High-precision forward method for electromagnetic field of fracture formation induction logging instrument based on born approximation

Abstract

The application discloses a high-precision forward method for electromagnetic field of a fracture formation induction logging instrument based on Born approximation, and comprises the following steps: 1) calculating electromagnetic field distribution in a formation without fractures; 2) calculating the influence of fractures on original electromagnetic field; and 3) superimposing the value of electromagnetic field in the formation without fractures and the influence of fractures on the original electromagnetic field to obtain an actual electromagnetic field. The method strictly considers the geometric shape of the fractures, and avoids directly carrying out grid division on the fracture area, so that the calculation result of the actual electromagnetic field is more accurate.

Description

Technical Field

[0001] This invention relates to well logging technology, and more particularly to a high-precision forward modeling method for electromagnetic fields in induction logging instruments for fractured formations based on the Born approximation. Background Technology

[0002] The core technology of induction logging forward modeling is electromagnetic field numerical simulation. Typically, when there are no small physical structures such as fractures or pores, the finite element method (FEM) is highly effective for electromagnetic field numerical simulation and has wide applications worldwide. However, it encounters difficulties in fractured formations. Because the scale of fractures is very small compared to the undisturbed formation, if the numerical simulation software strictly follows the Delauny mesh generation criterion, it will generate a very dense mesh, resulting in a huge computational burden. Even with only one fracture in the formation, a typical workstation computer (200GB RAM, 3.5GHz x 24CPU) is already unable to complete the mesh generation, let alone when multiple fractures exist. Theoretically, this scenario is only possible on a supercomputing platform. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a high-precision forward modeling method for electromagnetic fields of induction logging instruments in fractured formations based on the Born approximation, addressing the deficiencies in the existing technology.

[0004] The technical solution adopted by this invention to solve its technical problem is: a high-precision forward modeling method for electromagnetic fields of induction logging instruments in fractured formations based on the Born approximation, comprising the following steps:

[0005] 1) Calculate the electromagnetic field distribution in the formation without cracks;

[0006] 2) Calculate the effect of the crack on the original electromagnetic field;

[0007] 3) The actual electromagnetic field is obtained by superimposing the value of the electromagnetic field in the formation without cracks with the effect of the cracks on the original electromagnetic field.

[0008] According to the above scheme, in step 1), the electromagnetic field distribution in the formation without cracks is calculated as follows:

[0009] Let E (0) The electromagnetic field when the strata are intact and without fractures, thus E (0) satisfy:

[0010]

[0011] in, It is the complex wavenumber of the original strata. r0 and z0 are the coordinates of the cross-section of the transmitting coil; Iδ(r-r0)δ(r-z0) is the current density of the transmitting coil;

[0012] According to the above scheme, in step 2), the influence of the crack on the original electromagnetic field is calculated as follows:

[0013] Let E (i) Let i = 1, 2, 3, ...; represent the effect of the crack on the original electromagnetic field.

[0014] but

[0015] Where α is the change in the complex wavenumber of the fracture relative to the complex wavenumber of the undisturbed formation;

[0016] E (i) (i = 1, 2, 3, ...) can be solved using a set of recursive differential equations; that is:

[0017]

[0018]

[0019]

[0020] Solve for E using equation (5) (0) Then, substitute the result into equation (7.1) to solve for E. (1) Then E (1) Substitute into equation (7.2) to solve for E. (2)

[0021]

[0022] in: satisfy

[0023] In equations (8) and (9), This is called the dyadic Green's function. It is a unit tensor. It consists of three vector functions, which are the electric field distribution functions when the direction of the point AC power source at ρ0 points to the x, y, and z directions;

[0024] When the strata are infinitely homogeneous and isotropic, and there are no wellbore or formation interfaces, the solution to equation (9) is:

[0025]

[0026] Where φ(ρ,ρ0) is a scalar Green's function in free space, satisfying the equation:

[0027]

[0028] If the original area has a complex structure, resulting in no analytical solution for equation (9), a numerical solution can be obtained using finite element software. Then, the solution can be substituted into equation (8) to obtain E. (i) .

[0029] According to the above scheme, step 3) is:

[0030] E = E (0) +E (1) +E (2) +E (3) ...

