcy formula, wherein a stable flow state is formed in a tested rock core by a test fluid, the pressure difference at two ends of the rock core is recorded through testing, and meanwhile, the flow data passing through the rock core is monitored. The unsteady state method is characterized in that unstable seepage is formed in a tested rock core by test fluid, the flow is in an unsteady state, the pressure and the flow are in a gradual change process, and the permeability is calculated by a calculus method through the gradual change process of the test recorded pressure. For a low-permeability reservoir, it takes long time and is difficult to judge when the reservoir reaches a stable flowing stage, and the steady-state method test has great limitations.
The pulse decay method among the unsteady state methods was first proposed by Brace (1968) in testing granite permeability, by calculating the material balance of the upstream and downstream chambers and the core to obtain a response solution of dimensionless pressure time; lin et al (1977) analyzed the effect of the upstream and downstream chamber volumes on the experimental effect on this basis, and gave the optimized chamber volume for the experimental core; dicker et al (1988) designed a pulse attenuation method testing device, and adopted a response solution based on initial pressure, thereby greatly improving the testing efficiency; haskett et al (1988) improve the device based on a pulse attenuation method, can simultaneously measure the porosity and permeability of a rock core, has high test efficiency, and verifies a model result by adopting a method of rock core analysis and numerical simulation comparison; chen Zhi Ming et al (2013) design a set of experimental device based on a pulse attenuation method, can test the permeability of compact sandstone and shale, and further analyze the effect of the stress on the experimental result; yang (2017) and the like establish a semi-analytic solution model under the radial coordinate of the rock core, and the porosity and the permeability of the rock core can be obtained through experimental data fitting. The traditional pulse attenuation method is improved by Bryant (2018) and the like, the number of pressure sensors in the device is reduced, the cost of the testing device is greatly reduced, and meanwhile, the device has the advantages of convenience in operation, high testing efficiency and the like, and the defect that the influences of multi-scale seepage and adsorption desorption in a rock core are not considered is overcome.
The method can be only used for parameter testing of homogeneous cores and cannot be used for shale cores with shear seams, so that a testing method suitable for the shale cores with shear seams is urgently needed.
Disclosure of Invention
Aiming at the problems, the invention provides a method for testing the pulse attenuation permeability of a shear fracture shale core, which is used for testing the permeability of the shear fracture shale core.
The technical scheme of the invention is as follows:
a shear seam shale core pulse attenuation permeability test method comprises the following steps:
s1: establishing a shear joint shale core dual-medium physical model based on a multi-scale seepage model of gas in a shale shear joint core;
preferably, the multi-scale seepage model is a B-K model, and the mathematical description equation of the B-K model is as follows:
in the formula: k is a radical ofappAs apparent permeability, mD; k is a radical of∞As intrinsic permeability, mD; knIs the knudsen number and has no dimension; alpha is a sparse coefficient and is dimensionless; b is slip coefficient and is dimensionless.
Preferably, the shear fracture shale core dual-medium physical model comprises:
seepage equation of the fracture system:
in the formula: k is a radical offeFracture apparent permeability, mD; mu.sgIs the gas viscosity, mPas; b isgThe natural gas volume coefficient is dimensionless; p is a radical offFracture pressure, MPa; q. q.smThe flow rate of the unit volume fracture to the matrix is 1/s; t is time, s;
the crack porosity is zero dimension;
matrix system seepage equation:
in the formula: k is a radical ofmeAs the apparent permeability of the matrix, mD; p is a radical ofmIs the matrix pressure, MPa;
the porosity of the matrix is dimensionless; q. q.sdesThe unit core adsorption desorption gas amount is 1/s; vEIs the isothermal adsorbed gas volume per unit volume of core, cm3/cm3；
When the cross flow between the matrix and the fracture is quasi-steady state cross flow:
when the channeling between the matrix and the fracture is unsteady channeling:
in the formula: r is1Is the radius of the matrix particle, cm; r isbIs the integral radius, cm;
the boundary conditions are as follows:
an upstream chamber:
a downstream chamber:
in the formula: k is a radical ofmAs matrix permeability, mD; a is the surface area of the core in cm2(ii) a x is an abscissa value in the horizontal direction and is dimensionless; u is the x-axis coordinate of the upstream bottle position; k is a radical offFracture permeability, mD; vuIs the volume of the upstream chamber, cm3；VgIs a gas molar volume of 0.0224m3Per mol; z is a gas deviation factor and is dimensionless; r is a gas constant of 8.314J/(mol.K); t isduIs the upstream and downstream bottle temperature, K; p is a radical ofuThe upstream chamber pressure, MPa; cgIs a gas compression coefficient, MPa-1(ii) a d is the x-axis coordinate of the downstream bottle position; vdIs the volume of the downstream chamber, cm3；pdDownstream chamber pressure, MPa;
initial conditions:
in the formula: x and y are coordinates of a certain point in the system; p is a radical ofmiInitial pressure of matrix, MPa; p is a radical offiThe fracture initiation pressure, MPa; p is a radical ofu0Upstream initial pressure, MPa; p is a radical ofd0Downstream initial pressure, MPa.
