TY - JOUR
AU1 - Wu,, He-Zhen
AU2 - Fu,, Li-Yun
AU3 - Ge,, Hong-Kui
AB - Abstract A random medium is used to investigate reservoir heterogeneities in this study. Random media are characterized by autocorrelation functions that allow a construction of spatially anisotropic random structures with different correlation lengths and fluctuation standard deviations. Based on the analysis, we calculate a power spectrum using fast Fourier transform (FFT), which is observed in spatial wavelengths ranging from a few metres to a few thousand metres. Correlation distance and root mean square (RMS) height are directly obtained from the power spectrum. Numerical experiments show that the correlation length and fluctuation standard deviation can cause correlation distance and RMS height undergoing variations. Combining the characteristics of statistical parameters and sonic-log data, we quantitatively analyse the reservoir heterogeneities in the Yanchang Basin. The correlation distance and RMS height of coarse lithofacies in fluvial sandstones interpret a high-energy deposit and strong heterogeneity, affected by different lithological combinations. The correlation lengths decrease gradually from shales, tight sands to gas-bearing sands. Using the sonic-log data from 28 wells in the Yanchang Basin, we compute the isolines of correlation distances and RMS heights for both the He-8 and Shan-1 members in the studied area, which present a correlation with the distribution of gas. This offers an improved foundation for reservoir lateral prediction. reservoir heterogeneities, random media, correlation length, fluctuation standard deviation, correlation distance, RMS height Introduction Based on the classification and properties of diagenesis systems in sedimentary basins, the basin scales in geology could be defined as the scale 105–107 cm (Li and Sun 1997). Basin-scale heterogeneity contains information about the traces of the past sedimentary cycle and tectonic process and has been a major concern to geoscientists because of its importance in resource exploration and development. A sedimentary basin has heterogeneities on many scales, through smooth large-scale heterogeneous structures representing a layered model of the basin associated with small-scale random heterogeneous fluctuations existing at different levels of the basin. Wave scattering by such small-scale random heterogeneities leads to wave amplitude/phase fluctuations and wave attenuation. A number of methods for theoretical and numerical studies of large- and small-scale heterogeneities have been developed for different geological and geophysical environments during the past several decades. Seismic coda that is composed of a superposition of incoherent scattered waves has been proposed as direct evidence of random heterogeneities in the lithosphere (Aki 1969, Sato 1977, Wu and Aki 1985, Frankel and Clayton 1986). The statistical parameters of heterogeneities can be inferred from the coda excitation, scattering attenuation or transmission fluctuation across an array using local or teleseismic events (Wu 1986). Flatté and Wu (1988) combined the ACF (angular coherence functions) and the TCF (transverse coherence functions) of phase and log amplitude fluctuations of seismic waves across the NORSAR array to estimate the statistical distribution of small-scale heterogeneities under NORSAR. Levander and Holliger (1992) showed that coherent interference of randomly scattered waves can lead to misidentification of the crustal reflected events. Holliger and Levander (1992) directly measured the characteristic scale of heterogeneities by examining geological maps and an exposed section of lower crustal rocks. Sonic-log data represent velocity–depth profiles and provide an opportunity to examine the nature of the small-scale random heterogeneities of the upper crust (Levander et al1994, Wu et al1994, Holliger 1996). Source frequencies of sonic logs generally lie between 10 and 50 kHz and correspond to wavelengths of about 0.10–0.50 m in rocks. The source frequencies are significantly higher than the dominant frequencies (<50 Hz) of the seismic signal. The high frequencies and resolution of sonic-log data provide the information of small-scale velocity fluctuations. Sonic-log data thus represent 1D measurements of velocities with a resolution that is at least one order of magnitude higher than estimates obtained from the inversion of surface seismic data or from geological studies (Holliger 1997). Well-log data are reliable for short scale lengths of 1–100 m in the shallow crust (Wu and Aki 1988) and the velocity perturbation index of the heterogeneities are between 0.01 and 0.1. They have been widely used for statistical characteristics of basin-scale heterogeneities. Wu et al (1994) conducted the statistical characteristics of upper crustal heterogeneities using the sonic-log data from the German Continental deep drilling project (KTB). Numeral experiments of wave propagation through the random media built by sonic-log data from boreholes show more or less similarity between seismograms and real data (Levander et al1994, Kneib 1995, Holliger 1997, Frenje and Juhlin 1998). In general, sonic-log data can be divided into a smooth long-wavelength component, which characterizes deterministic large-scale geological structures, and a short-wavelength random component, which is often assumed a Gaussian distribution. Most well-log measurements are limited in sedimentary basins. The nature of depositional processes in a sedimentary basin tends to produce a system of multilayered heterogeneous media where each geological sequence is characteristic of relatively consistent lithologic constitution, spreading over a relatively large tectonic region. Both sonic and exploration seismic frequencies are sensitive to the