[0031]

[0032] Where L represents the integration path along the receiving coil, and E is the actual electromagnetic field including the effect of the crack.

[0033] The beneficial effects of this invention are:

[0034] The method of this invention not only strictly considers the geometry of the fracture, but also avoids directly meshing the fracture region; therefore, it will not assign an equivalent value to the fracture-developed stratum, resulting in a very coarse calculation result, nor will it generate distorted mesh cells in the fracture region, resulting in a more accurate calculation result. Attached Figure Description

[0035] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0036] Figure 1 This is a flowchart of a method according to an embodiment of the present invention;

[0037] Figure 2 This is a flowchart illustrating the calculation of the effect of cracks on the original electromagnetic field according to an embodiment of the present invention.

[0038] Figure 3 This is a schematic diagram of the point arrangement on the crack surface according to an embodiment of the present invention;

[0039] Figure 4 This is a schematic diagram of the finite element mesh near the point AC source on the crack surface according to an embodiment of the present invention;

[0040] Figure 5 This is a schematic diagram of the three components of the dyadic Green's function calculated by the finite element method in an embodiment of the present invention;

[0041] Figure 6 This is a schematic diagram of the finite element mesh under multiple cracks in an embodiment of the present invention. Detailed Implementation

[0042] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0043] like Figure 1 As shown, the high-precision forward modeling method for electromagnetic fields of induction logging instruments in fractured formations based on the Born approximation includes the following steps:

[0044] 1) Calculate the electromagnetic field distribution in the formation without cracks;

[0045] 2) Calculate the effect of the crack on the original electromagnetic field;

[0046] 3) The actual electromagnetic field is obtained by superimposing the value of the electromagnetic field in the formation without cracks with the effect of the cracks on the original electromagnetic field.

[0047] The differential equation satisfied by the electromagnetic field is decomposed into two parts: one part is the differential equation without cracks, and the other part is the effect of cracks on the original electromagnetic field.

[0048]

[0049] Where: J=Iδ(r-r0)δ(r-z0) is the current density of the transmitting coil, and r0 and z0 are the coordinates of the cross-section of the transmitting coil.

[0050] make Where K 2 It is the complex wave number. α is the complex wave number of the undisturbed strata, and α is the change in the complex wave number of the fracture relative to the complex wave number of the undisturbed strata.

[0051]

[0052] in: It is the complex wave number of the medium in the crack.

[0053] Then equation (2) becomes:

[0054]

[0055] According to the Born approximation method, let (i = 1, 2, 3, ...), then equation (4) becomes:

[0056] Let E (0) The electromagnetic field in a undisturbed formation without fractures satisfies the following equation:

[0057]

[0058] Then there is

[0059]

[0060] The effect of the crack on the original electromagnetic field can be solved by a set of differential equations.

[0061] E in the summation sign in equation (6) (i) The terms (i = 1, 2, 3, ...) represent the effect of the crack on the original electromagnetic field. It is easy to see from this equation that E (i) (i = 1, 2, 3, ...) can be solved using a set of recursive differential equations. That is:

[0062]

[0063]

[0064]

[0065] …

[0066] As long as the series (i = 1, 2, 3, ...) converges. It is the solution to the electromagnetic field in fractured strata.

[0067] Solve step by step (7.1), (7.2), (7.3), ...

[0068] Solve for E using equation (5) (0) Then, substitute the result into equation (7.1) to solve for E. (1) Then E (1) Substitute into equation (7.2) to solve for E. (2) And so on.

[0069] Based on current calculations, for high resistivity cracks, as long as the total width of the cracks in the crack development section is less than 1 cm, it is only necessary to calculate up to E. (1) This can meet the precision requirements in production.

[0070] According to Green's theorem for vectors, the solutions to equations (7.1), (7.2), (7.3), ... can be expressed in the following form:

[0071]

[0072] in: satisfy

[0073] In equations (8) and (9), This is called the dyadic Green's function. It is a unit tensor. It consists of three vector functions, which are the electric field distribution functions when the direction of the point AC power source at ρ0 points to the x, y, and z directions.

