CN112287545B - Time-space division order conductivity modeling and simulating method for two-phase conductive medium - Google Patents
Time-space division order conductivity modeling and simulating method for two-phase conductive medium Download PDFInfo
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Abstract
The invention relates to a time-space fractional order conductivity modeling and simulating method for a two-phase conductive medium. Introducing the newly constructed conductivity model into an electromagnetic diffusion equation, solving time and space fractional order differential terms in a frequency domain by adopting a finite difference and non-grid combined method, and finally completing numerical simulation of a time domain multi-scale induction-polarization symbiosis effect of an electromagnetic field through frequency-time conversion. The invention has the beneficial effects that the multi-scale time-space fractional order conductivity model of the complex rock structure is provided, the induction-polarization symbiosis effect of the actual geological complex geometric structure can be accurately described, and a theoretical basis is provided for researching the electromagnetic wave propagation mechanism of the complex geological structure.
Description
Technical Field
The invention relates to a time-space division order conductivity modeling and simulating method for a two-phase conductive medium, which is suitable for three-dimensional simulation of time domain electromagnetic multi-scale diffusion, in particular to high-precision three-dimensional numerical simulation of induction effect and polarization effect generated by a complex geometric structure of an actual geodetic medium.
Background
The Time domain Transient electromagnetic method (Time domain Transient electromagnetic methods) utilizes a long wire source or a loop wire source to output Time-varying current to the underground, excites the ground medium to generate an induction electromagnetic field, and detects the electrical property difference and the structure of the underground medium by measuring an electric field or magnetic field signal. The method has the advantages of strong resolving low resistance, good electromagnetic interference resistance and the like, and plays an important role in the fields of engineering geological exploration, resource detection, tectonic zone exploration and the like. As a non-uniform and strong dissipation medium, the underground lithology and physical property of the earth present strong non-uniformity and nonlinearity, particularly resources such as hidden or impregnated polymetallic ores, oil and gas reservoirs, composite oil and gas reservoirs, geothermy and the like, which belong to composite multi-phase conductive media, the multi-scale measurement of complex physical property characteristics or parameters becomes particularly important. The low-resistance and high-polarization abnormity is one of important indication marks for detecting polymetallic ores such as sulfide type and lead, zinc, silver and the like by a geophysical method, and the high-resistance and high-polarization abnormity is an important indication characteristic for distinguishing oil and gas reservoirs. The induction and polarization effects in the multiphase conductive medium exist simultaneously and accompany with each other under the excitation of the alternating field, the induction response can better distinguish the lithology of the stratum, and the polarization response can effectively identify the abnormity of favorable oil and gas reservoirs and metal ores. The method has great significance for detecting and distinguishing economic mineral deposits with economic value or mineralization zones without economic benefit, improving the accuracy and reliability of reservoir parameter prediction and oil gas detection, and deducing and explaining underground mineral resource distribution and energy occurrence state.
At present, the research on polarization effect at home and abroad is mainly based on the calculation of the electromagnetic response value of a complex polarizer of a three-dimensional Cole-Cole model, Marchant et al (2015) performs ohm's law dispersion in a frequency domain, then performs frequency-time transformation by adopting a digital filtering method, and finally calculates the response of an induction field and a polarization field of a porous polarization medium based on an integer order traditional finite difference method; the Mark E.Everett team (2009, 2012 and 2015) derives a frequency domain fractional order diffusion equation of the stochastic medium model based on a controllable source electric induction magnetic method, and calculates the electromagnetic response of the time domain stochastic medium model based on a G-S transformation algorithm. The grids and the like design a conductivity model presenting random characteristics along with space coordinates, perform conductivity assignment through a three-dimensional geometric model, and still adopt an integer order electromagnetic diffusion equation to perform time domain electromagnetic response numerical simulation of a random medium (grids, 2018); at present, electromagnetic simulation based on a random medium only assigns randomly changed conductivity values to space nodes or grids of a geometric model, and does not introduce a fractional order micro-integral term to characterize the spatial scale diffusion characteristics of an electromagnetic field. Wuqiong, Zhao Xue jiao and the like research the characteristics of roughness on the influence of electromagnetic field diffusion and electromagnetic slow diffusion based on a time fractional order electromagnetic diffusion equation, extract roughness information to perform resistivity imaging, and effectively improve the resistivity-depth imaging interpretation precision;
chinese patent CN 104392127a discloses an abnormal diffusion simulation method based on discrete fractional order difference, which adopts a one-dimensional classical diffusion equation describing diffusion phenomenon: defining discrete fractional order difference, discretizing the classical diffusion equation by using the discrete fractional order difference, and carrying out numerical simulation according to the initial boundary condition.