stratified medium system. Literatures abound with works aimed at describing the deterministic large-scale layered geological sequences as a multiscale self-similar complex structure. O'Doherty and Anstey (1971) proposed a frequency-dependent attenuation attributable to intrabed multiples in a layered medium system, and further elaborated their earlier work by relating seismic frequencies to the nature of sedimentation (Anstey and O'Doherty 2002). The reflectivity series of a sedimentary basin bears the short-period and long-period variations of sea level. Evidence from the analyses of well logs (e.g. Todoeschuck et al1990, Saggaf and Robinson 2000) reveals that the sedimentation cycles of the earth and the corresponding seismic signature possess strikingly similar structures of multiscale self-similarity. Both vertical and lateral small-scale random heterogeneities, such as those found in well-log data, that superimpose on a deterministic large-scale layered geological sequence are conveniently described by mathematical tools such as autocorrelation functions and power spectral density functions. The geological process that forms the short-wavelength random heterogeneities is poorly understood, possibly related to local variations in lithology, porosity, fluid, saturation, etc. The variability of sonic-log data can be described using a statistical approach. It is not the first time that a statistical characterization is used to describe heterogeneities in seismological studies (Aki and Richards 1980, Frankel and Clayton 1984, 1986, Ikelle et al1993, Roth and Korn 1993, Wu et al1994, Fu et al2002). The statistical studies and stochastic models of the small-scale random heterogeneities from well logs may infer some information about the petrophysical properties of a sedimentary basin and particularly provide an effective approach of directly identifying oil/gas reservoirs. In this study, we conduct an estimation of heterogeneity spectra of sonic logs from 28 wells over the Yanchang Basin, a complex continental deposit basin in Western China. The paper is organized as follows. Section 2 gives a description of the all sonic-log data analysed in this study. In section 3, we separate a small-scale stochastic component and a large-scale, deterministic component from the observed velocity–depth function, and then we describe random media for the small-scale fluctuation structure with some examples. In section 4, inversion of sonic-log velocities for small-scale seismic structure are achieved and compared with inversion of synthetic sonic-log data, and then quantitatively reservoir heterogeneity is analysed from different aspects. Finally, section 5 presents some conclusions. Sonic-log data The geology in the Yanchang Basin is well known from many wells drilled in the area. The basin is a deep basin with Paleozoic sediments. Gas fields are widely distributed over the basin, with a low pressure and low saturation. The main reservoirs in the basin lie in the upper Paleozoic associated with fluvial channel sands. The main gas-producing intervals in boreholes are the He-8 and Shan-1 members. These stratigraphic traps, each small in extent, thin, and with poor internal connection, are scattered along the deposit belts of braided and meandering streams in this region. Strong heterogeneities in the fluvial deposit environment are visible along both lateral and vertical directions, posing a challenge for reservoir geologists to locate gas-bearing sand bodies. Severe diagenesis in the upper Paleozoic sediments compacts many primary pores. As a result, the relatively high permeable sand bodies are mainly developed in coarse sedimentary facies with secondary pores. In this study, we analyse sonic logs only. The sonic logs from 28 wells in the basin are sampled in depth at 0.125 m. The intervals including the He-8 and Shan-1 members are picked up for each well, ranging from a few hundred metres to a few kilometres in length. Figure 1 shows velocity–depth curves calculated from the raw sonic logs for four wells. Figure 1 Open in new tabDownload slide Velocity–depth curves calculated from the raw sonic logs for four wells: (a) W1, (b) W15, (c) W7, (d) W25. Note the different scales of the plots. Figure 1 Open in new tabDownload slide Velocity–depth curves calculated from the raw sonic logs for four wells: (a) W1, (b) W15, (c) W7, (d) W25. Note the different scales of the plots. Stochastic analysis methods Removal of large-scale components The removal of deterministic trends is a common practice in statistics. This could be due to the maximum frequency being contained in the deterministic component (Chatfield 1980). Trend removal is to make the small-scale stochastic process stationary from a statistical point of view. However, from a seismological point of view, it is natural to associate the stochastic and deterministic parts of upper crustal velocity structure with the scattered and specular parts of seismic wavefields (Holliger 1997). The deterministic trends primarily depend on the dominant frequency of the seismic signal considered and are associated with low-to-intermediate seismic frequencies. They can be approximated by a polynomial fit to sonic logs. Figure 2 plots the fluctuations of the sonic-log data (previously shown in figure 1) after removing deterministic trends. These curves consist of the small-scale stochastic portion of raw sonic logs. Figure 3 shows the histograms computed from the small-scale fluctuation components shown in figure 2, respectively. We see that the velocity fluctuations can be characterized by Gaussian probability distributions (red lines in figure 3). The data sets of four wells shown in figure 1 are found to be representative in all the sonic logs. Figure 2 Open in new tabDownload slide Small-scale fluctuation components