CN113887150A - Method for estimating length of characteristic jet flow of compact sandstone - Google Patents
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Abstract
The invention discloses a method for estimating the length of a characteristic jet flow of tight sandstone. The research on the distribution characteristics of the cracks in the rock and the characteristic jet length characteristics is carried out, the influence of parameters such as permeability and crack density on the characteristic jet length is analyzed, and finally, a quantitative expression of the characteristic jet length is obtained, so that an effective method is provided for determining the characteristic jet length of the compact sandstone. The invention fully considers the relation between the characteristic jet flow length of the compact sandstone and the fracture distribution characteristic in the rock and forms a method for determining the characteristic jet flow length. The method characterizes the characteristic jet flow length through measurable parameters, so that the characteristic jet flow length is converted to be acquired through an indirect measurement mode.
Description
Technical Field
The invention relates to the field of seismic rock physics, in particular to a method for estimating the characteristic jet flow length of tight sandstone.
Background
The tight sandstone pore permeability is low, the micro-fracture in the rock develops, strong heterogeneity is presented, and the difficulty of reservoir prediction and fluid identification is greatly increased. The micro-pore structure of the reservoir is not only closely related to the rock physical characteristics, but also is a main factor influencing the oil and gas distribution of the reservoir. Therefore, the research on the characteristics of the micro-pore structure can not only define the basic characteristics and the reservoir performance of the reservoir, but also explain the basis of the reservoir oil-gas enrichment rule (white bin, etc., 2013). The presence of fissures inside the rock has an effect not only on the elasticity of the rock, but also on the flow of fluids in the rock (Carcione, 2007; Muller et al, 2010). In order to study the influence of fractures on the elastic properties of rocks, researchers studied equivalent medium theory and petrophysical experiments (Song et al, 2016; Yin et al, 2017). Based on the method, David and Zimmmerman (2012) predict the characteristics of pore distribution (DZ model for short) corresponding to fractures with different aspect ratios from the ultrasonic velocity of dry rock by using an MT model (Mori and Tanaka,1973) and a DEM model (Berrayman,1992), and analyze the rule that the characteristic of the longitudinal and transverse waves of saturated rock changes along with the effective pressure.
In order to study the wave response characteristics of seismic waves as they pass through complex pore media, researchers have proposed a series of theoretical models (Ba et al, 2011; 2017; Muller et al, 2010; Zhang et al, 2019; Zhang et al, 2020). With the progress of theoretical research and the accumulation of data measured in the field, it is gradually recognized that local flow caused by heterogeneity is the main cause of elastic wave attenuation frequency dispersion (yankui et al, 2000). The BISQ model (Dvorkin,1993) is based on the Biot model (Biot,1956a,1956b,1962), a new petrophysical parameter 'characteristic jet length' is introduced to describe a jet flow (Squirt flow), and the combination of macro scale and micro scale is realized. The fast longitudinal wave velocity frequency dispersion predicted by the model is obviously larger than the calculation result of the Biot model. However, the fast longitudinal wave velocity estimated by the model is smaller than the predicted value of the Gassmann model (Gassmann,1951) at lower frequencies, and is consistent with the prediction result of the Biot model at higher frequencies, which are not consistent with the physical fact. Mavko and Jizba (1991) propose a jet relation (M-J model) for calculating the elastic modulus of a high frequency unrelaxed wet rock skeleton based on a modified rock skeleton. The M-J model predicts a higher frequency compressional velocity value than that calculated by the Biot model, but the model predicts a compressional velocity value greater than that predicted by the Gassmann model at lower frequencies, the jet characteristic frequency is consistent with that of the Biot flow, and the formula is not applicable to the case where the pores contain gas or dry rock. Aiming at the limitation of the M-J model, Gurevich et al (2010) applies a pore structure model proposed by Murphy et al (1986), considers the pores between the contact surfaces of adjacent particles as soft pores, generalizes the M-J model to be suitable for the gas saturation condition and establishes a jet flow model with intermediate frequency dispersion, so that the model is expanded to the whole frequency range. Dvorkin et al (1995) rebuilds a complex modulus model of the fluid saturated rock jet based on the BISQ model principle in combination with a modified rock skeleton jet theoretical model, and finds an expression of the complex elastic modulus by considering one-dimensional radial flow. The predicted longitudinal wave velocity of the deduced complex modulus at the low-frequency limit is consistent with the result calculated by the Gassmann theory. However, this model does not consider the effect of Biot flow on the propagation characteristics of elastic waves, and the longitudinal wave velocity calculated by this model is larger than the theoretical maximum value at high frequencies (the longitudinal wave velocity calculated by the Biot model when all rock fractures are closed) (Wu et al, 2020).
Dvorkin et al (1993) first introduced a characteristic jet length to study the propagation characteristics of elastic waves in a reservoir, a parameter that cannot be obtained by experimental measurements and is on the order of magnitude of the rock particle size. Dvorkin et al (1994) then gives the physical meaning of the characteristic jet length, geometrically, this parameter represents the radius of the cylinder axially aligned with the wave propagation axis. Marketons and Best (2010) rely on ultrasonic experimental data of Best (1992) to study that a characteristic jet flow length has a certain power series relation with the viscosity of liquid. After the BISQ model was proposed, the researchers performed a lot of research work on the model, but focused on improving the BISQ model (Diallo and Appel., 2000; Diallo et al, 2003; Dvorkin, 1995; Wu et al, 2019; Wu et al,2020), jet anisotropy (Yang and Zhang,2000,2002) and wave field simulation (Yanwidend et al, 2002; Shenqing and Yankee, 2004), etc., without proposing a method for determining the characteristic jet length.
