CN116227139B - Method for measuring and calculating transmission properties of hole-crack rock based on two-part Hooke's law - Google Patents
Method for measuring and calculating transmission properties of hole-crack rock based on two-part Hooke's law Download PDFInfo
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Abstract
The invention discloses a method for measuring and calculating the transmission property of a hole fracture rock based on two Hooke's law, which specifically comprises the following steps: (1) Acquiring porosity, permeability and conductivity data under different effective stresses; (2) Based on two-part hooke's law, establishing a quantitative relationship between porosity and effective stress; (3) Based on two-part hooke's law, establishing a quantitative relationship between permeability and effective stress; (4) Based on two part hooke's law, establishing a quantitative relationship between conductivity and effective stress; (5) Based on the established quantitative relationship, the characteristics of rock transmission properties under different effective stresses are analyzed in combination with experimental data. The invention fully considers that the nonlinear change of the transmission characteristics of the pore-crack rock is often related to the nonuniform deformation of the pore structure in the pore-crack rock, and adopts two Hooke's law to design a measuring and calculating method for predicting the rock transmission properties under different effective pressures.
Description
Technical Field
The invention relates to the field of exploration and development of tight sandstone oil and gas reservoirs, in particular to a method for measuring and calculating transmission properties of hole-crack rock based on two-part Hooke's law.
Background
Subsurface reservoir rock is heterogeneous in nature, comprising a number of pores and fissures. The degree of deformation of the micro-cracks under the action of stress is obviously different, so that the elastic property of the rock can be influenced, the fluid flow in holes/cracks can be influenced, and particularly in low-permeability rock with micro-crack development, the existence of the micro-cracks can provide a high-permeability path for the flow of reservoir fluid. Therefore, the change characteristics of the permeability and the conductivity of the underground reservoir rock along with the effective stress are fully known, and scientific basis can be provided for the prediction and the monitoring of the underground oil and gas reservoir.
The holes/cracks in the rock deform to different degrees after being stressed, so that the influence on the porosity is different, and the rock mass is abstracted into two parts which respectively obey natural strain and engineering strain, namely a long narrow pore and micro-crack which generate obvious deformation form a soft part and a residual structure with smaller deformation form a hard part. The scholars Liu et al have introduced a new theoretical relationship (two-part hooke's law) to characterize the stress-strain relationship of the hole-crack rock and point out that the micro-cracks corresponding to the soft parts are the main cause of nonlinear deformation of the rock. The scholars Zheng and the like deduce the theoretical relationship among the porosity, the permeability and the effective stress of the low-permeability rock based on the model, and carry out fitting verification on the theoretical relationship by utilizing experimental data. Many scholars popularize and apply the stress-strain relation established by the model, obtain good practical application effect and indicate that the soft part inside the rock can also influence the conductivity characteristics of the rock. This is also confirmed by Watanabe et al, which subsequently analyzed the conductivity of the saturated brine granite. They found that the closure of the narrow pore size fraction (i.e. the fissures) resulted in a sharp drop in conductivity. Pang et al analyzed the effects of pore, fracture and clay content on elastic wave velocity and conductivity by constructing a sono-electro-mechanical joint petrophysical model. Meanwhile, walsh and Brace et al indicate that the fluid flow and current flow inside the rock have similarities, following similar migration paths. Moreover, a large number of studies indicate that there is a positive correlation between the permeability and conductivity of rock, so it is feasible to extend the two-part hooke's law to the study of the electrical characteristics of hole-fractured rock. And the application of the model can be used to establish a cross-property relationship between permeability and conductivity.
Disclosure of Invention
The purpose of the invention is that: the method is characterized by providing a method for measuring and calculating the transmission properties of the hole and crack rock based on two Hooke's law, distinguishing pores and cracks in the rock, and establishing a theoretical equation of the relation between the electrical properties and the rock mechanics.
