The flow of homogeneous fluids through porous media: analogies with other physical problems

http://cds.cern.ch/record/994225/files/0610327.pdf

Four-dimensional string compactifications with D-branes, orientifolds and fluxes

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

/pdf/the-topological-vertex-3lrad1kwkt.pdf

The Topological Vertex

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.

/pdf/moduli-spaces-of-local-systems-and-higher-teichmuller-theory-36h32nsutb.pdf

Moduli spaces of local systems and higher Teichmüller theory

We discuss systematic approaches to the classification of string/M theory vacua, and physical questions this might help us resolve. To this end, we initiate the study of ensembles of effective Lagrangians, which can be used to precisely study the predictive power of string theory, and in simple examples can lead to universality results. Using these ideas, we outline an approach to estimating the number of vacua of string/M theory which can realize the Standard Model.

https://iopscience.iop.org/article/10.1088/1126-6708/2003/05/046/pdf

The statistics of string/M theory vacua

Recently, certain higher-dimensional complex manifolds were obtained by S. Govindarajan [1] by associating a higher dimensional uniformisation to the generalised Teichmuller spaces of Hitchin. The extra dimensions are provided by the ‘times’ of the generalised KdV hierarchy. In this Letter, we complete the proof that these manifolds provide the analog of superspace for W-gravity and that W-symmetry linearises on these spaces. This is done by explicitly constructing the relationship between the Beltrami differentials which naturally occur in the higher-dimensional manifolds and the Beltrami differentials which occur in W-gravity. This also resolves an old puzzle regarding the relationship between KdV flow. and W-diffeomorphisms.

Covariantising the Beltrami equation in W-gravity

We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If $p_3(n)$ denotes the number of solid partitions of an integer $n$, we show that $\lim_{n\rightarrow\infty} n^{-3/4} \log p_3(n)\sim 1.822\pm 0.001$. This shows clear deviation from the value $1.7898$, attained by MacMahon numbers $m_3(n)$, that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in $\log p_3(n)$. In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to $n^{1/4}$, the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.

Estimating the asymptotics of solid partitions

Coal reservoir productions depend entirely on cleat permeability and porosity values. The models used to date to calculate cleat parameters are either stress-dependent or strain-dependent. A study is carried to analyze the individual effects of stress and strain on the reservoir and fluid properties. A one-dimensional cleat for an under saturated low permeable coal bed methane reservoir is semi-analytically solved. Standard stress models, Shi-Durucan and Cui-Bustin, predict a range within which field values lie. A comparative study between strain and stress model behavior is carried out. The strain model considers matrix swell and shrink but ignores stress effects due to them, while a stress model considers them all. The stress-model is dependent on effective horizontal stresses in the reservoir. Stress models are preferred to strain model. Stress-dependent permeability being closer to actual values captures the field in a better manner. Though standard stress models are more accurate than the strain model, yet they cannot determine a specific permeability value at any given point in time. Emphasis is, therefore, laid on developing a new stress-dependent model. An iterative combination study of the standard models provides the new model. An empirical equation to calculate cleat properties, approximate to the field, is developed from the research. The new permeability model is substituted in the fluid production equation to obtain cumulative gas/water produced at any time interval. The results lie in the range predicted by the standard stress models and match the field observations, thus more reliable. The stress model is simple to use and is mathematically easy to formulate. It is flexible and can accommodate reservoir temperature and sorption strain changes.

Semi-analytic analysis and optimization of stress-dependent permeability model for the coal bed methane gas reservoir

This manuscript primarily focuses on the constraints associated with the extended version of Darcy’s law that is used to describe the multiphase flow through a porous media; and in particular, a petroleum reservoir. This manuscript clearly brings out the basics associated with the usage of Darcy’s law, and reasons out the inapplicability of the Navier-Stokes Equation in order to describe the momentum conservation in a typical petroleum reservoir. Further, this work highlights the essence of continuum-based Darcy’s macroscopic-scale equation with that of Navier-Stokes’s microscopic-scale equation. Further, the absence of capillary forces in original Darcy’s equation and extending the same by considering the concept of ‘capillary pressure’ in order to accommodate the multi-phase flow has several critical constraints associated with it. In this manuscript, all these constraints or limitations have been posed in the form of a list of basic queries that need to be addressed or at least to be understood with clarity, when applying the multi-phase fluid flow equations associated with a petroleum reservoir. This study is limited to an oil-water two-phase system.

Darcy-and pore-scale issues associated with multi-phase fluid flow through a petroleum reservoir

We study a one-parameter family ($\ell=1,2,3,\ldots$) of configurations that are square-ice analogues of plane partitions. Using an algorithm due to Bratley and McKay, we carry out exact enumerations in order to study their asymptotic behaviour and establish, via Monte Carlo simulations as well as explicit bounds, that the asymptotic behaviour is similar to that of plane partitions. We finally carry out a series analysis and provide independent estimates for the asymptotic behaviour.

Suresh Govindarajan

Papers

Covariantising the Beltrami equation in W-gravity

Estimating the asymptotics of solid partitions

Semi-analytic analysis and optimization of stress-dependent permeability model for the coal bed methane gas reservoir

Darcy-and pore-scale issues associated with multi-phase fluid flow through a petroleum reservoir

On a square-ice analogue of plane partitions