Reduced Order Geomechanics Models

Abstract

Computational techniques are commonly used for real-time simulation of complex geomechanics problems, such as hydraulic dilation stimulation. A significant challenge in this realm is that high-fidelity mathematical models or full order models (FOMs) are computationally expensive as they must span multiple spatial and temporal length scales, often including nonlinearities and thermo-hydro-mechanical processing. The computationally intensive nature of these simulations continues to pose challenges in parameter estimation, uncertainty quantification, and optimization applications, where hundreds to thousands of simulations are required to achieve a solution. Intrusive reduced order models (ROMs) have emerged as a method to derive and train a computationally efficient surrogate/proxy model using the FOM. This thesis seeks to bridge the gap in existing intrusive ROMs in reservoir engineering by introducing efficient ROMs that are capable of capturing hydro-mechanical coupling behavior and path-dependent plastic deformation of rocks. A complex case involving hydraulic dilation stimulation is used to show the efficiency and accuracy of the ROM in addressing coupling, plasticity, and permeability enhancement features.

First, an efficient and accurate ROM is proposed for nonlinear porous media flow problems, with specific application to a two-dimensional layered reservoir with a two-well system. Standard projection-based intrusive ROMs without hyper-reduction, such as proper orthogonal decomposition-Galerkin (POD-Galerkin), have not demonstrated efficacy in reducing the computational cost of the ROM for nonlinear problems. In this context, we combine POD-Galerkin with discrete empirical interpolation method (DEIM) as a hyper-reduction technique to reduce the size of the system of equations and accelerate the computation of nonlinear terms (residual force vector and its Jacobian). Column-reduced Jacobian DEIM technique is employed to interpolate the Jacobian, leading to a significant reduction in the computational time of the online stage. The ROM is parameterized for the nonlinear transient injection rate (pumping schedule). Offline, training data is generated by the FOM runs with simple constant injection rates. Online, the ROM demonstrates high accuracy and efficiency for complex and time-varying pumping schedules, including sinusoidal, high-frequency, and time-discontinuous pumping schedules that are located outside of the training regime. It is shown that POD-DEIM ROM has about 10^3 times fewer degrees of freedom (DoFs) and is approximately 190 times faster than the FOM for a reservoir model with 3*10^4 DoFs, while maintaining an accurate solution in the online stage. The accuracy and efficiency of the POD-DEIM motivate its potential use as a surrogate model in the real-time control and monitoring of fluid injection processes.

Intrusive ROMs have faced considerable difficulties in accurately capturing the history-dependent nonlinear evolution of plastic strain. In the second objective, an intrusive ROM is developed and evaluated for a Drucker-Prager plasticity model, in which material properties and cyclic load path are parametric inputs. By constructing multiple local DEIM (LDEIM) approximations in combination with clustering and classifier techniques, a fast and accurate ROM is achieved. The FOM consists of a two-dimensional finite element analysis (FEA) of a deformable solid with Drucker-Prager plasticity. Offline, the temporal and parameterized training data generated from FOM runs are classified using the k-means clustering algorithm, whereby LDEIM basis vectors are computed. Online, a nearest neighbor classifier identifies the appropriate LDEIM. The ROM has three hyper-parameters (the size of the ROM, the number of clusters, and the number of DEIM measurement points per cluster), influencing both accuracy and speed-up. In a micromechanics porous media problem, parameterized by Young’s modulus and hardening modulus, the ROM’s performance is demonstrated for inputs within and outside of the training domain; error and speed-up vary with inputs - accuracy is highest for inputs within the training domain (Error: 1.0-3.5% vs 1.0-9.2%), while speed-up varies from 106 to 134 times. For a cyclic plasticity problem, parameterized by load path, the ROM exhibits stable and accurate online performance with a substantial speed-up for test load paths. Under FOMs with ~10^3 and ~5*10^4 DoFs, speed-ups are 11 and 770 times, respectively. Larger speed-ups seem likely for larger FOMs.

Finally, the ROM for nonlinear transient porous media flow as a diffusion problem is coupled with the ROM for plasticity to develop a novel ROM formulation for poroplasticity problems. This ROM aims to significantly reduce the computational costs for nonlinear and fully-coupled hydro-mechanical simulations in large-scale reservoirs. The developed mathematical model integrates a coupled system of equations from a two-dimensional FEA of momentum and mass balance equations equipped with Drucker-Prager plasticity and stress-dependent permeability enhancement models. The proposed ROM combines various ROMs, including POD-Galerkin to reduce the number of DoFs, DEIM to accelerate the computation of nonlinear terms, and local POD and local DEIM (LPOD/LDEIM) for further reductions in poroplasticity problems. LPOD and LDEIM classify the parameterized training data, obtained from offline FOM runs, into multiple subspaces with similar dynamic features. A new strategy for clustering and classification techniques tailored for the coupled formulation framework is introduced. The advantages of this ROM are demonstrated in a large-scale application involving hydraulic dilation stimulation of a reservoir with a horizontal well pair. The ROM is parameterized not only by the material properties but also by the injection rate. Its effectiveness is evaluated for more realistic use cases, where ROM remains efficient for injection rates that extend beyond the training data.

In large-scale subsurface flow modeling of hydraulic dilation stimulation, a speed-up of ~400 times is achieved, with a ROM reducing the model dimension from 10^5 DoFs to 100 DoFs. This substantial computational saving results from a real-time analysis of the ROM and becomes even more highlighted in multi-query problems, where the model must be executed for multiple inputs and system configurations. This ROM has a high potential for accelerating various problems, such as uncertainty quantification, design, history matching, and well control optimization. It is also recommended that the proposed ROM be adopted for other real-world subsurface applications, including conventional and unconventional oil and gas production, hydraulic fracturing, and carbon storage.