Inverse calculation of in situ stress in rock mass using the Surrogate-Model Accelerated Random Search algorithm

Abstract

A new approach is presented that incorporates the Surrogate Model Accelerated Random Search (SMARS) Algorithm to inversely determine the overall stress state in a rock mass based on sparse stress measurements. The SMARS algorithm relies on a random search component to maintain global search capabilities, while using the surrogate-model method to accelerate convergence to a solution estimate. Two examples sets are carried out in this presentation to display the utility of the proposed SMARS-based inverse characterization process. The first example set compares the SMARS algorithm with two other popular methods, namely a multiple linear regression analysis method and a neural network method, to estimate the in situ stress field for a simple numerically simulated test case. The results of the numerical testing verify that SMARS provides a relatively stable approach and gives rise to a relatively high accuracy and efficiency (i.e., with less computational expense) compared to the other contemporary approaches considered. Finally, an example is shown for utilizing the SMARS approach for in situ stress estimation based on an actual underground mine located in Pennsylvania. The SMARS results are shown to produce a realistic estimate of the distribution of stress within the area investigated, and overall, the approach has potential for practical use in realistic scenarios to efficiently and accurately estimate in situ stresses in rock mass.

Introduction

Rock at depth is subjected to a substantial amount of in situ stresses that result from the weight of the overlying strata and from locked in stresses of tectonic origin, and exist prior to any excavation or other construction operation. When an opening is excavated in rock, the stress field is locally disrupted and a new set of stresses is induced in the rock surrounding the opening, which may affect the construction operation. As such, knowledge of the magnitudes and directions of these in situ and induced stresses is an essential component of underground excavation design [1], [2], [3]. In particular, if the strength of the rock is exceeded the resulting instability can have serious consequences on the behavior of the excavations [4], [5], [6]. Therefore, the question of whether an assumed state of initial in situ stress is reliable and the selection of rock parameters is logical will directly and significantly affect the reliability of engineering design and the security of construction. As for the example of mine design, only when the stress state of a specific project area is known can an expert determine the overall layout of the mine and the optimal cross-sectional shape and size of the tunnel or quarry.

Several techniques have been developed over the years to obtain direct field measurements of in situ stress information [6], [7]. However, these field tests are often costly and time-consuming with dispersive data and/or covering small geological domains. Moreover, in situ stress fields are often complex and exist within geological systems with relatively high amounts of uncertainty. Thus, in order to obtain a relatively more comprehensive understanding of the in situ stress at an engineering site under consideration an inverse problem must often be formed and a suitable inverse solution method derived. In recent years, a number of methods to predict in situ stress have been proposed and applied in design and construction of various engineering sites, such as multiple linear regression analysis [8], stress (displacement) function methods [9], [10], boundary load (displacement) methods [11], and others. There have also been several “sub-“ solution strategies that have been utilized within these stress prediction algorithms, such as the direct surrogate inverse strategies, including neural network methods [12], as well as optimization strategies, including genetic algorithm optimization approaches [13], which have been used extensively within the boundary load (displacement) method framework in particular. Yet, the existing methods have shown some limitations, most commonly relating to limitations on the speed and accuracy of the inverse solution algorithms.

The multiple linear regression analysis method for in situ stress estimation has been used in many construction sites for simple stress estimates, but makes substantial assumptions about linearity and continuity, which diverge significantly from the real rock mass systems, and may therefore lead to significant inaccuracy. The stress (displacement) function method uses a multi-point fitting to provide a simple way to obtain a stress field with minimal computational cost. Furthermore, the stress function method has shown satisfactory results with uniform lithology and relatively simple geological conditions. However, if the geological conditions are complex and/or the lithology and rock structure are substantially heterogeneous, a higher-order stress or displacement function may be needed that satisfies the continuity conditions on the interface, and this method may not be suitable anymore. The boundary load (displacement) method casts the problem to determine the in situ stress field as an inverse problem to determine the boundary load or displacement on a finite section of the rock mass from measured stresses within this mass, often incorporating computational tools such as finite element method. The boundary load methods requires some type of parameterization of the boundary conditions to be determined and several different approaches have been utilized to inversely estimate the boundary loads given stress measurements with varying degrees of success. Furthermore, by casting the problem as an inverse problem, the boundary load method suffers from the typical challenges of ill-posedness, including non-uniqueness, non-existence, and instability. Some approaches that have been tested recently include surrogate direct inversion, such as the neural network approaches that attempt to use machine learning to map the inverse relationship between the stress measurements (output) and the boundary conditions (input), but often struggle due to the aforementioned ill-posedness and require large amounts of training data. Alternatively, optimization-based approaches, such as genetic algorithms, have been used to solve the inverse problem to determine the boundary loads, but these approaches often have difficulty in traversing the typically non-convex optimization search spaces and/or require a large number of forward analyses (typically computationally expensive) to estimate a solution. As a result, one logical approach is to combine various methods to solve the inverse problem and estimate a stress field that more effectively address the needs for both accuracy and computational efficiency, potentially utilizing one of a variety of works existing in the literature [14], [15], [16] combining global and local optimization to accelerate solution convergence and reduce computational cost.

The present work focuses on one particularly promising hybrid global–local optimization approach that combines a random search method with a surrogate-model technique, known as the Surrogate-Model Accelerated Random Search (SMARS) algorithm [17], applied to the problem of inversely calculating the in situ stress field in rock mass. In the SMARS algorithm the random search method is applied to search the optimization domain globally by randomly generating solution parameter estimates over the entire search area, while the surrogate-model is used in the locally optimal domain to accelerate the convergence to a global solution estimate for the inverse problem. The outline of the SMARS algorithm and its incorporation into an inverse solution method for in situ stress in rock mass in provided in the following section. Then, numerical examples that display the efficacy of the SAMRS approach are shown in Section 3, which are followed by the concluding remarks in Section 4.