CN104990853B - The Forecasting Methodology of the full rank permeability tensor of porous media - Google Patents
The Forecasting Methodology of the full rank permeability tensor of porous media Download PDFInfo
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Abstract
The invention provides a kind of Forecasting Methodology of the full rank permeability tensor of porous media, the Forecasting Methodology includes：The Controlling model of flow of fluid in porous media is set up, and porous media is divided into multiple grids staggeredly；Fluid is flowed in the first direction in porous media, Controlling model is solved using finite difference calculus, to determine the local velocity component u in grid at difference1i,j、v1i,j；According to local velocity component u1i,j、v1i,j, utilize the first Darcy velocity u for being segmented second-degree parabola numerical integrating and determining fluid1 D、v1 D；Fluid is flowed in porous media along the second direction vertical with first direction, Controlling model is solved using finite difference calculus, to determine the local velocity component u in grid at difference2i,j、v2i,j；According to local velocity component u2i,j、v2i,j, utilize the second Darcy velocity u for being segmented second-degree parabola numerical integrating and determining fluid2 D、v2 D；According to the first Darcy velocity and the second Darcy velocity, the full rank permeability tensor of fluid in porous media is determined.
Description
Technical field
The present invention relates to the prediction of permeability determination technical field, more particularly to a kind of full rank permeability tensor of porous media
Method.
Background technology
Key characteristic permeability is influence porous media (such as oil reservoir) seepage flow performance, dielectric strength, thermophysical property
Important parameter, basic research and the Process Design of the occasions such as oil reservoir migration, fibre strengthening, cement mortar are affected strongly.Especially
In oil reservoir migration, the change of permeability can play an important role across several magnitudes, the characteristics of motion to underground petroleum resources.
Therefore, accurate prediction permeability is always a heat subject of industrial quarters and academia.Study to focus mostly on and surveyed in experiment
Examination, such as it is Free-space measurement technology, coaxial probe technology, cavity resonant technology, transmission line technology and gas Motion Technology, same
E measurement technology is walked, although these technologies have obtained many permeability test datas from different perspectives, often by various factors
Influence and cause result reliability it is not strong.For example, test flow, test pressure, the property for testing fluid, laboratory apparatus
Knot is tested between edge effect and wall effect, the otherness of laboratory technician's operation, the experimental repeatability of same instrument, different instruments
Homogeneity of fruit etc., the fluctuation within a narrow range of these factors can have a significant impact to permeability survey result.This has been resulted in i.e.
Make measurement is the permeability of same porous media, and the data that different researchers obtain often have very big difference, and these data
It is all based on correct experimental principle and operating process draws.Oozed because the unified constant benchmark of shortage carrys out regulation porous media
The actual value of saturating rate so that the numerical value of porous media permeability has very big artificial property, have impact on scientific research and engineering design
Accuracy.
In order to solve the above-mentioned technical problem, diagonal permeability tensor is asked in the prior art generally to adopt with the following method：
First, the Controlling model for setting up fluid in porous media using stable state RANS is as follows：
Wherein, u, v are the local velocity of pure fluid mass, and p is the local pressure of fluid.Equation group is solved to use as schemed
Staggered-mesh shown in 1.
Following discretization is made to formula (1)~(3) using traditional finite difference method：
Wherein, the velocity amplitude on porous media solid is set to 0, and simultaneous formula (4)~(6) are solved, you can obtain porous
The flow velocity of each local location in medium.
By taking Fig. 1 as an example, dash area is solid area in figure, and remainder is fluid mass.Ask for it is porous shown in Fig. 1
In medium during local velocity at difference, above-mentioned traditional finite difference mode does not consider depositing for solid area in porous media
In the influence of fluid motion, difference directly is carried out with the velocity component of Neighbor Points, this leverages the accurate of flow field simulation
Degree.In poiseuille is flowed, the numerical solution and the average deviation of analytic solutions of local velocity's component in the case of different mesh generations
As shown in table 1.
The numerical solution and the average deviation of analytic solutions for local velocity's component that poiseuille is flowed under the different mesh generations of table 1
| Grid number | 20×20 | 40×40 | 60×60 |
| Average deviation | 44.8% | 25.4% | 18.2% |
, it is necessary to try to achieve whole stream using numerical integration method after local velocity's component of porous media flow field is obtained
The average speed of field.Conventional numerical integrations method is rectangular formula method, as shown in Fig. 2 assuming that local flow is local velocity
Its circulation area is multiplied by, then local flow addition is obtained into total flow, whole field average speed is finally given.Total flow expression formula is such as
Under：
I=I1+I2+I3+…+In-2+In-1+In=(v1+v2+v3+…+vn-2+vn-1+vn)Δx (7)
Total flow is only decomposed into the sum of products of local velocity and its circulation area by traditional numerical integration method, does not have
Have in view of influence of the solid wall surface to nowed forming.In fact, because the presence of wall limits the flowing of its neighbouring fluid,
So that the velocity gradient of near wall is larger and small away from the velocity gradient at wall, necessarily cause to flow everywhere when calculating total flow
The weight of speed is different.Conventional numerical integrations method shown in Fig. 2 do not account for this weight difference, therefore it utilizes upper
The accuracy for stating the average speed that numerical integrating is obtained is very low (being shown in Table 2).
