CN115508891A - Reservoir parameter prediction method based on Bayesian rock physics inversion of linear Gaussian distribution - Google Patents
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Abstract
The invention relates to a reservoir parameter prediction method based on Bayesian rock physical inversion of linear Gaussian distribution. A novel Bayes rock physics inversion technology is characterized in that seismic data are used as input, a Gray reflection coefficient approximation formula, a Gassmann equation and a Nur critical porosity model are combined, linear relations among the fluid saturated rock volume modulus, the shear modulus and the density, the matrix volume modulus, the matrix density, the fluid volume modulus, the fluid density and the porosity are constructed under first-order Taylor series approximation, stable rock physics inversion of linear Gaussian distribution is developed by utilizing an iterative algorithm under the constraint of a Bayes frame, and stable prediction of reservoir rock physics parameters such as clay content and water saturation is realized by combining a Voigt-Reuss-Hill average formula, a density weighted average formula and a Wood formula. The method has low dependency on the initial model, and the posterior mean value is assigned to the initial model through continuous iteration until the model converges.
Description
Technical Field
The invention relates to the technical field of seismic reservoir prediction and seismic rock physics, in particular to a reservoir parameter prediction technology based on Bayesian rock physics inversion of linear Gaussian distribution.
Background
Reservoir parameters of a formation mainly include elastic parameters, fluid types, petrophysical properties, and the like. The elastic parameters include compressional wave velocity, shear wave velocity, density, bulk modulus, shear modulus, etc. of the formation, the fluid types include oil, gas, water, etc. in the reservoir pores, and the petrophysical properties include porosity, permeability, fluid saturation, mineral composition, etc.
Generally, assuming that the physical relationship between the model properties and the observations is known, a geophysical inverse problem is given a set of geophysical observations and estimates of the subsurface model properties from the observations. The most common inversion in reservoir geophysical is seismic inversion, which is widely used for reservoir parameter prediction, where the velocity or impedance of a subsurface medium is predicted from seismic data through seismic amplitude and travel time information, which is called elastic parameter inversion. Similarly, direct prediction of rock and fluid properties from seismic data, such as porosity, mineral composition, and fluid saturation, may also be expressed as an inversion, commonly referred to as a petrophysical inversion.
The seismic wave propagation theory and the rock physical model are bridges for establishing the connection between rock physical properties and seismic observation data. Due to measurement errors and approximate uncertainty of seismic observation data, the solution to inversion is typically not unique. Geophysicists have proposed many methods of solving geophysical inversion problems, including deterministic methods and probabilistic methods. Most of the methods are applied to the inversion solving process of earthquake, and then the method is expanded to the inversion solving based on the rock physical property.
Disclosure of Invention
In view of the above problems, the present invention provides a linear gaussian distribution based bayesian rock physics inversion method, which is based on a new linearized forward model based on a linear approximation formula with amplitude varying with offset and a linearized rock physics model, and solves the inversion by using a convolution forward model of seismic wavelets and the linearized approximation equation.
In order to achieve the purpose, the invention adopts the following technical scheme:
in order to solve the inversion, firstly, the seismic forward model needs to be linearized, and the forward model in the invention is a convolution model based on seismic wavelets and reflection coefficients. Seismic reflection coefficients are based primarily on the Zoeppritz equation, which many expert scholars apply to linear approximations characterized by a number of different elastic parameters, including Aki-Richards approximations, fatti approximations, gray approximations, and the like. Since the invention develops a linearized petrophysical inversion, in order to establish a connection with a petrophysical model, the invention uses the reflection coefficient and the bulk modulus K of the fluid saturated rock sat Shear modulus mu sat And the Gray reflection coefficient approximation formula for the linear characteristic between the densities ρ.
Second, reservoir elastic parameters and pores are less common in reservoir characterization due to petrophysical model linearizationThe relationship between degree and mineral volume is generally close to a linear relationship, but the relationship between water saturation and saturation is generally non-linear, particularly when the pore fluid is homogeneously mixed in the pore space. The Gassmann equation gives the elastic modulus (K) of the fluid saturated rock sat And mu sat ) Modulus of elasticity (K) of dry rock skeleton dry And mu dry ) The relationship between them, therefore, a model is also needed in the Gassmann equation to calculate the modulus of elasticity (K) of the dry rock dry And mu dry ) While the Nur critical porosity model is a model for characterizing the dry rock elastic modulus (K) dry And mu dry ) Modulus of elasticity (K) with the matrix m And mu m ) Exhibiting a linear variation. In the invention, in the process of rock physical modeling, a Gassmann equation is combined with a Nur critical porosity model to obtain a rock physical model, the rock physical model is linearized by utilizing the first-order expansion approximation of the Taylor series, and the linearized rock physical model shows the volume modulus K of the fluid saturated rock sat Shear modulus mu sat And density ρ and bulk modulus K of the matrix m Matrix shear modulus μ m Matrix density ρ m Bulk modulus K of fluid f Fluid density ρ f And porosity phi is linear.
