Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks

Saturation of porous rocks with a mixture of two fluids (known as partial saturation) has a substantial effect on the seismic waves propagating through these rocks. In particular, partial saturation causes significant attenuation and dispersion of the propagating waves, due to wave-induced fluid flow. Such flow arises when a passing wave induces different fluid pressures in regions of rock saturated by different fluids. As partial fluid saturation can occur on different length scales, attenuation due to wave-induced fluid flow is ubiquitous. In particular, mesoscopic fluid flow due to heterogeneities occurring on a scale greater than porescale, but less than wavelength scale, is responsible for significant attenuation in the frequency range from 10 to 1000 Hz.Most models of attenuation and dispersion due to mesoscopic heterogeneities imply that fluid heterogeneities are distributed in a periodic/regular way. In 1D this corresponds to periodically alternating layering, in 3D as periodically distributed inclusions of a given shape (usually spheres). All these models yield very similar estimates of attenuation and dispersion.Experimental studies show that mesoscopic heterogeneities have less idealized distributions and that the distribution itself affects attenuation and dispersion. Therefore, theoretical models are required which would simulate the effect of more general and realistic fluid distributions.We have developed two theoretical models which simulate the effect of random distributions of mesoscopic fluid heterogeneities. The first model assumes that one fluid forms a random ensemble of spherical inclusions in a porous medium saturated by the other fluid. The attenuation and dispersion predicted by this model are very similar to those predicted for 3D periodic distribution. Attenuation (inverse quality factor) is proportional to ω at low frequencies for both distributions. This is in contrast to the 1D case, where random and periodically alternating layering shows different attenuation behaviour at low frequencies. The second model, which assumes a 3D continuous distribution of fluid heterogeneities, also predicts the same low-frequency asymptote of attenuation. However, the shapes of the frequency dependencies of attenuation are different. As the 3D continuous random approach assumes that there will be a distribution of different patch sizes, it is expected to be better suited to modelling experimental results. Further research is required in order to uncover how to relate the random functions to experimentally significant parameters.     

Estimating azimuthal stress‐induced P‐wave anisotropy from S‐wave anisotropy using sonic log or vertical seismic profile data

Most sedimentary rocks are anisotropic, yet it is often difficult to accurately incorporate anisotropy into seismic workflows because analysis of anisotropy requires knowledge of a number of parameters that are difficult to estimate from standard seismic data. In this study, we provide a methodology to infer azimuthal P‐wave anisotropy from S‐wave anisotropy calculated from log or vertical seismic profile data. This methodology involves a number of steps. First, we compute the azimuthal P‐wave anisotropy in the dry medium as a function of the azimuthal S‐wave anisotropy using a rock physics model, which accounts for the stress dependency of seismic wave velocities in dry isotropic elastic media subjected to triaxial compression. Once the P‐wave anisotropy in the dry medium is known, we use the anisotropic Gassmann equations to estimate the anisotropy of the saturated medium. We test this workflow on the log data acquired in the North West Shelf of Australia, where azimuthal anisotropy is likely caused by large differences between minimum and maximum horizontal stresses. The obtained results are compared to azimuthal P‐wave anisotropy obtained via orthorhombic tomography in the same area. In the clean sandstone layers, anisotropy parameters obtained by both methods are fairly consistent. In the shale and shaly sandstone layers, however, there is a significant discrepancy between results since the stress‐induced anisotropy model we use is not applicable to rocks exhibiting intrinsic anisotropy. This methodology could be useful for building the initial anisotropic velocity model for imaging, which is to be refined through migration velocity analysis.     

Pore fluid viscosity effects on P‐ and S‐wave anisotropy in synthetic silica‐cemented sandstone with aligned fractures