The numerical solution for the poiseuille flow field average speed that conventional numerical integrations method is obtained and parsing under the different mesh generations of table 2
The deviation of solution
| Grid number | 20×20 | 40×40 | 60×60 |
| Deviation | 24% | 12% | 9% |
Darcy's law has exactly carried out Rational Simplification with this key parameter of permeability to problem, and description fluid is by underground
The overall handling capacity of porous media, therefore permeability is the basis of reservoir numerical simulation.But permeability survey result is not true
Qualitative also to have certain artificial property from its definition, definition general at present is that permeability is assumed to be into symmetrical isotropism,
Its mathematical form is diagonal tensor, must first go through when so requiring experiment to measure and continuously attempts to find out main infiltration direction,
So that it is determined that the value of diagonal tensor, but this continuously attempt to that larger error can be introduced.Try to achieve porous media flow field
After bulk velocity, using the Classical Equation-Darcy's law for describing reservoir fluid flowing, to the diagonal permeability of porous media
Amount is solved：
Wherein, uD、vDFor overall rate of the fluid by porous media region, i.e. Darcy velocity；kxxAnd kyyFor permeability
Component in x and y directions；gx、gyFor acceleration of gravity x and y directions component；For porous media two
The overall pressure gradient at end；ρ is fluid density；μ is the dynamic viscosity of fluid.However, the traditional form of permeability is diagonal
Amount (counter-diagonal element be 0), that is, assume the symmetrical isotropism of permeability, but any component actually in permeability tensor is all
May not be 0, therefore, although the definition of permeability is simple, it is extremely difficult to carry out Accurate Prediction to it, existing experiment
Method is difficult to accomplish.
The content of the invention
In order to avoid the data reliability brought of experiment measurement it is poor the problem of, obtain various porous medias it is unique it is constant can
The permeability data leaned on is, it is necessary to which the limitation that breakthrough permeability is defined first, using the new ideas of the full rank tensor of permeability, that is, is abandoned
Symmetrical isotropic tradition of permeability is it is assumed that any component in permeability tensor all may not be 0.Secondly, in view of real
The limitation of test amount and the development of computational science, the parameter such as flow process, physical properties of fluids can be held very much in numerical simulation
It is easy to get to control, so as to overcome influence of many uncertain factors to measurement result in previous experiments measurement.Specifically,
Numerical method, which has, separates a variety of physical processes being coupled, the advantage (variables separation studied respectively
Method), and what this separation was not accomplished almost in an experiment.Therefore, numerically modeling is easily formed the measuring method of unified standard,
And obtain uniquely constant predict the outcome.Numerical method is combined with the advanced Detection Techniques such as digital rock can more form height
Effect, economic permeability survey method.
The invention provides a kind of Forecasting Methodology of the full rank permeability tensor of porous media, the Forecasting Methodology includes：
Set up the Controlling model of flow of fluid in porous media using stable state RANS, and by the porous media
It is divided into multiple grids staggeredly；
Fluid is flowed in the first direction in the porous media, the Controlling model is carried out using finite difference calculus
Difference is solved, to determine the local velocity component u in the grid at difference1i,j、v1i,j；
According to local velocity's component u1i,j、v1i,j, porous media is determined using second-degree parabola numerical integrating is segmented
First Darcy velocity u of interior fluid1 D、v1 D；
Fluid is flowed in the porous media along the second direction vertical with the first direction, utilize finite difference
Method carries out difference solution to the Controlling model, to determine the local velocity component u in the grid at difference2i,j、v2i,j；
According to local velocity's component u2i,j、v2i,j, porous media is determined using second-degree parabola numerical integrating is segmented
Second Darcy velocity u of interior fluid2 D、v2 D；
According to the first Darcy velocity u1 D、v1 DAnd the second Darcy velocity u2 D、v2 D, determine fluid in the porous media
Full rank permeability tensor K：
Wherein, μ is the dynamic viscosity of fluid in porous media；ρ is the density of fluid in porous media；G accelerates for gravity
Degree.