Thirdly, gray's approximation formula gives the reflection coefficient as a function of the bulk modulus K of the fluid saturated rock sat Shear modulus K sat And the linear variation relation of the density rho, and the Gassmann equation and the Nur critical porosity model give the bulk modulus K of the fluid saturated rock sat Shear modulus mu sat And density ρ and matrix bulk modulus K m Shear modulus mu m Density rho m Bulk modulus K of fluid f Fluid density ρ f And a linear variation of the porosity phi. Therefore, the Gray approximate formula, the Gassmann equation and the Nur critical porosity model are integrated, the model parameterization is carried out on the reflection coefficient formula again, and the reflection coefficient K along with the matrix bulk modulus is obtained m Shear modulus mu m Density ρ m Bulk modulus K of fluid f Fluid density ρ f And porosity phi-like reservoir parameter lineExpressions of sexual variation.
Finally, carrying out stable rock physics inversion of linear Gaussian distribution by using reservoir parameters and a reflection coefficient linearization expression and using an iterative algorithm under the constraint of a Bayesian framework to obtain reservoir parameters (K) m 、μ m 、ρ m 、K f 、ρ f Phi) in the same manner. While the Voigt-reus-Hill average formula, the density weighted average formula and the Wood formula respectively give reservoir parameters (K) m 、μ m 、ρ m 、K f 、ρ f Phi) and water saturation s w And clay volume fraction v c The relationship between the porosity phi and the water saturation s of the rock is finally realized by combining a Bayesian rock physics inversion formula w Clay content v c And predicting the rock physical parameters of the reservoir.
Due to the adoption of the technical scheme, the invention has the following advantages:
1. the Gassmann equation gives the elastic modulus (K) of the fluid saturated rock sat And mu sat ) Modulus of elasticity (K) of dry rock skeleton dry And mu dry ) The relation between the two, which is characterized by weak nonlinearity, and the Nur critical porosity model gives the elastic modulus (K) of the dry rock dry And mu dry ) Modulus of elasticity (K) with the matrix m And mu m ) The relationship between the two is weak nonlinear relationship after being combined, in order to solve the problem, the rock physical model has better linear characteristic, the invention utilizes first-order Taylor series expansion to linearize the weak nonlinear rock physical model to obtain the elastic parameter (K) of the fluid saturated rock sat And mu sat ) And reservoir parameter (K) m 、μ m 、ρ m 、K f 、ρ f Phi) is determined.
2. The seismic data are used as input, in the Bayesian rock physical inversion process, the dependence on a rock physical model is strong, the rock physical model depends on the initial value of a linearized model expanded by a Taylor series, and in the inversion process, the prior value of reservoir parameters is unknown, so the initial value is assumed to be constant and equal to the mean value of the reservoir parameters, but the initial model may have a larger difference with the true value. In order to solve the problem, an iterative method is proposed in the patent, which has low dependency on an initial model and assigns posterior means to the initial model through continuous iteration until the model converges.
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Various other advantages and benefits will become apparent to those of ordinary skill in the art upon reading the following detailed description of the preferred embodiments. The drawings are only for purposes of illustrating the preferred embodiments and are not to be construed as limiting the invention. Like parts are designated with like reference numerals throughout the drawings. In the drawings:
FIG. 1 is a schematic view of a well logging curve of a working area A in the east;
FIG. 2 is a schematic diagram of a synthetic seismic record partially stacked at three angles using A-well data and wavelet convolution;
FIG. 3 is a diagram illustrating Bayesian petrophysical inversion results of linear Gaussian distribution of a well A;
FIG. 4 is a schematic view of a seismic section superimposed on three angle portions of an actual work area in the east;
FIG. 5 is a schematic diagram of a Bayesian reservoir parameter inversion profile of linear Gaussian distribution in an actual work area of the east; and
FIG. 6 is a schematic diagram of a Bayesian petrophysical parameter inversion profile of linear Gaussian distribution in an actual work area of the east.
The reference symbols in the drawings denote the following:
Detailed Description
Exemplary embodiments of the present invention will be described in more detail below with reference to the accompanying drawings. While exemplary embodiments of the invention are shown in the drawings, it should be understood that the invention can be embodied in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.
A reasonable inversion method should target the estimation of the most likely model while quantifying the uncertainty in the model prediction. The probabilistic method of inverting the problem provides a natural framework for seismic and petrophysical inversion. In the probabilistic approach, the inverted solution may be represented as a Probability Density Function (PDF) of model variables or a set of model realizations sampled from the Probability Density Function (PDF), resulting in uncertainty in the model.
Therefore, the Bayesian inversion method is introduced in the invention to evaluate the posterior distribution of the model variables of a given set of measured geophysical data, and an analytic solution of the seismic inversion problem based on the seismic forward model linearization and the prior distribution of the model variables and the measurement errors under the Gaussian distribution assumption is provided. The Bayesian linearization method is subsequently extended to seismic reservoir prediction and fluid identification, the conventional reservoir prediction and fluid identification can only predict reservoir thickness, reservoir fluid type, reservoir scale and other parameters and cannot exactly characterize reservoir parameters such as reservoir porosity, reservoir fluid saturation, reservoir mineral volume and the like, and seismic petrophysical modeling is to establish a relationship between petrophysical and elastic variables through a petrophysical model.
