Ideas and Governing Equations

To practitioners in reservoir engineering and well test analysis, the state-of-the-art has bifurcated into two divergent paths. The first searches for simple closed-form solutions. These are naturally restricted to simplified geometries and boundary conditions, but analytical solutions, many employing method of images techniques, nonetheless involve cumbersome infinite series. More recent solutions for transient pressure analysis, given in terms of Laplace and Fourier transforms, tend to be more computational than analytical they require complicated numerical inversion, and hence, shed little insight on the physics. 

It seems, very often, that all of the analytical solutions that can be derived, have been derived. Thus, the seeond path described above falls largely in the realm of supercomputers, high-powered workstations, and brute force numerieal analysis it is the science, or more appropriately the art, that we call reservoir simulation, requiring industrywide eomparison projects for validation. There has been no middle ground for smart solutions that solve difficult problems, that is, for solutions that provide physical insight and are in themselves useful, models that can be used for calibration purposes to keep numerical solutions honest. This dearth of truly useful real world examples lends credence to the 

This book introduces classes of steady-state solutions that the interested reader can extend and generalize. They are particularly meaningful to reservoirs that produce under near-steady conditions at high rates, typical of many oil fields outside the United States. The solutions are useful in studies related to flow heterogeneities, hydraulic fractures, nonlinear gas flows, horizontal drilling, infill drilling, and formation evaluation. The analytical techniques used are described in detail, applied to nontrivial flow problems, and extensions are outlined in the Problems and Exercises sections at the end of each chapter. 

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Let N2 = 2" for the clarity only. The main idea behind the decomposition method is the further successive elimination from the governing equations of the vectors Yj with odd numbers and, after this, with even numbers divisible 2, 4, 8 etc. Other ideas are connected with setting the following equations for j = 2, 4, 6,.. ., N2 — 2, where N2 = 2 ... 

Reynolds [127] provided the fundamental ideas about averaging and was the first to accomplish the formulation of the governing equations for turbulent flows in terms of mean and fluctuating flow quantities rather than instantaneous quantities. Reynolds stated the mathematical rules for forming mean values. That is, he suggested splitting a turbulent velocity field into its mean and fluctuating components, and wrote down the equations of motion for these two velocity quantities. 

To reformulate the governing equations in terms of mean flow quantities rather than instantaneous quantities, Reynolds postulated the fundamental ideas of averaging. In the averaging procedure devised by Reynolds [127] the instantaneous quantities are decomposed into the sum of mean and fluctuating quantities. 

The role of constitutive equations is to instruct us in the relation between the forces within our continuum and the deformations that attend them. More prosaically, if we examine the governing equations derived from the balance of linear momentum, it is found that we have more unknowns than we do equations to determine them. Spanning this information gap is the role played by constitutive models. From the standpoint of building effective theories of material behavior, the construction of realistic and tractable constitutive models is one of our greatest challenges. In the sections that follow we will use the example of linear elasticity as a paradigm for the description of constitutive response. Having made our initial foray into this theory, we will examine in turn some of the ideas that attend the treatment of permanent deformation where the development of microscopically motivated constitutive models is much less mature. 

Nonconserved Fields and the Allen-Cahn Equation. Nonconserved order parameters (sueh as the state of order itself as introduced in eqn (12.22)) can have complex spatial distributions that evolve in time. The governing equation in this case is provided by the Allen-Cahn equation. To obtain this equation using the ideas introduced above, we hark back to the L2 norm. It is assumed that the instantaneous free energy of the system can be written down as a functional of the order parameter and its gradients. For example, in the context of the Allen-Cahn equation, the relevant free energy functional is... 

With a first-order reaction, the governing equation is linear and could thus be solved without any use of scaling or asymptotic methods. However, we could just as easily assume that the reaction rate is second order in c or add other complications that do not so easily allow an exact analytic solution. The point here is to illustrate the idea of the asymptotic approximation technique, which is easily generalizable to all of these problems. 

To determine the appropriate scaling, and the correct form for the governing equation in this boundary layer region, we use the idea of rescaling that was introduced in Section A. First, for convenience, we follow the analysis from that section, and redefine the dimensionless radial variable as... 

These examples illustrate the interaction of composition distribution and stress field in a deformed solid solution. The mathematical structure exists for analysis of such phenomena, but the governing equations are inherently nonlinear analysis is very difficult if shape changes are taken into account and systematic study has not yet been undertaken. The purpose here is to formulate and analyze a physical situation involving coupled deformation and composition evolution that serves as a reasonably transparent vehicle for presenting the underl5dng ideas, but that avoids the complexity of evolution of shape. 

To fix ideas, we will formulate the transformed problem for a single-phase, eompressible flow with a eonstant horizontal permeability kh in the areal (x,y) eoordinates, and a variable permeability kv(z) in the layered, vertieal z direetion. The matrix porosity z) may vary with z. The fluid viseosity p is assumed to be eonstant. The governing equation then takes the following form,... 

Reynolds [128] postulated that the Navier-Stokes equations are still valid for turbulent flows, but recognized that these equations could not be applied directly due to the complexity and irregularity of the fluid dynamic variables. A true description of these flows at all points in time and space was not feasible, and probably not very useful at the time. Instead, Reynolds proposed to develop equations governing the mean quantities that were actually measurable. To reformulate the governing equations in terms of mean flow quantities rather than instantaneous quantities, Reynolds postulated the fundamental ideas of averaging. 

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