In one embodiment, if point to be solved is located at fluid mass and its Neighbor Points of porous media in the grid
Positioned at fluid mass, difference solution is carried out to the Controlling model using finite difference calculus, to determine difference in the grid
The local velocity component u at place1i,j、v1i,j, including：
The Controlling model is carried out using the velocity component at the Neighbor Points in porous media fluid mass
Difference is solved, to determine the local velocity component u in the grid at difference1i,j、v1i,j。
In one embodiment, if point to be solved is located at the fluid mass and its Neighbor Points position of porous media in the grid
In solid area, difference solution is carried out to the Controlling model using finite difference calculus, to determine in the grid at difference
Local velocity component u1i,j、v1i,j, including：
The velocity component of the Neighbor Points positioned at porous media solid area is replaced with into the Liu Gujie away from its nearest neighbours
Velocity component at face；
1/4 grid is moved along the direction perpendicular with the velocity component flowed at liquid/solid interface toward fluid mass；
Difference solution is carried out to the Controlling model, to determine the local velocity component u at the point to be solved1i,j、
v1i,j。
In one embodiment, according to local velocity's component u1i,j、v1i,j, using being segmented second-degree parabola numerical integration
Method determines the first Darcy velocity u of fluid in porous media1 D、v1 D, including：
According to local velocity's component u1i,j、v1i,j, utilize the numerical integrating based on segmentation second-degree parabola, difference
Determine the local flow of fluid in the direction of the x axis and the local flow in the y-axis direction in porous media；
Local flow on the x-axis direction is added to the total flow obtained on x-axis direction, by the y-axis direction
Local flow is added the total flow obtained on y-axis direction；
According to the circulation area of the total flow and porous media on the x-axis direction in the direction of the x axis, and the y-axis
The circulation area of total flow and porous media in the y-axis direction on direction determines the first darcy of fluid in the porous media
Speed u1 D、v1 D。
In one embodiment, according to local velocity's component u1i,j、v1i,j, utilize the number based on segmentation second-degree parabola
It is worth integration method, the local flow of fluid in the direction of the x axis in porous media and local flow in the y-axis direction is determined respectively
Amount, including：
In the grid, a second-degree parabola is determined using three adjacent points；
Definite integral is carried out to the second-degree parabola, to determine the local flow of the fluid in porous media in the direction of the x axis
Amount or local flow in the y-axis direction.
In one embodiment, if point to be solved is located at fluid mass and its Neighbor Points of porous media in the grid
Positioned at fluid mass, difference solution is carried out to the Controlling model using finite difference calculus, to determine difference in the grid
The local velocity component u at place2i,j、v2i,j, including：
The Controlling model is carried out using the velocity component at the Neighbor Points in porous media fluid mass
Difference is solved, to determine the local velocity component u in the grid at difference2i,j、v2i,j。
In one embodiment, if point to be solved is located at the fluid mass and its Neighbor Points position of porous media in the grid
In solid area, difference solution is carried out to the Controlling model using finite difference calculus, to determine in the grid at difference
Local velocity component u2i,j、v2i,j, including：
The velocity component of the Neighbor Points positioned at porous media solid area is replaced with into the Liu Gujie away from its nearest neighbours
Velocity component at face；
1/4 grid is moved along the direction perpendicular with the velocity component flowed at liquid/solid interface toward fluid mass；
Difference solution is carried out to the Controlling model, to determine the local velocity component u at the point to be solved2i,j、
v2i,j。
In one embodiment, according to local velocity's component u2i,j、v2i,j, using being segmented second-degree parabola numerical integration
Method determines the second Darcy velocity u of fluid in porous media2 D、v2 D, including：
According to local velocity's component u2i,j、v2i,j, utilize the numerical integrating based on segmentation second-degree parabola, difference
Determine the local flow of fluid in the direction of the x axis and the local flow in the y-axis direction in porous media；
Local flow on the x-axis direction is added to the total flow obtained on x-axis direction, by the y-axis direction
Local flow is added the total flow obtained on y-axis direction；
According to the circulation area of the total flow and porous media on the x-axis direction in the direction of the x axis, and the y-axis
The circulation area of total flow and porous media in the y-axis direction on direction determines the second darcy of fluid in the porous media
Speed u2 D、v2 D。
In one embodiment, according to local velocity component u2i,j、v2i,j, accumulated using the numerical value based on segmentation second-degree parabola
Point-score, determines the local flow of fluid in the direction of the x axis and local flow in the y-axis direction in porous media, bag respectively
Include：
In the grid, a second-degree parabola is determined using three adjacent points；
Definite integral is carried out to the second-degree parabola, to determine the local flow of the fluid in porous media in the direction of the x axis
Amount or local flow in the y-axis direction.
In one embodiment, if point to be solved is located at the solid area of porous media in the grid, this is waited to ask
The velocity component of solution point is set to 0.
The present invention solve conventional finite difference algorithm and numerical integration algorithm accuracy is relatively low, grid convergence rate it is slow with
And the problem of be difficult to obtain accurately with grid independent solutions, it is applied to the exigent porous media of logarithm value computational accuracy
The calculating of permeability, can obtain high-precision Flow Field Distribution and high-precision permeability data, open and be completely dependent on height
Imitate the new method that numerical computations carry out porous media permeability high-precision forecast, it is to avoid the knot obtained in traditional experimental method
The problem of fruit reliability is not high.
Brief description of the drawings
In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing
There is the accompanying drawing used required in technology description to be briefly described, it should be apparent that, drawings in the following description are only this
Some embodiments of invention, for those of ordinary skill in the art, on the premise of not paying creative work, can be with
Other accompanying drawings are obtained according to these accompanying drawings.