Therefore, a Bayesian rock physical reservoir parameter inversion method based on linear Gaussian distribution is developed by combining seismic rock physical modeling under the framework of linear Gaussian distribution Bayesian distribution, and the rock physical inversion technology for directly predicting reservoir parameters is realized.
The method comprises the steps of firstly parameterizing a model of a reflection coefficient on the basis of research results of predecessors, representing the model into a linear reflection coefficient approximate formula corresponding to a rock physical model, then constructing a linear relation between an elastic parameter and a difference parameter through rock physical modeling, combining the linear reflection coefficient approximate formula with the rock physical model, re-modeling the reflection coefficient formula into linear representation of reservoir parameters, and finally developing linear Gaussian distribution Bayesian rock physical inversion prediction under the constraint of a Bayesian framework.
The whole calculation and implementation process can be divided into four steps:
step 1: linear reflection coefficient approximation formula derivation based on elastic parameters
Seismic data can be approximately represented as a convolution of seismic wavelets and a series of reflection coefficients, wherein the reflection coefficients are the elastic parameter difference of each layer interface, and therefore the seismic reflection coefficients depend on the elastic characteristics of each layer above and below the underground interface. The elastic property of the isotropic medium can be defined by three elastic parameters, namely longitudinal wave velocity V P Transverse wave velocity V S And the density p. Thus, seismic data can be characterized as:
d(t,θ)＝w(t,θ)*r PP (t,θ)＝∫w(u,θ)r PP (t-u,θ)d u (1)
where d (t, θ) is the resulting synthetic seismic record, w (t, θ) is the seismic wavelet, x is the convolution operator, r PP And (t, theta) is a PP reflection coefficient, and when t is earthquake two-way travel, theta is an earthquake incidence angle. Reflection coefficient r PP (t, θ) as a function of time t and angle of incidence θ can be calculated accurately using the Zoeppritz equation or can be obtained using the Aki-Richards linear approximation.
In the invention, the reflection coefficient is linearly approximated by Aki-Richards, namely, the intercept R, the gradient G and the curvature F can be expressed by the longitudinal wave velocity, the transverse wave velocity and the density as follows:
wherein:
wherein R represents intercept, G represents gradient, F represents curvature, and Δ V P Showing the difference of longitudinal wave speeds of the upper and lower layers of medium,
means, deltaV, representing the longitudinal wave velocities of the upper and lower layers of the medium S Showing the difference of the transverse wave speeds of the medium of the upper layer and the medium of the lower layer,
the mean value of the transverse wave speeds of the upper and lower layers of medium is shown, the DeltaP represents the difference of the densities of the upper and lower layers of medium,
mean value, t, representing the density of the medium of the upper and lower layers i And t i+1 Respectively, when two consecutive trips through the interface are made. Therefore, the reflection coefficient r PP The approximation of (θ) is:
theta is the seismic angle of incidence and the travel time variable t is omitted to simplify the notation. The reflectance formula, which can extend the single interface reflectance in equation (4) to continuous time, can be expressed as:
expanding the formula (5) to obtain the reflection coefficient and the saturated rock elastic modulus (K) sat And mu sat ) In relation to the density ρ, AVO expressed as Gray et al is approximated as follows:
wherein K is sat 、μ sat And ρ represent the bulk modulus, shear modulus, and density of the saturated rock, respectively.
Thus, if the elastic properties of the formation are known, the seismic response can be calculated:
d(t,θ)＝w(t,θ)*r PP (θ) (7)
wherein the function r PP (θ) can be calculated according to equations (5) and (6).
And 2, step: constructing a rock physical model, and obtaining the relation between the elastic parameter and the reservoir parameter
Seismic petrophysics is the study of the relationship between petrophysical properties (such as porosity, mineral composition, and fluid saturation) and elastic properties (such as elastic modulus, velocity, density, and impedance). In this patent, a linear petrophysical model is constructed based primarily on the Gassmann equation and the critical porosity model of Nur. The premise of the Gassmann equation is to estimate the elastic modulus of a fluid saturated rock, assuming the subsurface medium is isotropic and assuming the pore spaces are connected and in pressure equilibrium:
wherein K sat Bulk modulus, K, of fluid saturated rock dry Is the bulk modulus, K, of the dry rock skeleton m Is the bulk modulus of the rock matrix, K f Is the bulk modulus of the fluid,. Phi. sat Shear modulus, μ, of fluid saturated rock dry The shear modulus mu of the fluid saturated rock is that of the dry rock skeleton, since the presence of the fluid has no influence on the shear modulus of the rock sat Shear modulus mu with dry rock skeleton dry Are equal.
The modulus of elasticity of the dry rock skeleton is either measured in the laboratory or calculated from petrophysical models. Nur (1992) proposed a critical porosity model for estimating the relationship between the dry rock skeleton modulus and the matrix modulus, where the dry rock elastic modulus is expressed as a linear function of the porosity φ, and the linear regression coefficient depends on the matrix modulus and the critical porosity φ 0 The expression is as follows:
rock density ρ is typically expressed as a linear combination of matrix density and fluid density, expressed as:
ρ＝ρ m (1-φ)+ρ f φ (10)
where ρ is m Is rock baseDensity of matter, p f Is the density of the pore fluid.