In order to achieve the functions, the invention designs a method for measuring and calculating the transmission properties of rock with hole and fissure based on two-part Hooke' S law, which comprises the steps S1-S4 are executed for rock samples with the hole and fissure, the quantitative relation between each transmission property of the rock samples and effective stress is established, the step S5 is executed for the rock to be measured, and the quantitative relation is applied to finish measuring and calculating the transmission properties of the rock to be measured under different effective stresses:
step S1: actually measuring to obtain the porosity, permeability and conductivity of the rock sample under different effective stresses;
step S2: based on two-part hooke's law, establishing a quantitative relationship between porosity and effective stress;
step S3: based on two-part hooke's law, establishing a quantitative relationship between permeability and effective stress;
step S4: based on two part hooke's law, establishing a quantitative relationship between conductivity and effective stress;
step S5: aiming at the rock to be measured, according to different given effective stresses, the porosity, the permeability and the conductivity of the rock to be measured are obtained based on the quantitative relationships established in the steps S2-S4, and the transmission property measurement and calculation of the rock to be measured are completed.
As a preferred technical scheme of the invention: the specific method of step S2 is as follows:
based on two-part Hooke's law, dividing the rock into a soft part and a hard part, wherein the soft part is a hole crack which can generate obvious deformation in the rock, and the hard part is the rest part of the rock; the strain for the soft portion is as follows:
wherein K is t Bulk modulus of soft portion, σ is effective stress, V t Epsilon is the volume of the soft portion v,t For natural volumetric strain, equation (1) is integrated, and when σ=0, V t ＝V 0,t ，V 0,t Representing the volume of the soft portion in the unstressed state, V t The formula is as follows:
the strain for the hard portion is as follows:
wherein K is e Bulk modulus of hard part, V e Epsilon is the volume of the hard fraction v,e Is engineering volume strain; integrate equation (3), and when σ=0, V e ＝V 0,e ，V 0,e Representing the volume of the hard portion in the unstressed state, V e The formula is as follows:
thus, the effective stress σ of the rock versus strain ε is expressed as:
wherein V is 0 ＝V 0,t +V 0,e ，γ t ＝V 0,t /V 0 ，γ e ＝1-γ t ；
Based on the two-part hooke's law, the porosity of the rock is expressed as:
wherein phi is the total porosity of the rock, C e ＝1/K e Is the compression coefficient of the hard portion pore, phi e,0 For hard part porosity under no stress, gamma t,0 Is soft part porosity under no stress and phi e,0 +γ t,0 ＝φ 0 ，φ 0 To the rock porosity under no stress, phi e ＝φ e,0 (1-C e Sigma) is the porosity of the hard part, phi t ＝γ t,0 exp(-σ/K t ) Is soft fraction porosity;
substituting the measured minimum effective stress value for the zero effective stress value, substituting the zero effective stress value into formula (6) to obtain the following formula:
wherein Δσ=σ - σ 1 ，σ 1 Is the minimum effective stress value of actual measurement phi e,1 For the measured porosity of the hard portion at minimum effective stress, gamma t,1 Soft portion porosity at minimum effective stress measured.
As a preferred technical scheme of the invention: the specific method of step S3 is as follows:
based on the two-part hooke law, the soft part permeability k is calculated t And the porosity phi of t The relation between them is k t ＝α(φ t ) m The rock permeability k is expressed as:
wherein k is e,0 Is the permeability of the hard part under no stress, beta is the material constant of the hard part, and alpha and m are the material constants of the soft part;
substituting the measured minimum effective stress value for the zero effective stress value, substituting the zero effective stress value into formula (8) to obtain the following formula:
wherein Δσ=σ - σ 1 ，σ 1 Is the measured minimum effective stress value, k e,1 Is the measured permeability of the hard fraction at the minimum effective stress.