Fig. 1 show the staggered-mesh solved in the prior art used in stable state Stokes governing equation；
Fig. 2 show traditional numerical integrating based on rectangular formula in the prior art and solves embodiment；
Fig. 3 show the schematic flow sheet of the Forecasting Methodology of the full rank permeability tensor of porous media of the embodiment of the present invention；
Fig. 4 show the staggered-mesh used in solution porous media Controlling model of the embodiment of the present invention；
Difference when Fig. 5 show the embodiment of the present invention Neighbor Points used are located at solid area when carrying out difference solution
Schematic flow sheet；
Fig. 6 show the embodiment of the present invention and determines fluid in porous media using second-degree parabola numerical integrating is segmented
The schematic flow sheet of Darcy velocity；
Fig. 7 A~Fig. 7 G show the embodiment of the present invention and are based on segmentation second-degree parabola numerical integrating solution fluid along y-axis
The embodiment of the local circulation area in direction；
Fig. 8 show poiseuille flow schematic diagram of the embodiment of the present invention；
Fig. 9 A~Fig. 9 C show the flow field of the porous media of " one " of embodiment of the present invention character form structure；
Figure 10 A~Figure 10 C show the flow field of the porous media of " ten " of embodiment of the present invention character form structure；
Figure 11 A~Figure 11 C show the flow field of the porous media of the short ladder-type structure of the embodiment of the present invention；
Figure 12 A~Figure 12 C show the flow field of the porous media of the long ladder-type structure of the embodiment of the present invention；
Figure 13 A~Figure 13 C show the flow field of the porous media of " C " of embodiment of the present invention character form structure；
Figure 14 A~Figure 14 C show the flow field of the porous media of " G " of embodiment of the present invention character form structure；
Figure 15 A~Figure 15 C show the flow field of the porous media of irregular structure of the embodiment of the present invention；
Figure 16 show the structural representation of porous media of the embodiment of the present invention.
Embodiment
Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete
Site preparation is described, it is clear that described embodiment is only a part of embodiment of the invention, rather than whole embodiments.It is based on
Embodiment in the present invention, it is every other that those of ordinary skill in the art are obtained under the premise of creative work is not made
Embodiment, belongs to the scope of protection of the invention.
Fig. 3 show the schematic flow sheet of the Forecasting Methodology of the full rank permeability tensor of porous media of the embodiment of the present invention, should
Forecasting Methodology comprises the following steps：
Step 1, the Controlling model for setting up using stable state RANS flow of fluid in porous media, and this is porous
Medium is divided into multiple staggered-meshes.
When it is implemented, the fluid in porous media bypasses the flowing of porous media solid wall surface very under the force of gravity
Slow and convection action is negligible, therefore can be reduced to the incompressible Stokes flow of stable state under this yardstick, profit
The Controlling model of flow of fluid in porous media is set up with stable state RANS, as shown in formula (1)~(3), but with showing
Have unlike technology, in the embodiment of the present invention, u and v in formula (1)~(3) represent the local velocity in porous media respectively
(may be the flow velocity in fluid mass, it is also possible to be the flow velocity in solid area).
To solve the Controlling model of porous media, porous media has been divided into multiple grids staggeredly, such as Fig. 4 by the present invention
It is shown.In Fig. 4, grey grid representation porous media solid area, remaining grid representation fluid mass, black round dot represents net
Point in lattice everywhere, arrow represents the signal direction of velocity component at difference, u, v represent respectively fluid along the x-axis direction, y-axis
Velocity component on direction.In the present embodiment, the velocity component at difference, such as u are distinguished using subscripti,j+3/2Represent net
The velocity component of lattice midpoint (i, j+3/2) place along the x-axis direction.
Step 2, fluid is set to be flowed in the first direction in above-mentioned porous media, using finite difference calculus to the control mould
Type carries out difference solution, to determine local velocity's component in the grid at difference.
For clear explanation technical scheme, the present invention is illustrated so that first direction is x-axis direction as an example, but
The selection of first direction is not limited during specific implementation.
Control fluid is flowed along the x-axis direction in porous media, and the Controlling model is carried out using finite difference calculus
, it is necessary to which it is positioned at fluid mass or positioned at solid area to prejudge point and its Neighbor Points to be solved in grid before difference is solved
Domain.
If point to be solved is located at solid area, without solving, the velocity component of fluid at the point is directly set to 0.
If point to be solved is located at the fluid mass of porous media in the grid and its Neighbor Points also is located at fluid mass,
Then using the velocity component at consecutive points in fluid mass to the Controlling model carry out difference solution, i.e., respectively to formula (1),
(2) carry out discretization with (3) and obtain formula (9), (10) and (11), simultaneous formula (9), (10) and (11) is to solve in the grid not
With local velocity's component at point.