A petrophysical model is obtained by combining the Gassmann equation of equation (8), the critical porosity model of equation (9), and the density model of equation (10). This petrophysical model considers the elastic properties (K) in equation (6) sat 、μ sat And ρ) and reservoir properties (K) m 、μ m 、ρ m 、K f 、ρ f And phi) in which the bulk modulus K of the matrix m Shear modulus mu m And density ρ m Depending on the mineral composition of the porous rock, the bulk modulus K of the same pore fluid f And density ρ f Depending on the composition of the pore fluid.
Shear modulus mu for fluid saturated rock sat And density p, equations (8) and (9) indicate μ sat Relative to μ m Is linear, and equation (10) indicates ρ relative to ρ m Is linear. The elastic model of saturated rock is therefore linearly dependent on the matrix elastic model. The major non-linear changes in this petrophysical model are usually due to the fluid bulk modulus K f Therein, a fluid volume model K f The expression of (c) is:
wherein K w Denotes the bulk modulus of water in the pores, K g Denotes the bulk modulus, s, of the gas in the pores w Indicating the water saturation. It is shown by the expressions (8) to (11) that the saturated rock elastic modulus (K) is obtained in the presence of a plurality of pore fluids sat And mu sat ) Saturated with water w The degree of sum shows a non-linear relationship with the bulk modulus K of the fluid f Is a linear relationship.
Bulk modulus K for fluid saturated rock sat Since the Gassmann equation shows weak non-linear characteristics, the invention provides a mathematical method for linearizing a weak non-linear petrophysical model by using a first-order taylor series approximation. The purpose of the Taylor series is to supply the independent variableAt constant value, an arbitrary function is approximated using a polynomial in which the coefficients depend on the derivative of the function. For simple functions, a good approximation can be obtained using finite terms. Taylor series are used in geophysics to approximate complex nonlinear functions, and in petrophysical inversion we are interested in linear approximations of petrophysical models, so the taylor series expansion of the petrophysical equations retains only the first term.
Therefore, reservoir parameters (matrix bulk modulus K) for linearization are needed m Matrix shear modulus μ m Matrix density ρ m Fluid bulk modulus K f Fluid density ρ f And porosity phi) to develop model parameterization. Using the reservoir parameters as input to calculate the fluid saturation rock bulk modulus K sat Saturated rock shear modulus μ sat And elastic parameters such as rock density ρ.
And 3, step 3: model parameterization reflection coefficient, and linear relation between reservoir parameters and reflection coefficient is obtained
The primary goal of seismic reservoir characterization is to predict models of rock and fluid properties from well log data and seismic data. The formation elasticity parameter depends on the rock and fluid properties, and as porosity increases, the velocity generally decreases; as the gas saturation increases, the corresponding water saturation decreases, and the velocity generally decreases. If d represents the observed seismic data and q represents the elastic parameters to be characterized in the real subsurface, we can associate the elastic parameters q with the observed seismic data d by a set of geophysical operators f:
d＝f(q)+ε (12)
where epsilon represents random noise in the data. The geophysical operator f can comprise a petrophysical model and a seismic model, wherein the petrophysical model comprises an equivalent medium theory, a inclusion model, an empirical model for connecting reservoir parameters with elastic parameters and the like, the seismic model mainly refers to a geophysical equation for connecting the elastic parameters with seismic response, such as a Zoeppritz equation and an Aki-Richards linear approximation equation, and the elastic parameters q are used for solving as an inversion.
Based on the invention, a petrophysical inversion is carried out, q represents elastic parameters such as longitudinal wave velocity, transverse wave velocity and density, m represents reservoir parameters such as porosity phi and clay content v c And the water saturation s w And the like. Thus, the petrophysical model is represented by the function g:
q＝g(m)+e (13)
where e represents the reservoir parameter data error. Substituting equation (13) into equation (12) can result in a petrophysical inversion that can be written as follows:
d＝f(q)+ε＝f[g(m)]+ε (14)
in step 2, a relation between the elastic parameter q and the reservoir parameter m is constructed through rock physics modeling, and the reservoir parameter m = [ K ] m ,μ m ,ρ m ,K f ,ρ f ,φ] T Elastic parameter q = [ K ] sat ,μ sat ,ρ] T 。
After taking logarithms at two sides of the formula (13) and taking the first order of the logarithms by Taylor series expansion, a linearized expression is obtained as follows:
wherein
Is the average value of the reservoir parameters,
is m 0 The constant of (a) is constant (c),
is the Jacobian of the rock physical model at m 0 The value at (a) can be written as follows:
bringing formulae (15) and (16) into formula (6), the reservoir parameter m = [ K ] can be obtained m ,μ m ,ρ m ,K f ,ρ f ,φ] T The expression linearized with reflection coefficient is as follows:
wherein
Wherein
Wherein
The mean value of the bulk modulus of the matrix is expressed,
the mean value of the shear modulus of the matrix is shown,
the mean value of the density of the matrix is expressed,
the mean value of the bulk modulus of the fluid is expressed,
which represents the average value of the density of the fluid,
denotes the mean value of the porosity, phi 0 Representing the critical porosity value.