Disclosure of Invention
Aiming at the fact that the characteristic jet flow length is an important parameter of a relevant jet flow theoretical model, an effective establishing method is not formed at present, the invention provides a method for estimating the characteristic jet flow length of the compact sandstone, aiming at the compact sandstone, a fracture distribution parameter and a characteristic jet flow length value are calculated, the relation between the fracture distribution parameter and the characteristic jet flow length value is analyzed, and a quantitative relation formula of the characteristic jet flow length is established.
In order to solve the problems of the prior art, the invention adopts the technical scheme that:
a method of estimating a tight sandstone characteristic jet length, comprising the steps of:
and 7, selecting the main fracture aspect ratio, permeability, matrix modulus, fluid volume modulus and fracture density to characterize the characteristic jet flow length to form a characteristic jet flow length quantitative relation, wherein the characteristic jet flow length quantitative relation of the compact sandstone is as follows:
and fitting values of the parameters a and b according to the measurement result of the ultrasonic experiment, wherein the values have certain difference for different rock samples.
The improvement is that step 1 obtains ultrasonic velocity experimental data of pressure change of the tight sandstone, and the steps are as follows: and testing physical property parameters of the rock of the compact sandstone sample under the pressure change in a dry state and a water-saturated state by using an ultrasonic experimental measurement method, and calculating longitudinal wave speed and transverse wave speed in the dry and water-saturated states.
As an improvement, the step 2 of estimating the density of the tight sandstone fracture comprises the following steps:
the model establishes a quantitative relationship between the elastic modulus and the hard pores in the rock, and the equivalent elastic modulus can be expressed as:
wherein the content of the first and second substances,
and
the equivalent bulk modulus and the equivalent shear modulus of the rock, K, are respectively when only the hard pores are contained0And mu0Respectively the bulk modulus and shear modulus of the rock particles, phisIs the hard porosity of the rock, and M and N are the shape factors of the hard porosity, defined as follows:
wherein the shape of the hard pores is default to elliptical pores, γ is the aspect ratio of the elliptical pores, upsilon is the poisson ratio of the rock particles, and upsilon ═ 3K0-2μ0)/(6K0+2μ0) In the formula, the expression of g is as follows:
if the fractures are added to a pore medium containing hard pores, the content of the fractures is small, no interaction exists between the fractures and the pores, and the equivalent elastic modulus of the rock can be expressed as:
wherein the content of the first and second substances,
and
respectively the equivalent bulk modulus and the equivalent shear modulus when the rock contains cracks and hard pores,
is the rock poisson's ratio with only hard pores, Γ is the cumulative fracture density; because the fracture is approximately and completely closed under higher effective pressure, and the interior of the rock only contains hard pores, the optimal aspect ratio of the hard pores can be obtained by a formula (1) and a least square method;
the pressure dependence of the effective bulk modulus of the rock is related to the fracture density, so when the fracture density is known, the elastic modulus of the rock can be obtained through the formulas (5) and (6), and conversely, the fracture density at each pressure can be calculated through the elastic modulus at each effective pressure by using a least square method.
As an improvement, the step 3 of estimating the aspect ratio of the main fractures of the tight sandstone comprises the following steps:
quantitative relationship of cumulative fracture density with effective pressure change
In the formula, gammaiIs the initial fracture density at zero pressure, p is the pressure,
is a pressure constant of the same order of magnitude as the pressure;
dividing the pressure interval delta p of the dry rock test into n parts, and solving the fracture density of each pressure by using a formula (7):
wherein k is more than or equal to 0 and less than or equal to n;
dividing the pressure interval of the dry rock test at equal intervals delta p into n parts, wherein each part is delta p, and the fracture aspect ratio at each pressure point is as follows:
wherein the content of the first and second substances,
is the effective poisson's ratio at high pressure,
is the effective young's modulus at high pressure;
the relationship between fracture porosity and fracture density is:
and (3) solving the fracture aspect ratio corresponding to the maximum fracture density by using the formula (10), wherein the fracture aspect ratio is the main fracture aspect ratio.
Preferably, the step of deriving the MFS model at step 4 is as follows:
assuming an ideal cylinder of radius R, the relationship between the deformation of the framework (e ═ du/dx) and the fracture and pore pressures is as follows:
wherein u is the solid displacement, c0Is the acoustic velocity, p, of the liquidfIs the fluid density, Qc＝K0/(1-Kmsd/K0-φc)，Kmsd＝(1/K0-1/Khp+1/Kd)-1，KdIs the bulk modulus, phi, of the dry rockcIs the fracture porosity, KhpIs the bulk modulus of the dry rock when the fracture is fully closed;
assuming that the liquid inside the fracture can only flow in the direction perpendicular to the wave propagation direction, the fluid conservation of mass equation is:
wherein t is time and w is displacement of the liquid;
substituting the equations (11) and (12) into the equation (13) can obtain:
the relationship between fluid velocity and pore pressure gradient can be derived:
where η is the fluid viscosity and κ is the permeability.