[0074] When the strata are infinitely homogeneous and isotropic, and there are no wellbore or formation interfaces, the solution to equation (9) is:

[0075]

[0076]

[0077] Where φ(ρ,ρ0) is called the scalar Green's function in free space, satisfying the equation:

[0078]

[0079] If the original area has a complex structure, resulting in no analytical solution for equation (9), a numerical solution can be obtained using finite element software. Then, the solution can be substituted into equation (8) to obtain E. (i) .

[0080] All E (i) Add the values ​​of (i = 0, 1, 2, 3...) and calculate the induced electromotive force at the receiving coil.

[0081] E = E (0) +E (1) +E (2) +E (3) ...

[0082]

[0083] Where: L represents the integration path along the receiving coil.

[0084] The solution for the dyadic Green's function in step 2) is as follows;

[0085] 2.1) Select a set of points on the crack surface according to certain rules.

[0086] It is recommended to define the crack surface as a 50m × 50m rectangle. Following the principle that points closer to the transmitting coil should have smaller intervals, and points farther from the transmitting coil should have larger intervals, a set of points should be set. The number of points in the vertical direction is M, and the number of points in the horizontal direction is N, for a total of M × N points. The values ​​of M and N can be arbitrarily chosen; the larger the values, the more accurate the calculation results. In this example, both M and N are set to 41.

[0087] 2.2) Select a point and treat it as a point source of the dyadic Green's function;

[0088] Starting from any point, let its coordinates be ρ0. Imagine there is a point AC power source at this location with a source strength of δ(ρ-ρ0).

[0089] 2.3) Determine the dyadic Green's function at this point using the finite element numerical method or analytical method.

[0090] Solve the vector Helmholtz equation (9):

[0091]

[0092] The boundary conditions for this equation are the same as those for the undisturbed formation without fractures. That is, they satisfy the electromagnetic field continuity boundary conditions at the wellbore surface or formation interface surface.

[0093]

[0094] in, This is the normal vector of the boundary surface;

[0095] If the boundary structure is too complex, for example, if both the wellbore and the formation interface exist in the undisturbed formation, then equation (9) can be solved by the finite element numerical method.

[0096] 2.4) Find value

[0097] Multiply the E at the currently selected AC source ρ0 by the dyadic Green's function obtained in the previous step. (i-1) The value is fine. It should be noted that... It is a dual arrow, E (i-1) It is a vector. Therefore It is a right dot product of a dyadic vector and a vector (not commutative). The result is... The value is a vector.

[0098] 2.5) Select the next point

[0099] If not all points on the crack surface have been calculated... If the value is not specified, then select the next one. Then return to step 2.2). If the above process is completed, proceed to the next step.

[0100] 2.6) Calculate the integral

[0101] The integration region is the interior of the crack. Since the scale of the crack surface is much larger than its thickness, it is more convenient to calculate the integral value of the crack surface first, and then integrate over the crack width. That is:

[0102]

[0103] Where: V' is the internal region of the crack, S is the surface of the crack, H is the direction perpendicular to the surface of the crack, and h is the width of the crack.

[0104] Due to the calculation in the previous step The points located on the crack surface are discrete, therefore they need to be transformed into a smooth, continuous function before integration can be performed. The PCHIP interpolation method can be used to calculate all the points in step 5. The value is transformed into a continuous and smooth binary vector function F(x,y) on the crack surface. Substituting this interpolation function into equation (14), we get...

[0105]

[0106] The calculation process of equation (14) should use the numerical integration method. The surface integral can be obtained by trigonometric isoparametric element integration, and the integral in the H direction can be obtained by the second-order Simpson formula.

[0107] An example of a crack (finite element numerical solution)

[0108] 1. Mesh generation

[0109] When using the method of this invention, it is only necessary to calculate the dyadic Green's function at different locations within the crack region. Thus, only mesh refinement is needed near the point AC power source; the crack itself does not need to appear in the meshed model. Figure 5 ).

[0110] The red dot indicates the currently selected AC power source.

[0111] The black dashed line indicates the crack area.