Ultrasonic (500 kHz) P‐ and S‐wave velocity and attenuation anisotropy were measured in the laboratory on synthetic, octagonal‐shaped, silica‐cemented sandstone samples with aligned penny‐shaped voids as a function of pore fluid viscosity. One control (blank) sample was manufactured without fractures, another sample with a known fracture density (measured from X‐ray CT images). Velocity and attenuation were measured in four directions relative to the bedding fabric (introduced during packing of successive layers of sand grains during sample construction) and the coincident penny‐shaped voids (fractures). Both samples were measured when saturated with air, water (viscosity 1 cP) and glycerin (100 cP) to reveal poro‐visco‐elastic effects on velocity and attenuation, and their anisotropy. The blank sample was used to estimate the background anisotropy of the host rock in the fractured sample; the bedding fabric was found to show transverse isotropy with shear wave splitting (SWS) of 1.45 ± 1.18% (i.e. for S‐wave propagation along the bedding planes). In the fractured rock, maximum velocity and minimum attenuation of P‐waves was seen at 90° to the fracture normal. After correction for the background anisotropy, the fractured sample velocity anisotropy was expressed in terms of Thomsen's weak anisotropy parameters ε, γ & δ. A theory of frequency‐dependent seismic anisotropy in porous, fractured, media was able to predict the observed effect of viscosity and bulk modulus on ε and δ in water‐ and glycerin‐saturated samples, and the higher ε and δ values in air‐saturated samples. Theoretical predictions of fluid independent γ are also in agreement with the laboratory observations. We also observed the predicted polarisation cross‐over in shear‐wave splitting for wave propagation at 45° to the fracture normal as fluid viscosity and bulk modulus increases.     

Laboratory measurements of seismic attenuation and Young's modulus dispersion in a partially and fully water‐saturated porous sample made of sintered borosilicate glass

We measured the extensional‐mode attenuation and Young's modulus in a porous sample made of sintered borosilicate glass at microseismic to seismic frequencies (0.05–50 Hz) using the forced oscillation method. Partial saturation was achieved by water imbibition, varying the water saturation from an initial dry state up to ～99%, and by gas exsolution from an initially fully water‐saturated state down to ～99%. During forced oscillations of the sample effective stresses up to 10 MPa were applied. We observe frequency‐dependent attenuation, with a peak at 1–5 Hz, for ～99% water saturation achieved both by imbibition and by gas exsolution. The magnitude of this attenuation peak is consistently reduced with increasing fluid pressure and is largely insensitive to changes in effective stress. Similar observations have recently been attributed to wave‐induced gas exsolution–dissolution. At full water saturation, the left‐hand side of an attenuation curve, with a peak beyond the highest measured frequency, is observed at 3 MPa effective stress, while at 10 MPa effective stress the measured attenuation is negligible. This observation is consistent with wave‐induced fluid flow associated with mesoscopic compressibility contrasts in the sample's frame. These variations in compressibility could be due to fractures and/or compaction bands that formed between separate sets of forced‐oscillation experiments in response to the applied stresses. The agreement of the measured frequency‐dependent attenuation and Young's modulus with the Kramers–Kronig relations and additional data analyses indicate the good quality of the measurements. Our observations point to the complex interplay between structural and fluid heterogeneities on the measured seismic attenuation and they illustrate how these heterogeneities can facilitate the dominance of one attenuation mechanism over another.     

P‐ and S‐wave anisotropy to characterise and quantify damage in media: laboratory experiment using synthetic sample with aligned microcracks

Damage characterisation in solid media is studied in this work through ultrasonic measurements. A synthetic three‐dimensional printed sample including a system of horizontally aligned microcracks is used. In contrast to other manual fabrication methods presented in the literature, the construction process considered here ensures a better control and accuracy of size, shape, and spatial distribution of the microcrack network in the synthetic sample. The acoustic measurements were conducted through a specific device using triple acoustic sensors, which allows capturing at each incident direction three wave modes. The evolution of the ultrasonic velocities with respect to incident angle accounted for the damage‐induced anisotropy. The experimental results are then compared with some well‐known effective media theories in order to discuss their potential use for the following studies. Finally, we highlighted and compared the accuracy of these theories used for inversion procedure to quantify damage in the medium.     

不同压力下部分饱和砂岩纵波衰减的理论及实验研究

深入了解不同压力、频率、流体含量和流体分布对岩石中弹性波传播特性的影响,对指导油气勘探开发具有重要意义.不同尺度下的波致流效应,是声波传播过程中产生速度频散和衰减的重要原因.本文以不同压力下水饱和区域改进的骨架模量为纽带,建立了联合介观尺度斑块饱和效应与微观尺度喷射流效应的部分饱和岩石声学理论模型.开展针对性声学实验,根据不同压力下部分饱和砂岩纵波速度测量数据,确定理论模型中的相关参数,从而实现对不同压力下部分饱和岩石纵波衰减的定量表征.在此基础上,通过理论与实验测量的纵波衰减的对比,分析不同压力、含水饱和度以及频率对岩石纵波衰减的影响.研究结果表明,在较低压力,较高含水饱和度以及较高频段,喷射流效应较强,因此新建模型计算的衰减明显大于斑块饱和模型的衰减.由于新建模型体现了斑块饱和效应与喷射流效应的综合影响,相比于斑块饱和模型,新建模型计算的部分饱和岩石的纵波衰减更接近于实测衰减,但受到岩石自身因素影响,新建模型计算的衰减仍略小于实测衰减.

