CN109116420B - Method for predicting longitudinal wave velocity and attenuation of pore medium containing fractures - Google Patents

Method for predicting longitudinal wave velocity and attenuation of pore medium containing fractures Download PDF

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CN109116420B
CN109116420B CN201811200996.5A CN201811200996A CN109116420B CN 109116420 B CN109116420 B CN 109116420B CN 201811200996 A CN201811200996 A CN 201811200996A CN 109116420 B CN109116420 B CN 109116420B
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吴春芳
巴晶
张琳
檀文慧
程卫
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Hohai University HHU
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Abstract

The invention provides a method for predicting the longitudinal wave velocity and attenuation of a pore medium containing fractures, and belongs to the field of seismic rock physics. According to the invention, according to the BISQ model principle, the oscillation of the fluid in the fracture in the direction vertical to the wave propagation direction is considered. Starting from the description of two important characteristic parameters of fracture aspect ratio and fracture density, the BISQ model is improved based on the law of conservation of fluid mass, an improved theoretical model (Re-BISQ model for short) is deduced, the characteristics of the improved theoretical model are analyzed by numerical simulation, and experimental data of sandstone in Nanampton, England and compact sandstone in Su Li gas field in Ordos basin are used for verifying the effect of the invention. The result shows that the influence of the internal fracture of the underground rock in the wave propagation process is fully considered in the prediction of the longitudinal wave velocity and the attenuation, and an important basis is provided for reservoir prediction.

Description

Method for predicting longitudinal wave velocity and attenuation of pore medium containing fractures
Technical Field
The invention relates to the field of seismic rock physics, in particular to a method for predicting the longitudinal wave velocity and attenuation of a pore medium containing fractures.
Background
Reservoir media elastic wave propagation theory has undergone a move from single pore media to more complex pore media theory. Research on the wave propagation theory of single-pore media began at the first 50 s of the 20 th century, and Gassmann (1951) discussed the effect of the presence of pore fluid in single-pore media on the elastic properties of the media, without considering the relative motion between the pore fluid and the scaffold. Biot (1956 a; 1956 b; 1962) proposed the elastic wave kinetic equation of a fluid-solid coupled biphasic system, which lays the foundation of theoretical studies on biphasic media and predicts the existence of slow longitudinal waves in biphasic media. However, the Biot theoretical model also has the disadvantage (White, 1975; Mavko and Nur,1975) that it is difficult to explain the phenomena of high dispersion and strong attenuation of the waves (Dvorkin and Nur, 1993). The main reason for this is that the theoretical model assumes that all pores inside the rock are uniform (i.e. single pore medium model), and that the fluid under the excitation of the wave only undergoes pipe-in-pipe laminar flow in the wave propagation direction without other reactions, and these basic assumptions do not correspond to the complex situation inside the actual heterogeneous rock.
To address the above problems, some scholars consider the effect of the micro-flow of the pore fluid on wave propagation. Mavkoand Nur (1975) proposed a jet mechanism based on pore micro-geometry, successfully explaining the phenomena of high frequency dispersion and strong attenuation of waves, but this theory relies heavily on pore micro-geometry and is thus difficult to popularize and apply, thus also causing the cleavage of Biot flow mechanism and jet mechanism, which is not in accordance with physical reality. Dvorkin and Nur (1993) have established a BISQ (Biot/Squick) model based on the assumption that the pores contain saturated fluid, taking into account the Biot flow and the squrit flow in a mechanical model, so that the phenomena of high dispersion and strong attenuation of the waves are well explained. The BISQ model is improved by avoiding the characteristic jet flow length, but the predicted longitudinal wave velocity dispersion and attenuation of the BISQ model are shifted to a high frequency band along with the shift of the rock permeability, which is contrary to the result predicted by the BISQ model and is not consistent with the general knowledge of local fluid flow of scholars.
