The accurate description of fluid flow through porous media allows an engineer to properly analyze past behavior and predict future reservoir performance. In particular, appropriate mathematical models which describe fluid flow through porous media can be applied to well test and production data analysis. Such applications result in estimating important reservoir properties such as formation permeability, skin-factor, reservoir size, etc. "Real gas" flow problems (i.e., problems where the gas properties are specifically taken as implicit functions of pressure, temperature, and composition) are particularly challenging because the diffusivity equation for the "real gas" flow case is strongly non-linear. Whereas different methods exist which allow us to approximate the solution of the real gas diffusivity equation, all of these approximate methods have limitations. Whether in terms of limited applicability (say a specific pressure range), or due to the relative complexity (e.g., iterative character of the solution), each of the existing approximate solutions does have disadvantages. The purpose of this work is to provide a solution mechanism for the case of timedependent real gas flow which contains as few "limitations" as possible. In this work, we provide an approach which combines the so-called average pressure approximation, a convolution for the right-hand-side non-linearity, and the Laplace transformation (original concept was put forth by Mireles and Blasingame). Mireles and Blasingame used a similar scheme to solve the real gas flow problem conditioned by the constant rate inner boundary condition. In this work we provide solution schemes to solve the constant pressure inner boundary condition problem. Our new semi-analytical solution was developed and implemented in the form of a direct (non-iterative) numerical procedure and successfully verified against numerical simulation. Our work shows that while the validity of this approach does have its own assumptions (in particular, referencing the right-hand-side non-linearity to average reservoir pressure (similar to Mireles and Blasingame)), these assumptions are proved to be much less restrictive than those required by existing methods of solution for this problem. We believe that the accuracy of the proposed solution makes ituniversally applicable for gas reservoir engineering. This suggestion is based on the fact that no pseudotime formulation is used. We note that there are pseudotime implementations for this problem, but we also note that pseudotime requires a priori knowledge of the pressure distribution in the reservoir or iteration on gas-in-place. Our new approach has no such restrictions. In order to determine limits of validity of the proposed approach (i.e., the limitations imposed by the underlining assumptions), we discuss the nature of the average pressure approximation (which is the basis for this work). And, in order to prove the universal applicability of this approach, we have also applied this methodology to resolve the time-dependent inner boundary condition for real gas flow in reservoirs.

The exploitation of unconventional reservoirs goes hand in hand with the practice of hydraulic fracturing and, with an ever increasing demand in energy, this practice is set to experience significant growth in the coming years. Sophisticated analytic models are needed to accurately describe fluid flow in a hydraulic fracture and the problem has been approached from different directions in the past 3 decades - starting with the use of line-source functions for the infinite conductivity case, followed by the application of Laplace Transforms and the Boundary-Element Method for the finite-conductivity case. This topic remains an active area of research and, for the more complicated physical scenarios such as multiple transverse fractures in ultra-tight reservoirs, answers are presently being sought. Fractal theory has been successfully applied to pressure transient testing, albeit with an emphasis on the effects of natural fractures in pressure-rate behavior. In this work, we begin by performing a rigorous analytical and numerical study of the Fractal Diffusivity Equation and we show that it is more fundamental than the classic linear and radial diffusivity equations. Subsequently, we combine the Fractal Diffusivity Equation with the Trilinear Flow Model, culminating in a new semi-analytic solution for flow in a finite-conductivity vertical fracture which we name the "Fractal-Fracture Solution". This new solution is instantaneous and has an overall accuracy of 99.7%, thus making it comparable to the Trilinear Pseudoradial Solution for practical purposes. It may be used for pressure transient testing and reservoir characterization of hydrocarbon reservoirs being produced by a vertically fractured well. Additionally, this is the first time that fractal theory is used in fluid flow in porous media to address a problem not related to reservoir heterogeneity. Ultimately, this work is a demonstration of the untapped potential of fractal theory; our approach is very flexible and we believe that the same methodology may be extended to develop new reservoir flow solutions for pressing problems that the industry currently faces.

[pt] A Equação da Difusividade Hidráulica Não-Linear (EDHN) modela o escoamento monofásico de fluidos em meios porosos levando em conta a variação das propriedades da rocha e do fluido presente no interior de seus poros. Normalmente, a solução adimensional da linha-fonte pD(rD, tD) para escoamento de líquidos é encontrada por meio do uso da transformada de Laplace ou transformação de Boltzmann, o qual, o perfil transiente de pressões em coordenadas cartesianas é descrito pela função erro complementar erfc(xD, yD, tD) e, em coordenadas cilíndricas pela função integral exponencial Ei(rD, tD). Este trabalho propõe a solução analítica pelo método de expansão assíntotica de primeira ordem em séries, para solução de alguns problemas de escoamento de petróleo em meios porosos com permeabilidade dependente da pressão de poros e termo fonte. A solução geral será implementada no software Matlab (marca registrada) e a calibração do modelo matemático será realizada comparandose a solução obtida neste trabalho com a solução calculada por meio de um simulador de fluxo óleo em meios porosos denominado IMEX (marca registrada) , amplamente usado na indústria de petróleo e em pesquisas científicas e que usa o método de diferenças finitas. A solução geral da equação diferencial é dada pela soma da solução para escoamento de líquidos com permeabilidade constante e o termo de primeira ordem da expansão assintótica, composto pela não linearidade devido à variação de permeabilidade. O efeito da variação instantânea de permeabilidade em função da pressão de poros é claramente demonstrado nos gráficos diagnósticos e especializados apresentados. / [en] The Nonlinear Hydraulic Diffusivity Equation (NHDE) models the singlephase flow of fluids in porous media considering the variation in the properties of the rock and the fluid present inside its pores. Normally, the dimensionless linear solution for the flow of oil is performed using the Laplace and Fourier transform or Boltzmann transformation and provides the unsteady pressure profile in Cartesian coordinates given by complementary error function erfc(xD, yD, tD) and in cylindrical coordinates described by the exponential integral function Ei(rD, tD). This work develops a new analytical model based on an integro-differential solution to predict the formation mechanical damage caused by the permeability loss during the well-reservoir life-cycle for several oil flow problems. The appropriate Green s function (GF) to solve NHDE for each well-reservoir setting approached in this thesis is used. The general solution is implemented in the Matlab (trademark) and the mathematical model calibration will be carried out by comparing the solution obtained in this work to the porous media finite difference oil flow simulator named IMEX (trademark). The general solution of the NHDE is computed by the sum of the linear solution (constant permeability) and the first order term of the asymptotic series expansion, composed of the nonlinear effect of the permeability loss. The instantaneous permeability loss effect is clearly noticed in the diagnostic and specialized plots.