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SUMMARY This work presents a machine-learning-based framework to determine unknown coefficients in seismic wave equations for porous media saturated with fluids by using real data as labels, which are velocities of P and S waves. The coefficients are functions of basic rock physics parameters. By using this framework, the trained neural networks incorporate certain mathematical and physical constraints on the coefficients. Working on a single-fluid model, we train the networks with synthetic as well as real data sets. The prediction results show that the learned model is inherently stable, has good physical properties and can accurately predict synthetic data as well as real logging data of shale reservoirs with relative mean square errors less than 5 per cent. They also demonstrate that the wave propagation phenomenon corresponding to the logging data can be well described with the single-fluid model.

Reservoir parameter inversion from seismic data is an important issue in rock physics. The traditional optimisation-based inversion method requires high computational expense, and the process exhibits subjectivity due to the nonuniqueness of generated solutions. This study proposes a deep neural network (DNN)-based approach as a new means to analyse the sensitivity of seismic attributes to basic rock-physics parameters and then realise fast parameter inversion. First, synthetic data of inputs (reservoir properties) and outputs (seismic attributes) are generated using Biot's equations. Then, a forward DNN model is trained to carry out a sensitivity analysis. One can in turn investigate the influence of each rock-physics parameter on the seismic attributes calculated by Biot's equations, and the method can also be used to estimate and evaluate the accuracy of parameter inversion. Finally, DNNs are applied to parameter inversion. Different scenarios are designed to study the inversion accuracy of porosity, bulk and shear moduli of a rock matrix considering that the input quantities are different. It is found that the inversion of porosity is relatively easy and accurate, while more information is needed to make the inversion more accurate for bulk and shear moduli. From the presented results, the new approach makes it possible to realise accurate and pointwise inverse modelling with high efficiency for actual data interpretation and analysis.

SUMMARY The stability of partial differential equations determines the properties of their solutions. This study focuses on the stability analysis of the equations describing wave propagation in fluids-saturated porous media. We briefly introduce the stability analysis method for the wave propagation equations and discuss the adverse effects on the solutions. In this way, the first part of this paper is mainly devoted to the analysis of the Tuncay and Corapcioglu's (TC) model, which describes the dynamic behaviour of porous media saturated with two immiscible fluids. It is pointed out that the TC model allows spatially bounded but time-exponentially exploding solutions and may yield unstable numerical results. Based on the deduced unstable factors, we construct a stable equivalent fluid (SEF) model. We rigorously analyse the stability of the SEF model using the energy method. For predicting the influence of saturation on wave velocity, the robustness of this model is preserved due to its consistency with the original TC model. Furthermore, the numerical simulations of the wavefields show that the results of the TC model exponentially increase with time after the initial effective wave signal, which does not occur in the SEF model curves. This indicates the necessity of considering the stability from a mathematical point of view during the construction of physical model. It could be useful to merge the mathematical stability theory with the geophysical wave propagation modelling theory.

Physics informed neural networks (PINNs) have been a well-known tool in solving forward and inverse partial differential equations (PDEs) problem. In particular, for the forward problem, only the initial/boundary conditions and the equation itself as the physical information are needed to obtain the predicted solution within the definition domain. However, some existing works show that the standard PINN method is insufficient for solving complex problems. For the relatively complex one-dimensional Euler equation problem, due to the limited accuracy of the standard PINN method, the algorithm is improved in two aspects in this study. First, a gradient-related weight function is introduced into the loss function and the framework of gradient-weighted physics-informed neural networks (gwPINNs) is proposed. In particular, the weight function is carefully designed. Second, the sequence-to-sequence (seq2seq) learning is applied in the training process to improve the precision. Then, four cases of Riemann problems are solved using the improved algorithm. The results show that the improved algorithm can obtain high accuracy in solving these problems, and verify the feasibility of solving one-dimensional Euler equation based on the improved PINN method.

Poro-elastic wave equations are one of the fundamental problems in seismic wave exploration and applied mathematics. In the past few decades, elastic wave theory and numerical method of porous media have developed rapidly. However, the mathematical stability of such wave equations have not been fully studied, which may lead to numerical divergence in the wave propagation simulation in complex porous media. In this paper, we focus on the stability of the wave equation derived from Tuncay’s model and volume averaging method. By analyzing the stability of the first-order hyperbolic relaxation system, the mathematical stability of the wave equation is proved for the first time. Compared with existing poro-elastic wave equations (such as Biot’s theory), the advantage of newly derived equations is that it is not necessary to assume uniform distribution of pores. Such wave equations can spontaneously incorporate complex microscale pore/fracture structures into large-scale media, which is critical for unconventional oil and gas exploration. The process of proof and numerical examples shows that the wave equations are mathematically stable. These results can be applied to numerical simulation of wave field in reservoirs with pore/fracture networks, which is of great significance for unconventional oil and gas exploration.