S2: obtaining a finite volume discrete format corresponding to the physical model by using a finite volume method, solving a full implicit solving format of the physical model by combining a Newton-Laplacian iterative algorithm, and programming and solving the model; the method specifically comprises the following substeps:
s21: the seepage equation is processed based on the finite volume theory: firstly, dispersing a whole continuous solution domain to obtain a finite volume discrete format; secondly, processing a nonlinear partial differential equation set by adopting a Newton-Laplacian iterative algorithm on the finite volume discrete format to obtain a fully implicit solving format; finally, assembling the research units into an overall balance matrix to obtain an overall balance matrix equation;
preferably, the finite volume discrete format is:
in the formula: t iseIs a cell conductivity matrix; peIs a cell grid pressure matrix;
porosity, dimensionless;
the fully implicit solution format:
in the formula: n and k are iteration times; n is a radical ofeAccumulating an entry matrix for the cell;
the overall balance matrix equation is:
in the formula: t is the overall conductivity matrix; p is the overall pressure matrix; n is a total accumulated item matrix; rrIs a residual matrix.
S22: dispersing the adsorption and desorption items according to a finite volume method, introducing a Newton-Laplacian iteration algorithm to obtain a shale core unit discrete iteration format, sorting out corresponding partial derivative matrixes according to a unit conductivity matrix, a unit cumulative item matrix and a unit adsorption item matrix, and combining the three unit partial derivative matrixes to obtain a shale single dense medium model comprehensive discrete equation; processing a source-sink phase in a shale single dense medium model comprehensive discrete equation;
preferably, the discrete iteration format of the shale core unit is as follows:
δpe＝(δp1 δp2 δp3)T；Pe＝(p1 p2 p3)T；Re＝(R1 R2 R3)T
in the formula: subscript e represents a unit; veIs a unit adsorption and desorption term matrix; reIs a unit residue matrix; 1. and 2, 3 are three nodes of a triangular mesh.
The shale single dense medium model comprehensive discrete equation is as follows:
in the formula: v is an adsorption and desorption term matrix;
when the mesh has NN cells and MM nodes after being discretized, the formula (51):
overall residual matrix Rr＝(R1 R1 … RMM)T；
δP＝(δp1 δp2 … δpMM)T；P＝(p1 p2 … pMM)T；
Processing the source and sink items in the upstream grid:
in the formula: q. q.sgThe flow rate of the unit volume fracture to the matrix is 1/s; vpIs the core pore volume, cm3；k1As first node permeability, mD; p is a radical of1Is the first node pressure, MPa;
processing the source and sink items in the downstream grid:
in the formula: k is a radical ofnx(ii) the last mesh permeability, mD; p is a radical ofnxThe final mesh pressure, MPa.
S23: dispersing the channeling item according to a finite volume method, obtaining a shale dual medium model comprehensive discrete equation by adopting a solving method of a single medium model in S22, and simultaneously forming the equation into a synchronous solving format to obtain an implicit solving format of the shale core dual medium seepage differential equation.
Preferably, the shale dual medium model comprehensive discrete equation is as follows:
(Tf+δTf-W-δW-δNf)k(δPf)k+(W+δW)k(δPm)k＝-(Rf)k (66)
(Tm+δTm-W-δW-δNm-δVE)k(δPm)k+(W+δW)k(δPf)k＝-(Rm)k (67)
in the formula: t isfIs a fracture bulk conductivity matrix; t ismA bulk conductivity matrix for the matrix; w is a cross-flow item matrix; n is a radical offAccumulating an item matrix for the fracture population; n is a radical ofmAccumulating the term matrix for the population of matrices; rfThe crack residue matrix is used for measuring whether the operation is finished or not; rmIs matrix residue matrix, which is used to measure whether the operation is finished.
When the mesh has a total of NN units and MM nodes after being discretized, the equations (66) and (67) are as follows:
matrix of total residuals
Pressure total increment matrix
Pressure overall matrix
The implicit solving format of the shale core dual medium seepage differential equation is as follows:
A11＝Tf+δTf-W-δW-δNf；
A12＝A21＝W+δW；
A22＝Tm+δTm-W-δW-δNm-δVE
in the formula: a. the11、A12、A21、A22An intermediate matrix defined for ease of computation;
matrix array
The dimension 2MM by 2MM,
and
dimension 2MM x 1.
S3: and performing a pulse attenuation experiment on the shear seam shale core by using a pulse attenuation method experiment device, and measuring pressure and time relation curves of an upstream chamber and a downstream chamber by using experiment data.
S4: the method is characterized by combining a shear fracture shale core dual-medium physical model and adopting a curve fitting method to obtain the matrix permeability and the fracture permeability of the shear fracture shale core, and specifically comprises the following substeps:
s41: estimating matrix permeability and crack permeability parameters, specifically:
the permeability of the matrix is as follows:
in the formula: c is the slope of the dimensionless pressure time curve; l is the core length, cm;
crack permeability:
in the formula: ctIs the system comprehensive compression coefficient, MPa-1；tDcIs the slope of the dimensionless pressure time curve; t is tcConvergence time of experimental data, s;
s42: fitting experimental data by using pre-estimated data and an optimization algorithm to finally obtain matrix permeability and crack permeability; the optimization algorithm is a genetic algorithm, matrix permeability and crack permeability are determined by simultaneously fitting the pressure response curves of the upper and lower bottles, and in order to increase the accuracy of data fitting, a least square method is adopted to establish an upstream and downstream chamber pressure objective function:
in the formula: p is a radical ofuiCalculating the obtained upstream pressure, MPa, for the model; p is a radical ofdiCalculating the downstream pressure, MPa, obtained for the model; p is a radical ofeuiActual upstream pressure, MPa; p is a radical ofediActual downstream pressure, MPa.
