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**Table of contents**

- Other Resources
- Panagiotis Neofytou
- Authors are invited to submit abstracts covering, but not limited to, the following areas:
- Computational fluid dynamics - Wikipedia

Description of motion for Lagrangian:. If the density is constant, the flow is assumed to be incompressible and then continuity reduces it to:. Therefore, many terms vanished through equation results in a much simpler Navier-Stokes Equation:. Conservation of Energy is the first law of thermodynamics which states that the sum of the work and heat added to the system will result in the increase of the energy in the system:. One of the common types of an energy equation is:. The Mathematical model merely gives us interrelation between the transport parameters which are involved in the whole process, either directly or indirectly.

Even though every single term in those equations has relative effect on the physical phenomenon, changes in parameters should be considered simultaneously through the numerical solution which comprises differential equations, vector and tensor notations. Heat transfer, fluid dynamics, acoustic, electronics and quantum mechanics are the fields that PDEs are highly used to generate solutions.

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What is the significance of PDEs to seeking a solution on governing equations? To answer this question, we initially examine the basic structure of some PDEs as to create connotation. For instance:. Comparison between equation 5 and equation 13 specifies the Laplace part of the continuity equation.

What is the next step? What does this Laplace analogy mean? To start solving these enormous equations, the next step comes through discretization to ignite the numerical solution process.

### Other Resources

The numerical solution is a discretization-based method used in order to obtain approximate solutions to complex problems which cannot be solved with analytic methods because of complexity and ambiguities. As seen in Figure 3, solution processes without discretization merely give you an analytic solution which is exact but simple. Moreover, the accuracy of the numerical solution highly depends on the quality of the discretization. Broadly used discretization methods might be specified such as finite difference, finite volume, finite element, spectral element methods and boundary element.

Figure 4: The panoramic structure of a CFD project and its stages. Multitasking is one of the plagues of the century that generally ends up with procrastination or failure. Therefore, having planned, segmented and sequenced tasks is much more appropriate to achieving goals: this has also been working for CFD.

## Panagiotis Neofytou

In order to conduct an analysis, the solution domain is split into multiple sub-domains which are called cells. The combination of these cells in the computational structure is named mesh. The mesh as simplification of the domain is needed, because it is only possible to solve the mathematical model under the assumption of linearity. This means that we need to ensure that the behavior of the variables we want to solve for can assumed to be linear within each cell. This requirement also implies that a finer mesh generated via mesh refinement steps is needed for areas in the domain where the physical properties to be predicted are suspected to be highly volatile.

Errors based on mesh structure are an often encountered issue which results in the failure of the simulation. Therefore, a study of independency needs to be carried out. The accuracy of the solution enormously depends on the mesh structure. To conduct accurate solutions and obtaining reliable results, the analyst has to be extremely careful on the type of cell, the number of cell and the computation time.

The optimization of those restrictions is defined as mesh convergence which might be sorted as below:. Therefore, errors, based on mesh structure, can be eliminated and optimum value for number of elements might virtually be achieved as to optimize calculation time and necessary computation resources. An illustration is shown in Figure 4 that looks into static pressure change at imaginary region X through increase in number of elements. According to Figure 4, around 1,, elements would have been sufficient to conduct a reliable study. Creating a sculpture requires a highly talented artist with the ability to imagine the final product from the beginning.

Yet a sculpture can be, for example, a simple piece of rock in the beginning, but might become an exceptional artwork in the end. A completely gradual processing throughout carving is an important issue to obtain the desired unique shape. Keep in mind that in every single process, some of the elements, such as stone particles, leftovers, are thrown away from the object.

### Authors are invited to submit abstracts covering, but not limited to, the following areas:

CFD also has a similar structure that relies on gradual processing during the analysis. In regions that are highly critical to the simulation results for example a spoiler on a Formula 1 car the mesh is refined into smaller elements to make the simulation more accurate. Natural fractures in rock have complex geometries.

Over the past few decades, numerous measurements of fracture-wall roughness have been performed to characterize the geom- etry of fractures. Fracture roughness has been obtained from measurements using stylus profilometers Tsang ; Poon et al. All of these methods have shown that the fracture geometry is highly complex, with many showing that the structure of fractures is well-described by non-Euclidian fractal geometry Odling ; Kulatilake and Um ; Lanaro Self-affine synthetic fracture geometries have been created using this knowl- edge and used for experimental and numerical studies e.

The fracture profile geometries used for this study were obtained from sections of a micro-CT-scanned fracture within Berea sandstone Karpyn et al. Numerous numerical models have been used to describe the flow through a single fracture. Often the fracture is described using a fairly simple geometry, such as a composite set of perpendicular channels Ingham et al. When the area of study is on the reservoir scale, a high level of approximation is required due to the inherent computational cost of modeling flow through thousands of fractures McKoy and Sams Discrete fracture simulators often assume fractures as having closely spaced parallel walls.