And 4, step 4: linear Gaussian distribution rock physics inversion development based on Bayesian framework
Under a bayesian framework, the posterior distribution P (m | d) under the conditions of the observed seismic data d with the reservoir parameter m can be expressed as:
where P (m) is the prior distribution of reservoir parameters, P (d | m) is the likelihood function of the observed seismic data, and P (d) is the probability of the observed seismic data, also known as the normalization constant. The formula in equation (20) can be applied to both discrete random variables and continuous random variables. In the application of geophysical inversion, the continuous random variable inversion problem comprises elastic parameter inversion and petrophysical inversion of seismic data, wherein variables of the elastic parameter inversion mainly comprise longitudinal wave velocity, transverse wave velocity, density and the like, and variables of the petrophysical inversion mainly comprise porosity, mineral volume, fluid saturation and the like; the discrete random variable inversion mainly comprises lithofacies or rock type classification of seismic data and the like.
In the case of continuous reservoir parameter variables m, the most common PDF of continuous variables is a gaussian distribution, often referred to as a normal distribution. We say that the random variable X is distributed according to Gaussian
Distribution if its PDFf X (x) Can be written as:
gaussian distribution
The mean is 0 and the variance equal to 1 is also referred to as a standard gaussian distribution (or normal distribution). Gaussian distributions are symmetric and unimodal and can be used to describe many phenomena in nature. A gaussian distribution is defined over the entire set R of real numbers and it is always positive.
In order to solve the problem of inversion of reservoir parameters m, the invention provides a methodBayes inversion method of linear gaussians. Suppose reservoir parameter m = [ K ] m ,μ m ,ρ m ,K f ,ρ f ,φ] T Subject to a priori mean value xi m And the prior covariance matrix is ∑ m Of (e) a Gaussian distribution of (m: N (m; xi) m ,∑ m ) (ii) a Error term ε is mean 0 and covariance matrix is ∑ ε Of [ epsilon ]: N (epsilon; 0, [ sigma ]) in a Gaussian distribution of [ epsilon ] ε ) And is independent of the reservoir parameter m; the posterior distribution m | d also obeys the conditional mean value xi m|d In a Gaussian distribution of (m) d: N (m; xi) m|d ,∑ m|d ) Posterior mean xi m|d Can be expressed in the following form:
ξ m|d ＝ξ m +∑ m G T (G∑ m G T +∑ ε ) -1 (d-Gξ m ) (22)
conditional covariance matrix ∑ m|d The following:
∑ m|d ＝∑ m -∑ m G T (G∑ m G T +∑ ε ) -1 G∑ m (23)
wherein G represents a matrix associated with a linear petrophysical model that depends on a linearized initial model m developed in a Taylor series 0 The value of (c). In equations (17) and (18), we assume an initial model
Is constant and equal to the average value of the reservoir parameter because the a priori value of the reservoir parameter is unknown. If the difference between the initial value and the actually inverted reservoir parameter value is larger, the rock physical model has stronger nonlinear characteristics, and linearization may be inaccurate. To solve this problem, an iterative method is proposed in this patent, assuming initial values of the model
Calculating the solution of inversions (22) and (23), and then calculating m 0 ＝ξ m|d And iterate until convergence.
The main iterative algorithm steps are as follows:
step 1: let the initial model parameters
And 4, step 4: if yes, finishing the calculation; if not, let m 0 ＝ξ m|d ；
And 5: updating the matrix G related to the linear rock physical model, and jumping to the step 2;
calculating reservoir parameters m =of the rock by a Bayesian inversion algorithm]K m ,μ m ,ρ m ,K f ,ρ f ,φ] T And calculating other petrophysical parameters such as mineral volume fraction, water saturation and the like by combining the petrophysical models such as a Voigt-reus-Hill average formula, a weighted average formula, a Wood formula and the like. Assuming that the rock composition contains two minerals, quartz and clay, and two fluids, gas and water, the Voigt-reus-Hill average formula, the density weighted average formula, and the Wood formula are characterized as follows:
ρ m ＝v c ρ c +(1-v c )ρ q (26)
ρ f ＝s w ρ w +(1-s w )ρ g (27)
wherein v is c Denotes the clay content, K c Denotes the bulk modulus, K, of the clay q Denotes the bulk modulus, μ, of quartz c Denotes the shear modulus, μ, of the clay q Denotes the shear modulus, ρ, of quartz c Denotes the density, ρ, of the clay q Denotes the density, s, of quartz w Indicating water saturation, p w Denotes the density of water, p g Denotes the density of gas, K w Denotes the bulk modulus, K, of water g Indicating the bulk modulus of the gas.
