CN111474603B - Method and system for detecting conductivity of transverse isotropic rock containing inclined cracks - Google Patents
Method and system for detecting conductivity of transverse isotropic rock containing inclined cracks Download PDFInfo
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Abstract
The invention relates to a method and a system for detecting the conductivity of transverse isotropic rock containing inclined cracks, which comprises the following steps: measuring the conductivity parameter to obtain the conductivity tensor K of the background rock 0 And fracture conductivity tensor K;carrying out CT scanning on transverse isotropic rock containing cracks to obtain the volume content phi of the cracks, the aspect ratio alpha of the cracks and the included angle theta between the cracks and a background isotropic surface; characterizing the shape of the crack through the crack aspect ratio alpha and the included angle theta between the crack and the background isotropic surface, obtaining an oriented crack shape tensor H, and calculating the conductivity contribution tensor A of the crack according to the crack shape tensor H; and acquiring the conductivity tensor of the transverse isotropic rock of the inclined fracture based on the electrical sliding theory, and acquiring the conductivity of the rock according to the conductivity tensor. The fracture is obliquely crossed with the isotropic surface of the background medium at a certain angle, so that the condition of a real stratum containing the fracture can be more met, and the conductivity characteristic of an actual rock can be more truly reflected.
Description
Technical Field
The invention relates to a method and a system for detecting the conductivity of a transverse isotropic rock containing oblique fracture, belonging to the technical field of exploration geophysics.
Background
Electrical prospecting is a common method of evaluating fractured formations. In practical application, a proper electrical rock physical model is required to be selected to establish the connection between the anisotropic conductivity obtained by electrical prospecting measurement and the formation fracture characteristic parameters so as to accurately evaluate the fracture characteristics in the formation. A plurality of electrical rock physical models are established to research the electrical properties of the rock containing the cracks. Studies by Hashin and Shtrikman show that the electrical conductivity, thermal conductivity and permeability of the medium have similar mathematical expressions, so that the expression for the electrical conductivity can be obtained according to a calculation method for the thermal conductivity of the rock containing the cracks. Hatta and Taya investigated the thermal conductivity properties of isotropic rock containing ellipsoidal fractures and gave a closed form of the thermal conductivity solution in the case of parallel fractures. Shafir and Kachanov introduced the Oriented Distribution Function (ODF) to study the thermal conductivity properties of isotropic rocks containing multiple distributed fractures. However, the above models all assume that the background medium of the fractured rock is isotropic, unlike the background medium of the actual fractured formation which exhibits significant transverse isotropy. For the transverse isotropic rock containing the ellipsoidal cracks, giraud et al deduces an expression of effective thermal conductivity when the cracks are parallel to the background isotropic face by solving a Green function containing a single transverse isotropic background of the ellipsoidal cracks. Giraud et al subsequently investigated the electrical properties of transversely isotropic rock containing fractures and gave a closed solution of the effective conductivity in both the special cases of fractures parallel and perpendicular to the isotropic face of the background rock. Although Giraud et al have been able to characterize well the electrical properties of transversely isotropic rock with specially distributed fractures, in actual fractured formations, affected by complex geological effects, the fractures are not generally parallel to the isotropic plane, but rather are oblique to the isotropic plane of the formation at an angle. Therefore, the existing rock conductivity model is difficult to truly reflect the conductivity characteristics of the actual rock.
Disclosure of Invention
In view of the defects of the prior art, the invention aims to provide a method and a system for detecting conductivity of transverse isotropic rock containing inclined cracks, wherein the cracks are obliquely crossed with the isotropic surface of a background medium at a certain angle, so that the method and the system can better meet the condition of a real stratum containing cracks and can more truly reflect the conductivity characteristics of actual rock.