The simple harmonic expression is:
exp(-iωt) (16)
substituting equation (16) into equations (14) and (15) yields:
wherein, KfIs the bulk modulus of the fluid;
from equations (17) and (18), the differential equation of the pressure is derived as follows:
at the fracture boundary, the pore pressure is always constant, so at R, p, dp, 0, the result of solving equation (19) is:
wherein R is the characteristic jet length, λ2＝iωηφc/κ(1/Kf+1/(φcQc))，J0Is a zero order bessel function;
the average fluid pressure is:
wherein, J1Is a first order Bessel function;
in the modified rock, the external stress and displacement and pore mean pressure are as follows:
dσ＝Kmsdde-(1-Kmsd/K0)dpav (22)
modified bulk modulus of solids Kms:
Substituting equations (21) and (22) into equation (23) yields:
the following relationship exists between the dry pore volume, matrix modulus and dry rock modulus according to the definition of bulk modulus, and the dry bulk modulus of the modified rock can be derived.
The expression for modifying the dry shear modulus of the rock is as follows:
wherein, mudIs the shear modulus of dry rock according to
and Johnston,1981) relationship between wavenumber and wave velocity and attenuation, and expression of longitudinal wave velocity and attenuation factorFormula (II) is shown.
ρaFor fluid-solid coupling density, ρsIs the density of the matrix and is,
for a characteristic frequency, MdIs the uniaxial stress modulus of the rock skeleton,
as a refinement, the step 5 of calculating the characteristic jet length using MFS comprises the steps of: and (4) calculating the characteristic jet flow lengths of different pressures by using the MFS model obtained in the step 4, and preferably selecting the most appropriate characteristic jet flow length of each sample by using a least square method.
As an improvement, the source steps of the compact sandstone characteristic jet flow length quantitative relation formula in the step 7 are as follows:
the critical diffusion length of the fluid is given according to the characteristic frequency of the jet flow as follows:
from the equations (31) and (32), the critical diffusion length of the fluid is related to the fracture aspect ratio, permeability and other parameters, and is expressed as:
when the fluid diffusion length is less than the critical diffusion length, the pore fluid pressure has sufficient time to equilibrate with the fluid flow, whereas the pore fluid pressure has insufficient time to equilibrate.
The method is characterized in that the main fracture aspect ratio, permeability, matrix modulus, fluid volume modulus and fracture density are selected to represent the characteristic jet flow length, a characteristic jet flow length quantitative relation is formed, and the dense sandstone characteristic jet flow length quantitative relation is as follows:
and fitting values of the parameters a and b according to the measurement result of the ultrasonic experiment, wherein the values have certain difference for different rock samples.
Has the advantages that:
compared with the existing theory, the method for estimating the characteristic jet flow length of the tight sandstone combines the ultrasonic experimental data of the tight sandstone, adopts a DZ model to analyze the fracture distribution characteristics in the tight sandstone rock, adopts an MFS model to calculate the characteristic jet flow length, determines the influence factors of the characteristic jet flow length, researches the relationship between the characteristic jet flow length and the fracture distribution, mainly analyzes the relationship between the characteristic jet flow length and parameters such as fracture density, fracture aspect ratio, permeability and the like, and finally forms a quantitative relational expression of the characteristic jet flow length of the tight sandstone. The invention provides a certain basis for establishing the characteristic jet flow length of the compact sandstone, and the specific effects are as follows:
a) estimating fracture distribution parameters in the rock based on the ultrasonic experimental data of the compact sandstone with pressure change;
b) calculating the characteristic jet flow length by combining an MFS model;
c) and analyzing the relation between the characteristic jet flow length and the fracture parameters through compact sandstone experimental data, and providing an expression of the characteristic jet flow length.
Drawings
FIG. 1 is a schematic flow diagram of a method of estimating a characteristic jet length of tight sandstone in accordance with the present invention;
FIG. 2 is the longitudinal and transverse wave velocity ratio of dry and water-saturated state of 12 pieces of tight sandstone;
FIG. 3 is a graph of the relationship between the density and pressure of 12 tight sandstone fractures;
FIG. 4 is a plot of fracture density versus most dominant fracture aspect ratio for sample GAR 7;
FIG. 5 is a diagram of a modified rock skeleton;
FIG. 6 is the predicted compressional velocity of the MFS model;
FIG. 7 is a graph of model calculated values versus experimental values for sample GAR7 at various pressures;
FIG. 8 is a graph of the results of comparing the predicted compressional velocity of different characteristic jet lengths of GAR7 with experimental values;
FIG. 9 is a graph of characteristic jet length versus various parameters;
FIG. 10 is a plot of characteristic jet length versus fracture density, major fracture aspect ratio, and permeability.
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 aims at the fact that the characteristic jet flow length is an important factor of a jet flow theoretical model, firstly, fracture parameters (fracture density, fracture porosity and main fracture aspect ratio) are calculated by using a DZ model, secondly, a derived MFS model is adopted, the characteristic jet flow length is calculated by using the model, and finally, influence factors of the characteristic jet flow length are analyzed, and an expression for representing the characteristic jet flow length through the fracture parameters is formed by combining the physical significance of the characteristic jet flow length.
