Biot's equations-based reservoir parameter inversion using deep neural networks

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.

Explicit meshfree $${{\varvec{u}}}-{{\varvec{p}}}_\mathbf{\mathrm{w}}$$ solution of the dynamic Biot formulation at large strain

In this paper, an efficient and robust methodology to simulate saturated soils subjected to low-medium frequency dynamic loadings under large deformation regime is presented. The coupling between solid and fluid phases is solved through the dynamic reduced formulation $$u-p_\mathrm{w}$$ u - p w (solid displacement – pore water pressure) of the Biot’s equations. The additional novelty lies in the employment of an explicit two-steps Newmark predictor-corrector time integration scheme that enables accurate solutions of related geomechanical problems at large strain without the usually high computational cost associated with the implicit counterparts. Shape functions based on the elegant Local Maximum Entropy approach, through the Optimal Transportation Meshfree framework, are considered to solve numerically different dynamic problems in fluid saturated porous media.

Accurate discretization of poroelasticity without Darcy stability

In this manuscript we focus on the question: what is the correct notion of Stokes–Biot stability? Stokes–Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes–Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes–Biot stable Euler–Galerkin discretization schemes.

Stability of discrete schemes of Biot’s poroelastic equations

Summary The efficient and accurate numerical modeling of Biot’s equations of poroelasticity requires the knowledge of the exact stability conditions for a given set of input parameters. Up to now, a numerical stability analysis of the discretized elastodynamic Biot’s equations has been performed only for a few numerical schemes. We perform the von Neumann stability analysis of the discretized Biot’s equations. We use an explicit scheme for the wave propagation and different implicit and explicit schemes for Darcy’s flux. We derive the exact stability conditions for all the considered schemes. The obtained stability conditions for the discretized Biot’s equations were verified numerically in one-, two- and three-dimensions. Additionally, we present von Neumann stability analysis of the discretized linear damped wave equation considering different implicit and explicit schemes. We provide both the Matlab and symbolic Maple routines for the full reproducibility of the presented results. The routines can be used to obtain exact stability conditions for any given set of input material and numerical parameters.