For effective reservoir management and production optimization, it is important to understand drained volumes, pressure depletion and reservoir well rates at all flow times. For conventional reservoirs, this behavior is based on the concepts of reservoir pressure and energy and convective flow. But, with the development of unconventional reservoirs, there is increased focus on the unsteady state transient flow behavior. For analyzing such flow behaviors, well test analysis concepts are commonly applied, based on the analytical solutions of the diffusivity equation. In this thesis, we have proposed a novel methodology for estimating the drainage volumes and utilizing it to obtain the pressure and flux at any location in the reservoir. The result is a semi-analytic calculation only, with close to the simplicity of an analytic approach, but with significantly more generality. The approach is significantly faster than a conventional finite difference solution, although with some simplifying assumptions. The proposed solution is generalized to handle heterogeneous reservoirs, complex well geometries and bounded and semi-bounded reservoirs. Therefore, this approach is particularly beneficial for unconventional reservoir development with multiple transverse fractured horizontal wells, where limited analytical solutions are available. To estimate the drainage volume, we have applied an asymptotic solution to the diffusivity equation and determined the diffusive time of flight distribution. For the pressure solution, a geometric approximation has been applied within the drainage volume to reduce the full solution of the diffusivity equation to a system of decoupled ordinary differential equations. Besides, this asymptotic expression can also be extended to obtain the well rates, producing under constant bottomhole pressure constraint. In this thesis, we have described the detailed methodology and its validation through various case studies. We have also studied the limits of validity of the approximation to better understand the general applicability. We expect that this approach will enable the inversion of field performance data for improved well and/or fracture characterization, and similarly, the optimization of well trajectories and fracture design, in an analogous manner to how rapid but approximate streamline techniques have been used for improved conventional reservoir management.

[pt] Os dados adquiridos nos experimentos físicos e nas imagens geométricas ou médicas são tipicamente discretas. Esses dados são interpretados como amostras de uma função desconhecida, porém cujas derivadas servem para caracterizar o dado. Por exemplo, o movimento de um fluido é descrito por um campo de velocidades, uma curva é caracterizada pela evolução da sua curvatura, as imagens médicas são geralmente segmentadas por estimativas de gradiente, entre outros. É possível obter derivadas coerentes a partir de filtragem dos dados. Porém, em dados multi-dimensionais, os filtros usuais privilegiam direções alinhadas com os eixos, o que pode gerar problemas quando essas derivadas são interpretadas geometricamente. Por exemplo, a curvatura estimada dependeria da orientação da curva, perdendo o sentido geométrico da curvatura. O objetivo do presente trabalho é melhorar a invariância geométrica dos filtros de derivadas. / [en] Typical data acquired in physical experiments or in geometrical or medical imaging are discrete. This data is generally interpreted as samples of an unknown function, whose derivatives still serve for the data characterisation. For example, the movement of a fluid is described as a velocity field, a curve is characterised by the evolution of its curvature, images used in medical sciences are usually segmented by estimates of their gradients, among others. It is possible to obtain coherent derivatives by filtering the data. However, with multidimensional data, the usual filters present a bias towards to favor directions aligned with the axis, which may induce problems when the derivatives are interpreted geometrically. For example, the estimated curvature would depend on the orientation of the curve, loosing the geometric meaning of the curvature. The goal of the present work is to improve the geometric invariance of derivative filters.