A method for estimating the characteristic jet flow length of tight sandstone comprises the following specific steps as shown in figure 1:
The invention aims at a work area of a three-fold-system beard family river group on a Guangan gas field in the middle of a Sichuan basin, and 12 compact sandstone samples are selected in the work area. The compact sandstone sample is mainly rock debris quartz sandstone and a small amount of siliceous quartz sandstone, and consists of quartz, feldspar, rock debris, cement and the like, wherein the porosity is mainly distributed in a range of 3-14%, and the modulus of the compact sandstone matrix is 39 GPa.
And (3) processing the compact rock sample into cylinders with the diameters of 25mm and the heights of 25-50mm, and polishing and grinding two ends of each cylinder. The ultrasonic testing device disclosed by the invention is used for carrying out ultrasonic experimental measurement on 12 compact sandstone samples by using an ultrasonic pulse testing device, adopts an experimental device disclosed in the literature (Guomeishiu et al, 2009; Yan et al, 2011), and consists of a reaction container, a temperature control unit, a confining pressure control unit, a pore pressure control unit, a sound wave testing unit and the like.
For experimental observation of dried samples, samples were dried by the method of the literature (wei yi jun et al,2020), the rock samples were sealed with rubber sleeves and placed in an experimental apparatus, effective pressures (5MPa, 10MPa, 15MPa, 20MPa, 25MPa, 30MPa and 35MPa) were applied to each sample (here, 12 samples were applied in sequence, and 7 pressures were applied to each sample), the fluid for applying confining pressure was heated by an electric heating wire in the apparatus to 80 ℃ and maintained for half an hour, ultrasonic experimental observation was performed, and the ultrasonic waveform passing through the rock samples was recorded. And (4) observing a water saturation experiment, namely saturating the compact sandstone sample with water by using a vacuumizing and pressurizing saturation method, performing ultrasonic experiment measurement according to a drying experiment process, and recording the ultrasonic waveform passing through the rock sample. And calculating the longitudinal and transverse wave velocities of the compact sandstone under different effective pressures based on the extracted initial values of the longitudinal and transverse wave waveforms. Specific petrophysical parameters of the samples are shown in table 1.
TABLE 1 physical Properties of tight sandstone rock
| Sample number | Porosity (%) | Permeability (mD) | Particle Density (kg/m)3) |
| GAR11 | 3.03 | 0.001 | 2687 |
| GA3 | 3.46 | 0.005 | 2694 |
| GAR1 | 4.19 | 0.0005 | 2676 |
| GA10 | 4.89 | 0.004 | 2691 |
| GA6 | 6.26 | 0.046 | 2672 |
| GAR6 | 6.33 | 0.047 | 2665 |
| GA8 | 8.55 | 0.082 | 2670 |
| GAR7 | 8.65 | 0.028 | 2668 |
| GAR12 | 8.97 | 0.14 | 2662 |
| GA1 | 13.26 | 1.21 | 2659 |
| GAR8 | 13.35 | 1.320 | 2653 |
| GA2 | 13.91 | 1.370 | 2660 |
The relationship between the pressure and the shear wave velocity for the dry and water-saturated states of 12 groups of tight sandstone rock samples is reflected in fig. 2. The longitudinal and transverse wave velocities in the saturated water state are greater than those in the dry state, and in the two different states, similar changes are observed between the longitudinal and transverse wave velocities and the ambient pressure, both the longitudinal and transverse wave velocities increase with increasing pressure, the rate of change of velocity with increasing pressure is greater when the pressure is lower, and the velocity slows with increasing pressure (Carcione and Quiroga-Goode, 1995; Dengdeng et al 2015; Song Negteng et al 2015). The main reason for this is that since the fractures close successively during the pressure increase, the effect of pressure on the hard pores is very small and negligible (Chen 39065and Huang Fang, 2001; David and Zimmerman, 2012; Shapiro, 2003). Based on the method, the fracture distribution characteristics can be obtained by utilizing the speed-pressure change relation.
Establishing a quantitative relation between the elastic modulus and hard pores in the rock based on a Mori-Tanaka model, wherein the equivalent elastic modulus can be expressed as:
wherein the content of the first and second substances,
and
the equivalent bulk modulus and the equivalent shear modulus of the rock, K, are respectively when only the hard pores are contained0And mu0Respectively the bulk modulus and shear modulus of the rock particles, phisIs the hard porosity of the rock, and M and N are the shape factors of the hard porosity, defined as follows:
wherein the shape of the hard pores is default to elliptical pores, gamma is the aspect ratio of the elliptical pores, upsilon is the Poisson ratio of rock particles, and upsilon is (3K)0-2μ0)/(6K0+2μ0) In the formula, the expression of g is as follows:
if the fractures are added to a pore medium containing hard pores, the content of the fractures is small, no interaction exists between the fractures and the pores, and the equivalent elastic modulus of the rock can be expressed as:
wherein the content of the first and second substances,
is the poisson ratio of the rock when only hard pores are present,
and
the fracture density is the equivalent bulk modulus and the equivalent shear modulus when the rock contains fractures and hard pores respectively, gamma is the accumulated fracture density, and because the fractures are approximately completely closed under higher effective pressure and the rock only contains hard pores, the most suitable method can be adopted through the formula (1)The small two multiplication finds the optimal hard pore aspect ratio.