In order to achieve the aim, the invention provides a method for detecting the conductivity of transversely isotropic rock containing inclined cracks, which comprises the following steps: s1, measuring conductivity parameters to obtain background rock conductivity tensor K 0 And fracture conductivity tensor K * (ii) a S2, carrying out CT scanning on transverse isotropic rock containing cracks to obtain the volume content phi of the cracks, the aspect ratio alpha of the cracks and the included angle theta between the cracks and a background isotropic surface; s3, characterizing the shape of the crack through the crack aspect ratio alpha and the included angle theta between the crack and the background isotropic surface, obtaining an oriented crack shape tensor H, and calculating the conductivity contribution tensor A of the crack according to the crack shape tensor H, wherein the formula of the conductivity contribution tensor A is as follows:
A＝(I+P(K * -K 0 )) -1
wherein I is a second order unit tensor; p is the Hill tensor; s4, according to the conductivity tensor K of the background rock 0 Fracture conductivity tensor K * The volume content phi of the crack and the conductivity contribution tensor A of the crack are obtained on the basis of an electrical linear sliding theory, the conductivity tensor of the transverse isotropic rock containing the inclined crack is obtained, and the conductivity of the rock is obtained according to the conductivity tensor.
Further, the transverse isotropic rock conductivity tensor K of the inclined fracture is:
wherein k is x Horizontal conductivity of transverse isotropic background rock without cracks; k is a radical of z Vertical conductivity, k, of transversely isotropic background rock without cracks * Is the conductivity of the formation water in the fracture, phi is the volume content of the fracture, M 11 、M 22 、M 33 、M 23 And M 32 Are the non-zero elements in the inverse tensor M, which are both conductivity contribution tensors a.
Further, the inverse tensor M non-zero element of the conductivity contribution tensor A is calculated as:
wherein, T 22 、T 23 、T 32 And T 33 Are all elements of the transformed fracture shape tensor T, T 2 And t 3 Is the eigenvalue of the transformed fracture shape tensor T, λ 1 、λ 2 And λ 3 Are the diagonal elements of the Eshelby conduction tensor.
Further, the fracture shape tensor T is converted from the fracture shape tensor H and the conductivity tensor K of the background rock 0 Obtained by the following formula:
wherein the upper right-hand T in the formula represents a transposition operation; t is the tensor of the shape of the converted crack, and the calculation formula is as follows:
further, the eigenvalue T of the fracture shape tensor T is converted 1 、t 2 And t 3 The calculation formula of (2) is as follows:
t 1 ＝T 11 ，
wherein ε (x) is a Heaviside function, which can be expressed as
Further, a unit feature vector q corresponding to the fracture shape tensor T is converted 1 、q 2 And q is 3 ：
Further, the conductivity contribution tensor A has the formula:
A＝(I+P(K * -K 0 )) -1
wherein I is a second order unit tensor; p is the hilt tensor, and the hilt tensor has the formula:
wherein λ is 1 、λ 2 And λ 3 Are the diagonal elements of the Eshelby conduction tensor.
Further, λ 1 、λ 2 And λ 3 The expression of (c) is:
when t is 1 >t 2 >t 3 When the temperature of the water is higher than the set temperature,
λ 2 ＝1-λ 1 -λ 3 ，
wherein F and E are respectively:
when t is 1 >t 2 ＝t 3 When the temperature of the water is higher than the set temperature,
λ 1 ＝1-2λ 2 ；
when t is 1 ＝t 2 >t 3 When the temperature of the water is higher than the set temperature,
λ 3 ＝1-2λ 1 ；
when t is 1 ＝t 2 ＝t 3 When the temperature of the water is higher than the set temperature,
further, the fracture shape tensor H is:
the invention also discloses a transverse isotropic rock conductivity detection system containing the inclined cracks, which comprises the following steps: the conductivity tensor acquisition module is used for measuring the conductivity parameters to acquire the conductivity tensor K of the background rock 0 And fracture conductivity tensor K * (ii) a The crack parameter acquisition module is used for carrying out CT scanning on transverse isotropic rock containing cracks to obtain the volume content phi of the cracks, the aspect ratio alpha of the cracks and the included angle theta between the cracks and a background isotropic face; the conductivity contribution tensor acquisition module is used for representing the shape of the crack through the crack aspect ratio alpha and the included angle theta between the crack and the background isotropic surface, acquiring an oriented crack shape tensor H, and calculating a conductivity contribution tensor A of the crack according to the crack shape tensor H; a conductivity tensor acquisition module for acquiring the conductivity tensor K of the background rock 0 Fracture conductivity tensor K * The volume content phi of the crack and the conductivity contribution tensor A of the crack are obtained, the conductivity tensor of the transverse isotropic rock of the inclined crack is obtained on the basis of the electrical linear sliding theory, and the conductivity of the rock is obtained according to the conductivity tensor.