Pride (2004) indicates that the BISQ model defines pores in rock as being spatially continuous uniformly distributed pores, an assumption that is not consistent with physical reality. Dawn (2011) states that the BISQ theory does not relate to two important parameters in fractured media: fracture density and aspect ratio, and thus dawn derives the wave equation from two important parameters of the fracture. However, the model only increases the fracture parameters and cannot change the dispersion and attenuation of the fast longitudinal wave, so that the model parameters become more complex.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a method for predicting the longitudinal wave velocity and attenuation of a pore medium containing fractures.
The method is based on the BISQ model, combines the density and the aspect ratio characteristics of the fractures in the rock, mainly considers the influence of the coin-shaped fractures on wave propagation, and improves the BISQ model from the perspective of fluid mass conservation, so that the BISQ model is not complicated, and the wave propagation characteristics of the reservoir can be well reflected.
The invention provides a method for predicting the longitudinal wave velocity and attenuation of a pore medium containing fractures, which is characterized by comprising the following steps of:
step S1, characterizing the internal pore structure of the rock;
step S2, characterizing the wave propagation characteristics in the rock with cracks;
step S3, estimating the proportion of the porosity of the rock fractures to the total porosity;
step S4, deducing a Re-BISQ model;
step S5, predicting the longitudinal wave velocity and the inverse quality factor Q by using the Re-BISQ model-1；
Step S6, predicting the longitudinal wave velocity and the inverse quality factor Q by using the BISQ model-1；
Step S7, result analysis.
In step S1, the internal pore structure of the rock is characterized by using the morphology of the rock slice under a microscope.
In step S2, the propagation characteristics of the seismic waves inside the rock are represented by the propagation characteristic image of the elastic waves in the cylinder sponge.
In step S3, an expression of the ratio of fracture porosity to total porosity:
Figure BDA0001829977840000031
phi is the total porosity, phicIs the fracture porosity, D is the ratio of the fracture porosity to the total porosity; the fracture porosity expression isZeta is the aspect ratio of the fissures and ε is the fissure density.
Through steps S4-S6, longitudinal wave velocity and inverse quality factor of the two models BISQ and Re-BISQ predictions are calculated.
The differences between the two models BISQ and Re-BISQ in predicting the longitudinal wave velocity and the inverse quality factor are comparatively analyzed by step S7.
In step S4, in the case of one dimension, the equation for the oscillation of the fluid in the two-phase medium can be obtained as follows:
Figure BDA0001829977840000041
Figure BDA0001829977840000042
where t denotes time, κ and η denote porosity, permeability and fluid viscosity, respectively, u and w denote solid displacement and fluid displacement in the x-direction, respectively, and ρs、ρfAnd ρaRepresenting solids, fluids, and add-on densities; m is the Biot elasticity parameter; p is the fluid pressure;
Figure BDA0001829977840000043
is the pore elasticity parameter, KdryIs a solid skeleton bulk modulus, K0Is the matrix bulk modulus;
when the rock is squeezed, the fluid inside the rock fracture is sprayed to the hard pores, so that a spraying flow is generated, and the following equation can be obtained by combining the conservation of the mass of the fluid:
Figure BDA0001829977840000044
wherein the content of the first and second substances,v is the ratio of the fracture to the total porosity, v is the displacement of the fluid in the direction r; according to David and Zimmerman (2012), the expression of the fracture porosity isZeta is the aspect ratio of the crack, epsilon is the crack density;
the relationship between porosity and framework and fluid is as follows:
Figure BDA0001829977840000051
Figure BDA0001829977840000052
substituting (4) and (5) into (3),
wherein the content of the first and second substances,γ＝α-φ,and c is0Is the speed of sound in the liquid;
through Lagrange's equation and combining Biot (1956) dissipation function, the fluid dissipation function of the oscillation perpendicular to the propagation direction of the longitudinal wave is obtained, namely the fluid dissipation fracture dissipation function relationship is as follows:
Figure BDA0001829977840000056
the r-direction fluid displacement expression and the change form of the fracture fluid pressure are respectively set as follows:
