A Characterization of Circle Graphs in Terms of Multimatroid Representations

The isotropic matroid $M[IAS(G)]$ of a looped simple graph $G$ is a binary matroid equivalent to the isotropic system of $G$. In general, $M[IAS(G)]$ is not regular, so it cannot be represented over fields of characteristic $\neq 2$. The ground set of $M[IAS(G)]$ is denoted $W(G)$; it is partitioned into 3-element subsets corresponding to the vertices of $G$. When the rank function of $M[IAS(G)]$ is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted $\mathcal{Z}_{3}(G)$. In this paper we prove that $G$ is a circle graph if and only if for every field $\mathbb{F}$, there is an $\mathbb{F}$-representable matroid with ground set $W(G)$, which defines $\mathcal{Z}_{3}(G)$ by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.

Isotropic Matroids III: Connectivity

The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only if $G$ is prime (in the sense of Cunningham's split decomposition). We also show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is connected and has neither a pendant vertex nor a pair of twin vertices. Our most interesting theorem is that if $G$ has $n\geq7$ vertices then $M[IAS(G)]$ is not vertically $n$-connected. This abstract-seeming result is equivalent to the more concrete assertion that $G$ is locally equivalent to a graph with a vertex of degree $<\frac{n-1}{2}$.

Estimating the Pressure-Transient Response for a Horizontal or a Hydraulically Fractured Well at an Arbitrary Orientation in an Anisotropic Reservoir

Summary This article presents a method for finding the equivalent isotropic system for a horizontal well or a hydraulically fractured well at an arbitrary azimuth in an anisotropic reservoir. The equivalent isotropic system can then be used to calculate the pressure response of the original system. The horizontal well solution is compared with previous solutions reported in the literature. The general hydraulically fractured well case has not, to our knowledge, been presented before. This study confirms the observation by Yildiz and Ozkan1 that small deviations (up to 20°) from the optimal azimuth for a horizontal well do not have a major effect on the pressure response. Because this solution is obtained by finding an equivalent isotropic system, this study also confirms the observation by Earlougher2 that an anisotropic system cannot be identified from a single-well test. Introduction Permeability anisotropy may occur for a variety of reasons. One of the most important causes of permeability anisotropy is the presence of natural fractures. These reservoirs are often candidates for horizontal drilling. The optimal wellbore orientation is perpendicular to the direction of maximum permeability. Often, the principal axes of permeability are not well characterized during the initial stages of field development. Thus, the wellbore orientation may lie at any angle to the principal axes of permeability. A similar situation is that of a well with a hydraulic fracture in a naturally fractured reservoir. Although the fracture created will often be parallel to the primary set of natural fractures, this is not necessarily the case. Tectonic stresses control the direction of growth of both natural fracture systems and hydraulic fractures. If the tectonic stresses have changed since the formation of the natural fracture system, the hydraulic fracture may have a different orientation from the natural fractures. This may even occur because of production. Cases have been documented where repeat fracture treatments had a different orientation from the original fractures because of the change in the tectonic stresses caused by reservoir depletion.3 The case of a horizontal well in an anisotropic reservoir with the wellbore parallel to one of the principal axes of permeability has been studied by a number of authors. The case of a horizontal well having the wellbore at an arbitrary orientation with respect to the principal axes of permeability has previously been studied by Besson,4 by Zhang and Dusseault,5 and by Yildiz and Ozkan.1 Besson4 studied the case of a horizontal or slanted wellbore in a reservoir with horizontal to vertical anisotropy. Although he presented the transformations for the general case of kx?ky?kz, he did not consider the case of areal anisotropy further. Zhang and Dusseault5 presented a solution based on transforming the anisotropic system to an equivalent isotropic system. They also proposed graphical and numerical methods for determining the permeability anisotropy from analysis of tests in two horizontal wells having different azimuths. Yildiz and Ozkan1 likewise presented a solution by defining dimensionless variables that perform an implicit coordinate transformation. To the best of our knowledge, the fractured well case has been studied only for the situation where the hydraulic fracture is parallel to one of the principal axes of permeability.6 We present new analytical solutions for these two situations. The new solutions are obtained by transforming the original problem with anisotropic permeability into an equivalent problem with isotropic permeability. Thus, the pressure transient response in an anisotropic system cannot be distinguished from that of an isotropic system from the shape of the pressure response alone. The major contribution of this article is to define the properties of the equivalent isotropic system in terms of the properties of the original anisotropic system. Any existing solution for an isotropic system can then be used to evaluate the pressure response. For the hydraulically fractured well case, we assume a finite-conductivity fracture. The infinite-conductivity fracture case may be obtained in the limit as the fracture conductivity increases. In the next section we discuss a solution for the horizontal well case, while in the following section we discuss the hydraulically fractured well case. The derivations of the equivalent systems for these two cases are given in Appendices B and C, respectively.