Chinese patent CN 106776478A discloses a discrete fractional order differential method based on step-by-step calculation in anomalous diffusion, which reduces the limit of gamma function calculation, expands the number of points for simulation, and improves the simulation efficiency.
The method discloses a research method for electromagnetic single-scale diffusion at home and abroad. But no relevant research is available at present in the aspect of electromagnetic multi-scale diffusion. The Cole-Cole or GEMTIP model can only represent induction-polarization effect caused by medium frequency dispersion characteristics, and the existing model cannot accurately extract resistivity information for induction effect generated by geometrical structures in oil and gas reservoirs and pore media. Therefore, the multi-scale conductivity model capable of simultaneously representing the induced polarization effect and the medium geometric structure characteristics is provided, time domain electromagnetic multi-scale diffusion simulation is achieved, and the multi-scale conductivity model has great significance for effective interpretation of later-stage measurement data.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a time-space fractional order conductivity modeling and simulation method for a two-phase conductive medium, wherein a space fractional order item is introduced into a conductivity model of the two-phase conductive medium, and a multi-scale time-space fractional order conductivity model is established, wherein the multi-time fractional order item represents a multi-capacitance polarization effect of the medium, and the space fractional order item represents an induction effect generated by a complex geometric structure. Introducing the new conductivity model into an electromagnetic diffusion equation, iteratively solving the frequency domain electromagnetic diffusion equation based on a non-grid method, and finally completing the numerical simulation of the time domain multi-scale induction-polarization symbiosis effect of the electromagnetic field through frequency-time conversion. The invention is realized in such a way that a time-space division order conductivity modeling and simulating method of a two-phase conductive medium comprises the following steps:
1) introducing a space fractional order term into the conductivity model of the two-phase conductive medium, and establishing a multi-scale time-space fractional order conductivity model, wherein the multi-time fractional order term represents a multi-capacitance polarization effect of the medium, and the space fractional order term represents an induction effect generated by a complex geometric structure;
2) substituting the frequency domain conductivity expression into a frequency domain diffusion equation to generate a fractional order Laplace operator;
3) iterative solution is carried out on the frequency domain electromagnetic diffusion equation based on a non-grid method;
4) and completing the numerical simulation of the time domain multi-scale induction-polarization symbiosis effect of the electromagnetic field through frequency-time conversion, storing data, drawing a graph, and analyzing results.
Further, in step 1, the established multi-scale time-space fractional order conductivity model expression is as follows:
in the formula (1), sigma (omega) is frequency domain conductivity, i is an imaginary part, omega is angular frequency, and sigma is0Taking the value of DC conductivity, flVolume fraction of type I particles, MlIs tensor of rock material property, taulIs the time constant of the type I particles, ClThe frequency dispersion coefficient of the type I particles, (iv)αThe operator corresponds to the spatial fractional derivative of the Fourier map, v is a dimensionless geometric factor, and α is the fractal dimension of the anomaly.
Further, in step 2, the conductivity expression (1) is substituted into the diffusion equation of the frequency domain magnetic field
Multiplying both ends of the formula (2) by (i ν)-αComprises the following steps:
wherein
Is the fractional order laplacian in dimensionless coordinates v:
in step 3, discrete approximation is performed on the spatial fractional order differential term in equation (4), taking the x component as an example:
wherein Γ is a gamma function, a is an x-direction integration lower limit, b is an x-direction integration upper limit, τ is an integration variable, and Γ (α) represents a gamma function.
Compared with the prior art, the invention has the beneficial effects that: the multi-scale time-space fractional order conductivity model of the complex rock structure can accurately describe the induction-polarization symbiotic effect of the actual geological complex geometric structure and provide a theoretical basis for researching the electromagnetic wave propagation mechanism of the complex geological structure.