v(x,r,t)＝v0(r)ei(lx-ωt), (8a)
P(x,r,t)＝P0(r)ei(lx-ωt). (8b)
wherein the content of the first and second substances,
Figure BDA0001829977840000059
is the wave number;
substituting the formula (8) into the formula (7), the fluid pressure gradient equation in the vertical longitudinal wave propagation direction is derived as follows:
Figure BDA0001829977840000057
wherein the content of the first and second substances,
Figure BDA0001829977840000058
the solid displacement and the displacement of the fluid in the x direction are set as follows:
u(x,t)＝C1ei(lx-ωt), (10a)
w(x,t)＝C2ei(lx-ωt), (10b)
wherein, C1And C2Is a constant;
substituting equations (7) - (9) into equation (5) yields the following relationship between fluid and pressure:
Figure BDA0001829977840000061
solving the equation using the constant voltage boundary results in:
wherein, J0Is a function of the zero order bessel function,
Figure BDA0001829977840000063
the average fluid pressure can be derived from equations (8a), (8b), (10a) and (10 b):
Figure BDA0001829977840000064
wherein, J1Is a first order Bessel function;
the partial derivative is calculated from the above equation (11), so that the following relation can be obtained:
assuming the relationship:
Figure BDA0001829977840000066
in step S5, the longitudinal wave velocity V is obtainedpAnd attenuation factor α expression;
a1,2＝ωIm(X1,2) (16)
where ω is the angular velocity, ωc＝ηφ/κρf，ρ1＝(1-φ)ρs,ρ2＝φρf，
Expression (16) is expressed in inverse quality factor form as:
Figure BDA0001829977840000071
Figure BDA0001829977840000072
Figure BDA0001829977840000073
Figure BDA0001829977840000074
compared with the prior art, the invention has the following beneficial effects:
a) considering the influence of the cracks in the rock on the wave propagation characteristics;
b) the BISQ theory is combined, and a wave propagation theoretical equation containing the fracture is deduced;
c) through analyzing the experimental data of the rock samples in the two regions, the method provided by the invention is further proved to be capable of reflecting the propagation characteristics of the internal waves of the rock.
Drawings
FIG. 1 is a schematic flow chart of a method for predicting longitudinal wave velocity and attenuation of a porous medium containing fractures according to the present invention;
FIG. 2 is a graph of the wave propagation characteristics characterizing the presence of fractures within the rock, where black represents the jet, white represents the solid skeleton, the cylinder represents the cylindrical sponge assumed by Dvorkin and Nur (1993), the transverse arrows in the cylinder represent the jet direction and are perpendicular to the wave propagation direction, the vertical single arrow represents the Biot flow direction and coincides with the wave propagation direction, and R represents the characteristic jet length;
FIG. 3 is graphs showing the changes of the longitudinal wave velocity (graphs a and c) and the attenuation (graphs b and d) predicted by the three models (Biot, BISQ and Re-BISQ) with frequency, wherein the broken line represents the longitudinal wave velocity or the attenuation value predicted by the Biot model, the double-dashed line represents the longitudinal wave velocity or the attenuation value predicted by the BISQ model, and the chain-dashed line represents the longitudinal wave velocity or the attenuation value predicted by the Re-BISQ model;
FIG. 4 is a graph showing the relationship between permeability and attenuation of conventional sandstone compressional waves in south Amphon, England predicted by two models (BISQ and Re-BISQ), wherein a dotted line represents compressional wave attenuation values experimentally measured by Kliomentos and McCann (1990), a solid line represents compressional wave attenuation values predicted by the Re-BISQ model, and a dotted line represents compressional wave attenuation values predicted by the BISQ model;
fig. 5 is a relation between the velocity and the porosity of the tight sandstone compressional predicted by two models, wherein a dotted line represents the compressional wave attenuation value measured by the experiment (wanda xing, 2016), a solid line represents the compressional wave attenuation value predicted by the Re-BISQ model, and a dotted line represents the compressional wave attenuation value predicted by the BISQ model.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
The method of the invention deduces a wave propagation theoretical equation aiming at the influence of the fracture on the wave propagation, compares the difference of the two models in predicting the velocity and attenuation of the longitudinal wave, and further elaborates the effectiveness of the invention by combining the conventional sandstone and the compact sandstone.