The invention has the beneficial effects that:
the method comprises the steps of establishing a shear joint shale core dual-medium finite volume model based on multi-scale seepage of gas in a shale shear joint core, obtaining a corresponding finite volume discrete format by using a finite volume method in combination with internal and external boundary conditions and initial conditions of the core in a pulse attenuation method, solving a full-implicit solution format of the finite volume discrete format in combination with a Newton-Laplacian iterative algorithm, and programming and solving the model to form a set of experimental data fitting method suitable for solving permeability of the shear joint shale core; and in the solving process, full implicit iteration is adopted, so that the result convergence is large and the error is small.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a schematic structural view of a shear joint shale core of the present disclosure;
FIG. 2 is a schematic view of the dual media core microfluidic of the present invention;
FIG. 3 is a schematic diagram of a triangular mesh finite volume control unit according to the present invention;
FIG. 4 is a schematic view of a discrete grid according to the present invention;
FIG. 5 is a schematic diagram of a pulse attenuation apparatus according to the present invention;
FIG. 6 is a flow chart of the genetic algorithm calculation of the present invention;
FIG. 7 is a graph of the pressure distribution in the core at 10 seconds for the present invention;
FIG. 8 is a graph of the pressure distribution in the core at 100 seconds for the present invention;
FIG. 9 is a graph of pressure distribution in the core at 1000 seconds according to the present invention;
FIG. 10 is a graph of time dependence of fracture core pressure in accordance with the present invention.
In the figure: the device comprises a core holder 1, a core holder 2, an upstream chamber 2, a downstream chamber 3, an upstream chamber pressure sensor 4, a downstream chamber pressure sensor 5, a valve I6, a valve II 7 and a valve III 8.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
As shown in fig. 1 to 10, a method for testing pulse attenuation permeability of a shear fracture shale core includes the following steps:
s1: establishing a shear joint shale core dual-medium physical model based on a multi-scale seepage model of gas in a shale shear joint core;
due to the multi-scale characteristics of the pore radius of the shale core, the gas flow in the shale core has various flow states, and the B-K model proposed by Beskok in 1996 writes three flow states into an equation so as to achieve the purpose of coupling the shale multi-scale migration mechanism, wherein the mathematical description equation of the B-K model is as follows:
in the formula: k is a radical ofappAs apparent permeability, mD; k is a radical of∞As intrinsic permeability, mD; knIs the knudsen number and has no dimension; alpha is a sparse coefficient and is dimensionless; b is the slip coefficient which is usually-1 and dimensionless.
The flow of gas in the core is shown in fig. 2, and the gas flow pattern includes: (1) flow within the matrix, (2) flow within the fracture, (3) cross-flow between the matrix and the fracture. According to the gas flow pattern, the following seepage equation is given:
a crack system:
in the formula: k is a radical offeFracture apparent permeability, mD; mu.sgIs the gas viscosity, mPas; b isgThe natural gas volume coefficient is dimensionless; p is a radical offFracture pressure, MPa; q. q.smThe flow rate of the unit volume fracture to the matrix is 1/s; t is time, s;
the crack porosity is zero dimension;
matrix system:
in the formula: k is a radical ofmeAs the apparent permeability of the matrix, mD; p is a radical ofmIs the matrix pressure, MPa;
the porosity of the matrix is dimensionless; q. q.sdesThe unit core adsorption desorption gas amount is 1/s;
in the formula: vEIs the isothermal adsorbed gas volume per unit volume of core, cm3/cm3；
VEDetermined by the langmuir isothermal adsorption relationship:
in the formula: vLVolume of Langmuir, cm3/cm3；PLLangmuir pressure, MPa.
When the cross flow between the matrix and the fracture is quasi-steady state cross flow:
when the channeling between the matrix and the fracture is unsteady channeling:
in the formula: r is1Is a matrix particleParticle radius, cm; r isbIs the integral radius, cm;
the boundary conditions are as follows:
an upstream chamber:
a downstream chamber:
in the formula: k is a radical ofmAs matrix permeability, mD; a is the surface area of the core in cm2(ii) a x is an abscissa value in the horizontal direction and is dimensionless; u is the x-axis coordinate of the upstream bottle position; k is a radical offFracture permeability, mD; vuIs the volume of the upstream chamber, cm3；VgIs a gas molar volume of 0.0224m3Per mol; z is a gas deviation factor and is dimensionless; r is a gas constant of 8.314J/(mol.K); t isduIs the upstream and downstream bottle temperature, K; p is a radical ofuThe upstream chamber pressure, MPa; cgIs a gas compression coefficient, MPa-1(ii) a d is the x-axis coordinate of the downstream bottle position; vdIs the volume of the downstream chamber, cm3；pdDownstream chamber pressure, MPa;
initial conditions:
in the formula: x and y are coordinates of a certain point in the system; p is a radical ofmiInitial pressure of matrix, MPa; p is a radical offiThe fracture initiation pressure, MPa; p is a radical ofu0Upstream initial pressure, MPa; p is a radical ofd0Downstream initial pressure, MPa.