     

Velocity dispersion and fluid substitution in sandstone under partially saturated conditions

The elastic moduli of four sandstone samples are measured at seismic (2?2000 Hz) and ultrasonic (1 MHz) frequencies and water- and glycerin-saturated conditions. We observe that the high-permeability samples under partially water-saturated conditions and the low-permeability samples under partially glycerin-saturated conditions show little dispersion at low frequencies (2?2000 Hz). However, the high-permeability samples under partially glycerin-saturated conditions and the low-permeability samples under partially water-saturated conditions produce strong dispersion in the same frequency range (2?2000 Hz). This suggests that fluid mobility largely controls the pore-fluid movement and pore pressure in a porous medium. High fluid mobility facilitates pore-pressure equilibration either between pores or between heterogeneous regions, resulting in a low-frequency domain where the Gassmann equations are valid. In contrast, low fluid mobility produces pressure gradients even at seismic frequencies, and thus dispersion. The latter shows a systematic shift to lower frequencies with decreasing mobility. Sandstone samples showed variations in V_p as a function of fluid saturation. We explore the applicability of the Gassmann model on sandstone rocks. Two theoretical bounds for the P-velocity are known, the Gassmann–Wood and Gassmann–Hill limits. The observations confirm the effect of wave-induced flow on the transition from the Gassmann–Wood to the Gassmann–Hill limit. With decreasing fluid mobility, the P-velocity at 2–2000 Hz moves from the Gassmann–Wood boundary to the Gassmann–Hill boundary. In addition,, we investigate the mechanisms responsible for this transition.     

Modelling the DC electrical response of fully and partially saturated Permo-Triassic sandstone

Results from a laboratory investigation into the electrical properties of fully and partially saturated Wildmoor Triassic Sandstone have been modelled using the Archie, Waxman–Smits and Hanai–Bruggeman equations. The results demonstrate the limitation of using simple relationships to describe samples when the matrix resistivity ρ_r is not significantly greater than the saturating electrolyte resistivity ρ_w. In these situations Archie's parameters m and n are not accurately determined. Conversely, the more sophisticated Waxman–Smits and Hanai–Bruggeman models provide parameters that better describe the electrical properties of the rock and are able to identify heterogeneity between samples that would otherwise be missed. The ranges of values for matrix resistivity (49 < ρ_r < 161 Ωm) and cementation factor (1.6 < m < 2.1) obtained from the Hanai–Bruggeman model indicate significant variation between samples. Comparison of laboratory‐determined values for cation exchange capacity (0.06 < Q_v < 0.51 meq/mL) and those obtained from the Waxman–Smits model (0.09 < Q_v < 0.55 meq/mL) indicates a very strong correlation, suggesting this model is appropriate for describing the rock. There is good agreement between parameters modelled using fully and partially saturated versions of both the Hanai–Bruggeman and Waxman–Smits equations, indicating that the data are consistent with these models and that the assumptions made are appropriate.     

Pseudo Rayleigh wave in a partially saturated non-dissipative porous solid

Propagation of surface waves is studied at the pervious boundary of a porous solid saturated with a mixture of two immiscible fluids. An approach, based on continuum mixture theory, is used to derive a secular equation for the propagation of harmonic waves at the stress-free plane surface of this non-dissipative medium. Numerical analysis shows that this secular equation may not represent the propagation of true surface wave in the porous aggregate. Then, this equation is solved numerically for the propagation of pseudo Rayleigh wave or the leaky surface waves. To ensure the existence of pseudo Rayleigh wave, capillary effect between two (wetting and non-wetting) pore-fluids is related to the partial saturation. Effects of porosity and partial saturation coupled with capillary effect are observed on the phase velocity of pseudo Rayleigh waves in sandstone saturated with water-CO₂ mixture.     

