Deep learning (DL) seismic simulations have become a leading-edge field that could provide an effective alternative to traditional numerical solvers. We have developed a small-data-driven time-domain method for fast seismic simulations in complex media based on the physics-informed Fourier neural operator (FNO). Unlike most DL-based modeling schemes that either solve wave equations by embedding the physical constraints into the cost function or conduct physics-informed learning by incorporating the wave functions into convolutional neural networks (CNNs), the FNO uses a learning architecture similar to the structure of split-step Fourier wave propagators, which is composed of two CNNs formulated in the space and wavenumber domains, respectively. The space-domain CNN acts as a local trainable phase-screen compensation. The wavenumber-domain CNN represents a nonlocal spatial convolutional operator acting as a trainable wavenumber filter for the phase-shift process. The FNO method approximates the mathematical-physical behavior of wave equations through learning the mapping between seismic wavefields at different time/locations from training seismic data. That is, the learning process parameterizes the integral kernel directly in the Fourier space, so that we can establish an expressive and efficient architecture for a better balance between accuracy and performance than the traditional spatial CNNs. Applications to gradient, layered, and Marmousi velocity models demonstrate its performance in accuracy and efficiency. The FNO seismic simulation is a data-driven method that needs a small amount of training data, especially when using blended source training data. It is a discretization-independent method that is not subject to the limitation of spatial sampling and time steps imposed on traditional numerical solvers, implying that the training data can be discretized arbitrarily. It also is a model-independent method that can include absorption attenuation into seismic modeling without the need of viscoelastic wave equations.

SUMMARY: Thermoelasticity extends the classical elastic theory by coupling the fields of particle displacement and temperature. The classical theory of thermoelasticity, based on a parabolic-type heat-conduction equation, is characteristic of an unphysical behaviour of thermoelastic waves with discontinuities and infinite velocities as a function of frequency. A better physical system of equations incorporates a relaxation term into the heat equation; the equations predict three propagation modes, namely, a fast P wave (E wave), a slow thermal P wave (T wave), and a shear wave (S wave). We formulate a second-order tensor Green's function based on the Fourier transform of the thermodynamic equations. It is the displacement–temperature solution to a point (elastic or heat) source. The snapshots, obtained with the derived second-order tensor Green's function, show that the elastic and thermal P modes are dispersive and lossy, which is confirmed by a plane-wave analysis. These modes have similar characteristics of the fast and slow P waves of poroelasticity. Particularly, the thermal mode is diffusive at low thermal conductivities and becomes wave-like for high thermal conductivities.

Carbonate reservoirs in the S area of the Tarim Basin (China) are ultradeep hydrocarbon resources, with low porosity, complex fracture systems, and dissolved pores. Microfracturing is a key factor of reservoir connectivity and storage space. We have performed measurements on limestone samples, under different confining pressures, and we used the self-consistent approximation model and the Biot-Rayleigh theory of double porosity to study the microfractures. We have computed the fluid properties (mainly oil) as a function of temperature and pressure. Using the dependence of seismic [ ] on the microfractures, a multiscale 3D rock-physics template (RPT) is built, based on the attenuation, P-wave impedance, and phase velocity ratio. We estimate the ultrasonic and seismic attenuation with the spectral-ratio method and the improved frequency-shift method, respectively. Then, calibration of the RPTs is performed at ultrasonic and seismic frequencies. We use the RPTs to estimate the total and microfracture porosities. The results indicate that the total porosity is low and the microfracture porosity is relatively high, which is consistent with the well log data and actual oil production reports. This work presents a method for identification of deep carbonate reservoirs by using the microfracture porosity estimated from the 3D RPT, which could be exploited in oil and gas exploration.

Thermoelasticity is important in seismic propagation due to the effects related to wave attenuation and velocity dispersion. We have applied a novel finite-difference (FD) solver of the Lord-Shulman thermoelasticity equations to compute synthetic seismograms that include the effects of the thermal properties (expansion coefficient, thermal conductivity, and specific heat) compared with the classic forward-modeling codes. We use a time splitting method because the presence of a slow quasistatic mode (the thermal mode) makes the differential equations stiff and unstable for explicit time-stepping methods. The spatial derivatives are computed with a rotated staggered-grid FD method, and an unsplit convolutional perfectly matched layer is used to absorb the waves at the boundaries, with an optimal performance at the grazing incidence. The stability condition of the modeling algorithm is examined. The numerical experiments illustrate the effects of the thermoelasticity properties on the attenuation of the fast P-wave (or E-wave) and the slow thermal P-wave (or T-wave). These propagation modes have characteristics similar to the fast and slow P-waves of poroelasticity, respectively. The thermal expansion coefficient has a significant effect on the velocity dispersion and attenuation of the elastic waves, and the thermal conductivity affects the relaxation time of the thermal diffusion process, with the T mode becoming wave-like at high thermal conductivities and high frequencies.