As a preferred technical scheme of the invention: the specific method of step S4 is as follows:
the conductivity of rock is expressed as a function of effective stress as:
d(lnS e )/dσ＝a(1/V 0 )(dV e p /dσ) (10)
wherein S is e Is the conductivity of the hard part, a is a constant, and dV e p ＝-C e V e,0 dσ, when the above expression is integrated, there is the following expression:
S e ＝S e,0 exp[-aφ e,0 C e σ] (11)
wherein S is e,0 Conductivity of the hard portion at zero effective stress;
conductivity S of soft portion t Subtracting the hard fraction conductivity S from the total conductivity S of the rock e The method comprises the following steps:
S t ＝S-S e (12)
the relation between the conductivity of the soft portion and the porosity of the soft portion is expressed by a power function:
S t ＝b[φ t ] n (13)
wherein b and n are material constants of the soft portion conductivity, and are represented by the following formulas (11), (13) and phi t ＝γ t,0 exp(-σ/K t ) The relationship between rock conductivity and effective stress based on two-part hooke's law can be derived as:
substituting the measured minimum effective stress value for the zero effective stress value, substituting the zero effective stress value into formula (14) to obtain the following formula:
wherein S is e,1 、φ e,1 The conductivity and porosity of the hard portion at the minimum measured effective stress, respectively.
As a preferred technical scheme of the invention: the specific method of step S5 is as follows:
for the rock to be tested, according to the given effective stress, based on equation (7) regarding the porosity of the rock, the hard portion porosity phi is within a high effective stress range not lower than a preset effective stress threshold e Is in linear relation with the effective stress sigma, and the slope of the fitted straight line is-phi e,1 C e Is at the measured minimum effective stress value sigma 1 The intercept at the position can determine the porosity phi of the hard part under the minimum effective stress of the actual measurement e,1 The value is further obtained to obtain the compression coefficient C of the corresponding hard part pore e ，γ t,1 Can pass phi 1 -φ e,1 Obtaining;
in a low effective stress range below a preset effective stress threshold, according to the relation k between the soft fraction permeability and its porosity t ＝α(φ t ) m Determination of soft fraction permeability k t Is fitted to log (S) according to equation (11) e ) Linear relation with effective stress sigma, the slope of the fitted straight line is-aC e φ e,1 Based on C obtained e 、φ e,1 Further obtaining the value of constant a, the fitted line being at the measured minimum effective stress value sigma 1 The intercept at the position can determine the measured conductivity S of the hard part under the minimum effective stress e,1 Based on equation (12) and material constants of the soft-fraction conductivity by a double log-coordinate linear fit according to equation (13)b and n, hard fraction permeability k according to formula (9) e ＝k e,1 exp[-βC e φ e,1 Δσ]Log (k) e ) There is a linear relationship with the effective stress sigma, the slope of the fitted linear equation is-beta C e φ e,1 According to C obtained e And phi e,1 Further, the value of beta can be determined, and sigma 1 Substituting the extrapolated straight line to obtain the permeability k of the hard part under the measured minimum stress e,1 The method comprises the steps of carrying out a first treatment on the surface of the Knowing the measured permeability data, soft fraction permeability is determined by subtracting hard fraction permeability from total permeability, and α and m are linearly fitted to soft fraction permeability by double log coordinatesObtained.
The beneficial effects are that: the advantages of the present invention over the prior art include:
based on the two-part Hooke's law, the invention considers that the fluid flow and the current flow characteristics in the rock are similar, establishes the quantitative relation between the conductivity and the effective stress, tests the compact sandstone sample for verifying the effectiveness of the theoretical equation, and shows that the equation can quantitatively describe the change rule of the conductivity along with the effective stress through the comparison of the predicted conductivity and the actual measurement. And analysis of equation parameters shows that there is a cross-property relationship between permeability and conductivity.
Drawings
FIG. 1 is a flow chart of a method for measuring and calculating the transmission properties of a hole fracture rock based on two-part Hooke's law according to an embodiment of the invention;
FIG. 2 is a graph of porosity as a function of effective stress for three densified sandstone samples provided according to embodiments of the present invention;
FIG. 3 is a graph of permeability as a function of effective stress for three tight sandstone samples provided according to embodiments of the present invention;
FIG. 4 is a graph of the log conductivity versus effective stress for three densified sandstone samples provided according to an embodiment of the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for more clearly illustrating the technical aspects of the present invention, and are not intended to limit the scope of the present invention.