obtained by removing the deterministic trends of the sonic logs shown in figure 1. Figure 2 Open in new tabDownload slide Small-scale fluctuation components obtained by removing the deterministic trends of the sonic logs shown in figure 1. Figure 3 Open in new tabDownload slide Histograms computed from the small-scale fluctuation components shown in figure 2, fitted by the Gaussian probability distributions (red lines). Figure 3 Open in new tabDownload slide Histograms computed from the small-scale fluctuation components shown in figure 2, fitted by the Gaussian probability distributions (red lines). Construction of random media Random models of the small-scale random heterogeneities from well logs may contain some information about the petrophysical properties of a sedimentary sequence. They provide an effective approach for directly identifying oil/gas reservoirs. We describe some random models for the small-scale fluctuation components. The random models of velocity and density fluctuations are formulated for the 1D and 2D cases in this section. The equation for the wavefunction ϕ in a homogeneous, isotropic medium is given (Aki and Richards 1980) by where ρ is the density, and λ and μ are the Lamé parameters. The compressional velocity vp is given by For generally inhomogeneous, isotropic, elastic media, the material parameters can be expressed as where the fluctuations δλ, δρ and δμ are small compared with the average values λ0, ρ0 and μ0. According to the Birch theory (Birch 1961), we assume that the compressional velocity fluctuations and shear wave velocity fluctuations are the same, but the relative density fluctuations are proportional to the proportionality factor k. Therefore, we can describe the heterogeneity of random media using the compressional velocity fluctuations (Korn 1993): where is the background velocity, σ is the relative velocity perturbation and k ≈ 0.3–0.8. From (4), we have We assume that σ is isotropic and stationary in space. The spatial average is defined as 〈σ〉 = 0, where 〈⋅⋅⋅〉 denotes the process of spatial average. For a mean value of free fluctuation, the variance of fluctuation ε2 is defined as ε2 = 〈σ2〉. We then define the normalized autocorrelation function by where r is the position of points. Following Holliger and Levander (1992), the most prominent analytical correlation functions are the Gaussian, exponential and von Karman correlation functions (Wu and Aki 1985, Frankel and Clayton 1986, Goff and Jordan 1988). It is common to use the von Karman, i.e., self-similar, as the autocorrelation function of fluctuation (Ikelle et al1993, Korn 1993, Roth and Korn 1993, Tobias and Serge 1999). Random media characterized by these functions have spatial isotropic structures. We consider the case of self-similar autocorrelation function where a denotes the correlation length and kr is the wavenumber. According to the random process theory, the Fourier transform of the autocorrelation function is the power spectrum density of the stationary random process σ (Roth et al 1993, Fu et al2002, Xi and Yao 2004a, 2004b). In the wavenumber domain, the power spectrum density and relative velocity perturbation have the following relationship: where Pσσ and σ′ are the Fourier transform of autocorrelation function and relative velocity perturbation with respect to the spatial coordinates. From (8), we have Different realizations of a medium with a given autocorrelation function a only differ in their phase ϕ(kx, kz) (Roth et al 1993). To construct a random medium, we design random values to the phase ϕ which are distributed between 0 and 2π. The Fourier transform of equation (7) yields the following 1D and 2D Pσσ expressions, respectively: and where kx and kz are the horizontal and vertical wavenumbers, respectively, and ax and az are the correlation lengths that are parallel and perpendicular to the propagation direction of the incident wave, respectively. H is the Hurst number which was found to range from 0 to 1. The inverse Fourier transform of equation (9) yields the relative fluctuation. From the above calculations, we can construct 1D and 2D random media models. We can investigate how the fluctuation standard deviation ε and the correlation lengths ax and az affect random media as follows. We use the equation (10) to model 1D random media by assuming the Hurst number H ≈ 0. Figure 4(a) shows two randomly relative fluctuations of the same correlation length (a = 2.6 m) but with different fluctuation standard deviations of 0.05 and 0.5. The profiles show similar spatial variation patterns but with different magnitudes. Figure 4(b) shows two randomly relative fluctuations of the same fluctuation standard deviation (ε = 0.05) but with different correlation lengths of 2.6 m and 26 m. The correlation length a carries information about how far away different parts of the variations are still correlated with each other (Fu et al2002). It describes the average scale of fluctuation, that is, the heterogeneity of model. Figure 4 Open in new tabDownload slide (a) Randomly relative fluctuations of the same correlation length (a = 2.6 m) but different standard deviations of 0.05 (red line) and 0.5 (blue line). (b) Randomly relative fluctuations with the same standard deviation (ε = 0.05) but different correlation lengths 2.6 m (red line) and 26 m (blue line). Figure 4 Open in new tabDownload slide (a) Randomly relative fluctuations of the same correlation length (a = 2.6 m) but different standard deviations of 0.05 (red line) and 0.5 (blue line). (b) Randomly relative fluctuations with the same standard deviation (ε = 0.05) but different correlation lengths 2.6 m (red line) and 26 m (blue line). We use equation (11) with anisotropic autocorrelation functions to model 2D random media by assuming the Hurst number H ≈ 0. Figure 