Due to the adoption of the technical scheme, the invention has the following advantages:
the method only considers the cracks parallel or vertical to the isotropic surface aiming at the electric rock physical model of the rock containing the cracks, and in the actual fractured stratum, due to the influence of the ground stress, the cracks are not parallel to the isotropic surface generally but are oblique to the isotropic surface at a certain angle.
Aiming at the defects of the existing electrical rock physical model containing a crack medium, the conductivity calculation model of the transverse isotropic rock containing the inclined coin-shaped crack is deduced based on the transverse isotropic background medium and the coin-shaped crack which is obliquely crossed with an isotropic surface and has a small aspect ratio based on the real characteristics of the crack-containing rock. The comparison result shows that the change of each element of the conductivity tensor obtained by the calculation of the invention along with the inclination angle and the aspect ratio is in accordance with the expectation. The method can more effectively predict the electrical properties of the transverse isotropic rock containing the inclined fractures, and can provide support for electrical prospecting, well logging identification and fracture reservoir evaluation.
Drawings
FIG. 1 is a flow chart of a conductivity detection method for a transverse isotropic rock with an inclined fracture according to an embodiment of the invention;
FIG. 2 is a schematic structural diagram of a transversely isotropic rock with inclined fractures according to an embodiment of the present invention;
FIG. 3 shows a crack aspect ratio of 10 in an embodiment of the invention -4 ，10 -3 ，10 -2 ，10 -1 The change of elements in the conductivity tensor of the transverse isotropic rock containing inclined cracks along with the inclination angle of the cracks, and K is shown in figure 3 (a) 11 A graph that varies with crack inclination angle; FIG. 3 (b) is K 22 A graph that varies with crack inclination angle; FIG. 3 (c) is K 23 A curve graph varying with crack inclination angle; FIG. 3 (d) is K 33 A curve graph varying with crack inclination angle;
FIG. 4 is a graph of the change of non-zero elements of the conductivity tensor of the inclined fracture transverse isotropic rock with the fracture aspect ratio when the fracture inclination angle is 0, pi/8, pi/4, 3 pi/8 and pi/2 in one embodiment of the invention, and FIG. 4 (a) is K 11 A curve graph varying with crack inclination angle; FIG. 4 (b) is K 22 A curve graph varying with crack inclination angle; FIG. 4 (c) is K 23 A graph that varies with crack inclination angle; FIG. 4 (d) is K 33 Graph of change in crack inclination angle.
Detailed Description
The present invention is described in detail by way of specific embodiments in order to better understand the technical direction of the present invention for those skilled in the art. It should be understood, however, that the detailed description is provided for purposes of illustration only and should not be construed to limit the invention. In describing the present invention, it is to be understood that the terminology used is for the purpose of description only and is not intended to be indicative or implied of relative importance.
Example one
The embodiment discloses a method for detecting conductivity of transverse isotropic rock containing inclined fractures, which comprises the following steps of:
s1, measuring a conductivity parameter, wherein the conductivity parameter comprises the horizontal conductivity k of the transverse isotropic background rock without cracks x Perpendicular conductivity k z Conductivity k of formation water in fracture * Obtaining the conductivity tensor K of the background rock 0 And fracture conductivity tensor K * Wherein the background rock conductivity tensor K 0 The formula of (1) is:
fracture conductivity tensor K * The formula of (1) is:
s2, carrying out CT scanning on the transverse isotropic rock containing the cracks to obtain the volume content phi of the cracks, the aspect ratio alpha of the cracks and the included angle theta between the cracks and the background isotropic face. The included angle θ between the crack and the background isotropic surface is the inclination angle of the inclined crack, as shown in fig. 2, the parallel line represents the background isotropic surface, and the background isotropic surface is the transverse tangent plane of the rock. The plane where the ellipse is located is the plane where the crack is located, and the included angle between the plane where the crack is located and the background isotropic face is the included angle theta between the crack and the background isotropic face. The direction of the arrow in the figure is the normal direction of the plane in which the slit is located. The slit in this embodiment is a slanted coin-like slit.
