An optimal analytic solution for flow around right-angle faults with a finite conductivity fracture

 

ABSTRACT

Fluid flow around right-angle faults with a finite conductivity fracture is not only a basis for understanding motion of underground fluids such as water, oil and carbon dioxide in carbon capture and storage, but also important, for example, planning and installation of an artificial fracture for the purpose of irrigation in the dry area. In this study we present an optimal analytic solution of the fluid flow as shown in Figure 1.

The flow consists of two parts, one is the flow around right-angle faults and the other is the flow around a finite conductivity fracture. The latter exhibits singular behavior near the edges, on the other hand the former non-singular everywhere. The solution is thus expressed as the sum of non-singular and singular solutions. Being analytic everywhere within and on a problem boundary thus Cauchy’s integral formula holds, the non-singular solution can be determined by the complex variable boundary element method (Sato 2015). The singular solution is expressed as a combination of partial sums of different Laurent series expansion with multiple poles. We solve the non-singular and singular solutions simultaneously by using the implicit singularity programming (Sato 2015).

There is arbitrariness in choosing positions of the multiple poles appeared in the singular solution. In order to find optimal positions of the poles, we evaluate the discharge at an arbitrary point, for example a black dot in Figure 1, in the solution for some given poles. Changing the positions of the poles in the singular solution, we examine the convergence of the discharges so as to find the optimal positions of the poles, that is, the optimal solution. We also try to explain the reason for the optimal positions of the poles from a mathematical point of view.

 

Figure 1. An example of the solution for flow around right-angle faults with a finite conductivity fracture. The right-angle faults are x and y axes intersecting the origin, the fracture a red bold line. Red thin lines represent streamlines, blue equipotential lines. A black dot is an arbitrary point at which the discharge is evaluated. Near the right-angle faults fluid flow goes along the faults, while around the finite conductivity fracture flow goes perpendicular to the fracture.

 

REFERENCES

Sato, K., 2015: Complex Analysis for Practical Engineering, Springer.