The invention provides a method for predicting the longitudinal wave velocity and attenuation of a pore medium containing fractures, which specifically comprises the following steps as shown in figure 1:
step S1, characterizing the internal pore structure of the rock:
the rock medium observed from the earth's surface layer contains a large number of pores and fissures inside, and the observation of rock slices by microscopy shows that the aspect ratio of the fissures is very small compared to the pores inside the rock, increasing the complexity of the elastic properties of the rock.
Step S2, characterizing the propagation characteristics of the internal wave of the rock containing the cracks:
before introducing the Re-BISQ model, we first recognized a schematic of the model, as shown in figure 2. Dvorkin and Nur (1993) first studied Biot flow and local fluid flow in the same mechanical model, creating an instructive BISQ (Biot/Squrit) model. As shown in fig. 2(a), the actual model considered by Dvorkin and Nur is a cylindrical sponge, and the fluid pressure outside the sponge is always a constant, which we set to zero directly for the convenience of the study. Under the action of plane waves, a characteristic cylinder is selected in the rock, the axial direction of which is parallel to the propagation direction of longitudinal waves, the radius R of which is a characteristic parameter which is an order of magnitude dependent on the rock pore size and is called the characteristic jet length, Dvorkin and Nur (1993) assume that this parameter is a fundamental parameter of the rock, independent of the frequency and the properties of the fluid. Macroscopic Biot flow occurs along the axial direction. When seismic waves press against the rock, the fluid in the pores also oscillates perpendicular to the direction of wave propagation, flowing from the fracture to the hard pores (Mavko and Nur,1975), generating jets. From the established conservation of fluid mass relationship, it can be seen that Dvorkin and Nur (1993) do not distinguish between fractures and pores in rock when calculating the jet flow. In fact, the part of the oscillating fluid is mainly the fluid in the fracture, and the specific flowing condition is shown in fig. 2 (b).
Step S3, estimating the ratio of the porosity of the rock fractures to the total porosity:
the BISQ model does not separate the pores and fractures inside the rock, and considers that the jet flow is generated in the total pores, but actually the jet flow only acts in the fractures, and in order to more accurately characterize the wave propagation characteristic, the ratio of the rock fracture porosity to the total porosity needs to be calculated, and the expression is as follows:
Figure BDA0001829977840000101
according to David and Zimmerman (2012), the expression of the fracture porosity is
Figure BDA0001829977840000102
Zeta is the aspect ratio of the fissures and ε is the fissure density.
Step S4, derivation of Re-BISQ model:
biot (1956a), in the one-dimensional case, the equation for the oscillation of a fluid-containing medium in a two-phase medium can be obtained as follows:
Figure BDA0001829977840000103
Figure BDA0001829977840000104
where t denotes time, κ and η denote porosity, permeability and fluid viscosity, respectively, u and w denote solid displacement and fluid displacement in the x-direction, respectively, and ρs、ρfAnd ρaRepresenting solids, fluids, and add-on densities; m is the Biot elasticity parameter; p is the fluid pressure;
Figure BDA0001829977840000105
is the pore elasticity parameter. Wherein KdryIs a solid skeleton bulk modulus, K0Is the matrix bulk modulus.
When the rock is squeezed, the fluid inside the rock fracture is ejected towards the hard pores, creating a jet (Mavkoand Nur,1975), which, combined with the conservation of fluid mass, yields the following equation:
Figure BDA0001829977840000106
wherein the content of the first and second substances,
Figure BDA0001829977840000107
v is the fraction of the total porosity of the fracture and v is the displacement of the fluid in the direction r. According to David and Zimmerman (2012), the expression of the fracture porosity isZeta is the aspect ratio of the fissures and ε is the fissure density.
Biot (1941), Rice and clean (1976) indicate that the relationship between porosity and scaffold and fluid is as follows:
Figure BDA0001829977840000111
Figure BDA0001829977840000112
substituting (4) and (5) into (3),
wherein the content of the first and second substances,
Figure BDA0001829977840000114
γ＝α-φ,
Figure BDA0001829977840000115
and c is0Is the speed of sound in the liquid. Q and F are intermediate variables only and have no practical meaning.