Modulus and Density (K) of all minerals and fluids c 、K q 、K w 、K g 、μ c 、μ q 、ρ c 、ρ q 、ρ w 、ρ g ) All of which are constant and known, the clay content v is known from the formula (24) to the formula (28) c K in the main formulas (24) to (26) m 、μ m And ρ m To determine the water saturation s w Mainly represented by ρ of the formulas (27) to (28) f And K f To decide. According to Bayesian linear rock physics inversion formulas (22) and (23), the clay content v c And the water saturation s w The posterior distribution of (A) can have a reservoir parameter m = [ K = [) m ,μ m ,ρ m ,K f ,ρ f ,φ] T The calculation was performed as follows:
wherein
Denotes the clay content v c Is determined as a function of the cumulative density of the particles,
indicating the water saturation s w Is determined as a function of the cumulative density of the particles,
and
the marginal probability density functions of the rock and fluid obtained by equations (22) and (23).
The invention also provides a practical application embodiment, which is as follows:
the well logging curves of certain areas in the east are known as shown in fig. 1, and comprise six curves of longitudinal wave velocity, transverse wave velocity, density, porosity, clay content and water saturation. Assuming that there are three partial angle stacks, small, medium and large, the corresponding angles of incidence are 10, 20 and 30 degrees, respectively. The corresponding seismic response is then calculated according to equation (1) using a 30Hz Rake wavelet. Firstly, the reflection coefficient of the linearized approximation is calculated by using equation (5), then corresponding seismic records are synthesized by using corresponding convolution models for different incidence angles in equation (1), and finally the obtained seismic data set is shown in fig. 2, wherein the vertical axis is the double-travel time.
A bayesian petrophysical inversion technique based on linear gaussian distribution is performed on the partial synthetic stacked seismic records in fig. 2, and corresponding petrophysical parameters can be obtained as shown in fig. 3. The black is a logging measured value, the gray is a Bayesian rock physical inversion result of linear Gaussian distribution of the A well predicted by the patent technology in the invention, and the filling between the gray and the black represents the probability of the inversion result falling in the interval.
In order to further prove the reliability of the inversion technology in the invention, the technology is applied to the actual work area of the east oil field, the earthquake profile is partially superposed by three angles of the east oil field in fig. 4, and the obvious difference of the earthquake data amplitudes under different incidence angles can be seen from the diagramAnd carrying out linear Gaussian distribution Bayesian rock physics inversion. FIG. 5 shows a Bayesian reservoir parameter (K) for developing a linear Gaussian distribution using equation (17) under a Bayesian framework m 、μ m 、ρ m 、K f 、ρ f Phi) inversion profile, from which the matrix bulk modulus K can be seen m Matrix shear modulus μ m Matrix density ρ m Fluid bulk modulus K f Fluid density ρ f And the inversion section of the porosity phi is well consistent with the measured value of the well logging. The inversion result of fig. 5, combined with the Voigt-Reuss-Hill average formula, the density weighted average formula, the Wood formula and the bayesian inversion formula, can finally obtain the bayesian rock physical parameter inversion result based on the linear gaussian distribution as shown in fig. 6, and it can be seen from the graph that the porosity phi and the clay content v are shown c And water saturation s w The method is well matched with measured values on the well, and further proves that the prediction method is a reliable inversion prediction technology.
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (10)
1. A reservoir parameter prediction method based on linear Gaussian distribution Bayesian rock physics inversion is characterized by comprising the following steps:
using Gray approximate formula to give a linear variation relation of the reflection coefficient with the volume modulus of the fluid saturated rock, the shear modulus of the fluid saturated rock and the density of the fluid saturated rock;
representing the relation between the elastic modulus of the fluid saturated rock and the elastic modulus of the dry rock skeleton through a Gassmann equation, representing the linear change relation between the elastic modulus of the dry rock skeleton and the elastic modulus of the matrix through a Nur critical porosity model, and combining the Gassmann equation with the Nur critical porosity model to obtain a rock physical model;
linearizing the rock physical model by using a first-order expansion approximation of a Taylor series, wherein the linearized rock physical model shows that the bulk modulus, the shear modulus and the density of the fluid saturated rock are in a linear relation with the bulk modulus of the matrix, the shear modulus of the matrix, the density of the matrix, the bulk modulus of the fluid, the density of the fluid and the porosity of the rock;
synthesizing a Gray approximate formula, a Gassmann equation and a Nur critical porosity model, and carrying out model parameterization on the reflection coefficient to obtain an expression that the reflection coefficient linearly changes along with the bulk modulus of the matrix, the shear modulus of the matrix, the density of the matrix, the bulk modulus of the fluid, the density of the fluid and the porosity of the fluid; and
and under the constraint of a Bayesian frame, carrying out stable rock physical model inversion of linear Gaussian distribution by using an iterative algorithm, and then combining a Voigt-reus-Hill average formula, a density weighted average formula and a Wood formula to realize prediction of reservoir parameters.