S2: obtaining a finite volume discrete format corresponding to the physical model by using a finite volume method, solving a full implicit solving format of the physical model by combining a Newton-Laplacian iterative algorithm, and programming and solving the model; the method specifically comprises the following substeps:
s21: the seepage equation is processed based on the finite volume theory: firstly, dispersing a whole continuous solution domain to obtain a finite volume discrete format; secondly, processing a nonlinear partial differential equation set by adopting a Newton-Laplacian iterative algorithm on the finite volume discrete format to obtain a fully implicit solving format; finally, assembling the research units into an overall balance matrix to obtain an overall balance matrix equation; the method specifically comprises the following steps:
the continuity equation and the motion equation are:
in the formula: rho is gas density under stratum conditions, g/cm3(ii) a v is the gas flow velocity, cm/s;
porosity, dimensionless; k is the permeability, mD; p is pressure, MPa;
the gas density equation is:
in the formula: rhoscIs the gas density under the ground condition, g/cm3；
The unstable seepage equation of the actual gas reservoir can be obtained by combining the vertical type (11) and the formula (12):
the formula (13) contains nonlinear parameters related to pressure, adopts a numerical solution, combines a finite volume theory to solve a seepage partial differential equation, and specifically comprises the following steps:
as shown in fig. 3, the finite volume control unit selects a control volume iacd of a mesh node i, and integrates the diffusion term in equation (13) in the local definition domain:
in the formula: vaThe volume integral upper and lower limit ranges; omega is an integration area; spiIs a matrix of diffusion terms;
the surface integral is processed as a line integral:
in the formula: gamma-shapediIs the upper and lower limit range of the line integral; s is an integration area;
the unit normal vector is defined as:
in the formula: a. b, c and d are four points in the volume respectively, and ad is the distance between two points ad; ya and yc are two vertical coordinates of a and c; xa and xc are two-point abscissa of a and c;
assume that the parameters are linearly distributed within the volume control unit, and:
wherein:
ai＝yj-yh,aj＝yh-yi,ah＝yi-yj
bi＝xh-xj,bj＝xi-xh,bh＝xj-xi
ci＝xjyh-xhyj,cj＝xhyi-xiyh,ch＝xiyj-xjyi
in the formula:
Peis a cell grid pressure matrix;
pi、pj、phthe pressure of three points i, j and h is MPa;
Nlis an interpolation function;
Ni、Nj、Nhan interpolation function of three points of i, j and h is taken;
al、bl、cl、ai、aj、ah、bi、bj、bh、ci、cj、chintermediate parameters defined for ease of calculation;
xi、xj、xh、yi、yj、yhthe horizontal and vertical coordinates of the three points i, j and h are shown;
expanding the control body equation along the boundaries ac and cd, and substituting the control body equation into a two-dimensional operator and a vector expression to obtain:
in the formula: kxxIs the x-direction permeability, mD; kyyIs the y-direction permeability, mD; x is the number ofa、xd、ya、ydA, d are horizontal and vertical coordinates;
the interpolation function definition by equation (18) yields:
Ni+Nj+Nh＝1 (20)
when formula (20) is introduced into formula (19), the expression is obtained as a matrix:
equation (21) is a unit internal flow term expression defining an internal conductivity matrix Tmn：
In the formula: n is a radical ofr、NsAn interpolation function of r, s; r and s are any two combinations of i, j and h.
Bringing equation (18) into equation (22), the finite volume unit iacd internal flow term can be written as:
in the formula: t isimIs a conductivity matrix;
the characteristics of the Delaunay triangular mesh can know that c is the intersection line of the midpoints of three sides, and the areas of 3 control bodies divided by the gravity center of a triangle are equal, namely:
Vi+Vj+Vh＝A/3 (24)
in the formula: vi、Vj、VhThree control units in a triangular mesh;
each Delaunay triangular grid has 3 finite volume control units, and the flow relationships of the three parts are respectively expressed as:
in the formula: f. ofVi、fVj、fVhFlow rates of i, j and h are in cm3/s；Tij、Tih、Tji、Tjh、Thi、ThjIs a conductivity matrix among the three points i, j and h;
writing equation (25) in matrix form yields:
in the formula: fi、Fj、FhIntermediate moments defined for convenience of calculationArraying;
through the derivation, a finite volume balance matrix with physical significance is obtained:
equation (27) is abbreviated as:
in the formula: t iseIs a cell conductivity matrix;
the formula (28) is a conventional gas reservoir finite volume discrete format, and each part has definite physical significance. Wherein part of parameters are related to pressure and need to be solved iteratively, specifically, a Newton-Laprison iterative algorithm is adopted to process a nonlinear partial differential equation set, and the physical significance of the method is that x is usedkThe tangent line of (2) continuously approaches to the real root of a nonlinear partial differential equation in the model, and the basic expression is as follows:
equation (29) is written in matrix form:
and (27) calculating the time step of n +1 by backward difference to construct an implicit format:
in the formula: n and n +1 are iteration times;
equation (31) is in simplified form:
in the formula: n is a radical ofeAccumulating an entry matrix for the cell;
according to the approximation idea of Newton-Laplacian iterative algorithm: p is a radical ofn+1≈pk+1＝pk+δpkEquation (32) can be written as an iterative format:
in the formula: k is the number of iterations;
if the high order infinity is omitted in the calculation process of equation (33), then:
equation (34) is abbreviated as:
similarly, the accumulated items are sorted and abbreviated as:
and (3) bringing the formula (35) and the formula (36) into the formula (33) to obtain a full implicit solving format of the conventional gas reservoir:
through the method, the discrete value of the control unit can be obtained by approaching solution, so that the solution of the target seepage equation cannot be completed, and the research unit needs to be further assembled into an overall balance matrix, specifically:
for two cell grids as shown in fig. 4, the corresponding cell stiffness matrix and load vector are:
assembling according to the node relationship can obtain:
in the formula: t is the overall conductivity matrix; f is an intermediate matrix defined for convenient calculation;
equation (37) can be assembled into an overall balanced matrix equation according to the assembly principle of the matrix:
in the formula: p is the overall pressure matrix; n is a total accumulated item matrix; rrIs a residual matrix.