Equation of dynamics
Through Lagrange's equation and combining Biot (1956) dissipation function, the fluid dissipation function of the oscillation perpendicular to the propagation direction of the longitudinal wave is obtained, namely the fluid dissipation fracture dissipation function relationship is as follows:
Figure BDA0001829977840000116
the r-direction fluid displacement expression and the change form of the fracture fluid pressure are respectively set as follows:
v(x,r,t)＝v0(r)ei(lx-ωt), (8a)
P(x,r,t)＝P0(r)ei(lx-ωt). (8b)
wherein the content of the first and second substances,
Figure BDA0001829977840000117
(both of equations 8a and 8b) is the wave number, i represents the imaginary unit, vttRepresenting one derivative of the displacement of the fluid in the r direction, vtRepresenting the quadratic derivative of the fluid displacement in the r direction. V is0Representing the mode of fluid displacement in the r direction.
Substituting the formula (8) into the formula (7), the fluid pressure gradient equation in the vertical longitudinal wave propagation direction is derived as follows:
Figure BDA0001829977840000121
wherein the content of the first and second substances,
Figure BDA0001829977840000122
is a characteristic angular frequency, P0Is a die of pressure.
Fluid pressure
We believe that the displacement of the solid and the displacement of the fluid in the x-direction are influenced by the average of the pressure and the displacement of the fluid in the r-direction, which is influenced only by the local flow in the spherical Biot flow. Thus, the solid displacement and the displacement of the fluid in the x direction are set to be:
u(x,t)＝C1ei(lx-ωt), (10a)
w(x,t)＝C2ei(lx-ωt), (10b)
wherein, C1And C2Is a constant.
Substituting equations (7) - (9) into equation (5) yields the following relationship between fluid and pressure:
Figure BDA0001829977840000123
solving the equation using the constant voltage boundary results in:
wherein, J0Is a function of the zero order bessel function,
Figure BDA0001829977840000125
(λReis an intermediate variable)
The average fluid pressure can be derived from equations (8a), (8b), (10a) and (10 b):
Figure BDA0001829977840000126
Figure BDA0001829977840000131
wherein, J1Is a first order bessel function.
The partial derivative is calculated from the above equation (11), so that the following relation can be obtained:
Figure BDA0001829977840000132
assuming the relationship:
Figure BDA0001829977840000133
(FReis an intermediate variable)
Step S5, predicting phase velocity and inverse quality factor Q by using Re-BISQ model-1：
Average local fluid pressure PavThe actual fluid pressure kinetic equation (1b) can be substituted for (Dvorkin and Chur, 1993). From the relationship between (Toksoz and Johnston,1981) wave number and wave velocity and attenuation, the velocity (V) of the longitudinal wave is obtainedp) And the expression of attenuation factor (α). The specific derivation process is described in (Dvorkin and Nur,1993) appendix。
Figure BDA0001829977840000134
a1,2＝ωIm(X1,2) (16)
Where ω is the angular velocity, ρ1＝(1-φ)ρs,ρ2＝φρf,ρ1,ρ2Are all density constants.
Expression (16) is expressed in inverse quality factor form as:
Figure BDA0001829977840000135
Figure BDA0001829977840000136
Figure BDA0001829977840000137
Figure BDA0001829977840000141
note: except for parameters with a definite meaning, other parameters are intermediate variables.
Step S6, predicting phase velocity and inverse quality factor Q by using BISQ model-1：
The expressions of the longitudinal wave velocity and the attenuation (i.e., the inverse quality factor) are respectively expressed by expressions (15) and (17), and F in expressions (18) and (19)ReAnd F is replaced.