[0112] 2. Solve equation (9)

[0113] The following is the solution to equation (9) obtained using the finite element method. The electric field distribution in the stratum is represented by g when the direction of the AC power source at point x, y, and z, respectively. (x) (ρ,ρ0), g (y) (ρ,ρ0) and g (z) (ρ,ρ0), these three solutions constitute the dyadic Green's function.

[0114] The matrix expression contains three vector functions arranged in rows, namely:

[0115]

[0116] in: This represents the b component of the induced electric field generated when the direction of the point AC source is in direction a. For example, This represents the z-component of the induced electric field generated when the direction of the point AC source points towards the y-axis.

[0117] 3. Calculate the integral

[0118] This value is calculated using an interpolation-based integral formula. As long as... Figure 2 Step 6 can be performed without further explanation.

[0119] An example with multiple cracks (finite element numerical solution)

[0120] The advantages of this method are even more pronounced when multiple cracks exist in the strata and when the crack shapes are complex. The process is repeated for each crack. Figure 2 The steps are the same, and there is still no difficulty in meshing the crack region.

[0121] 1. Mesh generation

[0122] When there are multiple cracks in the stratum and the cracks have complex shapes, it is still only necessary to refine the mesh at the selected points on the cracks, rather than directly meshing the entire crack.

[0123] 2. Solve equation (9)

[0124] The solution method is the same as that for the case of a single crack, and will not be repeated here.

[0125] 3. Calculate the integral

[0126] like Figure 6 In the case of multiple cracks, the integration interval V′ consists of all cracks. Therefore, it is necessary to sum the integral values ​​for each crack to obtain the final result, i.e.:

[0127]

[0128] Where: subscript i represents the i-th crack, V i ′ represents the region within the i-th crack.

[0129] Based on existing analytical solutions for electromagnetic fields, after removing the fractures, the formation structure can be considered as either radially or longitudinally layered media. Therefore, equations (5) and (7.1), (7.2), (7.3), ... all have analytical solutions. Thus, analytical solutions can be used in both radially and longitudinally layered media cases.

[0130] It should be understood that those skilled in the art can make improvements or modifications based on the above description, and all such improvements and modifications should fall within the protection scope of the appended claims.

Claims

1. A high-precision forward modeling method for electromagnetic fields in induction logging instruments for fractured formations based on the Born approximation, characterized in that, Includes the following steps: 1) Calculate the electromagnetic field distribution in the formation without cracks; set up The electromagnetic field when the strata are intact and without fractures, thus satisfy: ； in, It is the complex wavenumber of the original strata. ; and These are the coordinates of the cross-section of the transmitting coil's lead end; The current density of the transmitting coil; 2) Calculate the effect of the crack on the original electromagnetic field; set up The effect of the crack on the original electromagnetic field. i =1,2,3,…; but (1) in, It is the change in the complex wavenumber of the fracture relative to the complex wavenumber of the undisturbed strata; It can be solved using a set of recursive differential equations; that is: （2.1） （2.2） （2.3） Solve using equation (1) Then, substitute the result into equation (2.1) to solve. Then Substitute into equation (2.2) to solve. ; but in, satisfy ; In the formula, For the dyadic Green's function, It is a unit tensor; where, It consists of 3 vector functions, which are respectively when The electric field distribution function when the direction of the point AC power source at a given location points in the x, y, and z directions; 3) The actual electromagnetic field is obtained by superimposing the value of the electromagnetic field in the formation without cracks with the influence of cracks on the original electromagnetic field.

2. The high-precision forward modeling method for electromagnetic fields of induction logging instruments in fractured formations based on the Born approximation according to claim 1, characterized in that, In step 2), the effect of the crack on the original electromagnetic field is calculated, and the solution is as follows: When the strata are infinitely homogeneous and isotropic, and there are no wellbore or formation interfaces, the equations... The solution is: in: Let be a scalar Green's function in free space, satisfying the equation: If the original region has a complex structure, resulting in no analytical solution for the equation, then a numerical solution is obtained using finite element software, and then substituted into the equation in the same way. Find .

3. The high-precision forward modeling method for electromagnetic fields of induction logging instruments in fractured formations based on the Born approximation according to claim 1, characterized in that, Step 3) is Where L represents the integration path along the receiving coil, This represents the actual electromagnetic field including the effects of cracks.

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