5 shows four different 2D random models with different correlation lengths of ax and az. The fluctuation standard deviation of the models is fixed as 0.1, with the average velocity being 5000 m s-1. These random media models are similar to the slabbed cores (Xi and Yao 2004a). The average scale of heterogeneity changes with correlation lengths. Four different models range from nearly spatial isotropic fluctuations (ax = az = 1 m and ax = az = 40 m) to anisotropic (ax = 40 m, az = 1 m), and finally to the plane-layered structure (ax → ∞) shown in figure 5(d). The plane-layered structure changes from 2D to 1D cases and is similar to the continental thin bed formation. These random media models can be used to describe reservoir heterogeneities. Figure 5 Open in new tabDownload slide Four different 2D random media models simulated by equation (11) with double-exponential autocorrelation functions, where ax and az are the autocorrelation lengths. Figure 5 Open in new tabDownload slide Four different 2D random media models simulated by equation (11) with double-exponential autocorrelation functions, where ax and az are the autocorrelation lengths. Application to field data Velocity fluctuations from sonic logs in comparison with synthetic sonic-log data In this section, we compute the stochastic components from the observed sonic-log data. The statistical parameters (i.e., power-law exponent, correlation length and standard deviation) will be generated based on the spectral analysis (Walden and Hosken 1985, Holliger 1996). The statistical properties of the sonic-logs velocity fluctuations could be described by its power spectrum. The power spectrum is estimated by taking FFT analysis on the samples with the Hanning window (Todoeschuck et al1990) and then by taking the square of the amplitude spectrum. The form of the window makes little influence to the results. If the power spectrum of the velocity of well logs is proportional to kα, we then have where C is a constant and α is some power-law exponent of the wavenumber k that is characteristic of fractal phenomena. We see that the above power spectrum approaches a power-law form with a slope of α which is related to the Hurst number H by 2H + 1 for the 1D spectrum (Wu et al1994). A least-squares fit to a straight line is in log–log coordinates (Todoeschuck et al1990, Wu et al1994). The left panel in figure 6 shows the power spectra of sonic-log velocity fluctuations for four wells, with a least-squares fit for the slope values of α. The changes in slopes at high wavenumbers could be due to inherent averaging of the logging process over the active length of the tool used. The straight lines of the power-law fits are shifted by 2 for display purposes in the figure in the logarithmic wavenumber range between -2 and 0. Besides the slope α, two additional parameters can be obtained from the sonic-log velocity fluctuations: the correlation length a and the root mean square (RMS) height σ. The correlation length a of a self-similar medium corresponds to the spatial lag over which the autocorrelation function falls by 1/e in the right panel of figure 6. Figure 6 Open in new tabDownload slide Power spectra (left) and autocorrelation functions (right) of sonic-log velocity fluctuations from four wells. The straight lines in the power spectra are the least-squares fits that are shifted by 2 for display purposes. The straight lines in the autocorrelation functions mark a line with the spatial lag over which the autocorrelation function falls by 1/e. Figure 6 Open in new tabDownload slide Power spectra (left) and autocorrelation functions (right) of sonic-log velocity fluctuations from four wells. The straight lines in the power spectra are the least-squares fits that are shifted by 2 for display purposes. The straight lines in the autocorrelation functions mark a line with the spatial lag over which the autocorrelation function falls by 1/e. In table 1, we compare the statistical parameters of sonic-log velocity fluctuations, calculated from the observed and synthetic sonic logs, respectively. These parameters include the Hurst number H, the correlation length a and the RMS height σ. The values (H1, a1, σ1) are computed from the observed sonic logs and will be used for generation of synthetic sonic-log data. The values (H2, a2, σ2) are calculated with accounting for noise and system response from the synthetic sonic-log data by the following steps (Wu et al1994, Holliger 1996): (1) creating a 1D velocity fluctuation model using the von Karman autocorrelation function and the power spectral density function; (2) filtering the velocity variations by running-average filter; (3) adding white noises, (4) calculating the Hurst number from the power spectrum and (5) calculating the correlation length and the RMS height from the autocorrelation functions. The values (H3, a3, σ3) are obtained from the synthetic sonic-log data without accounting for noise and system response by the same steps as computing the (H2, a2, σ2). From the table, we see that H2 corresponds to relatively small values, whereas H3 corresponds to larger values. a2 and σ2 are comparable to a3 and σ3. Table 1 Comparisons of statistical parameters from observed and synthetic sonic logs. Well H1 σ1 a1 H2 σ2 a2 H3 σ3 a3 W1 0.15 0.64 0.23 0.17 0.53 0.25 0.53 0.58 0.27 W2 0.14 0.69 0.46 0.18 0.54 0.20 0.54 0.58 0.20 W3 0.13 0.68 0.17 0.15 0.50 0.33 0.49 0.54 0.33 W4 0.11 0.71 0.20 0.14 0.52 0.48 0.49 0.56 0.48 W5 0.22 0.72 0.14 0.12 0.51 0.57 0.48 0.57 0.61 W6 0.13 0.64 0.13 0.15 0.51 0.61 0.49 0.56 0.59 W7 0.15 0.67 0.20 0.12 0.51 0.68 0.48 0.56 0.70 W8 0.09 0.68 0.17 0.14 0.52 0.60 0.48 0.55 0.62 W9 0.17 0.70 0.19 0.14 0.52 0.58 0.48 0.56 0.61 W10 0.12 0.67 0.22 0.15 0.51 0.67 0.49 0.55 0.61 W11 0.16 0.76 0.17 0.15 0.52 0.58 0.50 0.55 0.58 W12 0.24 