Step S7, result analysis:
the numerical simulation calculation parameters of the present invention are derived from Berryman (1980). The values of all parameters are respectively phi is 0.15, and the volume modulus K of the solid framework isdry＝16×109Pa, poisson ratio υ 0.15, matrix density ρs＝2650kg/m3，
Matrix bulk modulus K0＝38×109Pa, permeability kappa of 1 × 10-15m2Fluid density ρf＝1000kg/m3Fluid viscosity η ═ 0.001Pa · s, bulk modulus K of the fluidf＝2.25×109Pa, fluid-solid coupling density rhoa＝420kg/m3The characteristic jet length R is 1mm, the fracture aspect ratio ζ is 0.02(Mavko et al, 1998; Dongning et al 2014), and the fracture density ε is 0.15. The velocity and attenuation of longitudinal waves are calculated through steps S4-S5, and a graph is drawn.
FIG. 3 shows the fast longitudinal wave velocity (graph a) and attenuation (graph b), and the slow longitudinal wave velocity (graph c) and attenuation (graph d) as a function of frequency predicted by the Biot model, the BISQ model and the Re-BISQ model. The attenuation peak frequencies of the Re-BISQ model and the BISQ model are equal, and under the frequency, the fast longitudinal wave speed and the attenuation calculated by the Re-BISQ model are both larger than the value calculated by the BISQ model; as shown in fig. 3(b), when the calculation result is satisfied with the angular frequency ω → ∞ as a result of the calculation, the fast longitudinal wave velocity and the attenuation value are the same as those calculated by the Biot model. It can be seen that the fast longitudinal wave velocity values calculated by the two models are not greatly different, the calculated value of the Re-BISQ model is slightly larger than the BISQ model value, and the difference value is not more than 1%. In the low frequency band (10)3-105Hz) the calculated attenuation value of the Re-BISQ model is larger than the calculated value of the BISQ model, namely the attenuation value of the low-frequency band is increased. In the high frequency range (greater than 10)5Hz), the attenuation value of the Re-BISQ model is slightly smaller than the value calculated by the BISQ model, namely the attenuation of a high frequency band is reduced, the value of the attenuation is closer to the attenuation value calculated by the Biot model along with the increase of the frequency, and the Re-BISQ model can distinguish squit attenuation from Biot attenuation. This is mainly because, in the low frequency band, due to the elongated fissures contained in the rock, when the seismic waves press against the rock, the fluid in the pores flows from the fissures towards the hard pores (Mavko and Nur,1975), generating jets. The jet stream causes the rock to produce a greater attenuation in the mesoscale range, i.e. the coin-type attenuation value is greater than the sphere-type attenuation value (Pride, 2004). At high frequencies, the fluid flow within the rock is dominated by macroscopic Biot flow, i.e. the fluid pressure equilibrium within the rock occurs in the wavelength scale range (Pride, 2004). Thus, at high frequencies within the rock, macroscopic Biot flow dominates, and at low frequencies, mesoscopic localization dominatesA stream is an squrit stream. Therefore, the Re-BISQ model is superior to the BISQ model and the Biot model in predicting the internal attenuation of the rock.
When the frequency band of the Re-BISQ model is lower, the calculated slow longitudinal wave speed and attenuation are larger than those of the BISQ model and the Biot model, and the difference between the calculated slow longitudinal wave speed and attenuation and the calculated slow longitudinal wave speed and attenuation is very small. And the Re-BSQ is very close to the calculated value of the BISQ model, and the calculated results of the three models are equal when the frequency is higher.
In order to further verify the effectiveness and the correctness of the method provided by the invention, two different experimental data of conventional sandstone and compact sandstone are respectively selected, and the relationship between the prediction results and the experimental values of the two models is analyzed. The following detailed description of specific embodiments of the present invention is made with reference to the accompanying drawings and the embodiments:
example one
In this example, 10 samples of saturated conventional sandstone samples from southampton, uk were compared with theoretical predicted values:
10 pieces of conventional sandstone with the experimental measurement frequency of 1MHz, the confining pressure of 40MPa and the bulk modulus K of pore fluidf＝2.25×109Pa, fluid density ρf＝1000Kg/m3. The porosity phi is 15 +/-2%, and the permeability is less than 100mD.