Referring to fig. 1, in the method for measuring and calculating the transmission properties of the rock with the hole and fissure based on the two-part hooke law provided by the embodiment of the invention, for a rock sample containing the hole and fissure, step S1-step S4 are executed, a quantitative relation between each transmission property of the rock sample and effective stress is established, step S5 is executed for the rock to be measured, and the quantitative relation is applied to finish measuring and calculating the transmission properties of the rock to be measured under different effective stresses:
step S1: the porosity, permeability and conductivity of the rock sample under different effective stresses are obtained through actual measurement.
Three compact sandstone samples are selected as rock samples, and the rock samples are all low-permeability rocks. The method for measuring the porosity and permeability of the rock sample under different effective stresses is as follows:
the porosity and permeability are measured using a pulse decay-based porosimeter under effective stress conditions of 2.07-58.61MPa, wherein the porosity is determined based on helium expansion, and the permeability is corrected using an unsteady pulse transient decay technique and using the Klenburg slip effect. The accuracy of the porosity and permeability measurements were + -0.5% and + -0.001 mD, respectively.
The method for measuring the conductivity of the rock sample under different effective stresses is as follows:
after the rock sample was dried, it was placed in a vacuum chamber filled with 5% brine and evacuated for 2h. Conductivity was then measured using an impedance-capacitance-resistance (LCR) meter at a confining pressure of 5-35MPa and a constant zero pore pressure. The relative error of the conductivity measurement was 0.5%.
Step S2: based on two-part hooke's law, establishing a quantitative relationship between porosity and effective stress;
the specific method of step S2 is as follows:
the hole/crack in the rock can deform to different degrees after being stressed, the two-part hooke law divides the low-permeability rock into a soft part and a hard part, wherein the soft part refers to a pore and a micro-crack which can generate obvious deformation in the rock, and the strain behavior of the part can be described by adopting natural strain; the hard portion refers to the remainder of the rock and the corresponding strain behavior may be described using engineering strain. The strain for the soft portion is as follows:
wherein K is t For the bulk modulus of the soft portion, σ is the effective stress (confining pressure minus pore pressure), V t Epsilon is the volume of the soft portion v,t For natural volumetric strain, equation (1) is integrated, and when σ=0, V t ＝V 0,t ，V 0,t Representing the volume of the soft portion in the unstressed state, V t The formula is as follows:
the strain for the hard portion is as follows:
wherein K is e Bulk modulus of hard part, V e Epsilon is the volume of the hard fraction v,e Is engineering volume strain; integrate equation (3), and when σ=0, V e ＝V 0,e ，V 0,e Representing the volume of the hard portion in the unstressed state, V e The formula is as follows:
thus, the effective stress σ of the rock versus strain ε is expressed as:
wherein V is 0 ＝V 0,t +V 0,e ，γ t ＝V 0,t /V 0 ，γ e ＝1-γ t ；
Based on the two-part hooke's law, the porosity of the rock is expressed as:
wherein phi is the total porosity of the rock, C e ＝1/K e Is the compression coefficient of the hard portion pore, phi e,0 For hard part porosity under no stress, gamma t,0 Is soft part porosity under no stress and phi e,0 +γ t,0 ＝φ 0 ，φ 0 To the rock porosity under no stress, phi e ＝φ e,0 (1-C e Sigma) is the porosity of the hard part, phi t ＝γ t,0 exp(-σ/K t ) Is soft fraction porosity;
considering the limitation of an experimental method, a measured value under zero effective stress cannot be obtained in the experimental process, and an actually measured minimum effective stress value is adopted to replace the zero effective stress value, and substituted into formula (6) to obtain the following formula:
wherein Δσ=σ - σ 1 ，σ 1 Is the minimum effective stress value of actual measurement phi e,1 For the measured porosity of the hard portion at minimum effective stress, gamma t,1 Soft portion porosity at minimum effective stress measured.