Groundwater is a valuable water source for drinking and irrigation purposes in semiarid regions. Groundwater pollution may affect human health if it is not pretreated and provided for human use. This study investigated the hydrochemical characteristics driving groundwater quality for drinking and irrigation purposes and potential human health risks in the Xinzhou Basin, Shanxi Province, North China. More specifically, we first investigated hydrochemical characteristics using a descriptive statistical analysis method. We then classified the hydrochemical types and analyzed the evolution mechanisms of groundwater using Piper and Gibbs diagrams. Finally, we appraised the groundwater quality for drinking and irrigation purposes using the entropy water quality index (EWQI). We assessed the associated human health risks for different age and sex groups through drinking intake and dermal contact pathways. Overall, we found that (1) Ca-HCO3 and Ca·Mg-HCO3 were the dominant hydrochemical types and were mainly governed by rock weathering and water–rock interactions. (2) Based on the EWQI classifications, 67.74% of the groundwater samples were classified as medium quality and acceptable for drinking purpose. According to the values of sodium adsorption ratio (SAR), residual sodium carbonate (RSC) and soluble sodium percentage (%Na), 90.32% of the samples were suitable for irrigation, while the remaining samples were unfit for irrigation because of the high salinity in the groundwater. (3) Some contaminants in the groundwater, such as NO3−, NO2− and F−, exceeded the standard limits and may cause potential risks to human health. Our work presented in this paper could establish reasonable management strategies for sustainable groundwater quality protection to protect public health.

Reservoir rocks are heterogeneous porous media saturated with multiphase fluids, in which strong wave dissipation and velocity dispersion are closely associated with fabric heterogeneities and patchy saturation at different scales. The irregular solid inclusions and fluid patches are ubiquitous in nature, whereas the impact of geometry on wave dissipation is still not well-understood. We have investigated the dependence of wave attenuation and velocity on patch geometry. The governing equations for wave propagation in a porous medium, containing fluid/solid heterogeneities of ellipsoidal triple-layer patches, are derived from the Lagrange equations on the basis of the potential and kinetic energies. Harmonic functions describe the wave-induced local fluid flow of an ellipsoidal patch. The effects of the aspect ratio on wave velocity are illustrated with numerical examples and comparisons with laboratory measurements. The results indicate that the P-wave velocity dispersion and attenuation depend on the aspect ratio of the ellipsoidal heterogeneities, especially in the intermediate frequency range. In the case of Fort Union sandstone, the P-wave velocity increases toward an upper bound as the aspect ratio decreases. The example of a North Sea sandstone clearly indicates that introducing ellipsoidal heterogeneities gives a better description of laboratory data than that based on spherical patches. The unexpected high-velocity values previously reported and ascribed to sample heterogeneities are explained by varying the aspect ratio of the inclusions (or patches).

We compare the structural and mechanical properties of mechanically stable (MS) packings of frictional disks in two spatial dimensions (2D) generated with isotropic compression and simple shear protocols from discrete element modeling (DEM) simulations. We find that the average contact number and packing fraction at jamming onset are similar (with relative deviations $$ 0$$ for MS packings generated via simple shear. To investigate the difference in the stress state of MS packings, we develop packing-generation protocols to first unjam the MS packings, remove the frictional contacts, and then rejam them. Using these protocols, we are able to obtain rejammed packings with nearly identical particle positions and stress anisotropy distributions compared to the original jammed packings. However, we find that when we directly compare the original jammed packings and rejammed ones, there are finite stress anisotropy deviations $$\varDelta {{\hat{\varSigma }}}_{xy}$$. The deviations are smaller than the stress anisotropy fluctuations obtained by enumerating the force solutions within the null space of the contact networks generated via the DEM simulations. These results emphasize that even though the compression and shear jamming protocols generate packings with the same contact networks, there can be residual differences in the normal and tangential forces at each contact, and thus differences in the stress anisotropy.