S22: dispersing the adsorption and desorption items according to a finite volume method, introducing a Newton-Laplacian iteration algorithm to obtain a shale core unit discrete iteration format, sorting out corresponding partial derivative matrixes according to a unit conductivity matrix, a unit cumulative item matrix and a unit adsorption item matrix, and combining the three unit partial derivative matrixes to obtain a shale single dense medium model comprehensive discrete equation; processing a source-sink phase in a shale single dense medium model comprehensive discrete equation; the method specifically comprises the following steps:
the dispersion is carried out according to a finite volume method, and the adsorption and desorption item unit discrete matrix can be converted into:
introducing a Newton-Laplacian iterative algorithm:
substituting the formula (43) into a unit discrete equation (41) of a conventional single heavy medium, and finishing to obtain a shale core unit discrete iteration format:
wherein: delta Pe＝(δp1 δp2 δp3)T；Pe＝(p1 p2 p3)T；Re＝(R1 R2 R3)T
In the formula: subscript e represents a unit; veIs a unit adsorption and desorption term matrix; reIs a unit residue matrix; 1. and 2, 3 are three nodes of a triangular mesh.
Cell conductivity matrix:
cell accumulation term matrix:
unit adsorption term matrix:
on the basis of the three unit matrix types (45), the formula (46) and the formula (47), the partial derivative matrix is respectively arranged:
cell conductivity deflector matrix:
element cumulative term partial derivative matrix:
unit adsorption term partial derivative matrix:
by adopting a matrix assembly method, combining three unit partial derivative matrix types (48), an expression (49) and an expression (50), thereby obtaining a shale single dense medium model comprehensive discrete equation:
in the formula: v is an adsorption and desorption term matrix;
when the mesh has NN cells and MM nodes after being discretized, the formula (51):
overall residual matrix Rr＝(R1 R1 … RMM)T；
δP＝(δp1 δp2 … δpMM)T；P＝(p1 p2 … pMM)T；
And implicit processing is adopted for model nonlinear parameters.
The source-sink term in the seepage flow equation exists in the grids contacted by the upstream and downstream chambers, and the source-sink term in the upstream grid is processed:
in the formula: q. q.sgThe flow rate of the unit volume fracture to the matrix is 1/s; vpIs the core pore volume, cm3；k1As first node permeability, mD; p is a radical of1Is the first node pressure, MPa;
source and sink processing in downstream grids:
in the formula: k is a radical ofnxPermeability of the last node, mD; p is a radical ofnxThe last node pressure, MPa.
S23: the flow of gas in a shear-seam-developing shale core is considered a dual media model. Dispersing the channeling item according to a finite volume method, obtaining a shale dual medium model comprehensive discrete equation by adopting a solution method of a single medium model in S22, and simultaneously forming the equation into a synchronous solution format to obtain an implicit solution format of the shale core dual medium seepage differential equation, wherein the implicit solution format specifically comprises the following steps:
when the cross flow between the matrix and the fracture is quasi-steady-state cross flow, the dispersion is carried out according to a finite volume method, and a cross flow item unit discrete matrix can be expressed as:
combining with a solution method of a single dense medium model discrete format, the discrete form of the dual medium model has the following form:
micro-crack system:
in the formula: n is a radical offAccumulating an item matrix for the fracture population;
matrix system:
wherein:
η＝m,f；
in the formula: n is a radical ofmAccumulating the term matrix for the population of matrices;
dual dielectric element conductivity matrix:
in the formula: taking f as the crack and m as the matrix; t isηA conductivity matrix that is a matrix or fracture; dual media element cumulative term matrix:
in the formula: n is a radical ofηThe matrix is a matrix of accumulated terms of the matrix or the crack;
intersystem cell cross-flow term matrix:
in the formula: w is a cross-flow item matrix;
matrix system unit adsorption term matrix:
and (3) adopting a Newton-Laplacian iterative algorithm to iteratively solve the seepage differential equation:
a fracture system;
in the formula: t isfIs a fracture bulk conductivity matrix;
matrix system:
in the formula: t ismA bulk conductivity matrix for the matrix;
the equations (62) and (63) are simplified to obtain the discrete matrix incremental form of the dual-medium system unit:
(Tf (e)+δTf (e)-W(e)-δW(e)-δNf (e))k(δPf (e))k+(W(e)+δW(e))k(δPm (e))k＝-(Rf (e))k (64)
in the formula: rfThe crack residue matrix is used for measuring whether the operation is finished or not; rmThe matrix residue matrix is used for measuring whether the operation is finished or not; and when the residual quantity is less than the set residual quantity, finishing the iterative calculation, wherein the set residual quantity is set in the programming process and can be adjusted according to the calculation time and the precision.