Fig. 4 is a graph showing the relationship between the actually measured values (Klimentos and McCann,1990) and the model-predicted longitudinal wave attenuation values, in which the solid line represents the longitudinal wave attenuation values predicted by the Re-BISQ model, the broken line represents the longitudinal wave attenuation values predicted by the BISQ model according to the steps, and the chain line represents the experimentally measured longitudinal wave attenuation values. The prediction result of the Re-BISQ model is closer to the experimental measurement value. Compared with measured values, the prediction result of the method is well matched with experimental data, and the method shows that the method more accurately reflects the attenuation characteristic of the conventional sandstone.
Example two
In this example, the experimental data and the theoretical predicted value of 10 tight sandstone samples in the tondors basin surrige gas field are compared:
experimental setup, experimental conditions, experimental protocol and literature for 10 tight sandstones (wushiyang et al, 2000; wang daxing et al,2006) substantially identical was used. The stratum conditions adopted in the experiment are 105 ℃ and 29 MPa. The experimental measurement frequency was 1 MHz. Bulk modulus K of pore fluidf＝2.25×109Pa, fluid density ρf＝1000Kg/m3。
Fig. 5 shows a schematic diagram of the relationship between the experimentally measured value (wang da xing, 2016) and the model-predicted longitudinal wave attenuation value, in which the solid line represents the Re-BISQ model-predicted longitudinal wave attenuation value, the broken line represents the BISQ model-predicted longitudinal wave attenuation value according to the procedure, and the dot-dash line represents the experimentally measured longitudinal wave attenuation value. The prediction result of the Re-BISQ model is closer to the experimental measurement value. Compared with measured values, the prediction result of the method is well matched with experimental data, and the method shows that the method more accurately reflects the attenuation characteristic of the compact sandstone.
The invention is researched on the basis of the research results of predecessors that the fracture has an important influence (pride, 2004; dawn, 2011) on the propagation of the elastic wave, a wave propagation theoretical equation is deduced, and the influence of the fracture on the wave propagation is analyzed on sandstone and tight sandstone samples in two different areas, so that the influence of rock heterogeneity on wave dispersion and attenuation is further explained.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.

Claims (7)

1. A method for predicting the longitudinal wave velocity and attenuation of a pore medium containing fractures is characterized by comprising the following steps:
step S1, characterizing the internal pore structure of the rock;
step S2, characterizing the wave propagation characteristics in the rock with cracks;
step S3, estimating the proportion of the porosity of the rock fractures to the total porosity;
in step S4, in the case of one dimension, the equation for the oscillation of the fluid in the two-phase medium can be obtained as follows:
Figure FDA0002252054270000011
Figure FDA0002252054270000012
where t denotes time, κ and η denote permeability and fluid viscosity, respectively, u and w denote displacement of solid and fluid in the x-direction, respectively, and ρs、ρfAnd ρaRepresenting solids, fluids, and add-on densities; m is the Biot elasticity parameter; p is the fluid pressure;
Figure FDA0002252054270000013
is the pore elasticity parameter; wherein KdryIs a solid skeleton bulk modulus, K0Is the matrix bulk modulus;
when the rock is squeezed, the fluid inside the rock fracture is sprayed to the hard pores, so that a spraying flow is generated, and the following equation can be obtained by combining the conservation of the mass of the fluid:
Figure FDA0002252054270000014
wherein the content of the first and second substances,
Figure FDA0002252054270000015
the ratio of the total porosity of the fracture, v is the displacement of the fluid in the direction r, and according to David and Zimmerman, the expression for the fracture porosity is
Figure FDA0002252054270000016
Zeta is the aspect ratio of the crack, epsilon is the crack density;
the relationship between porosity and framework and fluid is as follows:
Figure FDA0002252054270000021
substituting (4) and (5) into (3),
Figure FDA0002252054270000023