0.70 0.20 0.14 0.50 0.47 0.49 0.54 0.46 W13 0.02 0.71 0.18 0.16 0.52 0.23 0.55 0.58 0.25 W14 0.21 0.71 0.20 0.16 0.52 0.30 0.50 0.54 0.32 W15 0.16 0.76 0.19 0.15 0.51 0.54 0.49 0.54 0.56 W16 0.25 0.76 0.19 0.15 0.51 0.49 0.49 0.54 0.51 W17 0.18 0.68 0.15 0.13 0.51 0.59 0.48 0.57 0.61 W18 0.19 0.55 0.42 0.16 0.52 0.31 0.49 0.54 0.33 W19 0.24 0.67 0.19 0.22 0.60 0.18 0.53 0.58 0.18 W20 0.11 0.70 0.19 0.49 0.55 0.33 0.49 0.55 0.33 W21 0.35 0.74 0.73 0.17 0.58 0.17 0.54 0.60 0.17 W22 0.06 0.68 0.16 0.22 0.61 0.17 0.54 0.59 0.17 W23 0.15 0.70 0.23 0.14 0.49 0.31 0.49 0.54 0.33 W24 0.16 0.62 0.17 0.15 0.51 0.30 0.49 0.54 0.33 W25 0.16 0.69 0.17 0.14 0.51 0.30 0.50 0.55 0.33 W26 0.16 0.64 0.19 0.15 0.50 0.32 0.49 0.54 0.34 W27 0.13 0.69 0.16 0.15 0.51 0.30 0.49 0.54 0.33 W28 0.08 0.70 0.19 0.14 0.51 0.30 0.49 0.54 0.33 Well H1 σ1 a1 H2 σ2 a2 H3 σ3 a3 W1 0.15 0.64 0.23 0.17 0.53 0.25 0.53 0.58 0.27 W2 0.14 0.69 0.46 0.18 0.54 0.20 0.54 0.58 0.20 W3 0.13 0.68 0.17 0.15 0.50 0.33 0.49 0.54 0.33 W4 0.11 0.71 0.20 0.14 0.52 0.48 0.49 0.56 0.48 W5 0.22 0.72 0.14 0.12 0.51 0.57 0.48 0.57 0.61 W6 0.13 0.64 0.13 0.15 0.51 0.61 0.49 0.56 0.59 W7 0.15 0.67 0.20 0.12 0.51 0.68 0.48 0.56 0.70 W8 0.09 0.68 0.17 0.14 0.52 0.60 0.48 0.55 0.62 W9 0.17 0.70 0.19 0.14 0.52 0.58 0.48 0.56 0.61 W10 0.12 0.67 0.22 0.15 0.51 0.67 0.49 0.55 0.61 W11 0.16 0.76 0.17 0.15 0.52 0.58 0.50 0.55 0.58 W12 0.24 0.70 0.20 0.14 0.50 0.47 0.49 0.54 0.46 W13 0.02 0.71 0.18 0.16 0.52 0.23 0.55 0.58 0.25 W14 0.21 0.71 0.20 0.16 0.52 0.30 0.50 0.54 0.32 W15 0.16 0.76 0.19 0.15 0.51 0.54 0.49 0.54 0.56 W16 0.25 0.76 0.19 0.15 0.51 0.49 0.49 0.54 0.51 W17 0.18 0.68 0.15 0.13 0.51 0.59 0.48 0.57 0.61 W18 0.19 0.55 0.42 0.16 0.52 0.31 0.49 0.54 0.33 W19 0.24 0.67 0.19 0.22 0.60 0.18 0.53 0.58 0.18 W20 0.11 0.70 0.19 0.49 0.55 0.33 0.49 0.55 0.33 W21 0.35 0.74 0.73 0.17 0.58 0.17 0.54 0.60 0.17 W22 0.06 0.68 0.16 0.22 0.61 0.17 0.54 0.59 0.17 W23 0.15 0.70 0.23 0.14 0.49 0.31 0.49 0.54 0.33 W24 0.16 0.62 0.17 0.15 0.51 0.30 0.49 0.54 0.33 W25 0.16 0.69 0.17 0.14 0.51 0.30 0.50 0.55 0.33 W26 0.16 0.64 0.19 0.15 0.50 0.32 0.49 0.54 0.34 W27 0.13 0.69 0.16 0.15 0.51 0.30 0.49 0.54 0.33 W28 0.08 0.70 0.19 0.14 0.51 0.30 0.49 0.54 0.33 H1, a1, σ1 = Hurst number, correlation length and standard deviation of velocity fluctuations used for generation of synthetic sonic-log data. H2, a2, σ2 = Hurst number, correlation length and standard deviation of velocity fluctuations obtained from inversion of synthetic sonic-log data accounting for noise and system response. H3, a3, σ3 = Hurst number, correlation length and standard deviation of velocity fluctuations obtained from inversion of synthetic sonic-log data without accounting for noise and system response. Open in new tab Table 1 Comparisons of statistical parameters from observed and synthetic sonic logs. Well H1 σ1 a1 H2 σ2 a2 H3 σ3 a3 W1 0.15 0.64 0.23 0.17 0.53 0.25 0.53 0.58 0.27 W2 0.14 0.69 0.46 0.18 0.54 0.20 0.54 0.58 0.20 W3 0.13 0.68 0.17 0.15 0.50 0.33 0.49 0.54 0.33 W4 0.11 0.71 0.20 0.14 0.52 0.48 0.49 0.56 0.48 W5 0.22 0.72 0.14 0.12 0.51 0.57 0.48 0.57 0.61 W6 0.13 0.64 0.13 0.15 0.51 0.61 0.49 0.56 0.59 W7 0.15 0.67 0.20 0.12 0.51 0.68 0.48 0.56 0.70 W8 0.09 0.68 0.17 0.14 0.52 0.60 0.48 0.55 0.62 W9 0.17 0.70 0.19 0.14 0.52 0.58 0.48 0.56 0.61 W10 0.12 0.67 0.22 0.15 0.51 0.67 0.49 0.55 0.61 W11 0.16 0.76 0.17 0.15 0.52 0.58 0.50 0.55 0.58 W12 0.24 0.70 0.20 0.14 0.50 0.47 0.49 0.54 0.46 W13 0.02 0.71 0.18 0.16 0.52 0.23 0.55 0.58 0.25 W14 0.21 0.71 0.20 0.16 0.52 0.30 0.50 0.54 0.32 W15 0.16 0.76 0.19 0.15 0.51 0.54 0.49 0.54 0.56 W16 0.25 0.76 0.19 0.15 0.51 0.49 0.49 0.54 0.51 W17 0.18 0.68 0.15 0.13 0.51 0.59 0.48 0.57 0.61 W18 0.19 0.55 0.42 0.16 0.52 0.31 0.49 0.54 0.33 W19 0.24 0.67 0.19 0.22 0.60 0.18 0.53 0.58 0.18 W20 0.11 0.70 0.19 0.49 0.55 0.33 0.49 0.55 0.33 W21 0.35 0.74 0.73 0.17 0.58 0.17 0.54 0.60 0.17 W22 0.06 0.68 0.16 0.22 0.61 0.17 0.54 0.59 0.17 W23 0.15 0.70 0.23 0.14 0.49 0.31 0.49 0.54 0.33 W24 0.16 0.62 0.17 0.15 0.51 0.30 0.49 0.54 0.33 W25 0.16 0.69 0.17 0.14 0.51 0.30 0.50 0.55 0.33 W26 0.16 0.64 0.19 0.15 0.50 0.32 0.49 0.54 0.34 W27 0.13 0.69 0.16 0.15 0.51 0.30 0.49 0.54 0.33 W28 0.08 0.70 0.19 0.14 0.51 0.30 0.49 0.54 0.33 Well H1 σ1 a1 H2 σ2 a2 H3 σ3 a3 W1 0.15 0.64 0.23 0.17 0.53 0.25 0.53 0.58 0.27 W2 0.14 0.69 0.46 0.18 0.54 0.20 0.54 0.58 0.20 W3 0.13 0.68 0.17 0.15 0.50 0.33 0.49 0.54 0.33 W4 0.11 0.71 0.20 0.14 0.52 0.48 0.49 0.56 0.48 W5 0.22 0.72 0.14 0.12 0.51 0.57 0.48 0.57 0.61 W6 0.13 0.64 0.13 0.15 0.51 0.61 0.49 0.56 0.59 W7 0.15 0.67 0.20 0.12 0.51 0.68 0.48 0.56 0.70 W8 0.09 0.68 0.17 0.14 0.52 0.60 0.48 0.55 0.62 W9 0.17 0.70 0.19 0.14 0.52 0.58 0.48 0.56 0.61 W10 0.12 0.67 0.22 0.15 0.51 0.67 0.49 0.55 0.61 W11 0.16 0.76 0.17 0.15 0.52 0.58 0.50 0.55 0.58 W12 0.24 0.70 0.20 0.14 0.50 0.47 0.49 0.54 0.46 W13 0.02 0.71 0.18 0.16 0.52 0.23 0.55 0.58 0.25 W14 0.21 0.71 0.20 0.16 0.52 0.30 0.50 0.54 0.32 W15 0.16 0.76 0.19 0.15 0.51 0.54 0.49 0.54 0.56 W16 0.25 0.76 0.19 0.15 0.51 0.49 0.49 0.54 0.51 W17 0.18 0.68 0.15 0.13 0.51 0.59 0.48 0.57 0.61 W18 0.19 0.55 0.42 0.16 0.52 0.31 0.49 0.54 0.33 W19 0.24 0.67 0.19 0.22 0.60 0.18 0.53 0.58 0.18 W20 0.11 0.70 0.19 0.49 0.55 0.33 0.49 0.55 0.33 W21 0.35 0.74 0.73 0.17 0.58 0.17 0.54 0.60 0.17 W22 0.06 0.68 0.16 0.22 0.61 0.17 0.54 0.59 0.17 W23 0.15 0.70 0.23 0.14 0.49 0.31 0.49 0.54 0.33 W24 0.16 0.62 0.17 0.15 0.51 0.30 0.49 0.54 0.33 W25 0.16 0.69 0.17 0.14 0.51 0.30 0.50 0.55 0.33 W26 0.16 0.64 0.19 0.15 0.50 0.32 0.49 0.54 0.34 W27 0.13 0.69 0.16 0.15 0.51 0.30 0.49 0.54 0.33 W28 0.08 0.70 0.19 0.14 0.51 0.30 0.49 0.54 0.33 H1, a1, σ1 = Hurst number, correlation length and standard deviation of velocity fluctuations used for generation of synthetic sonic-log data. H2, a2, σ2 = Hurst number, correlation length and standard deviation