Assembling the unit matrixes (64) (65), thereby obtaining a shale dual medium model comprehensive discrete equation:
(Tf+δTf-W-δW-δNf)k(δPf)k+(W+δW)k(δPm)k＝-(Rf)k (66)
(Tm+δTm-W-δW-δNm-δVE)k(δpm)k+(W+δW)k(δPf)k＝-(Rm)k (67)
when the mesh has a total of NN units and MM nodes after being discretized, the equations (66) and (67) are as follows:
matrix of total residuals
Pressure total increment matrix
Pressure overall matrix
Equations (66) (67) are combined into a synchronous solution format:
wherein:
A11＝Tf+δTf-W-δW-δNf；
A12＝A21＝W+δW；
A22＝Tm+δTm-W-δW-δNm-δVE
in the formula: a. the11、A12、A21、A22An intermediate matrix defined for ease of computation;
the formula (68) is an implicit solving format of the shale core dual medium seepage differential equation;
matrix array
The dimension 2MM by 2MM,
and
dimension 2MM x 1.
S3: performing a pulse attenuation experiment on the shear seam shale core by using a pulse attenuation method experiment device, and measuring pressure and time relation curves of an upstream chamber and a downstream chamber by using experiment data;
the pulse attenuation method experimental device is shown in fig. 5 and comprises a core holder 1, wherein the left end and the right end of the core holder 1 are respectively connected with an upstream chamber 2 and a downstream chamber 3 through pipelines, an upstream pressure sensor 4 and a downstream pressure sensor 5 are respectively arranged on the upstream chamber 2 and the downstream chamber 3, and an air source is connected to the two ends of the core holder 1 through pipelines. The specific experimental process using the pulse attenuation experimental device is as follows:
1) numbering the rock core, testing the length L and the diameter D of the rock core, and checking whether the rock core is damaged;
2) connecting equipment according to a schematic diagram of a pulse attenuation method experimental device shown in FIG. 5, and using the sealing property of a metal core detection device, wherein if the reading of pressure sensors of upstream and downstream chambers is kept unchanged, the sealing property is good;
3) the rock core is loaded into a holder, the rock core is dried and vacuumized, the original phenomena of liquid interference and gas molecular diffusion are avoided, the whole device is arranged in an oven, the temperature of the oven is adjusted to 89 ℃, three valves are opened, a gas cylinder provides initial pressure of about 10MPa for a system through a valve 3, the gas is fully diffused into the rock core within the inflation time, and if the pressures of an upstream chamber and a downstream chamber are reduced simultaneously, the initial pressure is not saturated thoroughly;
4) keeping the temperature of the oven at 60 ℃ constant in the experimental process, closing the first valve 6 and the second valve 7 after the pressure of the system is stable, and slowly pressurizing the upstream cavity by about 30pis through the third valve 8;
5) after the pressure in the upstream cavity is stable, closing the third valve 8, opening the second valve 7, and recording the downstream pressure and the sensor differential pressure in real time by using an automatic acquisition system of the oven;
6) the pressure and time relationship curves of the upstream chamber and the downstream chamber were measured from the recorded experimental data.
S4: the method is characterized by combining a shear fracture shale core dual-medium physical model and adopting a curve fitting method to obtain the matrix permeability and the fracture permeability of the shear fracture shale core, and specifically comprises the following substeps:
s41: according to a method for estimating the sample permeability by combining a continuity equation, a state equation and the like, which is provided by Saghafi et al in 2011 on the basis of researching the core seepage mechanism, matrix permeability and fracture permeability parameters are estimated, and the method specifically comprises the following steps:
the permeability of the matrix is as follows:
in the formula: c is the slope of the dimensionless pressure time curve; l is the core length, cm;
crack permeability:
in the formula: ctIs the system comprehensive compression coefficient, MPa-1；tDcIs the slope of the dimensionless pressure time curve; t is tcConvergence time of experimental data, s;
s42: fitting experimental data by using pre-estimated data and an optimization algorithm to finally obtain matrix permeability and crack permeability; the optimization algorithm is a genetic algorithm, matrix permeability and crack permeability are determined by simultaneously fitting the pressure response curves of the upper and lower bottles, and in order to increase the accuracy of data fitting, a least square method is adopted to establish an upstream and downstream chamber pressure objective function:
in the formula: p is a radical ofeuiActual upstream pressure, MPa; p is a radical ofediActual downstream pressure, MPa.
In a specific embodiment, a pulse attenuation experiment is performed on a shear fracture shale core, the core has a shear fracture at a cross section, and the parameters measured in the experiment are shown in table 1:
TABLE 1 Experimental parameters
| L＝5.00cm | D＝2.50cm | Pu＝10.20MPa | Pd＝10.00MPa | VL＝3.71cm3/cm3 |
| PL＝5.72MPa | T＝60℃ | Φm＝0.05 | Φf＝0.0001 |
In the process of performing a pulse attenuation experiment on the shear fracture shale core, the pressure distribution in the core at 10 seconds is shown in fig. 7, the pressure distribution in the core at 100 seconds is shown in fig. 8, the pressure distribution in the core at 1000 seconds is shown in fig. 9, the pressure and time variation relationship is shown in fig. 10, and it can be seen from fig. 7-10 that: in the displacement of the shear seam shale core, the pressure difference between an upstream chamber and a downstream chamber quickly reaches a first balance state through the shear seam, the pressure at two ends of the core is higher than that of a core matrix, and the pressures are transmitted to the matrix together to reach a second balance state; the seepage flow in the shear joint single medium core is divided into three stages: a shear slot flow section, a matrix flow section, and a balance section.