含流体砂岩地震波频散实验研究

为了研究孔隙流体对不同渗透率岩石地震波速度的影响,在实验室利用跨频带岩石弹性参数测试系统得到了应变幅值 -6的2~2000 Hz频段下的地震波速度和1 MHz频率下的超声波速度,利用差分共振声谱法得到了频率600 Hz岩石干燥和完全饱水情况下岩石声学参数.实验表明,在低饱和度下,致密砂岩在地震和超声频段下没有明显的频散;在高饱和度下纵波速度的频散变得明显.从干燥到完全水饱和条件,不同频率测量的致密砂岩的体积模量随岩石孔隙度增高而降低,且体积模量的变化量受岩石微观孔隙结构的影响较大.高孔、高渗砂岩无论在低含水度下还是在高含水饱和度下频散微弱,并且在地震频段下围压对于岩石纵横波速度的影响要大于频率的影响.高孔、高渗砂岩和致密砂岩不同含水饱和度下的频散差异可应用于储层预测,油气检测等方面,同时该研究可以更好地帮助理解岩石的黏弹性行为,促进岩石物理频散理论的发展,提高地震解释的精度.     

Analytical model for fully three‐dimensional radial dispersion in a finite‐thickness aquifer

Analytical solutions for contaminant transport in a non‐uniform flow filed are very difficult and relatively rare in subsurface hydrology. The difficulty is because of the fact that velocity vector in the non‐uniform flow field is space‐dependent rather than constant. In this study, an analytical model is presented for describing the three‐dimensional contaminant transport from an area source in a radial flow field which is a simplest case of the non‐uniform flow. The development of the analytical model is achieved by coupling the power series technique, the Laplace transform and the two finite Fourier cosine transform. The developed analytical model is examined by comparing with the Laplace transform finite difference (LTFD) solution. Excellent agreements between the developed analytical model and the numerical model certificate the accuracy of the developed model. The developed model can evaluate solution for Peclet number up to 100. Moreover, the mathematical behaviours of the developed solution are also studied. More specifically, a hypothetical convergent flow tracer test is considered as an illustrative example to demonstrate the three‐dimensional concentration distribution in a radial flow field. The developed model can serve as benchmark to check the more comprehensive three‐dimensional numerical solutions describing non‐uniform flow contaminant transport. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison of P‐wave attenuation models of wave‐induced flow

Wave‐induced oscillatory fluid flow in the vicinity of inclusions embedded in porous rocks is one of the main causes for P‐wave dispersion and attenuation at seismic frequencies. Hence, the P‐wave velocity depends on wave frequency, porosity, saturation, and other rock parameters. Several analytical models quantify this wave‐induced flow attenuation and result in characteristic velocity–saturation relations. Here, we compare some of these models by analyzing their low‐ and high‐frequency asymptotic behaviours and by applying them to measured velocity–saturation relations. Specifically, the Biot–Rayleigh model considering spherical inclusions embedded in an isotropic rock matrix is compared with White's and Johnson's models of patchy saturation. The modeling of laboratory data for tight sandstone and limestone indicates that, by selecting appropriate inclusion size, the Biot‐Rayleigh predictions are close to the measured values, particularly for intermediate and high water saturations.     

Acoustic wave propagation in saturated porous media: reformulation of the Biot/Squirt flow theory

A model of wave propagation in fluid-saturated porous media is developed where the principal fluid/solid interaction mode affecting the propagation of the acoustic wave results from the conjunction of the Biot and the Squirt flow mechanism. The difference between the original Biot/Squirt (BISQ) flow theory and the new theory, which we call the reformulated BISQ, is that the average fluid pressure term appearing in the dynamic equation for a two component solid/fluid continuum is independent of squirt flow length. P-velocity and attenuation relate to measurable rock physical parameters: the Biot's poroelastic constants, porosity, permeability, pore fluid compressibility and viscosity. Modelling shows that velocity and attenuation dispersion obtained using the reformulated BISQ theory are of the same order of magnitude as those obtained using the original BISQ theory. Investigation on permeability effect on velocity and attenuation dispersion indicate that the transition zone in velocity and attenuation peak, occurring both at the relaxation frequency, shifts toward high frequency when permeability decreases. This behaviour agrees with Biot's theory prediction.     

