Comparison of seismic anisotropy of sedimentary strata at low- and high-frequency limits

The layer-induced seismic anisotropy of sedimentary strata is frequency-dependent. At the low-frequency limit, the effective anisotropic properties of the layered media can be estimated by the Backus averaging model. At the high-frequency limit, the apparent anisotropic properties of the layered media can be estimated by ray theory. First, we build a database of laboratory ultrasonic measurement on sedimentary rocks from the literature. The database includes ultrasonic velocity measurements on sandstones and carbonate rocks, and velocity-anisotropy measurements on shales. Then, we simulate the sedimentary strata by randomly selecting a certain number of rock samples and using their laboratory measurement results to parameterize each layer. For each realization of the sedimentary strata, we estimate the effective and apparent seismic anisotropy parameters using the Backus average and ray theory, respectively. We find that, relative to Backus averaging, ray theory usually underestimates the Thomsen parameters [Formula: see text] and [Formula: see text], and overestimates [Formula: see text]. For an effective layered medium consisting of isotropic sedimentary rocks, the differences are significant. These differences decrease when shales with intrinsic seismic anisotropy are included. For the same sedimentary strata, the seismic wave should perceive stronger seismic anisotropy than the ultrasonic wave.

Analysis of seismic anisotropy parameters for sedimentary strata

Based on a large quantity of laboratory ultrasonic measurement data of sedimentary rocks and using Monte Carlo simulation and Backus averaging, we have analyzed the layering effects on seismic anisotropy more realistically than in previous studies. The layering effects are studied for different types of rocks under different saturation conditions. If the sedimentary strata consist of only isotropic sedimentary layers and are brine-saturated, the [Formula: see text] value for the effective transversely isotropic (TI) medium is usually negative. The [Formula: see text] value will increase noticeably and can be mostly positive if the sedimentary strata are gas bearing. Based on simulation results, [Formula: see text] can be determined by other TI elastic constants for a layered medium consisting of isotropic layers. Therefore, [Formula: see text] can be predicted from the other Thomsen parameters with confidence. The theoretical expression of [Formula: see text] for an effective TI medium consisting of isotropic sedimentary rocks can be simplified with excellent accuracy into a neat form. The anisotropic properties of the interbedding system of shales and isotropic sedimentary rocks are primarily influenced by the intrinsic anisotropy of shales. There are moderate to strong correlations among the Thomson anisotropy parameters.

DYNAMIC YOUNG MODULUS VARIATION THROUGH A TURBIDITIC SEDIMENTARY INTERVAL

ABSTRACT. In reservoir development, the integration of results from distinct disciplines aims at increasing oil and gas production. In this context, geoscientistsand engineers rely mainly on the records from geophysical well logging for controlling the spatial and depth variation of petrophysical properties of the formations.In this paper, we use geophysical well logging data through the turbiditic Namorado reservoir in Campos basin for calculating the spatial and depth variation of theisotropic Young modulus. The reservoir formation is mainly formed by subhorizontal thin layers of sandstones and shales. Such structural feature of the reservoircorroborated for approximating the sedimentary interval to a vertically transverse isotropic (VTI) medium using the Backus-averaging technique. We then calculated thecorrespondent Young moduli, in order to construct depth maps showing the spatial variation of VTI Young moduli in the reservoir formation. In the construction of themaps along depths selected a priori, an interpolation process was applied, which incorporates an inverse-distance square-weighted interpolator into a “search radiusscheme”. We evaluated the uncertainties in the interpolated maps by calculating statistical measures, revealing the robustness of the inverse-distance square weightedinterpolator coupled to the proposed “search radius scheme”. For a selected depth, the resulting maps for the isotropic Young modulus show higher magnitudes ifcompared to the magnitudes of the VTI Young moduli (i.e., perpendicular and parallel) maps. This result confirms previous works which point out the importance ofincorporating anisotropy in geomechanical studies, particularly in the analysis of in situ stresses in which use of elastic moduli is crucial. On the other hand, comparingthe VTI Young moduli depth maps reveals that, in the Namorado reservoir, anisotropy induced by thin layers has weak influence on the calculation of Young modulus.Keywords: geophysical well logs, effective VTI elastic stiffnesses, Backus averaging, dynamic Young modulus, spatial interpolation, Namorado reservoir. RESUMO. No desenvolvimento de reservatórios, a integração de resultados de diversas disciplinas visa ao aumento da produção de óleo e gás. Nesse contexto,geocientistas e engenheiros contam principalmente com os registros de perfilagem geofísica de poços para controlar as variações espacial e em profundidade daspropriedades petrofísicas das formações. Nesse artigo, usamos dados de perfilagem geofísica através do reservatório turbidítico de Namorado na bacia de Campospara calcular as variações espacial e em profundidade do módulo de Young isotrópico. A formação onde o reservatório se insere é composta principalmente porcamadas sub-horizontais delgadas de arenitos e folhelhos. Tal feição estrutural do reservatório corroborou para aproximar o intervalo sedimentar a um meio transversalmenteisotrópico vertical (TIV) usando a técnica de Backus. Calculamos então os módulos de Young correspondentes, a fim de construir mapas representativosda variação espacial dos módulos de Young TIV através da formação do reservatório. Na construção dos mapas ao longo das profundidades selecionadas a priori, aplicamos um processo de interpolação que incorpora um interpolador de distância inversa com pesos quadráticos num “esquema de raio de procura”. Avaliamos asincertezas nos mapas interpolados através do cálculo de medidas estatísticas, revelando a robustez do interpolador de distância inversa com pesos quadráticos acopladoao “esquema de raio de procura” proposto. Para uma profundidade selecionada, os mapas resultantes para o módulo de Young isotrópico mostram magnitudes maioresse comparadas às magnitudes dos mapas para o módulo de Young TIV (i.e., perpendicular e paralelo). Esse resultado confirma trabalhos anteriores que salientam aimportância de incorporar a anisotropia em estudos geomecânicos, particularmente em análise de tensões in situ onde o uso de módulos elásticos é crucial. Por outrolado, a comparação dos mapas em profundidade para os módulos de Young TIV revela que, no reservatório Namorado, a anisotropia induzida por camadas delgadastem fraca influência no cálculo do módulo de Young.Palavras-chave: perfis geofísicos de poços, rigidezas elásticas efetivas para meio TIV, média de Backus, módulo de Young dinâmico, interpolação espacial, reservatório Namorado.