According to the shear joint shale core pulse attenuation permeability test method, the apparent permeability k of the fracture in the embodiment is measuredfe10.00mD, apparent permeability k of matrixme＝0.0001mD。
Although the present invention has been described with reference to a preferred embodiment, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (8)
1. A shear fracture shale core pulse attenuation permeability test method is characterized by comprising the following steps:
s1: establishing a shear joint shale core dual-medium physical model based on a multi-scale seepage model of gas in a shale shear joint core; the shear fracture shale core dual-medium physical model comprises:
seepage equation of the fracture system:
in the formula: k is a radical offeFracture apparent permeability, mD; mu.sgIs the gas viscosity, mPas; b isgThe natural gas volume coefficient is dimensionless; p is a radical offFracture pressure, MPa; q. q.smIs the channeling of a unit volume fracture into the matrixAmount, 1/s; t is time, s;
the crack porosity is zero dimension;
matrix system seepage equation:
in the formula: k is a radical ofmeAs the apparent permeability of the matrix, mD; p is a radical ofmIs the matrix pressure, MPa;
the porosity of the matrix is dimensionless; q. q.sdesThe unit core adsorption desorption gas amount is 1/s; vEIs the isothermal adsorbed gas volume per unit volume of core, cm3/cm3；
When the cross flow between the matrix and the fracture is quasi-steady state cross flow:
in the formula: alpha is a sparse coefficient and is dimensionless;
when the channeling between the matrix and the fracture is unsteady channeling:
in the formula: r is1Is the radius of the matrix particle, cm; r isbIs the integral radius, cm;
the boundary conditions are as follows:
an upstream chamber:
a downstream chamber:
in the formula: k is a radical ofmAs matrix permeability, mD; a is the surface area of the core in cm2(ii) a x is an abscissa value in the horizontal direction and is dimensionless; u is the x-axis coordinate of the upstream bottle position; k is a radical offFracture permeability, mD; vuIs the volume of the upstream chamber, cm3；VgIs a gas molar volume of 0.0224m3Per mol; z is a gas deviation factor and is dimensionless; r is a gas constant of 8.314J/(mol.K); t isduIs the upstream and downstream bottle temperature, K; p is a radical ofuThe upstream chamber pressure, MPa; cgIs a gas compression coefficient, MPa-1(ii) a d is the x-axis coordinate of the downstream bottle position; vdIs the volume of the downstream chamber, cm3；pdDownstream chamber pressure, MPa;
initial conditions:
in the formula: x and y are coordinates of a certain point in the system; p is a radical ofmiInitial pressure of matrix, MPa; p is a radical offiThe fracture initiation pressure, MPa; p is a radical ofu0Upstream initial pressure, MPa; p is a radical ofd0Downstream initial pressure, MPa;
s2: obtaining a finite volume discrete format corresponding to the physical model by using a finite volume method, solving a full implicit solving format of the physical model by combining a Newton-Laplacian iterative algorithm, and programming and solving the model;
s3: performing a pulse attenuation experiment on the shear seam shale core by using a pulse attenuation method experiment device, and measuring pressure and time relation curves of an upstream chamber and a downstream chamber by using experiment data;
s4: and (3) combining a shear fracture shale core dual-medium physical model, and obtaining the matrix permeability and the fracture permeability of the shear fracture shale core by adopting a curve fitting method.
2. The shear fracture shale core pulse attenuation permeability test method according to claim 1, wherein the multi-scale seepage model in the step S1 is a B-K model, and the mathematical description equation of the B-K model is as follows:
in the formula: k is a radical ofappAs apparent permeability, mD; k is a radical of∞As intrinsic permeability, mD; knIs the knudsen number and has no dimension; alpha is a sparse coefficient and is dimensionless; b is slip coefficient and is dimensionless.
3. The shear fracture shale core pulse attenuation permeability test method as claimed in claim 1, wherein the step S2 of solving the shear fracture shale core dual medium physical model comprises the following sub-steps:
s21: the seepage equation is processed based on the finite volume theory: firstly, dispersing a whole continuous solution domain to obtain a finite volume discrete format; secondly, processing a nonlinear partial differential equation set by adopting a Newton-Laplacian iterative algorithm on the finite volume discrete format to obtain a fully implicit solving format; finally, assembling the research units into an overall balance matrix to obtain an overall balance matrix equation;
s22: dispersing the adsorption and desorption items according to a finite volume method, introducing a Newton-Laplacian iteration algorithm to obtain a shale core unit discrete iteration format, sorting out corresponding partial derivative matrixes according to a unit conductivity matrix, a unit cumulative item matrix and a unit adsorption item matrix, and combining the three unit partial derivative matrixes to obtain a shale single dense medium model comprehensive discrete equation; processing a source-sink phase in a shale single dense medium model comprehensive discrete equation;
s23: dispersing the channeling item according to a finite volume method, obtaining a shale dual medium model comprehensive discrete equation by adopting a solving method of a single medium model in S22, and simultaneously forming the equation into a synchronous solving format to obtain an implicit solving format of the shale core dual medium seepage differential equation.