The porosity of saturated shallow sediments from seismic compressional and shear wave velocities

This study devises a new analytical relationship to determine the porosity of water-saturated soils at shallow depth using seismic compressional and shear wave velocities. Seismic refraction surveys together with soil sample collection were performed in selected areas containing water-saturated clay–silt, sand and gravely soils. Classification of clay–silt, sand and gravel dense soils provided the coefficient of experimental equation between the data sets, namely, Poisson's ratio, shear modulus and porosity values. This study presents a new analytical relationship between Poisson's ratio and shear modulus values, which are obtained from seismic velocities and porosity values of water-saturated material computed from water content and grain densities, which are determined by laboratory analysis of disturbed samples. The analytical relationship between data sets indicates that when the shear modulus of water-saturated loose soil increases, porosity decreases logarithmically. If shear modulus increases in dense or solid saturated soils, porosity decreases linearly.     

部分饱和孔隙岩石中声波传播数值研究

利用基于Biot理论的孔隙弹性介质的高阶交错网格有限差分算法,模拟了具有随机分布特征的多种流体饱和岩石中声波在中心频率分别为25,50,75,100kHz时的声场特点. 对于一个由两种成分（气和水）饱和的岩石模型, 假设含不同流体的孔隙介质随机分布在不同的宏观区域,该区域尺度远小于计算的声波波长;组成模型的两种随机分布介质具有相同的固体骨架参数、渗透率和孔隙度,但分别被具有不同压缩性、密度和黏滞系数特性的水和气饱和. 计算和统计分析结果表明,在两种孔隙成分随机分布的部分饱和条件下纵波速度比较复杂,除骨架参数外,其变化主要依赖于中心频率、各种孔隙成分饱和度及饱和介质的速度. 比较该随机分布模型、Gassmann理论模型和White的“气包”模型,发现三种模型得到的纵波速度和衰减规律有较好的定性对应关系. 其次,按照这种随机计算模型的处理方法,本文还首次计算了一个三种流体成分充填饱和的例子,即岩石模型中的孔隙被水、油和气部分饱和,计算时保持模型含水饱和度不变而只改变含油和含气饱和度. 在这种计算条件下,纵波速度随中心频率呈增大的趋势但有起伏变化. 声场快照显示了各种转换波在多种孔隙成分充填（两种和三种孔隙成分）岩石中的声场特征,复杂的水-油-气界面的非均匀分布对声场有重要影响,纵波能量主要转换形成了较为复杂的多种慢纵波和横波.     

Back‐projection stacking of P‐ and S‐waves to determine location and focal mechanism of microseismic events recorded by a surface array

We present an automatic method of processing microseismic data acquired at the surface by a star‐like array. The back‐projection approach allows successive determination of the hypocenter position of each event and of its focal mechanisms. One‐component vertical geophone groups and three‐component accelerometers are employed to monitor both P‐ and S‐waves. Hypocenter coordinates are determined in a grid by back‐projection stacking of the short‐time‐average‐to‐long‐time‐average ratio of absolute amplitudes at vertical components and polarization norm derived from horizontal components of the P‐ and S‐waves, respectively. To make the location process more efficient, calculation is started with a coarse grid and zoomed to the optimum hypocenter using an oct‐tree algorithm. The focal mechanism is then determined by stacking the vertical component seismograms corrected for the theoretical P‐wave polarity of the focal mechanism. The mechanism is resolved in the coordinate space of strike, dip, and rake angles. The method is tested on 34 selected events of a dataset of hydraulic fracture monitoring of a shale gas play in North America. It was found that, by including S‐waves, the vertical accuracy of locations improved by a factor of two and is equal to approximately the horizontal location error. A twofold enhancement of horizontal location accuracy is achieved if a denser array of geophone groups is used instead of the sparse array of three‐component seismometers. The determined focal mechanisms are similar to those obtained by other methods applied to the same dataset.     

3D P‐wave traveltime computation in transversely isotropic media using layer‐by‐layer wavefront marching

Subsurface rocks (e.g. shale) may induce seismic anisotropy, such as transverse isotropy. Traveltime computation is an essential component of depth imaging and tomography in transversely isotropic media. It is natural to compute the traveltime using the wavefront marching method. However, tracking the 3D wavefront is expensive, especially in anisotropic media. Besides, the wavefront marching method usually computes the traveltime using the eikonal equation. However, the anisotropic eikonal equation is highly non‐linear and it is challenging to solve. To address these issues, we present a layer‐by‐layer wavefront marching method to compute the P‐wave traveltime in 3D transversely isotropic media. To simplify the wavefront tracking, it uses the traveltime of the previous depth as the boundary condition to compute that of the next depth based on the wavefront marching. A strategy of traveltime computation is designed to guarantee the causality of wave propagation. To avoid solving the non‐linear eikonal equation, it updates traveltime along the expanding wavefront by Fermat's principle. To compute the traveltime using Fermat's principle, an approximate group velocity with high accuracy in transversely isotropic media is adopted to describe the ray propagation. Numerical examples on 3D vertical transverse isotropy and tilted transverse isotropy models show that the proposed method computes the traveltime with high accuracy. It can find applications in modelling and depth migration.     