S3, the crack shape is represented through the crack aspect ratio alpha and the included angle theta between the crack and the background isotropic face, and the oriented crack shape tensor H is obtained. The formula of the fracture shape tensor H is:
based on fracture shape tensor H and background rock conductivity tensor K 0 A transformed fracture shape tensor T is obtained by:
t has the following form:
wherein the content of the first and second substances,
after obtaining the transformed fracture shape tensor T, it is necessary to make a decisionConverting the fracture shape tensor T to obtain an eigenvalue T of the converted fracture shape tensor T 1 、t 2 、t 3 And corresponding unit feature vector q 1 、q 2 And q is 3 ：
t 1 ＝T 11 ，
Where ε (x) is a Heaviside function, which may be expressed as
By obtaining the eigenvalues T of the transformed fracture shape tensor T 1 、t 2 And t 3 And corresponding unit feature vector q 1 、q 2 And q is 3 And further obtaining the conductivity contribution tensor A of the fracture:
A＝(I+P(K * -K 0 )) -1
wherein I is a second order unit tensor; p is the hill tensor.
The calculation formula of the hill tensor is as follows:
wherein λ is 1 、λ 2 And λ 3 Are the diagonal elements of the Eshelby conduction tensor.
When t is 1 >t 2 >t 3 When the temperature of the water is higher than the set temperature,
λ 2 ＝1-λ 1 -λ 3 ，
wherein F and E are respectively:
when t is 1 >t 2 ＝t 3 When the utility model is used, the water is discharged,
λ 1 ＝1-2λ 2 ；
when t is 1 ＝t 2 >t 3 When the temperature of the water is higher than the set temperature,
λ 3 ＝1-2λ 1 ；
when t is 1 ＝t 2 ＝t 3 When the temperature of the water is higher than the set temperature,
to calculate the conductivity contribution tensor A of the fracture, the inverse tensor M of A is first calculated. The non-zero elements of the inverse tensor M are as follows:
s4, according to the conductivity tensor K of the background rock 0 Fracture conductivity tensor K * The volume content phi of the crack and the conductivity contribution tensor A of the crack are obtained, the conductivity tensor of the transverse isotropic rock of the inclined crack is obtained on the basis of the electrical linear sliding theory, and the conductivity of the rock is obtained according to the conductivity tensor. The formula of the theory of electrical linear sliding is as follows:
K＝K 0 +φ(K * -K 0 )A
tensor K is used for conductivity of background rock 0 Fracture conductivity tensor K * Substituting the volume content phi of the crack and the conductivity contribution tensor A of the crack into the formula to obtain the conductivity tensor K of the transverse isotropic rock of the inclined crack and the electrical conductivity of the transverse isotropic rock of the inclined crackThe conductivity tensor K is calculated as:
wherein k is x Horizontal conductivity of transverse isotropic background rock without cracks; k is a radical of z Is transversely isotropic background rock vertical conductivity, k, without cracks * Is the conductivity of the formation water in the fracture, phi is the volume content of the fracture, M 11 、M 22 、M 33 、M 23 And M 32 Are elements of the inverse tensor M, which is the conductivity contribution tensor a.
Example two
Based on the same inventive concept, the embodiment discloses a conductivity detection system for a transverse isotropic rock containing an inclined fracture, which comprises:
the conductivity tensor acquisition module is used for measuring the conductivity parameters to acquire the conductivity tensor K of the background rock 0 And fracture conductivity tensor K * ；
The crack parameter acquisition module is used for carrying out CT scanning on transverse isotropic rock containing cracks to obtain the volume content phi of the cracks, the aspect ratio alpha of the cracks and the included angle theta between the cracks and a background isotropic surface;
the conductivity contribution tensor acquisition module is used for representing the shape of the crack through the crack aspect ratio alpha and the included angle theta between the crack and the background isotropic surface, acquiring an oriented crack shape tensor H, and calculating a conductivity contribution tensor A of the crack according to the crack shape tensor H;
a conductivity tensor acquisition module for acquiring the conductivity tensor K of the background rock 0 Fracture conductivity tensor K * The volume content phi of the fracture and the conductivity contribution tensor A of the fracture, the conductivity tensor of the transverse isotropic rock of the inclined fracture is obtained on the basis of the electrical linear sliding theory, and the conductivity of the rock is obtained according to the conductivity tensor.