If point to be solved is located at the fluid mass of porous media in the grid and its Neighbor Points is located at solid area, such as
Shown in Fig. 5, comprise the following steps：
S501：The velocity component of the Neighbor Points positioned at porous media solid area is replaced with into the stream away from its nearest neighbours
Velocity component at liquid/solid interface；
S502：1/4 grid is moved along the direction perpendicular with the velocity component flowed at liquid/solid interface toward fluid mass；
S503：Again, difference solution is carried out to the Controlling model, to determine the local velocity minute at the point to be solved
Amount.
Only said in the embodiment of the present invention with carrying out difference solution progress exemplary to the velocity component at several points in Fig. 4
It is bright.For example, to point (i-1, j+1/2), (i, j-1/2), (i, j+1/2), (i, j+1), (i, j+3/2), (i+1/2, j-1), (i+
1/2, j-1/2), (i+1/2, j), (i+1/2, j+1/2), (i+1/2, j+1), (i+1, j+1/2), (i+3/2, speed j) etc.
Spend component carry out difference to ask for during velocity component at above-mentioned point, due to point (i+3/2, j), (i, j+3/2) in solid area,
Therefore by the velocity component u at point (i, j+3/2) placei,j+3/2Replace with the velocity component flowed at liquid/solid interface recently apart from the point
ui,j+1, and along with velocity component ui,j+11/4 grid is moved in perpendicular direction toward fluid mass, i.e., will be poor in formula (10)
Denominator (y used in timesharingj+1-yj) replace with (yj+3/4-yj)；By point (i+3/2, j) place velocity component vi+3/2,jReplace with away from
From the velocity component v at the stream liquid/solid interface of the point recentlyi+1,j, and along with velocity component vi+1,jPerpendicular direction is toward fluid
1/4 grid is moved in region, i.e., by denominator (x used during difference in formula (11)i+1-xi) replace with (xi+3/4-xi), it can obtain
Controlling model at stream liquid/solid interface：
Simultaneous formula (12)~(14) are solved, and can obtain the velocity component at difference in grid.
Step 3, the local velocity's component tried to achieve according to step 2, using being segmented, the determination of second-degree parabola numerical integrating is more
First Darcy velocity u of fluid in the medium of hole1 D、v1 D。
When it is implemented, as shown in fig. 6, comprising the following steps：
S601：According to local velocity's component u1i,j、v1i,j, using the numerical integrating based on segmentation second-degree parabola,
The local flow of fluid in the direction of the x axis and local flow in the y-axis direction in porous media are determined respectively；
When it is implemented, in the grid, a second-degree parabola is determined using three adjacent points, secondly to above-mentioned
Second-degree parabola carries out definite integral, to determine the local flow in the direction of the x axis of the fluid in porous media or in y-axis direction
On local flow.
S602：Local flow on above-mentioned x-axis direction is added to the total flow obtained on x-axis direction, by above-mentioned y-axis direction
On local flow be added and obtain total flow on y-axis direction；
S603：Circulation area according to the total flow and porous media on the x-axis direction in the direction of the x axis respectively, with
And the circulation area of the total flow and porous media on the y-axis direction in the y-axis direction determines fluid in the porous media
The first Darcy velocity u1 D、v1 D。
Fig. 7 A~Fig. 7 G show the embodiment of the present invention and are based on segmentation second-degree parabola numerical integrating solution fluid along y-axis
The embodiment of the local flow in direction.For example, calculating local flow I1When, used x0、x1、x23 points of coefficient it is undetermined two
Secondary parabola (see Fig. 7 A) approximately to replace the speed in this regional area (region between Fig. 7 A second-degree parabolas and x-axis)
Distribution, by this 3 points position and flow velocity, (this 3 points of corresponding flow velocitys along the y-axis direction are respectively 0, v1、v2) can solve
To the parabolical undetermined coefficient.Secondly, the parabola tried to achieve is done into definite integral in dash area can obtain local flow
I1, as shown in Figure 7 B.The integration of other parts is similarly carried out, as shown in Fig. 7 C~7G., can be with using above-mentioned numerical integrating
Obtain each local flow as follows：
…
Formula (15)~(20), which are added, can obtain the total flow expression formula of fluid in the y-axis direction in porous media：
When the total flow in the y-axis direction of fluid in porous media divided by porous media fluid are flowed in the y-axis direction
Total circulation area, Darcy velocity v when fluid flows along the y-axis direction in porous media can be obtained1 D。
Similarly, the local flow of fluid along the x-axis direction, then root first can be asked based on segmentation second-degree parabola numerical integrating
According to the local flow of fluid in porous media along the x-axis direction, the total flow of fluid in the direction of the x axis in porous media is asked for, most
Total flow and its total circulation area according to fluid in the direction of the x axis afterwards, when determining that fluid flows along the x-axis direction in porous media
Darcy velocity u1 D。
Step 4, fluid is flowed in the porous media along the y-axis direction vertical with the x-axis, utilize finite difference
Method carries out difference solution to the Controlling model, to determine local velocity's component in the grid at difference.
When it is implemented, control fluid flows in porous media along the y-axis direction perpendicular with x-axis direction, with step 2
It is similar, it is necessary to prejudge to be solved in grid before difference solution is carried out to the Controlling model using finite difference calculus
Point is positioned at fluid mass or positioned at solid area.