The accuracy of such simulations may be improved by inclusion of phenome- nological equations for flow through real fractures.

## Computational fluid dynamics - Wikipedia

For single-fracture studies that focus on describing fluid flow within a fractured piece of rock, a geometry that captures the compli- cated structure of naturally occurring fractures is desired. Similar to the treatment of flow through porous media, various simplified mathemati- cal relationships have been examined to describe flow through fractures. This relationship has been shown to be quite useful in predicting fluid transport through highly fractured reservoirs McKoy and Sams , but does not account for the small-scale roughness of the fracture or the possibility that flow adjacent to the fracture walls exist due to the rock permeability.

A relationship with the simplicity of the cubic-law, while incorporating the complexity of the fracture geometry and including flow contributions from the surrounding permeable medium, would have obvious benefits for use in reservoir-scale discrete fracture simulators. Numerical studies using complex geometries include the study of Piri and Karpyn Karpyn and Piri ; Piri and Karpyn where an invasion percolation pore-throat model was used to model multiphase immiscible flow through a CT-scanned fracture. This modeling was shown to compare favorably to multiphase experiments Piri and Karpyn Tracer dis- persion was modeled using lattice-Boltzmann techniques by Drazer and Koplik within a two-dimensional 2D rough-walled fracture.

Numerical simulations of flow through a natural fracture in sandstone were per- formed by Al-Yaarubi et al. These simulations were shown to compare well to experiments performed through similar 2 cm square fracture geometries, and the deviations from the Forchheimer equations at low flow rates were examined. The goal of this study was to perform simulations similar to these studies and to obtain a relationship that is appropriate to describe laminar, single-phase fluid flow in a fracture for use in a reservoir-scale, discrete fracture simulator.

While fractures often play a crucial role in determining the flow distribution though frac- tured rock, fluid can also move through the surrounding permeable media Rangel-German et al. A friction-factor correlation for flow through rock fractures has been proposed by Nazri- doust et al. Their simulations were performed through 2D fracture profiles within Berea sandstone where the fracture walls were assumed to be well-described by no-slip boundary conditions, i.

The fracture geometries used were obtained from slices of a micro-CT-scanned fracture in Berea sandstone by Karpyn et al. Nazridoust et al. An analysis of the roughness and the fractal properties of these 2D profiles is first presented. A review of the friction-factor correlation proposed by Nazridoust et al. The flow through the fracture and surrounding rock matrix was solved with a constant pressure gradient across the length of the domain. Differ- ent aspects of the flow solutions, including the transport of fluid between the fracture and medium, as well as the applicability of the new friction-factor, are presented in Sect.

This new friction-factor may be useful in simulations of highly fractured reservoirs where the formation permeability is appreciable. Micro-CT scanning was performed on this fracture, embedded within the rock, with a voxel resolution of Information about the micro-scanning techniques and detailed three- dimensional 3D fracture properties can be found in Karpyn et al. The fracture geometries, described in this article, were obtained by transforming the 3D data obtained from the CT-scans into four separate, completely open, 2D profiles by Nazridoust et al.

These profiles were used to cre- ate the current fracture models, and are shown in Fig. The length of the fracture profiles, As can be seen, these profiles are completely open no zero apertures , and there are no apertures greater than 1. The total open area of each slice of the fracture was measured, and as shown in Table 1, is around 60 mm2. Fractures have been shown to be well-described by fractals, since seminal the study of Mandelbrot et al. Numerous studies have measured the Df of fracture profiles in concrete Dougan et al.

From this and Eq. The high values of R 2 for the power-law fittings of the logarithmic plots in Fig. As is shown in both Fig. These Hurst exponents are lower than the average H observed by Dougan et al.

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The self-affine Brownian fractal structure has been used as a fracture geometry analogy in numerous studies of flow through fractures e. The results of the single-phase flow simulations through fractures with impermeable walls presented by Nazridoust et al. The pressure losses across the fractures were reported for different flow rates of air and water. With this information, and Eq.

It is important to note that using Eq. The value of tortuosity is difficult to measure in physical situations, therefore we used CFD to directly measure the length of flow path lines along the fracture length for low velocity flows within the fracture profiles. The details of the computational methods used to solve the fluid flow with the fracture are given in the next section.

For these low velocity flows, the fracture walls were modeled as no-slip boundaries [unlike in the following sections discussing flow through permeable matrix discussion, but similar to the models discussed by Nazridoust et al. The length of these particle trajectories were evaluated and averaged to determine the length that a particle travels within the fracture. This averaged valued was used as an estimate of L e. The particles tended to meander in large aperture locations and constrict into a smaller area within small aperture regions. In summary, the fracture geometries used in this study were obtained from the thres- holding of micro-CT-scanned data of a real fracture in Berea sandstone reported by Karpyn et al.