EXAMPLE III
In order to better illustrate the technical solution of the present invention, this embodiment takes the actual formation rock conductivity as an example for illustration. In this embodiment, the horizontal conductivity k of the background rock is obtained according to the detection result of detecting the rock in the stratum of the certain place x =0.0524S/m, vertical conductivity k z And the volume content of cracks is 0.0025 and is not less than 0.0117S/m.
FIG. 3 shows a crack aspect ratio of 10 -4 ，10 -3 ，10 -2 ，10 -1 The conductivity tensor of the transverse isotropic rock containing the inclined fracture changes along with the inclination angle of the fracture. FIG. 3 (a) is K 11 A graph that varies with crack inclination angle; FIG. 3 (b) is K 22 A graph that varies with crack inclination angle; FIG. 3 (c) is K 23 A graph that varies with crack inclination angle; FIG. 3 (d) is K 33 Graph of change with crack inclination angle.
FIG. 4 shows the variation of conductivity tensor non-zero elements of a transversely isotropic rock with inclined fractures with an inclination angle of 0, pi/8, pi/4, 3 pi/8, pi/2 of the fracture as a function of the fracture aspect ratio, and FIG. 4 (a) shows K 11 A graph that varies with crack inclination angle; FIG. 4 (b) is K 22 A graph that varies with crack inclination angle; FIG. 4 (c) is K 23 A curve graph varying with crack inclination angle; FIG. 4 (d) is K 33 Graph of change in crack inclination angle.
Comparing the detection results in the graph 3 and the graph 4 with the resistivity detected by the actual stratum, the fact that the non-zero elements of the conductivity tensor are consistent with the actual measurement results can be found, the detection results in the invention accord with the expected results, and the fact that the electrical rock physical model provided at this time can accurately calculate the conductivity of the rock containing the inclined fractures is shown.
Aiming at the defects of the existing electrical rock physical model containing a crack medium, the conductivity calculation model of the transverse isotropic rock containing the inclined coin-shaped crack is deduced based on the real characteristics of the crack-containing rock and based on the transverse isotropic background medium and the coin-shaped crack which is obliquely crossed with the isotropic surface and has a small vertical and horizontal ratio, so that the electrical property of the rock containing the inclined crack can be more effectively predicted, and support can be provided for electrical prospecting, well logging identification and evaluation of a crack reservoir.
The above description is only for the specific embodiments of the present application, but the scope of the present application is not limited thereto, and any person skilled in the art can easily conceive of the changes or substitutions within the technical scope of the present application, and shall be covered by the scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.
Claims (7)
1. The method for detecting the conductivity of the transverse isotropic rock containing the inclined fractures is characterized by comprising the following steps of:
s1, measuring the conductivity parameter to obtain the conductivity tensor K of the background rock 0 And fracture conductivity tensor K * ；
S2, carrying out CT scanning on transverse isotropic rock containing cracks to obtain the volume content phi of the cracks, the aspect ratio alpha of the cracks and the included angle theta between the cracks and a background isotropic surface;
s3, characterizing the shape of the crack through the crack aspect ratio alpha and the included angle theta between the crack and the background isotropic surface, obtaining an oriented crack shape tensor H, and calculating a conductivity contribution tensor A of the crack according to the crack shape tensor H:
A＝(I+P(K * -K 0 )) -1
wherein I is a second order unit tensor; p is the Hill tensor;
the calculation formula of the Hill tensor is as follows:
wherein λ is 1 、λ 2 And λ 3 Are diagonal elements of the Eshelby conduction tensor;
unit feature vector q 1 、q 2 And q is 3 ：
Converting the fracture shape tensor T from the fracture shape tensor H and combining the background rock conductivity tensor K 0 Obtained by the following formula:
wherein, T at the upper right corner of the formula represents transposition operation;
the calculation formula of the converted fracture shape tensor T is as follows:
s4, according to the conductivity tensor K of the background rock 0 Fracture conductivity tensor K * The volume content phi of the fracture and the conductivity contribution tensor A of the fracture are obtained, the conductivity tensor of the transverse isotropic rock of the inclined fracture is obtained on the basis of the electrical linear sliding theory, and the conductivity of the rock is obtained according to the conductivity tensor.