If point to be solved is located at solid area, without solving, the velocity component of fluid at the point is directly set to 0.
If point to be solved is located at the fluid mass of porous media in the grid and its Neighbor Points also is located at fluid mass,
Local velocity's component at the point to be solved then is asked for using formula (9), (10) and (11), to determine difference in the grid
Local velocity's component at place.
If point to be solved is located at the fluid mass of porous media in the grid and its Neighbor Points is located at solid area,
Will positioned at the velocity component of the Neighbor Points of solid area replace with away from its nearest neighbours stream liquid/solid interface at velocity component, and along with
1/4 grid is moved in the perpendicular direction of velocity component at the stream liquid/solid interface toward fluid mass, recycle formula (12)~
(14) difference solution is carried out to the Controlling model, to determine local velocity's component in grid at difference.
Step 5, the local velocity's component tried to achieve according to step 4, using being segmented, the determination of second-degree parabola numerical integrating is more
Second Darcy velocity u of fluid in the medium of hole2 D、v2 D。
When it is implemented, step 5 is similar with step 3, here is omitted.
Step 6, the first Darcy velocity u tried to achieve according to step 31 D、v1 DAnd the second Darcy velocity u that step 5 is tried to achieve2 D、
v2 D, determine the full rank permeability tensor K of fluid in the porous media：
Wherein, μ is the dynamic viscosity of fluid in porous media；ρ is the density of fluid in porous media；G accelerates for gravity
Degree.
The permeability tensor of flow of fluid in porous media is expanded to full rank tensor form by the present invention by diagonal tensor, i.e.,
Do not do any to the counter-diagonal element in permeability tensor it is assumed that the elements in a main diagonal and counter-diagonal element amount to four points
Amount can use arbitrary value, to adapt it to any situation.Therefore, formula (8) is rewritable is：
Wherein, uD、vDFor overall rate of the fluid by porous media region, i.e. Darcy velocity；kxxAnd kyyFor permeability
Component in x and y directions；kyxAnd kyxFor the counter-diagonal element of permeability；gx、gyFor acceleration of gravity x and y directions point
Amount；For the overall pressure gradient at porous media two ends；ρ is fluid density；μ is the dynamic viscosity of fluid.
Due to when it is implemented, employ periodic boundary condition to porous media, therefore the stagnation pressure at porous media two ends
Strong gradientIt is 0, then formula (23) is rewritable is：
If acceleration of gravity is along x-axis positive direction, i.e. gx=g, gy=0, then formula (23) be changed into：
If acceleration of gravity is along y-axis positive direction, i.e. gx=0, gy=g, then formula (23) be changed into：
The full rank permeability tensor K of fluid in porous media can be obtained according to formula (25) and formula (26)：
From formula (27), pass through above-mentioned sample example twice, you can try to achieve by solving four obtained Darcy velocities
Institute in full rank permeability tensor is important.Coordinate system of the full rank permeability tensor tried to achieve using the present invention with being set up is had
Pass, x-axis direction and y-axis direction, which are differed, is set to the minimum and maximum direction of permeability.In order to obtain fluid by porous media knot
The substantive characteristics of structure seepage flow, the present invention is tried to achieve unrelated and only relevant with porous media structure with coordinate system by formula (28)~(30)
Four parameters such as maximum permeability, minimum permeability, main infiltration direction and anisotropy rate.
Using formula (27) to above-mentioned full rank permeability tensor K, Orthogonal Decomposition is carried out：
K=VKeffV-1 (28)
Wherein,For equivalent permeability tensor, the substantive characteristics of full rank permeability tensor is embodied,
kmaxFor maximum permeability, kminFor minimum permeability；It is the matrix of orthogonal eigenvectors composition.
Other specification is defined as follows：
ξ=kmax/kmin (29)
α=arctan (v2,1/v1,1) (30)
Wherein, ξ is anisotropy rate, represents the anisotropic degree of porous media；α is kmaxBetween direction and x-axis direction
Angle, represent main infiltration direction.
Apply the present invention to typical poiseuille flowing, as shown in Figure 8.Fluid is under gravity along the y-axis direction
Make laminar motion, the endless property in y-axis direction is simulated using periodic boundary condition, design parameter is：L=1m, gy=
9.8m/s2.The method provided using the present invention calculates the numerical solution and solution of the full rank permeability tensor of obtained poiseuille flowing
The average deviation of analysis solution is shown in Table 3.
The numerical solution and the average deviation of analytic solutions of the poiseuille flowing obtained under the different mesh generations of table 3 using the present invention
The Forecasting Methodology provided using the present invention can be carried out to the full rank permeability tensor of the porous media of different structure
Prediction, such as Fig. 9 A~Fig. 9 C show the flow field of the porous media of " one " character form structure, and Figure 10 A~Figure 10 C are shown " ten "
The flow field of the porous media of character form structure, Figure 11 A~Figure 11 C show the flow field of the porous media of short ladder-type structure, Figure 12 A
~Figure 12 C show the flow field of the porous media of long ladder-type structure, and Figure 13 A~Figure 13 C show the porous of " C " character form structure
The flow field of medium, Figure 14 A~Figure 14 C show the flow field of the porous media of " G " character form structure；Figure 15 A~Figure 15 C are shown not
The flow field of the porous media of regular texture.The full rank permeability of the porous media of the above-mentioned seven kinds of structures obtained using the present invention
Amount is shown in Table 4.