4. The shear fracture shale core pulse attenuation permeability test method according to claim 3, wherein the finite volume discrete format in step S21 is:
in the formula: t iseIs a cell conductivity matrix; peIs a cell grid pressure matrix;
porosity, dimensionless;
the fully implicit solution format is:
in the formula: k. n is the number of iterations; n is a radical ofeAccumulating an entry matrix for the cell;
the overall balance matrix equation is:
in the formula: t is the overall conductivity matrix; p is the overall pressure matrix; n is a total accumulated item matrix; rrIs a residual matrix.
5. The shear fracture shale core pulse attenuation permeability test method according to claim 3, wherein the shale core unit discrete iteration format in step S22 is as follows:
δPe＝(δp1 δp2 δp3)T；Pe＝(p1 p2 p3)T；Re＝(R1 R2 R3)T
in the formula: subscript e represents a unit; t iseIs a cell conductivity matrix; peIs a cell grid pressure matrix; n is a radical ofeAccumulating an entry matrix for the cell; veIs a unit adsorption and desorption term matrix; reIs a unit residue matrix; k is the number of iterations; 1. 2, 3 are three nodes of a triangular mesh;
the shale single dense medium model comprehensive discrete equation is as follows:
in the formula: t is the overall conductivity matrix; p is the overall pressure matrix; n is a total accumulated item matrix; v is an adsorption and desorption term matrix; rrIs a residual matrix;
when the mesh has NN cells and MM nodes after being discretized, the formula (51):
overall residual matrix Rr＝(R1 R1 … RMM)T；
δP＝(δp1 δp2 … δpMM)T；P＝(p1 p2 … pMM)T；
Processing the source and sink items in the upstream grid:
in the formula: q. q.sgThe flow rate of the unit volume fracture to the matrix is 1/s; vpIs the core pore volume, cm3；k1As first node permeability, mD; p is a radical of1Is the first node pressure, MPa;
processing the source and sink items in the downstream grid:
in the formula: k is a radical ofnxPermeability of the last node, mD; p is a radical ofnxThe first node pressure, MPa.
6. The shear fracture shale core pulse attenuation permeability test method according to claim 3, wherein when the cross flow between the matrix and the fracture is a quasi-steady state cross flow, the shale dual medium model comprehensive discrete equation in the step S23 is as follows:
(Tf+δTf-W-δW-δNf)k(δPf)k+(W+δW)k(δPm)k＝-(Rf)k (66)
(Tm+δTm-W-δW-δNm-δVE)k(δPm)k+(W+δW)k(δPf)k＝-(Rm)k (67)
in the formula: t isfIs a fracture bulk conductivity matrix; t ismA bulk conductivity matrix for the matrix; w is a cross-flow item matrix; n is a radical offAccumulating an item matrix for the fracture population; n is a radical ofmAccumulating the term matrix for the population of matrices; pfIs a fracture overall pressure matrix; pmIs a matrix overall pressure matrix; rfThe crack residue matrix is used for measuring whether the operation is finished or not; rmThe matrix residue matrix is used for measuring whether the operation is finished or not; k is the number of iterations;
when the mesh has a total of NN units and MM nodes after being discretized, the equations (66) and (67) are as follows:
matrix of total residuals
Pressure total increment matrix
Pressure overall matrix
The implicit solving format of the shale core dual medium seepage differential equation is as follows:
A11＝Tf+δTf-W-δW-δNf；
A12＝A21＝W+δW；
A22＝Tm+δTm-W-δW-δNm-δVE
in the formula: a. the11、A12、A21、A22An intermediate matrix defined for ease of computation;
matrix array
The dimension 2MM by 2MM,
and
dimension 2MM x 1.
7. The shear fracture shale core pulse attenuation permeability test method as claimed in claim 1, wherein step S4 includes the following sub-steps:
s41: estimating matrix permeability and crack permeability parameters, specifically:
the permeability of the matrix is as follows:
in the formula: k is a radical ofmAs matrix permeability, mD; c is the slope of the dimensionless pressure time curve; mu.sgIs the gas viscosity, mPas; l is the core length, cm; a is the surface area of the core in cm2；pmIs the matrix pressure, MPa; vuIs the volume of the upstream chamber, cm3；VdIs the volume of the downstream chamber, cm3；
Crack permeability:
in the formula: k is a radical offFracture permeability, mD;
the crack porosity is zero dimension; ctIs the system comprehensive compression coefficient, MPa-1；tDcIs the slope of the dimensionless pressure time curve; t is tcConvergence time of experimental data, s;
s42: and fitting experimental data by using the pre-estimated data and an optimization algorithm to finally obtain the matrix permeability and the crack permeability.
8. The shear fracture shale core pulse attenuation permeability test method as claimed in claim 7, wherein in step S42, the optimization algorithm is a genetic algorithm, matrix permeability and fracture permeability are determined by simultaneously fitting the upstream and downstream bottle pressure response curves, and in order to increase the accuracy of data fitting, a least square method is used to establish the upstream and downstream chamber pressure objective functions:
in the formula: p is a radical ofuiCalculating the obtained upstream pressure, MPa, for the model; p is a radical ofdiCalculating the downstream pressure, MPa, obtained for the model; p is a radical ofeuiActual upstream pressure, MPa; p is a radical ofediActual downstream pressure, MPa.
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