of velocity fluctuations obtained from inversion of synthetic sonic-log data accounting for noise and system response. H3, a3, σ3 = Hurst number, correlation length and standard deviation of velocity fluctuations obtained from inversion of synthetic sonic-log data without accounting for noise and system response. Open in new tab Reservoir heterogeneity and distribution of gas-bearing sands With the statistical parameters of sonic-log velocity fluctuations in the Yanchang Basin, we quantitatively analyse reservoir heterogeneities from the aspects of deposit environment, lithological combination and distribution of gas-bearing sands. We use the He-8 member as an example to describe geological and geophysical characteristics. The depositional facies of the He-8 member in the Yanchang Basin belongs to the branch channel of fluvial delta plain (see figure 7). The complex sand bodies extend along both the south and north directions. The thickness of single sand bodies varies greatly with obvious heterogeneities. The values in natural gamma ray, density and compensated neutron logging are low, whereas those in resistivity and acoustic logging are comparatively high. It is difficult to trace the anomalies in seismic profiles. The characteristics of the geological and geophysical responses of the He-8 member are typical and are different from other sections. Quartz and lithic quartz sandstones are main types of reservoir rock in the He-8 member. The sandstones with a high content of quartz, especially coarse quartz, are well graded and become the foundation for good reservoir. The development of secondary pore is the key for building highly permeable reservoirs. Quantitative analyses of reservoir heterogeneities focus on reservoir structure and lithological characteristic using observed sonic logs. Figure 7 Open in new tabDownload slide Sedimentological pattern of the He-8 member (revised from the China Scientific Research Institute of Petroleum Exploration and Development). Figure 7 Open in new tabDownload slide Sedimentological pattern of the He-8 member (revised from the China Scientific Research Institute of Petroleum Exploration and Development). Heterogeneities in fluvial sandstones can be classified using third-order hierarchy. The channel deposits in the Yanchang Basin with fluvial hierarchies can be described by a sedimentological pattern. The effective reservoirs in the Yanchang Basin are such sandstones with coarse lithofacies belts. Sedimentary microfacies consist of high-energy channel bars and planar-flow bars. In general, the high-energy channel bars are dominated by coarse lithofacies. Two typical wells, W23 and W25, are selected for a detailed analysis of lithological and petrophysical properties using sonic logs. The computed β and γ parameters are shown in table 2 where we can see that different lithologies lead to different solutions. The correlation distances β of both wells are largest for shale layers, turn medium for gas-bearing sands and become least for tight sand layers. The RMS heights γ of both wells are smaller for shale layers and tight sands than for gas-bearing sands. The velocity fluctuations for both wells undergo strong variations along the vertical direction. Table 2 Correlation distance β and RMS height γ of different lithologies from W23 and W25. W23 W25 Different lithologies β γ β γ Shale layer 2.26 1.80 5.95 1.35 Tight sand layer 1.17 1.97 0.23 1.21 Gas-bearing sand layer 2.17 2.01 5.89 1.42 W23 W25 Different lithologies β γ β γ Shale layer 2.26 1.80 5.95 1.35 Tight sand layer 1.17 1.97 0.23 1.21 Gas-bearing sand layer 2.17 2.01 5.89 1.42 Open in new tab Table 2 Correlation distance β and RMS height γ of different lithologies from W23 and W25. W23 W25 Different lithologies β γ β γ Shale layer 2.26 1.80 5.95 1.35 Tight sand layer 1.17 1.97 0.23 1.21 Gas-bearing sand layer 2.17 2.01 5.89 1.42 W23 W25 Different lithologies β γ β γ Shale layer 2.26 1.80 5.95 1.35 Tight sand layer 1.17 1.97 0.23 1.21 Gas-bearing sand layer 2.17 2.01 5.89 1.42 Open in new tab Summary results of investigation of all wells We analyse all sonic logs from 28 wells in the Yanchang Basin. The lithological combination for the He-8 and Shan-1 members in these wells is composed mainly of quartzite clastic rocks. The entire analysis is divided into three steps for each well. First, the power spectra of the upper He-8, the lower He-8 and Shan-1 members are estimated from sonic logs for measuring reservoir heterogeneities. Then, the statistical parameters are calculated for each member from the power spectra. Finally, we map these parameters in isoline with well locations marked to associate with the distribution of gas-bearing sands in the basin. Figure 8 shows the β (left panel) and the γ (right panel) isolines of all wells in the studied area, respectively, for the upper He-8 (figures 8(a) and (b)), the lower He-8 (figures 8(c) and (d)) and the Shan-1 (figures 8(e) and (f)) members. In these figures, the blue area represents the least value. Figure 9 illustrates the distribution characteristics of the gas-content index, resulting from integrating Poisson's ratio properties and the thickness parameters of the favourable reservoirs based on 3D seismic data. It presents the gas-content characteristics of effective reservoirs. By comparing figures 8 and 9, we see that the gas distribution is consistent between the β and γ isolines in figure 8 and the gas-content index in figure 9. Figure 8 Open in new tabDownload slide Isolines maps of β for (a) the upper He-8 section, (c) the lower He-8 section and (e) Shan1 member. Isolines maps of γ for (b) the upper He-8 section; (d) the lower He-8 section and (f) Shan1 member. The wells locations were marked by black stars. Figure 8 Open in new tabDownload slide Isolines maps of β for (a) the upper