Step S3: based on two-part hooke's law, establishing a quantitative relationship between permeability and effective stress;
the specific method of step S3 is as follows:
based on the two-part hooke law, the soft part permeability k is calculated t And to thisPorosity phi t The relation between them is k t ＝α(φ t ) m The rock permeability k is expressed as:
wherein k is e,0 Is the permeability of the hard part under no stress, beta is the material constant of the hard part, and alpha and m are the material constants of the soft part;
substituting the measured minimum effective stress value for the zero effective stress value, substituting the zero effective stress value into formula (8) to obtain the following formula:
wherein Δσ=σ - σ 1 ，σ 1 Is the measured minimum effective stress value, k e,1 Is the measured permeability of the hard fraction at the minimum effective stress.
Step S4: based on two part hooke's law, establishing a quantitative relationship between conductivity and effective stress;
the specific method of step S4 is as follows:
the change in conductivity with effective stress is also related to deformation of the pores/fissures inside the rock. Therefore, the relation between the conductivity and the effective stress can be established through the porosity, and the pore space in the rock can be divided into a soft part and a hard part, so that the contribution of different parts to the rock conductivity can be analyzed. First, the effect of the hard portion on the conductivity of the low permeability rock was analyzed, considering that the pores or fissures of the soft portion are substantially completely closed in the high effective stress range of not less than 15MPa, the contribution to the conductivity was negligible, and the relationship of the conductivity of the rock to the effective stress was expressed as:
d(lnS e )/dσ＝a(1/V 0 )(dV e p /dσ) (10)
wherein S is e Is the conductivity of the hard part, a is a constant, and dV e p ＝-C e V e,0 dσ, when the above expression is integrated, there is the following expression:
S e ＝S e,0 exp[-aφ e,0 C e σ] (11)
wherein S is e,0 Conductivity of the hard portion at zero effective stress;
in the low effective stress range below 15MPa, the decrease in rock conductivity is mainly related to the greater deformation of the pores or fissures of the soft fraction, so the soft fraction conductivity S t Subtracting the hard fraction conductivity S from the total conductivity S of the rock e The method comprises the following steps:
S t ＝S-S e (12)
the relation between the conductivity of the soft portion and the porosity of the soft portion is expressed by a power function:
S t ＝b[φ t ] n (13)
wherein b and n are material constants of the soft portion conductivity, and are represented by the following formulas (11), (13) and phi t ＝γ t,0 exp(-σ/K t ) The relationship between rock conductivity and effective stress based on two-part hooke's law can be derived as:
substituting the measured minimum effective stress value for the zero effective stress value, substituting the zero effective stress value into formula (14) to obtain the following formula:
wherein S is e,1 、φ e,1 The conductivity and porosity of the hard portion at the minimum measured effective stress, respectively.
Step S5: aiming at the rock to be measured, according to different given effective stresses, the porosity, the permeability and the conductivity of the rock to be measured are obtained based on the quantitative relationships established in the steps S2-S4, and the transmission property measurement and calculation of the rock to be measured are completed.
The specific method of step S5 is as follows:
for the rock to be tested, according to the given effective stress, based on equation (7) regarding the porosity of the rock, the hard portion porosity phi is within a high effective stress range not lower than a preset effective stress threshold e Is in linear relation with the effective stress sigma, and the slope of the fitted straight line is-phi e,1 C e Is at the measured minimum effective stress value sigma 1 The intercept at the position can determine the porosity phi of the hard part under the minimum effective stress of the actual measurement e,1 The value is further obtained to obtain the compression coefficient C of the corresponding hard part pore e ，γ t,1 Can pass phi 1 -φ e,1 Obtaining;
in a low effective stress range below a preset effective stress threshold, according to the relation k between the soft fraction permeability and its porosity t ＝α(φ t ) m Determination of soft fraction permeability k t Is fitted to log (S) according to equation (11) e ) Linear relation with effective stress sigma, the slope of the fitted straight line is-aC e φ e,1 Based on C obtained e 、φ e,1 Further obtaining the value of constant a, the fitted line being at the measured minimum effective stress value sigma 1 The intercept at the position can determine the measured conductivity S of the hard part under the minimum effective stress e,1 Based on equation (12) and material constants b and n for soft fraction conductivity by a double log-coordinate linear fit according to equation (13), hard fraction permeability k according to equation (9) e ＝k e,1 exp[-βC e φ e,1 Δσ]Log (k) e ) There is a linear relationship with the effective stress sigma, the slope of the fitted linear equation is-beta C e φ e,1 According to C obtained e And phi e,1 Further, the value of beta can be determined, and sigma 1 Substituting the extrapolated straight line to obtain the permeability k of the hard part under the measured minimum stress e,1 The method comprises the steps of carrying out a first treatment on the surface of the Knowing the measured permeability data, soft fraction permeability is the total permeability minus hard fraction permeability, α and m are linearly fitted to the soft fraction by a double log-coordinatesPermeability ofObtained.