wherein the content of the first and second substances,and c is0Being the speed of sound in the liquid, Q and F are intermediate variables; through Lagrange's equation and combining with Biot dissipation function, obtain the fluid dissipation function of the oscillation perpendicular to the propagation direction of the longitudinal wave, namely the fluid dissipation function relation is as follows:
Figure FDA0002252054270000025
the r-direction fluid displacement expression and the change form of the fracture fluid pressure are respectively set as follows:
v(x,r,t)＝v0(r)ei(lx-ωt)(8a)
P(x,r,t)＝P0(r)ei(lx-ωt)(8b)
wherein the content of the first and second substances,is the wavenumber, i denotes the imaginary unit, vttRepresenting one derivative of the displacement of the fluid in the r direction, vtRepresenting the secondary derivation of the r-direction fluid displacement; v is0A mode representing a fluid displacement in the r direction;
substituting the formula (8) into the formula (7), the fluid pressure gradient equation in the vertical longitudinal wave propagation direction is derived as follows:
Figure FDA0002252054270000026
wherein the content of the first and second substances,
Figure FDA0002252054270000027
is a characteristic angular frequency, P0A die that is a press;
the solid displacement and the displacement of the fluid in the x direction are set as follows:
u(x,t)＝C1ei(lx-ωt), (10a)
w(x,t)＝C2ei(lx-ωt),(10b)
wherein, C1And C2Is a constant;
substituting equations (7) - (9) into equation (5) yields the following relationship between fluid and pressure:
Figure FDA0002252054270000031
solving the equation using the constant voltage boundary results in:
Figure FDA0002252054270000032
wherein, J0Is a function of the zero order bessel function,
Figure FDA0002252054270000033
λReis an intermediate variable;
the average fluid pressure can be derived from equations (8a), (8b), (10a) and (10 b):
Figure FDA0002252054270000034
wherein, J1Is a first order Bessel function;
the partial derivative is calculated from the above equation (11), so that the following relation can be obtained:
assuming the relationship:
Figure FDA0002252054270000036
FReis an intermediate variable;
step S5, predicting the longitudinal wave velocity and the inverse quality factor Q by using the Re-BISQ model-1；
Step S6, predicting the longitudinal wave velocity and the inverse quality factor Q by using the BISQ model-1；
Step S7, result analysis.
2. The method for predicting the longitudinal wave velocity and the attenuation of the pore medium containing the fractures according to claim 1, wherein in the step S1, the internal pore structure characteristics of the rock are characterized by utilizing the shape of the rock slice under a microscope.
3. The method for predicting the velocity and attenuation of longitudinal waves of a pore medium containing fractures according to claim 1, wherein in step S2, the propagation characteristics of seismic waves inside rocks are characterized by using the propagation characteristic image of elastic waves in a cylinder sponge.
4. The method for predicting the longitudinal wave velocity and attenuation of a pore medium containing fractures as claimed in claim 1, wherein in the step S3, the expression of the ratio of fracture porosity to total porosity is as follows:
Figure FDA0002252054270000041
phi is the total porosity, phicIs the fracture porosity, D is the ratio of the fracture porosity to the total porosity; the fracture porosity expression is
Figure FDA0002252054270000042
Zeta is the aspect ratio of the fissures and ε is the fissure density.
5. The method for predicting the longitudinal wave velocity and the attenuation of the pore medium containing the fissure as claimed in claim 1, wherein the longitudinal wave velocity and the inverse quality factor predicted by the two models BISQ and Re-BISQ are calculated through steps S4-S6.
6. The method as claimed in claim 1, wherein the difference between the two models of BISQ and Re-BISQ in predicting the longitudinal wave velocity and the inverse quality factor is comparatively analyzed by step S7.
7. The method for predicting the longitudinal wave velocity and attenuation of a pore medium containing fractures according to claim 1, wherein in step S5, the longitudinal wave velocity V is obtainedpAnd attenuation factor α expression;
Figure FDA0002252054270000051
a1,2＝ωIm(X1,2) (16)
where ω is the angular velocity, ωc＝ηφ/κρf，ρ1＝(1-φ)ρs,ρ2＝φρf，
Expression (16) is expressed in inverse quality factor form as:
Figure FDA0002252054270000052
Figure FDA0002252054270000053
Figure FDA0002252054270000054
Figure FDA0002252054270000055
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