Comparison of seismic upscaling methods: From sonic to seismic

The term “upscaling” used here means a prediction of elastic-wave velocities at lower frequencies from the velocities at higher frequencies. Three different methods of upscaling are considered, including the simple averaging, Backus averaging, and pair correlation function methods. These methods are applied to upscale the elastic-wave velocities measured at sonic frequencies ([Formula: see text], logging data) available for a well penetrating layers of gas-bearing shales and carbonates. As a result, a velocity distribution over depth for [Formula: see text] and [Formula: see text] is found in the frequency range of [Formula: see text]. The difference in the results obtained for a particular depth by the three theoretical methods in the surface seismic frequency bandwidth [Formula: see text] is [Formula: see text] for P-wave and [Formula: see text] for S-wave velocity. This difference is attributed to different theoretical backgrounds underlying these methods.

Cross-property relations between electrical conductivity and the seismic velocity of rocks

Cross-property relations are useful when some rock properties can be measured more easily than other properties. Relations between electrical conductivity and seismic velocity, stiffness moduli, and density can be obtained by expressing the porosity in terms of those properties. There are many possible ways to combine the constitutive equations to obtain a relation, each one representing a given type of rock. The relations depend on the assumptions to obtain the constitutive equations. In the electromagnetic case, the equations involve Archie’s law and its modifications for a conducting frame, the Hashin-Shtrikman (HS) bounds, and the self-similar and complex refraction-index method (CRIM) models. In the elastic case, the stress-strain relations are mainly based on the time-average equation, the HS bounds, and the Gassmann equation. Also, expressions for dry rocks and for anisotropic media, using Backus averaging, are analyzed. The relations are applied to a shale saturated with brine (overburden) and to a sandstone saturated with oil (reservoir). Tests with sections of a North Sea well log show that the best fit is given by the relation between the Gassmann velocity and the CRIM, self-similar, and Archie models for the conductivity.

Vertical propagation of low-frequency waves in finely layered media

Multiple scattering in finely layered sediments is important for interpreting stratigraphic data, matching well-log data with seismic data, and seismic modeling. Two methods have been used to treat this problem in seismic applications: the O’Doherty-Anstey approximation and Backus averaging. The O’Doherty-Anstey approximation describes the stratigraphic-filtering effects, while Backus averaging defines the elastic properties for an effective medium from the stack of the layers. It is very important to know when the layered medium can be considered as an effective medium. In this paper, we only investigate vertical propagation. Therefore, no anisotropy effect is taken into consideration. Using the matrix-propagator method, we derive equations for transmission and reflection responses from the stack of horizontal layers. From the transmission response, we compute the phase velocity and compare the zero-frequency limit with the effective-medium velocity from Backus averaging. We also investigate how the transition from time-average medium to effective medium depends on contrast; i.e., strength of the reflection-coefficient series. Using numerical examples, we show that a transition zone exists between the effective medium (low-frequency limit) and the time-average medium (high-frequency limit), and that the width of this zone depends on the strength of the reflection-coefficient series.

Effective media: A forward modeling view

The existing effective media theories, such as Backus averaging, can be only used in media that possess certain characteristics regarding to concentration, shape, or geometry of their heterogeneities. These limitations originate from necessity of having analytical description of complex stress and strain fields that normally arise in microheterogeneous solids. The need for explicit solutions can be eliminated by computing the stresses and strains numerically. As a result, effective media can be in principle constructed for solids of arbitrary complexity. This simple idea is tested on two 2D models, where conventional analytical effective media theories are likely to break down. The first model is an isotropic layered solid with cracks that intersect the layer interfaces, the second is a layered medium containing random inclusions. In both cases, some differences are observed between the effective stiffness coefficients obtained numerically and those derived using the existing effective media theories. For instance, it is demonstrated that Backus averaging (improperly) applied to horizontal isotropic layers with random inclusions leads to biases in the calculated vertical velocities. Overall, the proposed technique enables us to establish the limits of applicability of conventional effective media theories. It also makes it possible to obtain quantitative estimates of the errors incurred because of violating certain assumptions of a given theory.