The following is one embodiment of the present invention:
the present embodiment obtains experimental data of the porosity, permeability and conductivity of three tight sandstone samples as a function of effective stress based on the experimental measurement method in step S1, as shown in fig. 2-4. Points in the graph represent measured elasto-electric data of the sample, the curve is a fitted curve of each equation, the broken line is a fitted line of measured data of the hard part under high stress, the fitted line is in a nonlinear decrease along with the increase of stress in a low effective stress range (< 15 MPa), and the fitted line is in a linear decrease in the high effective stress range. Thus based on the two-part hooke's law, the rock mass can be abstracted into two parts subject to natural and engineering strain, respectively. And a theoretical equation of porosity, permeability, conductivity and effective stress is established, and the equation parameters are calculated and discussed in combination with the measured data, and the fitting parameters are shown in table 1.
TABLE 1
Fig. 2 and 3 are the results of the porosity and permeability fits, respectively, for three tight sandstone samples. In the figure, the measurement result is nonlinear reduced along with the increase of the effective stress in a low effective stress range, and the high effective stress section is linear change. The predictions of formulas (7) and (9) fit the experimental data well, where the porosity (permeability) of the TS1-TS3 samples fit the coefficient R 2 0.97 (0.98), 0.96 (0.96), 0.92 (0.85), respectively. From the table it can be seen that the porosity of the soft portion of sample TS2 is the largest and sample TS3 is the smallest. This also indicates that in the low effective stress range, the porosity drop of sample TS2 is greatest, resulting in a corresponding permeability drop also being greatest, while both the porosity and permeability drop of sample TS3 are smallest. And it was noted that in the high effective stress range, the porosity, permeability vs. stress exhibited good linear relationship, indicating that in this range the soft fraction vs. porosity and stressThe contribution of permeability is negligible.
Fig. 4 is a fit of the tight sandstone sample conductivity. It can be seen from the graph that the measured conductivity also features a non-linear change with increasing effective stress. The prediction result of the formula (15) has good fitting effect with experimental data, and the fitting coefficients R of three samples 2 0.99, 0.96, 0.99, respectively. In the low effective stress range, the rock conductivity change is remarkable, because the soft part has higher stress sensitivity, the pores and micro cracks of the soft part are stressed and compressed as key channels for fluid migration, and the conductive solution migrates, so that the current passing through the part is rapidly reduced, the conductivity is reduced along with the rapid reduction, and the nonlinear reduction trend is shown; when the stress is gradually increased, the soft part fluid channel is closed, the compression strength of the hard part is relatively large, the compression change of the pores is slow, the conductivity reduction trend is slow, and the obvious linear reduction trend is shown. Similarly to the permeability change, in the low effective stress range, since the sample TS2 soft portion porosity decreases the amplitude is greatest, the corresponding conductivity decreases the amplitude is also greatest, and the sample TS3 porosity and conductivity decreases the amplitude is also smallest. The significant change in conductivity of the soft fraction suggests that the soft fraction is not negligible as an important fluid migration channel in the low effective stress range during the study of fluid migration inside the low permeability rock.
