Generic Analytical Solution Research Articles

A Generic Analytical Elastic Solution for Excavation Responses of an Arbitrarily Shaped Deep Opening under Biaxial In Situ Stresses

This paper presents a generic analytical solution for stresses and displacements developed around an arbitrarily shaped underground opening excavated in an elastic soil/rock mass with biaxial in situ stresses. A series of separable functions satisfying the governing biharmonic equation is employed as the basic stress function for the problem considered. With the basic stress function and the stress rotation equation, the normal stress and the shear stress on the opening boundary surface are expressed in terms of separable function series with unknown coefficients to make use of the boundary condition. Based on the variational theory, the unknown coefficients are determined by minimizing the difference between the real stress boundary condition and the assumed stress function on the opening surface. The excavation responses around different shapes of openings, including elliptical-shaped, ovaloid-shaped, horseshoe-shaped, and D-shaped openings, are investigated and compared with those from the finite-element simulations. The results show that both the stress field and the displacement field agree well with those from the corresponding numerical model, which verify the proposed solving technique. The proposed solution provides an easy, generic, and feasible approach to assess the stresses and displacements developed around an arbitrarily shaped opening, which is simpler and more straightforward than the complex theory–based solutions.

Coupled poroelastic solutions for the reservoir and caprock layers with the overburden confinement effects

Coupled theory of poroelasticity is used to develop a generic analytical solution for the time-dependent solid stress and pore fluid pressure of the subsurface rocks. A uniaxial model comprising three layers of rock formations, namely, the reservoir, caprock and burden rock is considered. The model entails different mechanical and flow properties for the considered rock layers. A linear stress–displacement formulation at the interface of the caprock and overburden defined and is calibrated to account for the confinement effect of the burden rock. The closed-form solution to the considered coupled, poroelastic problem is derived in Laplace transform space. The time-domain solution is retrieved by numerical inversion of the solution in Laplace transform space.The obtained general solution is used to assess the coupled pore fluid pressure and stress evolution of rock layers in the following applied problems. (1) Caprock integrity upon fluid injection into the reservoir. (2) Seal rock efficiency of containing fluid transport from an abnormally pressured reservoir compartment. Findings indicate that the time evolution of pore fluid transport within and in between the rock layers is influenced by the relative magnitude of both flow and mechanical properties of the reservoir and caprock. In particular, the contrast between the elastic moduli of the reservoir and caprock are found to have substantial effect on the rate and magnitude of overpressure dissipation from the reservoir. Coulomb’s shear failure criterion, together with Terzaghi’s effective stress criterion for tensile failure, is used to assess the time-dependent failure tendency of the caprock upon exposure to excess pressure of the reservoir in the case of fluid injection problem. Findings suggest that higher injection rate, stiffer overburden, thinner reservoir and stiffer reservoir rock would enhance the failure tendency of the caprock.

A Generic analytical solution for modelling pumping tests in wells intersecting fractures

The behaviour of transient flow due to pumping in fractured rocks has been studied for at least the past 80 years. Analytical solutions were proposed for solving the issue of a well intersecting and pumping from one vertical, horizontal or inclined fracture in homogeneous aquifers, but their domain of application—even if covering various fracture geometries—was restricted to isotropic or anisotropic aquifers, whose potential boundaries had to be parallel or orthogonal to the fracture direction. The issue thus remains unsolved for many field cases. For example, a well intersecting and pumping a fracture in a multilayer or a dual-porosity aquifer, where intersected fractures are not necessarily parallel or orthogonal to aquifer boundaries, where several fractures with various orientations intersect the well, or the effect of pumping not only in fractures, but also in the aquifer through the screened interval of the well.Using a mathematical demonstration, we show that integrating the well-known Theis analytical solution (Theis, 1935) along the fracture axis is identical to the equally well-known analytical solution of Gringarten et al. (1974) for a uniform-flux fracture fully penetrating a homogeneous aquifer. This result implies that any existing line- or point-source solution can be used for implementing one or more discrete fractures that are intersected by the well. Several theoretical examples are presented and discussed: a single vertical fracture in a dual-porosity aquifer or in a multi-layer system (with a partially intersecting fracture); one and two inclined fractures in a leaky-aquifer system with pumping either only from the fracture(s), or also from the aquifer between fracture(s) in the screened interval of the well. For the cases with several pumping sources, analytical solutions of flowrate contribution from each individual source (fractures and well) are presented, and the drawdown behaviour according to the length of the pumped screened interval of the well is discussed. Other advantages of this proposed generic analytical solution are also given.The application of this solution to field data should provide additional field information on fracture geometry, as well as identifying the connectivity between the pumped fractures and other aquifers.

Front Tracking Using a Hybrid Analytical Finite Element Approach for Two-Phase Flow Applied to Supercritical CO2 Replacing Brine in a Heterogeneous Reservoir and Caprock

Predicting fluid replacement by two-phase flow in heterogeneous porous media is of importance for issues such as supercritical CO2 sequestration, the integrity of caprocks and the operation of oil water/brine systems. When considering coupled process modelling, the location of the interface is of importance as most of the significant interaction between processes will be happening there. Modelling two-phase flow using grid based techniques presents a problem as the fluid–fluid interface location is approximated across the scale of the discretisation. Adaptive grid methods allow the discretisation to follow the interface through the model, but are computationally expensive and make coupling to other processes (thermal, mechanical and chemical) complicated due to the constant alteration in grid size and effects thereof. Interface tracking methods have been developed that apply sophisticated reconstruction algorithms based on either the ratio of volumes of a fluid in an element (Volume of Fluid Methods) or the advective velocity of the interface throughout the modelling regime (Level set method). In this article, we present an “Analytical Front Tracking” method where a generic analytical solution for two-phase flow is used to “add information” to a finite element model. The location of the front within individual geometrical elements is predicted using the saturation values in the elements and the velocity field of the element. This removes the necessity for grid adaptation, and reduces the need for assumptions as to the shape of the interface as this is predicted by the analytical solution. The method is verified against a standard benchmark solution and then applied to the case of CO2 pooling and forcing its way into a heterogeneous caprock, replacing hot brine and eventually breaking through. Finally the method is applied to simulate supercritical CO2 injected into a brine saturated heterogeneous reservoir rock leading to significant viscous fingering and developement of preferential flow paths. The results are compared with to a finite volume simulation.

Site-Specific Probabilistic Seismic-Hazard Assessment: Direct Amplitude-Based Approach

Conventional probabilistic seismic-hazard assessment (PSHA) is difficult to apply in regions lacking sufficient information concerning geological setting, active faults, and so forth. Also, for a PSHA, site effects arising from both crustal rock and overlying soil sediments are generally not assessed rigorously. This is of particular importance for those metropolitan cities having a significant proportion of reclaimed land, because the site-to-site variability of such site effects can be very large. The objective of this article is to demonstrate an alternative procedure for assessing seismic hazard, developed from the conventional source-based Cornell-McGuire PSHA approach, based on considering an infinite number of sources. The proposed new procedure is termed the direct amplitude-based (DAB) approach. The major advantage of the proposed DAB approach is that it is not necessary to characterize any seismic sources. Moreover, if a site-specific and event- specific ground-motion attenuation model is available, a more accurate PSHA could be performed. Also, a generic analytical solution for the proposed procedure has been derived to avoid the need for a lengthy integration process. Using the proposed approach, peak ground velocities have been computed at different return periods, to form a seismic hazard curve, citing Hong Kong as a case study.