The full rank permeability tensor result of calculation of the porous media of several typical structures of table 4
The present invention solves traditional finite-difference algorithm and numerical integration algorithm accuracy is relatively low, grid convergence rate is slow
And the problem of be difficult to obtain accurately with grid independent solutions, the full rank permeability of fluid in the porous media that the present invention is provided
The Forecasting Methodology of tensor is applied to the calculating that logarithm value computational accuracy requires higher porous media permeability, can obtain high-precision
The Flow Field Distribution (see Fig. 9~Figure 15) of degree, and high-precision permeability data (being shown in Table 4), open complete complete dependence efficiently number
Value calculates the new method for carrying out porous media permeability high-precision forecast, it is to avoid the result obtained in traditional experimental method can
By property it is not high the problem of.
For the porous media of " one " character form structure shown in Fig. 9 A, the permeability tensor obtained using conventional method and profit
The result of calculation of the full rank permeability tensor obtained with the present invention is shown in Table 5.
The comparison of permeability and conventional method result of calculation that table 5 is obtained using the present invention
For the porous media shown in Figure 16, the permeability tensor obtained using conventional method using the present invention with being obtained
The result of calculation of full rank permeability tensor is shown in Table 6.
The comparison of the permeability that the new algorithm of table 6 is calculated and conventional method result of calculation
It can be seen that no matter traditional algorithm is in which kind of mesh generation from the calculating contrast of above two porous media permeability
Under, its result of calculation has very big error (being shown in Table the data in the 4th row in 5~6), or even distortion completely is (in table 6
200%)；And the Forecasting Methodology that the present invention is provided then remains that higher computational accuracy (is shown in Table the number in the 6th row in 5~6
According to), and increasing with grid number, tended towards stability quickly using the computing permeability result obtained by the present invention, that is,
Say grid convergence rate quickly, obtained reliable results degree is high.As can be seen here, the present invention is to be suitable for calculating porous media infiltration
The multiprecision arithmetic of rate.
The present invention solve conventional finite difference algorithm and numerical integration algorithm accuracy is relatively low, grid convergence rate it is slow with
And the problem of be difficult to obtain accurately with grid independent solutions, it is applied to the exigent porous media of logarithm value computational accuracy
The calculating of permeability, can obtain high-precision Flow Field Distribution and high-precision permeability data, open and be completely dependent on height
Imitate the new method that numerical computations carry out porous media permeability high-precision forecast, it is to avoid the knot obtained in traditional experimental method
The problem of fruit reliability is not high.
Apply specific embodiment in the present invention to be set forth the principle and embodiment of the present invention, above example
Explanation be only intended to help to understand the method and its core concept of the present invention；Simultaneously for those of ordinary skill in the art,
According to the thought of the present invention, it will change in specific embodiments and applications, in summary, in this specification
Appearance should not be construed as limiting the invention.
Claims (10)
1. a kind of Forecasting Methodology of the full rank permeability tensor of porous media, it is characterised in that the Forecasting Methodology includes：
The Controlling model of flow of fluid in porous media is set up using stable state RANS, and the porous media is divided
Into multiple grids staggeredly；
Fluid is flowed in the first direction in the porous media, difference is carried out to the Controlling model using finite difference calculus
Solve, to determine the local velocity component u in the grid at difference1i,j、v1i,j；
According to local velocity's component u1i,j、v1i,j, determine to flow in porous media using second-degree parabola numerical integrating is segmented
First Darcy velocity u of body1 D、v1 D；
Fluid is flowed in the porous media along the second direction vertical with the first direction, utilize finite difference calculus pair
The Controlling model carries out difference solution, to determine the local velocity component u in the grid at difference2i,j、v2i,j；
According to local velocity's component u2i,j、v2i,j, determine to flow in porous media using second-degree parabola numerical integrating is segmented
Second Darcy velocity u of body2 D、v2 D；
According to the first Darcy velocity u1 D、v1 DAnd the second Darcy velocity u2 D、v2 D, determine the complete of fluid in the porous media
Rank permeability tensor K：
Wherein, μ is the dynamic viscosity of fluid in porous media；ρ is the density of fluid in porous media；G is acceleration of gravity.