In summary, the method for measuring and calculating the transmission properties of the hole-crack rock based on the two-part hooke law provided by the embodiment of the invention has the following beneficial effects: based on two-part Hooke's law, the theoretical formula between the conductivity and the effective stress of the low-permeability rock is deduced, the porosity, the permeability and the conductivity data of the compact sandstone are selected for fitting analysis, and the result shows that: the model can not only explain the nonlinear characteristics of low permeability rock mechanics and permeability, but also describe the characteristic of conductivity. The model may establish a cross-relationship between permeability and conductivity.
The embodiments of the present invention have been described in detail with reference to the drawings, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the spirit of the present invention.
Claims (4)
1. A method for measuring and calculating transmission properties of a hole and crack rock based on two-part Hooke' S law is characterized in that for a rock sample containing the hole and crack, step S1-step S4 are executed, a quantitative relation between each transmission property of the rock sample and effective stress is established, step S5 is executed for the rock to be measured, and the quantitative relation is applied to finish measuring and calculating the transmission properties of the rock to be measured under different effective stresses:
step S1: actually measuring to obtain the porosity, permeability and conductivity of the rock sample under different effective stresses;
step S2: based on two-part hooke's law, establishing a quantitative relationship between porosity and effective stress;
the specific method of step S2 is as follows:
based on two-part hooke's law, dividing the rock into a soft part and a hard part, wherein the soft part is a hole crack which generates obvious deformation in the rock, and the hard part is the rest part of the rock; the strain for the soft portion is as follows:
wherein K is t Bulk modulus of soft portion, σ is effective stress, V t Epsilon is the volume of the soft portion v,t For natural volumetric strain, equation (1) is integrated, and when σ=0, V t ＝V 0,t ，V 0,t Representing the volume of the soft portion in the unstressed state, V t The formula is as follows:
the strain for the hard portion is as follows:
wherein K is e Bulk modulus of hard part, V e Epsilon is the volume of the hard fraction v,e Is engineering volume strain; integrate equation (3), and when σ=0, V e ＝V 0,e ，V 0,e Representing the volume of the hard portion in the unstressed state, V e The formula is as follows:
thus, the effective stress σ of the rock versus strain ε is expressed as:
wherein V is 0 ＝V 0,t +V 0,e ，γ t ＝V 0,t /V 0 ，γ e ＝1-γ t ；
Based on the two-part hooke's law, the porosity of the rock is expressed as:
wherein phi is the total porosity of the rock, C e ＝1/K e Is the compression coefficient of the hard portion pore, phi e,0 For hard part porosity under no stress, gamma t,0 Is soft part porosity under no stress and phi e,0 +γ t,0 ＝φ 0 ，φ 0 To the rock porosity under no stress, v e ＝φ e,0 (1-C e Sigma) is the porosity of the hard part, phi t ＝γ t,0 exp(-σ/K t ) Is soft fraction porosity;
substituting the measured minimum effective stress value for the zero effective stress value, substituting the zero effective stress value into formula (6) to obtain the following formula:
wherein Δσ=σ - σ 1 ，σ 1 Is the minimum effective stress value of actual measurement phi e,1 For the measured porosity of the hard portion at minimum effective stress, gamma t,1 Is the measured soft portion porosity at minimum effective stress;
step S3: based on two-part hooke's law, establishing a quantitative relationship between permeability and effective stress;
step S4: based on two part hooke's law, establishing a quantitative relationship between conductivity and effective stress;
step S5: aiming at the rock to be measured, according to different given effective stresses, the porosity, the permeability and the conductivity of the rock to be measured are obtained based on the quantitative relationships established in the steps S2-S4, and the transmission property measurement and calculation of the rock to be measured are completed.