He-8 section, (c) the lower He-8 section and (e) Shan1 member. Isolines maps of γ for (b) the upper He-8 section; (d) the lower He-8 section and (f) Shan1 member. The wells locations were marked by black stars. Figure 9 Open in new tabDownload slide The distribution characteristics of the gas-content index resulting from integrating the Poisson's ratio properties and the thickness parameters of the favourable reservoirs based on 3D seismic data (revised from the China Scientific Research Institute of Petroleum Exploration and Development). Figure 9 Open in new tabDownload slide The distribution characteristics of the gas-content index resulting from integrating the Poisson's ratio properties and the thickness parameters of the favourable reservoirs based on 3D seismic data (revised from the China Scientific Research Institute of Petroleum Exploration and Development). Conclusions and discussions The random medium allows us to investigate media with isotropic heterogeneities. We demonstrate that the random media with exponential autocorrelation function can be used to model reservoir heterogeneities. The correlation lengths a and b that are perpendicular and parallel to the propagation direction, respectively, carry some information on the average scales of fluctuations. The models are quite similar to continental thin formations in 1D random media and slabbed cores in 2D random media. The sonic-log data offer an opportunity to measure directly the properties of reservoir heterogeneities. We conduct a quantitative assessment of reservoir heterogeneities by studying the relation of the statistical characterization of random medium models and sonic-log data in the Yanchang Basin. Statistical parameters such as the correlation distance β and the RMS height γ are computed from the power-law spectra of sonic logs. Combining the characteristics of statistical parameters and sonic-log data in the Yanchang Basin, we analyse reservoir heterogeneities from different aspects, depending on lithological combinations. The correlation distance and RMS height of the coarse lithofacies in the fluvial sandstones are about 1.4 m and 1.1, respectively, corresponding to a high-energy deposit and a weak heterogeneity. The correlation length of shale, tight sand to gas-bearing sand decreases gradually. The isolines of the correlation distances and the RMS heights of the He-8 and Shan-1 members from 28 wells appear to be consistent with the distribution of gas discoveries in the Yanchang Basin. Sonic logs generally contain basin-wide deterministic trends that can be used to characterize basin-scale heterogeneities. Reservoir heterogeneities could be locally described by fine scale information contained in sonic logs, with some fine scale heterogeneities fitted into random medium models, but others controlled in sedimentary succession by non-random processes such as climate, base level variation and denudation rate and morphological evolution of the landscape of the hinterland. Acknowledgments This research was funded by the National High Technology Research and Development Program (863 Program) of China, grant no 2006AA06Z240 and the Central Level, Scientific Research Institutes' Basic R&D Operations Special Fund (DQJB09A01) of the institute of geophysics, China earthquake administration. References Aki K . , 1969 Analysis of seismic coda of local earthquakes as scattered waves , J. Geophys. Res. , vol. 74 (pg. 615 - 31 )http://dx.doi.org/10.1029/JB074i002p00615 01480227 Google Scholar Crossref Search ADS WorldCat Aki K , Richards P G . , 1980 , Quantitative Seismology , vol. vol 2 San Francisco Freeman Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Anstey N A , O'Doherty R F . , 2002 Cycles, layers, and reflections: part 1 , Leading Edge , vol. 21 pg. 44 http://dx.doi.org/10.1190/1.1445847 00168033 Google Scholar Crossref Search ADS WorldCat Birch F . , 1961 The velocity of compressional waves in rocks to 10 kb , J. Geophys. Res. , vol. 66 (pg. 2199 - 224 )http://dx.doi.org/10.1029/JZ066i007p02199 01480227 Google Scholar Crossref Search ADS WorldCat Chatfield C . , 1980 , The Analysis of Time Series: An Introduction London Chapman & Hall Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Flatté M , Wu R S . , 1988 Small scale structure in the lithosphere and asthenosphere deduced from arrival time and amplitude fluctuations , J. Geophys. Res. , vol. 93 pg. 6001 http://dx.doi.org/10.1029/JB093iB06p06001 01480227 Google Scholar Crossref Search ADS WorldCat Frankel A , Clayton R W . , 1984 Finite-difference simulation of wave propagation in two-dimensional random media , Bull. Seismol. Soc. Am. , vol. 74 (pg. 2167 - 86 ) OpenURL Placeholder Text WorldCat Frankel A , Clayton R W . , 1986 Finite-difference simulation of seismic scattering implications for the propagation of short-period seismic waves in the crust and models of crustal heterogeneity , J. Geophys. Res. , vol. 91 (pg. 6465 - 89 )http://dx.doi.org/10.1029/JB091iB06p06465 01480227 Google Scholar Crossref Search ADS WorldCat Frenje L , Juhlin C . , 1998 Scattering of seismic waves simulated by finite difference modelling in random media: application to the Gravberg-1 well, Sweden , Tectonophysics , vol. 293 (pg. 61 - 8 )http://dx.doi.org/10.1016/S0040-1951(98)00079-1 00401951 Google Scholar Crossref Search ADS WorldCat Fu L Y , Wu R S , Campillo M . , 2002 Energy partition and attenuation of regional phases by random free surface , Bull. Seismol. Soc. Am. , vol. 92 (pg. 1992 - 2007 )http://dx.doi.org/10.1785/0120000292 00371106 19433573 Google Scholar Crossref Search ADS WorldCat Goff J A , Jordan T H . , 1988 Stochastic modeling of seafloor morphology—inversion of sea beam data for second order statistics , J. Geophys. Res. , vol. 93 (pg. 13589 - 608 )http://dx.doi.org/10.1029/JB093iB11p13589 