2. Forecasting Methodology according to claim 1, it is characterised in that if point to be solved is located at porous be situated between in the grid
The fluid mass of matter and its Neighbor Points also is located at fluid mass, carries out difference to the Controlling model using finite difference calculus and asks
Solution, to determine the local velocity component u in the grid at difference1i,j、v1i,j, including：
Difference is carried out to the Controlling model using the velocity component at the Neighbor Points in porous media fluid mass
Solve, to determine the local velocity component u in the grid at difference1i,j、v1i,j。
3. Forecasting Methodology according to claim 1, it is characterised in that if point to be solved is located at porous be situated between in the grid
The fluid mass of matter and its Neighbor Points are located at solid area, and difference solution is carried out to the Controlling model using finite difference calculus,
To determine the local velocity component u in the grid at difference1i,j、v1i,j, including：
It will be replaced with positioned at the velocity component of the Neighbor Points of porous media solid area at the stream liquid/solid interface away from its nearest neighbours
Velocity component；
1/4 grid is moved along the direction perpendicular with the velocity component flowed at liquid/solid interface toward fluid mass；
Difference solution is carried out to the Controlling model, to determine the local velocity component u at the point to be solved1i,j、v1i,j。
4. the Forecasting Methodology according to Claims 2 or 3, it is characterised in that according to local velocity's component u1i,j、v1i,j,
Utilize the first Darcy velocity u for being segmented second-degree parabola numerical integrating and determining fluid in porous media1 D、v1 D, including：
According to local velocity's component u1i,j、v1i,j, using the numerical integrating based on segmentation second-degree parabola, determine respectively
The local flow of fluid in the direction of the x axis and local flow in the y-axis direction in porous media；
Local flow on the x-axis direction is added to the total flow obtained on x-axis direction, by the part on the y-axis direction
Flow is added the total flow obtained on y-axis direction；
According to the circulation area of the total flow and porous media on the x-axis direction in the direction of the x axis, and the y-axis direction
On total flow and porous media circulation area in the y-axis direction determine the first Darcy velocity of fluid in the porous media
u1 D、v1 D。
5. Forecasting Methodology according to claim 4, it is characterised in that according to local velocity's component u1i,j、v1i,j, profit
With the numerical integrating based on segmentation second-degree parabola, the local flow of fluid in the direction of the x axis in porous media is determined respectively
Amount and local flow in the y-axis direction, including：
In the grid, a second-degree parabola is determined using three adjacent points；
Definite integral is carried out to the second-degree parabola, with determine the local flow in the direction of the x axis of the fluid in porous media or
The local flow of person in the y-axis direction.
6. Forecasting Methodology according to claim 1, it is characterised in that if point to be solved is located at porous be situated between in the grid
The fluid mass of matter and its Neighbor Points also is located at fluid mass, carries out difference to the Controlling model using finite difference calculus and asks
Solution, to determine the local velocity component u in the grid at difference2i,j、v2i,j, including：
Difference is carried out to the Controlling model using the velocity component at the Neighbor Points in porous media fluid mass
Solve, to determine the local velocity component u in the grid at difference2i,j、v2i,j。
7. Forecasting Methodology according to claim 1, it is characterised in that if point to be solved is located at porous be situated between in the grid
The fluid mass of matter and its Neighbor Points are located at solid area, and difference solution is carried out to the Controlling model using finite difference calculus,
To determine the local velocity component u in the grid at difference2i,j、v2i,j, including：
It will be replaced with positioned at the velocity component of the Neighbor Points of porous media solid area at the stream liquid/solid interface away from its nearest neighbours
Velocity component；
1/4 grid is moved along the direction perpendicular with the velocity component flowed at liquid/solid interface toward fluid mass；
Difference solution is carried out to the Controlling model, to determine the local velocity component u at the point to be solved2i,j、v2i,j。
8. the Forecasting Methodology according to claim 6 or 7, it is characterised in that according to local velocity's component u2i,j、v2i,j,
Utilize the second Darcy velocity u for being segmented second-degree parabola numerical integrating and determining fluid in porous media2 D、v2 D, including：
According to local velocity's component u2i,j、v2i,j, using the numerical integrating based on segmentation second-degree parabola, determine respectively
The local flow of fluid in the direction of the x axis and local flow in the y-axis direction in porous media；
Local flow on the x-axis direction is added to the total flow obtained on x-axis direction, by the part on the y-axis direction
Flow is added the total flow obtained on y-axis direction；
According to the circulation area of the total flow and porous media on the x-axis direction in the direction of the x axis, and the y-axis direction
On total flow and porous media circulation area in the y-axis direction determine the second Darcy velocity of fluid in the porous media
u2 D、v2 D。
9. Forecasting Methodology according to claim 8, it is characterised in that according to local velocity component u2i,j、v2i,j, utilize base
In the numerical integrating of segmentation second-degree parabola, determine respectively fluid local flow in the direction of the x axis in porous media and
Local flow in the y-axis direction, including：
In the grid, a second-degree parabola is determined using three adjacent points；
Definite integral is carried out to the second-degree parabola, with determine the local flow in the direction of the x axis of the fluid in porous media or
The local flow of person in the y-axis direction.
10. Forecasting Methodology according to claim 1, it is characterised in that if point to be solved is positioned at porous in the grid
The solid area of medium, then be set to 0 by the velocity component of the point to be solved.
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