2. The reservoir parameter prediction method based on linear gaussian-distributed bayesian petrophysical inversion of claim 1, wherein the reflection coefficients are linearly approximated by Aki-Richards, and the intercept, gradient and curvature are expressed by compressional velocity, shear velocity and density as follows:
wherein:
wherein R represents intercept, G represents gradient, and F represents curvature，△V P Showing the difference of longitudinal wave speeds of the upper and lower layers of medium,
means, deltaV, representing the longitudinal wave velocities of the upper and lower layers of the medium S Showing the difference of the transverse wave speeds of the medium of the upper layer and the medium of the lower layer,
the average value of the transverse wave speeds of the upper and lower layers of medium is shown, the Deltarho represents the difference of the densities of the upper and lower layers of medium,
means, t, representing the density of the medium in the upper and lower layers i And t i+1 Respectively representing two consecutive trips through the interface;
reflection coefficient r PP The approximation of (θ) is:
wherein r is PP (t, theta) represents a PP reflection coefficient, t represents earthquake two-way travel time, and theta represents an earthquake incidence angle;
extending the single interface reflection coefficient in equation (4) to the continuous-time reflectivity equation:
equation (5) is expanded to relate the reflection coefficient to the saturated rock elastic modulus and density, expressed as follows:
wherein K sat Denotes the bulk modulus, μ, of the fluid saturated rock sat Indicating fluid saturationThe shear modulus of the rock, ρ, represents the density of the fluid saturated rock;
if the elastic properties of the formation are known, the seismic response can be calculated:
d(t,θ)＝w(t,θ)*r PP (θ) (7)
wherein the function r PP (θ) is calculated from the equations (5) and (6).
3. The reservoir parameter prediction method based on linear gaussian-distributed bayesian petrophysical inversion of claim 1, wherein said Gassmann equation assumes the subsurface medium is isotropic and the pore space is connected and in pressure equilibrium for estimating the elastic modulus of the fluid saturated rock:
wherein K sat Bulk modulus, K, of fluid saturated rock dry Is the bulk modulus, K, of the dry rock skeleton m Is the bulk modulus of the rock matrix, K f Is the bulk modulus of the fluid, phi is the porosity, mu sat Shear modulus, μ, of fluid saturated rock dry Shear modulus of dry rock skeleton, shear modulus of fluid saturated rock mu sat Shear modulus mu with dry rock skeleton dry Are equal.
4. The reservoir parameter prediction method based on linear gaussian distributed bayesian petrophysical inversion of claim 3, wherein the Nur critical porosity model is used to estimate the relationship between the skeleton modulus and the matrix modulus of the dry rock, wherein the elastic modulus of the dry rock is expressed as a linear function of the porosity, and the linear regression coefficient depends on the matrix modulus and the critical porosity and has the expression:
wherein phi is 0 Represents the critical porosity;
rock density is usually expressed as a linear combination of matrix density and fluid density, expressed as:
ρ＝ρ m (1-φ)+ρ f φ (10)
where ρ is m Is the density of the rock matrix, p f Is the density of the pore fluid;
by combining the formula (8), the formula (9) and the formula (10), a petrophysical model is obtained.
5. The reservoir parameter prediction method based on linear gaussian distributed bayesian petrophysical inversion of claim 4, wherein said obtaining a petrophysical model comprises:
for the shear modulus and density of the fluid saturated rock, equations (8) and (9) indicate that the shear modulus of the fluid saturated rock is linear with respect to the shear modulus of the matrix, equation (10) indicates that the density of the rock is linear with respect to the density of the matrix, the elasticity model of the saturated rock varies linearly with the elasticity model of the matrix, and the nonlinear variation in the petrophysical model is due to the bulk modulus of the fluid, wherein the expression of the fluid volume model is:
wherein:
K f the bulk modulus of the fluid is expressed,
K w which represents the bulk modulus of the water in the pores,
K g the bulk modulus of the gas in the pores is expressed,
s w indicating the water saturation.
6. The reservoir parameter prediction method based on linear Gaussian distributed Bayesian petrophysical inversion according to claim 2,
the elastic parameters are related to the observed seismic data by equation (12):
d＝f(q)+ε (12)
wherein,
d represents seismic data;
f represents a geophysical operator;
q represents an elasticity parameter;
ε represents the random noise in the data;
the petrophysical inversion comprises:
the petrophysical model is represented by the function g:
q＝g(m)+e (13)
wherein e represents a reservoir parameter data error, q represents an elastic parameter, and m represents a reservoir parameter;
substituting equation (13) into equation (12) yields the petrophysical inversion:
d＝f(q)+ε＝f[g(m)]+ε (14)
constructing a relation between the elastic parameter q and the reservoir parameter m through rock physics modeling,
let the reservoir parameter m = [ K = m ,μ m ,ρ m ,K f ,ρ f ,φ] T Elastic parameter q = [ K ] sat ,μ sat ,ρ] T ；
After taking logarithms at two sides of the formula (13) and taking the first order of the logarithms by Taylor series expansion, a linearized expression is obtained as follows:
wherein
Is the average value of the reservoir parameters,
is m 0 The constant of (a) is constant (c),
is the jacobian of the petrophysical model;
at m 0 The values at (a) are written as follows:
bringing formulas (15) and (16) into formula (6) to obtain
Reservoir parameter m = [ K = m ,μ m ,ρ m ,K f ,ρ f ,φ] T The expression linearized with the reflection coefficient is as follows:
wherein
Wherein
The mean value of the bulk modulus of the matrix is expressed,
the mean value of the shear modulus of the matrix is shown,
the mean value of the density of the matrix is expressed,
the mean value of the bulk modulus of the fluid is expressed,
which represents the average value of the density of the fluid,
means of porosity, phi 0 Representing the critical porosity value.