The pressure dependence of the effective bulk modulus of the rock is related to fracture density, so when the fracture density is known, the elastic modulus of the rock can be obtained through formulas (5) and (6), conversely, the fracture density can be obtained through the elastic modulus under each effective pressure, and the accumulated fracture density under each pressure can be calculated by using a least square method.
quantitative relationship of cumulative fracture density as a function of effective pressure:
in the formula, gammaiIs the initial fracture density at zero pressure, p is the pressure,
is a pressure constant of the same order of magnitude as the pressure.
Dividing pressure intervals of the dry rock test into n parts at equal intervals, and solving fracture density of each pressure by using a formula (7):
wherein k is more than or equal to 0 and less than or equal to n.
Dividing the pressure interval of the dry rock test at equal intervals delta p into n parts, wherein each part is delta p, and the fracture aspect ratio at each pressure point is as follows:
wherein the content of the first and second substances,
is the effective poisson's ratio at high pressure,
is the effective young's modulus at high pressure.
Fracture porosity is related to fracture density by (David and Zimmerman, 2012):
and (3) solving the fracture aspect ratio corresponding to the maximum fracture density by using the formula (10), wherein the fracture aspect ratio is the main fracture aspect ratio.
The aspect ratio of fractures in actual reservoir rock is not a fixed value, but is continuously distributed within a certain range (dunne, 2015). However, at a certain fixed pressure, the major fracture aspect ratio is a fixed value. By major fracture aspect ratio is meant the fracture aspect ratio corresponding to the maximum fracture porosity (Sun et al,2020), and thus the invention selects the major fracture aspect ratio, and figure 4 shows the fracture aspect ratio of the major fracture at different pressures for the sample GAR 7. In the pressure range of 5-35MPa, when the pressure is different, the fracture aspect ratio is similar to the distribution form of the fracture porosity, but the fracture porosity corresponding to each fracture aspect ratio is reduced when the pressure is higher, and the reduction of the fracture porosity is more obvious when the pressure is lower. Mainly due to the increased pressure, which closes part of the fracture.
And 4, deriving an MFS model:
walsh (1965), Mavko and Jizaba (1991) and Shapiro (2003) believe that the pores of a rock can be divided into hard and soft pores, with hard pores constituting the primary porosity of the rock. Soft pores (fissures) have a very small aspect ratio and a very narrow space compared to pores inside rocks, and easily expand and contract under external force, thereby forming a jet flow.
The MFS model considers the heterogeneity of pore media, divides pores into fractures and hard pores, meanwhile, incorporates fractures (jet flows) into a solid framework, has a structure shown in figure 5, corrects the rock framework (similar to a new rock framework), calculates the bulk modulus and shear modulus of the corrected rock, and analyzes the reservoir elastic wave propagation characteristics.
Using the Dvorkin and Nur (1993) concept, assuming an ideal cylinder of radius R, the relationship between the deformation of the framework (e ═ du/dx) and the fracture and pore pressures is as follows:
where u is the solid displacement and x is parallel to the wave propagation direction. c. C0Is the acoustic velocity, p, of the liquidfIs the fluid density, Qc＝K0/(1-Kmsd/K0-φc)，Kmsd＝(1/K0-1/Khp+1/Kd)-1，KdIs the bulk modulus, phi, of the dry rockcIs the fracture porosity, KhpIs the bulk modulus of the dry rock when the fracture is fully closed.
Assuming that the liquid inside the fracture can only flow perpendicular to the direction of wave propagation, the fluid conservation of mass equation is therefore:
where t is time and w is the displacement of the liquid.
Substituting the equations (11) and (12) into the equation (13) can obtain:
the relationship between fluid velocity and pore pressure gradient can be derived by Darcy's law:
where η is the fluid viscosity and κ is the permeability.
The simple harmonic expression is:
exp(-iωt) (16)
substituting equation (16) into equations (14) and (15) yields:
wherein, KfIs the bulk modulus of the fluid.
From equations (17) and (18), the differential equation of the pressure is derived as follows:
at the fracture boundary, the pore pressure is always constant, so at R, p, dp, 0, the result of solving equation (19) is:
wherein R is the characteristic jet length, λ2＝iωηφc/κ(1/Kf+1/(φcQc))，J0Is a zero order bessel function.
The average fluid pressure is:
wherein, J1Is a first order bessel function.
(Biot, 1941; Rice and Cleary, 1976; Dvorkin and Nur,1993) indicate the relationship between external stress and displacement and pore pressure in rock, and thus, in modified rock, external stress and displacement and pore mean pressure are as follows:
dσ＝Kmsdde-(1-Kmsd/K0)dpav (22)
modified bulk modulus of solids Kms：
Substituting equations (21) and (22) into equation (23) yields:
the dry bulk modulus of the modified rock can be derived from the definition of bulk modulus, the following relationship (Walsh,1965) between dry pore volume, matrix modulus and dry rock modulus.
Mavko and Jizba (1991) gives the expression for modifying the dry shear modulus of rock as follows:
wherein, mudIs the shear modulus of dry rock according to
and Johnston,1981) wave number versus wave velocity and attenuation, and deriving expressions for longitudinal wave velocity and attenuation factor.