2. The method for detecting the conductivity of the transversely isotropic rock with inclined fractures as claimed in claim 1, wherein the conductivity tensor K of the transversely isotropic rock with inclined fractures is as follows:
wherein k is x Horizontal conductivity of transverse isotropic background rock without cracks; k is a radical of z Is transversely isotropic background rock vertical conductivity, k, without cracks * Is the conductivity of the formation water in the fracture, phi is the volume content of the fracture, M 11 、M 22 、M 33 、M 23 And M 32 Are non-zero elements in the inverse tensor M, which is the conductivity contribution tensor A.
3. The method for detecting the conductivity of the transverse isotropic rock with the inclined fractures as claimed in claim 2, wherein the calculation formula of the inverse tensor M nonzero element of the conductivity contribution tensor A is as follows:
wherein, T 22 、T 23 、T 32 And T 33 Are all elements of the transformed fracture shape tensor T, T 2 And t 3 Is the eigenvalue of the transformed fracture shape tensor T, λ 1 、λ 2 And λ 3 Are the diagonal elements of the Eshelby conduction tensor.
4. The method for detecting the conductivity of the transversely isotropic rock with inclined fractures as claimed in claim 1, wherein the eigenvalue T of the transformed fracture shape tensor T 1 、t 2 And t 3 The calculation formula of (2) is as follows:
t 1 ＝T 11 ，
wherein ε (x) is a Heaviside function, expressed as
5. The method for detecting the conductivity of the transversely isotropic rock with inclined fractures as claimed in claim 1, wherein the lambda is 1 、λ 2 And λ 3 The expression of (a) is:
when t is 1 ＞t 2 ＞t 3 When the temperature of the water is higher than the set temperature,
λ 2 ＝1-λ 1 -λ 3 ，
wherein F and E are respectively:
when t is 1 ＞t 2 ＝t 3 When the temperature of the water is higher than the set temperature,
λ 1 ＝1-2λ 2 ；
when t is 1 ＝t 2 ＞t 3 When the temperature of the water is higher than the set temperature,
λ 3 ＝1-2λ 1 ；
when t is 1 ＝t 2 ＝t 3 When the temperature of the water is higher than the set temperature,
6. the method for detecting the conductivity of the transverse isotropic rock with the inclined fractures as claimed in any one of claims 1 to 5, wherein the fracture shape tensor H is as follows:
7. an inclined fracture-containing transversely isotropic rock conductivity detection system, comprising:
the conductivity tensor acquisition module is used for measuring the conductivity parameters to acquire the conductivity tensor K of the background rock 0 And fracture conductivity tensor K * ；
The crack parameter acquisition module is used for carrying out CT scanning on transverse isotropic rock containing cracks to obtain the volume content phi of the cracks, the aspect ratio alpha of the cracks and the included angle theta between the cracks and a background isotropic surface;
and the conductivity contribution tensor acquisition module is used for representing the shape of the crack through the crack aspect ratio alpha and the included angle theta between the crack and the background isotropic surface, acquiring an oriented crack shape tensor H, and calculating a conductivity contribution tensor A of the crack according to the crack shape tensor H:
A＝(I+P(K * -K 0 )) -1
wherein, I is a second order unit tensor; p is the Hill tensor;
the computation formula of the Hill tensor is as follows:
wherein λ is 1 、λ 2 And λ 3 Are the diagonal elements of the Eshelby conduction tensor;
unit feature vector q 1 、q 2 And q is 3 ：
Converting the fracture shape tensor T from the fracture shape tensor H in combination with the background rock conductivity tensor K 0 Obtained by the following formula:
wherein, T at the upper right corner of the formula represents transposition operation;
the calculation formula of the transformation crack shape tensor T is as follows:
a conductivity tensor acquisition module for acquiring the conductivity tensor K of the background rock 0 Fracture conductivity tensor K * Volume fraction of cracks phi and conductivity contribution of cracksTensor A, a transverse isotropic rock conductivity tensor of the inclined fracture is obtained based on an electrical linear sliding theory, and the conductivity of the rock is obtained according to the conductivity tensor.
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