Laboratory measurements of guided‐wave propagation within a fluid‐saturated fracture

A fluid‐saturated flat channel between solids, such as a fracture, is known to support guided waves—sometimes called Krauklis waves. At low frequencies, Krauklis waves can have very low velocity and large attenuation and are very dispersive. Because they propagate primarily within the fluid channel formed by a fracture, Krauklis waves can potentially be used for geological fracture characterization in the field. Using an analogue fracture consisting of a pair of flat slender plates with a mediating fluid layer—a trilayer model—we conducted laboratory measurements of the velocity and attenuation of Krauklis waves. Unlike previous experiments using ultrasonic waves, these experiments used frequencies well below 1 kHz, resulting in extremely low velocity and large attenuation of the waves. The mechanical compliance of the fracture was varied by modifying the stiffness of the fluid seal of the physical fracture model, and proppant (fracture‐filling high‐permeability sand) was also introduced into the fracture to examine its impact on wave propagation. A theoretical frequency equation for the trilayer model was derived using the poroelastic linear‐slip interface model, and its solutions were compared to the experimental results.     

Surface‐wave inversion for a P‐velocity profile with a constant depth gradient of the squared slowness

Surface waves are often used to estimate a near‐surface shear‐velocity profile. The inverse problem is solved for the locally one‐dimensional problem of a set of homogeneous horizontal elastic layers. The result is a set of shear velocities, one for each layer. To obtain a P‐wave velocity profile, the P‐guided waves should be included in the inversion scheme. As an alternative to a multi‐layered model, we consider a simple smooth acoustic constant‐density velocity model, which has a negative constant vertical depth gradient of the squared P‐wave slowness and is bounded by a free surface at the top and a homogeneous half‐space at the bottom. The exact solution involves Airy functions and provides an analytical expression for the dispersion equation. If the ratio is sufficiently small, the dispersion curves can be picked from the seismic data and inverted for the continuous P‐wave velocity profile. The potential advantages of our model are its low computational cost and the fact that the result can serve as a smooth starting model for full‐waveform inversion. For the latter, a smooth initial model is often preferred over a rough one. We test the inversion approach on synthetic elastic data computed for a single‐layer P‐wave model and on field data, both with a small ratio. We find that a single‐layer model can recover either the shallow or deeper part of the profile but not both, when compared with the result of a multi‐layer inversion that we use as a reference. An extension of our analytic model to two layers above a homogeneous half‐space, each with a constant vertical gradient of the squared P‐wave slowness and connected in a continuous manner, improves the fit of the picked dispersion curves. The resulting profile resembles a smooth approximation of the multi‐layered one but contains, of course, less detail. As it turns out, our method does not degrade as gracefully as, for instance, diving‐wave tomography, and we can only hope to fit a subset of the dispersion curves. Therefore, the applicability of the method is limited to cases where the ratio is small and the profile is sufficiently simple. A further extension of the two‐layer model to more layers, each with a constant depth gradient of the squared slowness, might improve the fit of the modal structure but at an increased cost.     

The offset‐midpoint travel‐time pyramid for P‐wave in 2D transversely isotropic media with a tilted symmetry axis

For pre‐stack phase‐shift migration in homogeneous isotropic media, the offset‐midpoint travel time is represented by the double‐square‐root equation. The travel time as a function of offset and midpoint resembles the shape of Cheops’ pyramid. This is also valid for transversely isotropic media with a vertical symmetry axis. In this study, we extend the offset‐midpoint travel‐time pyramid to the case of 2D transversely isotropic media with a tilted symmetry axis. The P‐wave analytical travel‐time pyramid is derived under the assumption of weak anelliptical property of the tilted transverse isotropy media. The travel‐time equation for the dip‐constrained transversely isotropic model is obtained from the depth‐domain travel‐time pyramid. The potential applications of the derived offset‐midpoint travel‐time equation include pre‐stack Kirchhoff migration, anisotropic parameter estimation, and travel‐time calculation in transversely isotropic media with a tilted symmetry axis.