ρaFor fluid-solid coupling density, ρsIs the density of the matrix and is,
is the characteristic frequency. MdIs the uniaxial stress modulus of the rock skeleton,
the numerical simulation calculation parameters of the present invention are according to Carcione and Gurenvich (2011). The values of all parameters are respectively porosity of 0.2, solid framework bulk modulus of 18GPa, Poisson ratio of 0.15 and matrix density of 2650kg/m3Matrix bulk modulus 50GPa, permeability 2X 10-15m2Fluid density 1040kg/m3The viscosity of the fluid is 0.1cp, the bulk modulus of the fluid is 2.25GPa, and the fluid-solid coupling density is 420kg/m3The characteristic jet length is 2mm, the fracture porosity is 0.0002, the bulk modulus of the dry rock is 20GPa when no fracture exists, and the aspect ratio of the fracture is 0.0008. And calculating the longitudinal wave velocity and drawing a curve graph.
Analysis of FIG. 6 reveals that the two models (MFS model and Dvorkin (1995)) in the graph are in agreement with the Biot model calculations of longitudinal wave velocity in the low frequency band. However, the calculated compressional velocity of the Dvorkin (1995) model exceeds the maximum value (Vpmax), thus limiting the application of the model. The main reason for this is that the boundary conditions are not set properly. Method for determining maximum valueThe dry bulk modulus of the corrected rock is set to a value corresponding to the bulk modulus (K) at which the fracture is completely closedmd＝Khp) Then, the velocity of the longitudinal wave is calculated by substituting the value into the Biot equation, and the result is the maximum value of the velocity of the longitudinal wave.
The MFS model calculations proposed by the present invention show that the longitudinal wave velocity approaches the maximum value and does not exceed the maximum value as the frequency increases, which is consistent with the theoretical research results in the industry (Dvorkin and Nur, 1993; Joge, 1997).
In conclusion, in the MFS model provided by the invention, the calculated value of the longitudinal wave velocity is consistent with the calculated value of the Gassmann-Biot model in the low frequency band, and the longitudinal wave velocity calculated in the high frequency band does not exceed the theoretical maximum value. Therefore, the model provided by the invention can reflect the reservoir elastic wave propagation characteristics more accurately and comprehensively.
And 5, calculating the characteristic jet flow length by using an MFS model:
based on the MFS model derived in step 4, the characteristic jet flow lengths for different pressures are calculated by the model, and the most suitable characteristic jet flow length for each sample is optimized by using the least square method.
FIG. 7 is a graph showing the dispersion characteristics of GAR7 samples at different pressures, wherein the pressure is in the range of 5-35MPa and the characteristic jet length values are: 0.055mm, 0.051mm, 0.049mm, 0.044mm, 0.041mm, 0.038mm and 0.030 mm. The relevant parameters of the fluid were calculated by the Batzle-Wang equation (Batzle and Wang, 1992). Based on the drying longitudinal and transverse wave speeds and densities measured by experiments, the shear modulus and the bulk modulus of the dry rock can be calculated.
As can be seen from fig. 7, a characteristic jet length can be found at each pressure, so that the predicted value of the model is consistent with the experimentally measured value, and therefore, the model can effectively describe the characteristics of the velocity and the pressure change of the tight sandstone compressional wave. The primary reason for the decrease in the dispersion of compressional velocity in tight sandstone with increasing pressure is the change in the internal micro-pore structure of the rock caused by the increase in pressure.
FIG. 8 shows the comparison of compressional wave velocity with experimental values for the jet length prediction samples GAR7 with different characteristics. The characteristic jet flow length is increased, the longitudinal wave speed predicted by the model is increased, the relation between the pressure and the predicted value and the experimentally measured value of the model is consistent, namely, the pressure is increased, the speed is increased rapidly when the pressure is low, and the speed is increased slowly when the pressure is high. With increasing pressure, the appropriate characteristic jet length decreases. The main reason for this is that when the pressure increases, the microcracks, which have small aspect ratios, close. As can be seen from the comparison of the model calculated values and the experimental measurements for tight sandstone samples in fig. 8, the rock samples can be characterized by a fixed characteristic jet length under different pressure conditions. It follows that the characteristic jet length is a fundamental property of rock (Dvorkin and Nur, 1993).
and (5) analyzing the influence of the fracture parameters on the length of the characteristic jet flow through the computed fracture distribution parameters and the characteristic jet flow length in the step 2-5.
Figure 9 is a graph of the characteristic jet length of 12 pieces of tight sandstone plotted against various parameters. The characteristic jet length increases with increasing permeability, increasing fracture density, and smaller major fracture aspect ratios.
fracture density, which is a parameter used to describe the number of fractures, and fracture aspect ratio, which is a parameter indicative of the fracture shape, are the main parameters that characterize the microstructure of the fracture, both parameters affecting the jet effect (dawn, 2010). The fracture density and fracture aspect ratio can be obtained through model calculation according to experimental data, and the characteristic jet flow length is a variable introduced for researching the local movement of fluid in the fracture. Analysis has shown that for rock samples, there is a characteristic jet flow length that is most appropriate over a range of pressures, while at a fixed pressure, the major fracture aspect ratio and fracture density are fixed values. Permeability is an indication of how easily fluid in the pore medium flows in its pores and can be measured directly by an instrument to obtain this parameter, and analysis has shown that permeability also affects the characteristic jet length. Thus, the characteristic jet length is related to parameters such as fracture density, major fracture aspect ratio, and permeability.
