Boundary Integral Element Method

Major faults are known to lie along material interfaces, including plate interfaces and tectonic lines. Recent high-precision observations have revealed that there are also faults in the vicinity of material interfaces. This suggests that slip on a fault and stress changes near one can be affected significantly by the presence of a material interface. In this paper, we develop a new mathematical framework to derive an expression for a static slip response function by drawing on properties of complex functions. Our method is characterized by its unique ability to analyse both a fault lying along a material interface and a fault intersecting a material interface under a unified framework. A few previous studies have theoretically dealt with the problem of slip on a fault that intersects a material interface, but their mathematical techniques were so complicated that they had only limited utility for seismological applications. By contrast, our mathematical framework has a simple structure thanks to the characteristics of the complex functions. Using our formula, one can easily calculate the fault slip and the static stress changes nearby, if only the stress drop on the fault is given. The static stress is represented as a sum of stress changes due to the slip and a background stress that is generally discontinuous across a material interface. Understanding of the static stress state around a fault has important implications for evaluating the likelihood of secondary failures. We find that, when the fault is located along, or in the vicinity of, a material interface, the background stress plays a key role in the formation of the stress state around a fault, and accordingly also in the triggering of secondary failures.

In general, potential flow describes the flow vector field as the gradient of some scalar potential function. Various definitions of potential function have been used in literature. In this study, the discharge potential was used to solve the problem of groundwater flow around wells. The advantage of discharge potential is that it is applicable to combined confined and unconfined groundwater flow. When the discharge potential satisfies Laplace’s equation, potential theory is applicable. When two or more discharge potentials each satisfy Laplace’s equation, their sum also satisfies it which means that the principle of superposition can be implemented. Using the latter, solutions to various groundwater flow problems were made. One advantage of analytical solution using discharge potentials is that the size of the computation domain does not influence the solution unlike numerical methods such as finite differences in which the entire boundary condition surrounding the computation doma...

Because the Earth's crust is quite heterogeneous and consists of various types of rocks, many faults are located near material interfaces. Recent observational studies have revealed that some faults cross the interfaces, but theoretical treatment of their dynamic behaviour is still undeveloped. In this paper, we develop a boundary integral equation method (BIEM) to analyse the dynamic behaviour of antiplane rupture propagation crossing a material interface. First, we obtain an exact solution of the stress response due to a rectangular slip rate function on one discretized crack element embedded in one of the elastic half-spaces welded along a planar interface. This solution is helpful not only in BIEM simulations of rupture propagation but also in benchmark testing or extension of other numerical schemes. Next, we simulate the dynamic propagation of an antiplane rupture crossing the material interface by using the exact solution for a stress response in a BIEM framework. We find that a shear wave reflected from the material interface generates a significant change in the slip rate on the crack depending on the angle between the interface and the crack, contrast in the elastic properties, and rupture velocity. Moreover, we infer from our numerical results and previous related works that a backward propagating healing front emerges under rate-weakening friction owing to interaction between the reflected wave and friction.

Major faults are known to lie along material interfaces, including plate interfaces and tectonic lines. Recent high-precision observations have revealed that there are also faults in the vicinity of material interfaces. This suggests that slip on a fault and stress changes near one can be affected significantly by the presence of a material interface. In this paper, we develop a new mathematical framework to derive an expression for a static slip response function by drawing on properties of complex functions. Our method is characterized by its unique ability to analyse both a fault lying along a material interface and a fault intersecting a material interface under a unified framework. A few previous studies have theoretically dealt with the problem of slip on a fault that intersects a material interface, but their mathematical techniques were so complicated that they had only limited utility for seismological applications. By contrast, our mathematical framework has a simple structure thanks to the characteristics of the complex functions. Using our formula, one can easily calculate the fault slip and the static stress changes nearby, if only the stress drop on the fault is given. The static stress is represented as a sum of stress changes due to the slip and a background stress that is generally discontinuous across a material interface. Understanding of the static stress state around a fault has important implications for evaluating the likelihood of secondary failures. We find that, when the fault is located along, or in the vicinity of, a material interface, the background stress plays a key role in the formation of the stress state around a fault, and accordingly also in the triggering of secondary failures.

Submarine groundwater discharge (SGD) is a key problem of coastal hydrogeological systems and one of the most difficult for assessment. Reliable estimation of this discharge is an important factor for considering environmental condition and processes of estuaries and bays. This estimation also is a precondition for achieving the sustainable exploration of the fresh water resources. The present paper offers a solution of the SGD into Moreton Bay, South-East Queensland that is based on a numerical-analytical approach. The numerical procedure incorporates 2-D hydraulic head and stream function models as well as a 2-D flow net, which represents the groundwater flow direction, streamlines and the equipotential lines with relevant hydraulic gradients of the aquifer. The Poisson’s partial differential equation (PDE) is used as a governing equation of the numerical modelling. The numerical solution is obtained with the implementation of the hybrid boundary integral equation/finite element method (BIEM/FEM) using a developed computer code. The estimation of the pollutant concentrations along the Moreton Bay and estuarine shorelines is achieved on the basis of analytical solution of the advection-dispersion PDE that includes Laplace’s integral transformations. A case study is provided to illustrate the applicability of the approach developed. The study investigates the effect of nitrate pollution. The results obtained reveal the potential of the approach for accurate prediction of the SGD and pollutant concentrations into Moreton Bay.

Submarine groundwater discharge (SGD) is a key problem of coastal hydrogeological systems and one of the most difficult for assessment. Reliable estimation of this discharge is an important factor for considering environmental condition and processes of estuaries and bays. This estimation also is a precondition for achieving the sustainable exploitation of the fresh water resources. The present paper offers a solution of the SGD into Moreton Bay, South-East Queensland that is based on a numerical-analytical approach. The numerical procedure incorporates 2-D hydraulic head and stream function models as well as a 2-D flow net, which represents the groundwater flow direction, streamlines and the equipotential lines with relevant hydraulic gradients of the aquifer. The Poisson's partial differential equation (PDE) is used as a governing equation of the numerical modeling. The numerical solution is obtained with the implementation of the hybrid boundary integral equation/finite element method (BIEM/FEM) using a developed computer code. The estimation of the pollutant concentrations along the Moreton Bay and estuarine shorelines is achieved on the basis of analytical solution of the advection-dispersion PDE that includes Laplace's integral transformations. A case study is provided to illustrate the applicability of the approach developed. The study investigates the effect of nitrate pollution. The results obtained reveal the potential of the approach for accurate prediction of the SGD and pollutant concentrations into Moreton Bay.