Drawings
FIG. 1 is a flow chart of a time-space division numerical order conductivity modeling and simulation method for a two-phase conductive medium;
FIG. 2 is the effect of fractal dimension on the received induced electromotive force;
FIG. 3 is the effect of polarizability on received induced electromotive force;
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Examples
Referring to fig. 1, a time-space division order conductivity modeling and simulation method for a two-phase conductive medium includes:
1) setting a calculation area as x: -40km to 40km, z: -40km to 40km, nodes in the calculation area are uniformly distributed, the node distance is 800 meters, and the total number of the nodes is 101101 to 10201; dirichlet boundary conditions are adopted for four sides of the calculation region, and the artificial current source is at (0m,0 m).
2) Electromagnetic parameters are set in the whole calculation area, the emission frequency is 2n Hz (n is 0,1,2, 10), and the magnetic permeability is 4 pi 10-7Dielectric constant of 1/36 pi x 10-9The earth conductivity is 0.01S/m and the air conductivity is 1 x 10-6S/m, c is 0.5, time constant is 0.01S, infinite frequency conductivity is 0.1, frozen soil layer is between 40m and 120m, transmitting-receiving distance is 20m
3) The method comprises the steps of setting parameters of a meshless method, including selection of shape function types and setting of shape function parameters and support domain parameters, initializing a large sparse matrix K, wherein the matrix scale is 1020110201, loading a first calculation point and searching nodes in the radius of the support domain, interpolating to obtain a shape function, dispersing fixed integrals by adopting a 4-point Gaussian integral formula, interpolating and summing to obtain fractional order derivatives of the shape function, assigning the shape function result to a corresponding position of the large sparse matrix, selecting the next calculation point until all calculation points are circulated completely, loading Dirichlet boundary conditions and current sources, solving a linear equation set by adopting an LU decomposition method to obtain an electric field value, and iteratively solving a frequency domain electromagnetic diffusion equation based on the meshless method.
4) And completing the numerical simulation of the time domain multi-scale induction-polarization symbiosis effect of the electromagnetic field through frequency-time conversion, storing data, drawing a graph, and analyzing results.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.
Claims (1)
1. A time-space division order conductivity modeling and simulating method of a two-phase conductive medium is characterized by comprising the following steps:
1) introducing a space fractional order term into the conductivity model of the two-phase conductive medium, and establishing a multi-scale time-space fractional order conductivity model, wherein the multi-time fractional order term represents a multi-capacitance polarization effect of the medium, and the space fractional order term represents an induction effect generated by a complex geometric structure;
2) substituting the frequency domain conductivity expression into a frequency domain diffusion equation to generate a fractional order Laplace operator;
3) iterative solution is carried out on the frequency domain electromagnetic diffusion equation based on a non-grid method;
4) completing numerical simulation of time domain multi-scale induction-polarization symbiosis effect of the electromagnetic field through frequency-time conversion, storing data, drawing a graph, and analyzing results;
in step 1, the established multi-scale time-space fractional order conductivity model expression is as follows:
in the formula (1), sigma (omega) is frequency domain conductivity, i is an imaginary part, omega is angular frequency, and sigma is0Taking the value of DC conductivity, f1Is the volume fraction of type 1 particles, f2Is the volume fraction of type 2 particles, M1Tensor of properties of rock material constituted by type 1 particles, M2Tensor of properties, tau, of rock material constituted by type 2 particles1Is the time constant of type 1 particles, τ2Is the time constant of type 2 particles, C1Is the dispersion coefficient of type 1 particles, C2The frequency dispersion coefficient of the type 2 particle, (iv)αThe operator corresponds to the space fractional order derivative of Fourier mapping, v is a dimensionless geometric factor, and alpha is the fractal dimension of the abnormal body;
in step 2, substituting the conductivity expression (1) into the diffusion equation of the frequency domain magnetic field
Wherein
The parameter is Laplace operator, H is magnetic field, i is imaginary part, omega is angular frequency, and the meaning of other parameters is the same as that of the parameter in the formula (1);
multiplying both ends of the formula (2) by (i ν)-αComprises the following steps:
wherein
Is the fractional order laplacian in dimensionless coordinates v:
wherein
Is a partial derivative symbol, and x, y and z represent the axial direction of partial derivative;
in step 3, discrete approximation is performed on the fractional order spatial differentiation term in equation (4), taking the x component as an example:
wherein gamma is gamma function, a is lower integration limit in x direction, b is upper integration limit in x direction, tau is integration variable, H(2)Is the second order partial derivative to the magnetic field.
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| US17/190,445 US20220137252A1 (en) | 2020-10-29 | 2021-03-03 | Space-time fractional conductivity modeling of two-phase conducting media and simulation method thereof |
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