In order to comprehensively clarify the relation between the characteristic jet flow length and each parameter, the invention introduces the characteristic frequency of the jet flow and the critical fluid diffusion length to more clearly characterize the characteristic of the characteristic jet flow length.
The authors indicate that jet eigenfrequency is closely related to bulk modulus, fracture aspect ratio and fluid viscosity (O' Connell and Budiansky, 1978; Jones et al, 1980; Li et al,2018), and that the expression:
the critical diffusion length of the fluid is given according to the characteristic frequency of the jet flow as follows:
from equations (31) and (32), the critical diffusion length of the fluid is related to the fracture aspect ratio and permeability parameters, and is expressed as:
when the fluid diffusion length is less than the critical diffusion length, the pore fluid pressure has sufficient time to equilibrate with the fluid flow, whereas the pore fluid pressure has insufficient time to equilibrate.
Based on previous studies (Pride, 2004; Dvorkin and Nur,1993) and the present invention analysis, it was suggested that parameters such as major fracture aspect ratio, permeability, matrix modulus, bulk modulus of fluid and fracture density, etc. characterize the characteristic jet length, the formula proposed in the present invention is:
and fitting values of the parameters a and b according to the measurement result of the ultrasonic experiment, wherein the values have certain difference for different rock samples. The present invention fits the values of a and b of 0.03865 and 63.42, respectively, based on experimental data for 12 tight sandstones. Accordingly, equation (34) can be expressed as:
figure 10 visually depicts the quantitative relationship between the characteristic jet length and various parameters.
In conclusion, the invention fully considers the relationship between the characteristic jet flow length of the tight sandstone and the fracture distribution characteristics in the rock, and forms a method for determining the characteristic jet flow length. The method characterizes the characteristic jet flow length through measurable parameters, so that the characteristic jet flow length is converted to be acquired through an indirect measurement mode.
The above description is only a preferred embodiment of the present invention, and the scope of the present invention is not limited thereto, and any simple modifications or equivalent substitutions of the technical solutions that can be obviously obtained by those skilled in the art within the technical scope of the present invention are within the scope of the present invention.
Claims (7)
1. A method of estimating the characteristic jet length of tight sand comprising the steps of:
step 1, acquiring ultrasonic velocity experimental data of pressure change of tight sandstone;
step 2, estimating the fracture density of the tight sandstone;
step 3, estimating the aspect ratio of main fractures of the compact sandstone;
step 4, deriving an MFS model;
step 5, calculating the characteristic jet flow length by using an MFS model;
step 6, analyzing the relation between the characteristic jet flow length and the fracture distribution parameters;
and 7, selecting the main fracture aspect ratio, permeability, matrix modulus, fluid volume modulus and fracture density to characterize the characteristic jet flow length to form a characteristic jet flow length quantitative relation, wherein the characteristic jet flow length quantitative relation of the compact sandstone is as follows:
and fitting the values of the parameters a and b according to the measurement result of the ultrasonic experiment, wherein the values have respective differences for different rock samples.
2. The method for estimating the characteristic jet flow length of tight sandstone according to claim 1, wherein the step 1 is to obtain ultrasonic velocity experimental data of the pressure change of tight sandstone, and comprises the following steps: and testing physical property parameters of the rock of the compact sandstone sample under the pressure change in a dry state and a water-saturated state by using an ultrasonic experimental measurement method, and calculating longitudinal wave speed and transverse wave speed in the dry and water-saturated states.
3. The method for estimating the tight sandstone characteristic jet flow length as claimed in claim 1, wherein the step 2 of estimating the tight sandstone fracture density comprises the following steps:
establishing a quantitative relationship between the modulus of elasticity and the hard porosity in the rock, the equivalent modulus of elasticity can be expressed as:
wherein the content of the first and second substances,
and
the equivalent bulk modulus and the equivalent shear modulus of the rock, K, are respectively when only the hard pores are contained0And mu0Respectively the bulk modulus and shear modulus of the rock particles, phisIs the hard porosity of the rock, and M and N are the shape factors of the hard porosity, defined as follows:
wherein the shape of the hard pores is default to elliptical pores, gamma is the aspect ratio of the elliptical pores, upsilon is the Poisson ratio of rock particles, and upsilon is (3K)0-2μ0)/(6K0+2μ0) In the formula, the expression of g is as follows:
if the fractures are added to a pore medium containing hard pores, the content of the fractures is small, no interaction exists between the fractures and the hard pores, and the equivalent elastic modulus of the rock can be expressed as:
wherein the content of the first and second substances,
and
respectively the equivalent bulk modulus and the equivalent shear modulus when the rock contains cracks and hard pores,
is the rock poisson's ratio with only hard pores, Γ is the cumulative fracture density; because the fracture is approximately and completely closed under the effective pressure, the rock only contains hard pores, and the optimal aspect ratio of the hard pores is obtained by adopting a least square method through a formula (1);
the pressure correlation of the effective bulk modulus of the rock is related to fracture density, when the fracture density is known, the equivalent bulk modulus and the equivalent shear modulus of the rock can be obtained through formulas (5) and (6), and conversely, the fracture density under each pressure can be calculated through the equivalent bulk modulus and the equivalent shear modulus under each effective pressure by using a least square method.