7. The reservoir parameter prediction method based on linear gaussian-distributed bayesian petrophysical inversion according to claim 1, characterized in that under bayesian framework, the posterior distribution P (m | d) of the reservoir parameter m under the condition of the observed seismic data d is expressed as:
wherein P (m) is prior distribution of reservoir parameters, P (d | m) is a likelihood function of observing seismic data, and P (d) is probability of observing seismic data;
the formula in equation (20) is applied to both discrete random variables and continuous random variables.
8. The reservoir parameter prediction method based on linear Gaussian distribution Bayesian petrophysical inversion according to claim 7, wherein in the case of reservoir parameter variable m continuous, the most common PDF of continuous variables is Gaussian distribution, commonly referred to as normal distribution;
distribution of random variable X according to Gauss
Distribution if its PDFf X (x) Writing into:
gaussian distribution
The mean is 0 and the variance is equal to 1, called the standard gaussian distribution.
9. The reservoir parameter prediction method based on linear gaussian-distributed bayesian petrophysical inversion of claim 8, wherein the solution of the inversion of the reservoir parameter m is solved by the linear gaussian bayesian inversion method assuming the reservoir parameter m = [ K ]) m ,μ m ,ρ m ,K f ,ρ f ,φ] T Subject to a priori mean value xi m And the prior covariance matrix is sigma m Gaussian distribution of (m) N (m; xi) m ,∑ m ) (ii) a Error term ε is mean 0 and covariance matrix ∑ ε Of [ epsilon ]: N (epsilon; 0, [ sigma ]) in a Gaussian distribution of [ epsilon ] ε ) And is independent of the reservoir parameter m; the posterior distribution m | d also obeys the conditional mean value xi m|d In a Gaussian distribution of (m) d: N (m; xi) m|d ,∑ m|d ) Posterior mean xi m|d Expressed in the following form:
ξ m|d ＝ξ m +∑ m G T (G∑ m G T +∑ ε ) -1 (d-Gξ m ) (22)
conditional covariance matrix sigma m|d The following were used:
∑ m|d ＝∑ m -∑ m G T (G∑ m G T +∑ ε ) -1 G∑ m (23)
wherein G represents a matrix related to a linear petrophysical model, which depends on a linearized initial model m developed by a Taylor series 0 The value of (c).
10. The reservoir parameter prediction method based on linear gaussian distributed bayesian petrophysical inversion of claim 9, wherein said iterative algorithm comprises the following steps:
step 1: let initial model parameters
Step 2: calculating posterior mean value xi m|d Sum conditional covariance matrix ∑ m|d ；
And step 3: judge | | | d-G (ξ) m|d ) Whether | is less than a set error;
and 4, step 4: if yes, finishing the calculation; if not, let m 0 ＝ξ m|d ；
And 5: updating a matrix G related to the linear rock physical model, and jumping to the step 2;
calculating reservoir parameters m = [ K ] of the rock by a Bayesian inversion algorithm m ,μ m ,ρ m ,K f ,ρ f ,φ] T Assuming that the rock composition contains two fluids, i.e., quartz and clay minerals and gas and water, the Voigt-reus-Hill average formula, the density weighted average formula, and the Wood formula are characterized as follows:
ρ m ＝v c ρ c +(1-v c )ρ q (26)
ρ f ＝s w ρ w +(1-s w )ρ g (27)
wherein v is c Denotes the clay content, K c Denotes the bulk modulus, K, of the clay q Denotes the bulk modulus, μ, of quartz c Denotes a clayShear modulus of (d), mu q Denotes the shear modulus, ρ, of quartz c Denotes the density of the clay, p q Denotes the density, s, of quartz w Representing the water saturation, p w Denotes the density of water, p g Denotes the density of gas, K w Denotes the bulk modulus, K, of water g Represents the bulk modulus of gas;
modulus and density (K) of all minerals and fluids c 、K q 、K w 、K g 、μ c 、μ q 、ρ c 、ρ q 、ρ w 、ρ g ) All of which are constant and known, the clay content v is known from the formula (24) to the formula (28) c K in the main formulas (24) to (26) m 、μ m And ρ m To determine the water saturation s w Mainly represented by ρ of the formulas (27) to (28) f And K f To determine;
according to Bayesian linear rock physics inversion formulas (22) and (23), the clay content v is known c And the water saturation s w The posterior distribution of (A) can have a reservoir parameter m = [ K = m ,μ m ,ρ m ,K f ,ρ f ,φ] T The calculation was performed as follows:
wherein
Denotes the clay content v c The function of the cumulative density of (a),
indicating water saturation s w The function of the cumulative density of (a),
and
the marginal probability density functions of the rock and fluid obtained by equations (22) and (23).
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