01480227 Google Scholar Crossref Search ADS WorldCat Holliger K . , 1996 Upper-crustal seismic velocity heterogeneity as derived from a variety of P-wave sonic logs , Geophys. J. Int. , vol. 125 (pg. 813 - 29 )http://dx.doi.org/10.1111/j.1365-246X.1996.tb06025.x 0956540X 1365246X Google Scholar Crossref Search ADS WorldCat Holliger K . , 1997 Seismic scattering in the upper crystalline crust based on evidence from sonic logs , Geophys. J. Int. , vol. 128 (pg. 65 - 72 )http://dx.doi.org/10.1111/j.1365-246X.1997.tb04071.x 0956540X 1365246X Google Scholar Crossref Search ADS WorldCat Holliger K , Levander A R . , 1992 A stochastic view of lower crustal fabric based on evidence from the Ivrea zone , Geophys. Res. Lett. , vol. 19 (pg. 1153 - 6 )http://dx.doi.org/10.1029/92GL00919 00948276 Google Scholar Crossref Search ADS WorldCat Ikelle L T , Yung S , Daube F . , 1993 2D random media with ellipsoidal autocorrelation function , Geophysics , vol. 58 (pg. 1359 - 72 )http://dx.doi.org/10.1190/1.1443518 1070485X Google Scholar Crossref Search ADS WorldCat Kneib G . , 1995 The statistical nature of the upper continental crystalline crust derived from in situ seismic measurements , Geophys. J. Int. , vol. 122 (pg. 594 - 616 )http://dx.doi.org/10.1111/j.1365-246X.1995.tb07015.x 0956540X 1365246X Google Scholar Crossref Search ADS WorldCat Korn M . , 1993 Seismic wave in random media , J. Appl. Geophys. , vol. 29 (pg. 247 - 69 )http://dx.doi.org/10.1016/0926-9851(93)90007-L 09269851 Google Scholar Crossref Search ADS WorldCat Levander A R , Hobbs R W , Smith S K , England R W , Snyder D B , Holliger K . , 1994 The crust as a heterogeneous ‘optical’ medium, or ‘crocodiles in the mist’ , Tectonophysics , vol. 232 (pg. 281 - 97 )http://dx.doi.org/10.1016/0040-1951(94)90090-6 00401951 Google Scholar Crossref Search ADS WorldCat Levander A R , Holliger K . , 1992 Small-scale heterogeneity and larger-scale velocity structure of the continental crust , J. Geophys. Res. , vol. 97 (pg. 8797 - 804 )http://dx.doi.org/10.1029/92JB00659 01480227 Google Scholar Crossref Search ADS WorldCat Li Z , Sun Y C . , 1997 Reservoir diagenesis sequences and framework in intracontinent rift basin, East China , J. China Univ. Geosci. , vol. 8 (pg. 68 - 72 ) OpenURL Placeholder Text WorldCat O'Doherty R F , Anstey N A . , 1971 Reflections on amplitudes , Geophys. Prospect. , vol. 19 (pg. 430 - 58 )http://dx.doi.org/10.1111/j.1365-2478.1971.tb00610.x 00168025 13652478 Google Scholar Crossref Search ADS WorldCat Roth M , Korn M . , 1993 Single scattering theory versus numerical modeling in 2D random media , Geophys. J. Int. , vol. 112 (pg. 124 - 40 )http://dx.doi.org/10.1111/j.1365-246X.1993.tb01442.x 0956540X 1365246X Google Scholar Crossref Search ADS WorldCat Saggaf M M , Robinson E A . , 2000 A unified framework for the deconvolution of traces of nonwhite reflectivity , Geophysics , vol. 65 (pg. 1660 - 76 )http://dx.doi.org/10.1190/1.1444854 1070485X Google Scholar Crossref Search ADS WorldCat Sato H . , 1977 Energy propagation including scattering effect: single isotropic scattering approximation , J. Phys. Earth. , vol. 25 (pg. 27 - 41 ) Google Scholar Crossref Search ADS WorldCat Tobias M M , Serge A S . , 1999 Green's function construction for 2D and 3D elastic random media 69th Ann. Int. Mtg (Soc. Expl. Geophys.) expanded abstracts Todoeschuck J P , Jensen O G , Labonte S . , 1990 Gaussian scaling noise model of seismic reflection sequences: evidence from well logs , Geophysics , vol. 55 (pg. 480 - 4 )http://dx.doi.org/10.1190/1.1442857 1070485X Google Scholar Crossref Search ADS WorldCat Walden A T , Hosken J W J . , 1985 An investigation of the spectral properties of primary reflection coefficients , Geophys. Prospect. , vol. 33 (pg. 400 - 35 )http://dx.doi.org/10.1111/j.1365-2478.1985.tb00443.x 00168025 13652478 Google Scholar Crossref Search ADS WorldCat Wu R S . , 1986 Fractal dimensions of fault surfaces and the inhomogeneity spectrum of the lithosphere revealed from seismic wave scattering Proceeding of the international symposium on multiple scattering of waves in random media and random rough surface 29 July to 2 Aug 2 1985, Pennsylvania, USA (pg. 929 - 40 ) Wu R S , Aki K . , 1985 The fractal nature of inhomogeneities in the lithosphere evidenced from seismic wave scattering , Pure Appl. Geophys. , vol. 123 (pg. 805 - 18 )http://dx.doi.org/10.1007/BF00876971 00334553 14209136 Google Scholar Crossref Search ADS WorldCat Wu R S , Aki K . , 1988 Introduction: seismic wave scattering in a three-dimensionally heterogeneous earth , Pure Appl. Geophys. , vol. 128 (pg. 1 - 6 )http://dx.doi.org/10.1007/BF01772587 00334553 14209136 Google Scholar Crossref Search ADS WorldCat Wu R S , Xu Z Y , Li X P . , 1994 Heterogeneity spectrum and scale-anisotropy in the upper crust revealed by the German continental deep-drilling (KTB) holes , Geophys. Res. Lett. , vol. 21 (pg. 911 - 4 )http://dx.doi.org/10.1029/94GL00772 00948276 Google Scholar Crossref Search ADS WorldCat Xi X , Yao Y . , 2004a The analysis of the wave field characteristics in 2-D viscoelastic random medium , Prog. Geophys. , vol. 19 (pg. 608 - 15 ) OpenURL Placeholder Text WorldCat Xi X , Yao Y . , 2004b The wave field characteristics of 2-D transversely isotropic elastic random medium , Prog. Geophys. , vol. 19 (pg. 924 - 32 ) OpenURL Placeholder Text WorldCat © 2010 Nanjing Institute of Geophysical Prospecting
TI - Quantitative analysis of basin-scale heterogeneities using sonic-log data in the Yanchang Basin
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2132/7/1/004
DA - 2010-03-11
UR - https://www.deepdyve.com/lp/oxford-university-press/quantitative-analysis-of-basin-scale-heterogeneities-using-sonic-log-q8AxoXLyqA
SP - 41
VL - 7
IS - 1
DP - DeepDyve
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