4. The method for estimating the characteristic jet length of tight sandstone according to claim 1, wherein the step 3 of estimating the aspect ratio of the main fracture of tight sandstone comprises the following steps:
quantitative relationship of cumulative fracture density as a function of effective pressure:
in the formula, gammaiIs the initial fracture density at zero pressure, p is the pressure,
is a pressure constant of the same order of magnitude as the pressure;
dividing the pressure interval delta p of the dry rock test into n parts, and solving the fracture density of each pressure by using a formula (7):
wherein k is more than or equal to 0 and less than or equal to n;
dividing the pressure interval of the dry rock test at equal intervals delta p into n parts, wherein each part is delta p, and the fracture aspect ratio at each pressure point is as follows:
wherein the content of the first and second substances,
is the effective poisson's ratio at high pressure,
is the effective young's modulus at high pressure;
the relationship between fracture porosity and fracture density is:
and (3) solving the fracture aspect ratio corresponding to the maximum fracture density by using a formula (10), namely the main fracture aspect ratio.
5. The method for estimating the tight sandstone characteristic jet flow length as claimed in claim 1, wherein the step of deriving the MFS model in the step 4 is as follows:
assuming an ideal cylinder of radius R, the relationship between the deformation of the framework and the fracture and pore pressure is as follows:
wherein u is the solid displacement, c0Is the acoustic velocity, p, of the liquidfIs the fluid density, Qc＝K0/(1-Kmsd/K0-φc)，Kmsd＝(1/K0-1/Khp+1/Kd)-1，KdIs the bulk modulus, phi, of the dry rockcIs the fracture porosity, KhpIs the bulk modulus of the dry rock when the fracture is fully closed;
assuming that the liquid inside the fracture can only flow in the direction perpendicular to the wave propagation direction, the fluid conservation of mass equation is:
wherein t is time and w is displacement of the liquid;
substituting the equations (11) and (12) into the equation (13) can obtain:
the relationship between fluid velocity and pore pressure gradient can be derived:
wherein η is fluid viscosity and κ is permeability;
the simple harmonic expression is:
exp(-iωt) (16)
substituting equation (16) into equations (14) and (15) yields:
wherein, KfIs the bulk modulus of the fluid;
from equations (17) and (18), the differential equation of the pressure is derived as follows:
at the fracture boundary, the pore pressure is always constant, so at R, p, dp, 0, the result of solving equation (19) is:
wherein R is the characteristic jet length, λ2＝iωηφc/κ(1/Kf+1/(φcQc))，J0Is a zero order Bessel function;
the average fluid pressure is:
wherein, J1Is a first order Bessel function;
in the modified rock, the external stress and displacement and pore mean pressure are as follows:
dσ＝Kmsdde-(1-Kmsd/K0)dpav (22)
modified bulk modulus of solids Kms:
Substituting equations (21) and (22) into equation (23) yields:
the following relationship exists between the dry pore volume, matrix modulus and dry rock modulus according to the definition of bulk modulus, and the dry bulk modulus of the modified rock can be derived.
The expression for modifying the dry shear modulus of the rock is as follows:
wherein, mudThe shear modulus of the dry rock is obtained, and longitudinal wave velocity and attenuation factor expressions are obtained according to the relationship between wave number and wave velocity and attenuation.
ρaFor fluid-solid coupling density, ρsIs the density of the matrix and is,
for a characteristic frequency, MdIs the uniaxial stress modulus of the rock skeleton,
6. the method for estimating the characteristic jet length of tight sandstone according to claim 1, wherein the step 5 of calculating the characteristic jet length by using MFS comprises the steps of: and (4) calculating the characteristic jet flow lengths of different pressures by using the MFS model obtained in the step 4, and preferably selecting the most appropriate characteristic jet flow length of each sample by using a least square method.
7. The method for estimating the length of the tight sandstone characteristic jet flow according to claim 1, wherein the quantitative length relation of the tight sandstone characteristic jet flow length in the step 7 is derived by the following steps: the jet characteristic frequency is closely related to the bulk modulus, the fracture aspect ratio and the fluid viscosity, and the expression is as follows:
the critical diffusion length of the fluid is given according to the characteristic frequency of the jet flow as follows:
from the equations (31) and (32), the critical diffusion length of the fluid is related to the fracture aspect ratio, permeability and other parameters, and is expressed as:
when the fluid diffusion length is less than the critical diffusion length, the pore fluid pressure has enough time to reach equilibrium through the fluid flow, otherwise, the pore fluid pressure has insufficient time to reach equilibrium;
the method comprises the following steps of selecting the main fracture aspect ratio, permeability, matrix modulus, fluid volume modulus and fracture density to characterize the characteristic jet flow length to form a characteristic jet flow length quantitative relation, wherein the characteristic jet flow length quantitative relation of the compact sandstone is as follows:
and fitting the values of the parameters a and b according to the measurement result of the ultrasonic experiment, wherein the values have respective differences for different rock samples.
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