2. The method for measuring and calculating the transmission properties of the hole and fissure rock based on the two-part hooke' S law according to claim 1, wherein the specific method of the step S3 is as follows:
based on the two-part hooke law, the soft part permeability k is calculated t And the porosity phi of t The relation between them is k t ＝α(φ t ) m The rock permeability k is expressed as:
wherein k is e,0 Is the permeability of the hard part under no stress, beta is the material constant of the hard part, and alpha and m are the material constants of the soft part;
substituting the measured minimum effective stress value for the zero effective stress value, substituting the zero effective stress value into formula (8) to obtain the following formula:
wherein Δσ=σ - σ 1 ，σ 1 Is the measured minimum effective stress value, k e,1 Is the measured permeability of the hard fraction at the minimum effective stress.
3. The method for measuring and calculating the transmission properties of the hole and fissure rock based on the two-part hooke' S law according to claim 2, wherein the specific method of the step S4 is as follows:
the conductivity of rock is expressed as a function of effective stress as:
wherein S is e Is the conductivity of the hard part, a is a constant, andintegrating the above formula, the following formula is given:
wherein S is e,0 Conductivity of the hard portion at zero effective stress;
conductivity S of soft portion t Subtracting the hard fraction conductivity S from the total conductivity S of the rock e The method comprises the following steps:
S t ＝S-S e (12)
the relation between the conductivity of the soft portion and the porosity of the soft portion is expressed by a power function:
S t ＝b[φ t ] n (13)
wherein b and n are material constants of the soft portion conductivity, and are represented by the following formulas (11), (13) and phi t ＝γ t,0 exp(-σ/K t ) The two-part hooke-based product is obtainedThe relationship between the rock conductivity and the effective stress of law is:
substituting the measured minimum effective stress value for the zero effective stress value, substituting the zero effective stress value into formula (14) to obtain the following formula:
wherein S is e,1 、φ e,1 The conductivity and porosity of the hard portion at the minimum measured effective stress, respectively.
4. The method for measuring and calculating the transmission properties of the hole and fissure rock based on the two-part hooke' S law according to claim 3, wherein the specific method of the step S5 is as follows:
for the rock to be tested, according to the given effective stress, based on equation (7) regarding the porosity of the rock, the hard portion porosity phi is within a high effective stress range not lower than a preset effective stress threshold e Is in linear relation with the effective stress sigma, and the slope of the fitted straight line is-phi e,1 C e Is at the measured minimum effective stress value sigma 1 The intercept at the position can determine the porosity phi of the hard part under the minimum effective stress of the actual measurement e,1 The value is further obtained to obtain the compression coefficient C of the corresponding hard part pore e ，γ t,1 Through phi 1 -φ e,1 Obtaining;
in a low effective stress range below a preset effective stress threshold, according to the relation k between the soft fraction permeability and its porosity t ＝α(φ t ) m Determination of soft fraction permeability k t Is fitted to log (S) according to equation (11) e ) Linear relation with effective stress sigma, the slope of the fitted straight line is-aC e φ e,1 Based on C obtained e 、φ e,1 Is used as a reference to the value of (a),further obtaining the value of the constant a, wherein the fitted straight line is at the measured minimum effective stress value sigma 1 The intercept at the position can determine the measured conductivity S of the hard part under the minimum effective stress e,1 Based on equation (12) and material constants b and n for soft fraction conductivity by a double log-coordinate linear fit according to equation (13), hard fraction permeability k according to equation (9) e ＝k e,1 exp[-βC e φ e,1 Δσ]Log (k) e ) There is a linear relationship with the effective stress sigma, the slope of the fitted linear equation is-beta C e φ e,1 According to C obtained e And phi e,1 Further determining the value of beta and sigma 1 Substituting the extrapolated straight line to obtain the permeability k of the hard part under the measured minimum stress e,1 The method comprises the steps of carrying out a first treatment on the surface of the Knowing the measured permeability data, soft fraction permeability is determined by subtracting hard fraction permeability from total permeability, and α and m are linearly fitted to soft fraction permeability by double log coordinatesObtained.
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| 含流体致密砂岩的纵波频散及衰减:基于双重双重孔隙结构模型描述的特征分析;巴晶 等;地球物理学报;第61卷(第03期);第1053-1068页 * |
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