A Semi-Analytic Solution for Flow in Finite-Conductivity Vertical Fractures Using Fractal Theory

Abstract

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 iv 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. v DEDICATIONThis thesis is dedicated to my parents, Fernando and Rosa. The effort that went into this thesis is a reflection of your unconditional love and support.In any field, find the strangest thing and then explore it.-John Archibald WheelerNo man should escape our universities without knowing how little he knows. ACKNOWLEDGEMENTSIt is difficult for me to imagine a more privileged set of circumstances under which to do one's graduate studies. The fact of the matter is that for the past 2 years I have been paid to learn from some of the brightest minds in petroleum engineering. I would like to thank Drs. Thomas A. Blasingame and George J. Moridis for believing in me, welcoming me in their research group and for funding my studies.Wherever my career takes me in the future, I will always hold them in the highest esteem.

Analysis of the Complex Fracture Flow in Multiple Fractured Horizontal Wells With the Fractal Tree-Like Network Models

“…Assume that we have N fractures and production simultaneously, due to the superposition principle, the pressure drop at the tip of the fracture is given by Eq. (19).…”

Simulation of the Production Performance of Fractured Horizontal Wells in Shale Gas Reservoirs Considering the Complex Fracture Shape

“…The fracture property parameters are simple to calculate and easy to establish analytical or semianalytical models, which are one of the more widely used fracture models in the early stage. Perkins–Kern–Nordgren and Khristianovic–Geertsma–DeKlerk models are the representatives of these kinds of models. , The overall and local composite area model uses the fractal theory to establish the hydraulic fracture model, which assumes that the hydraulic fracture shape conforms to the fractal structure characteristics . The hydraulic fractures will bifurcate twice during extension to form secondary fractures, and the secondary fractures will bifurcate three times during extension to develop tertiary fractures, and so on, creating a fractured shape with a dendritic structure. , This method usually simplifies the natural fracture characteristics of the reservoir into a dual medium model.…”

“…4,5 The overall and local composite area model uses the fractal theory to establish the hydraulic fracture model, 6 which assumes that the hydraulic fracture shape conforms to the fractal structure characteristics. 7 The hydraulic fractures will bifurcate twice during extension to form secondary fractures, and the secondary fractures will bifurcate three times during extension to develop tertiary fractures, and so on, creating a fractured shape with a dendritic structure. 8,9 This method usually simplifies the natural fracture characteristics of the reservoir into a dual medium model.…”

Novel Mathematical Model for Transient Pressure Analysis of Multifractured Horizontal Wells in Naturally Fractured Oil Reservoirs

“…23 − 25 Since it was first successfully introduced in the analysis of transient pressure behavior, 26 the fractal theory has been applied to the numerical model 27 and the analytical model 28 for the fractal-fracture network. This theory was then extended and combined with the trilinear flow model for the vertical well, 29 which defined fractal porosity and permeability as follows ( eqs 1 and 2 ). Here, ϕ r is the reference porosity (dimensionless).…”

“…High heterogeneity is a predominant problem in naturally fractured reservoirs, and the fractal theory has been verified as a valid approach for the scenario of high heterogeneity. − Since it was first successfully introduced in the analysis of transient pressure behavior, the fractal theory has been applied to the numerical model and the analytical model for the fractal-fracture network. This theory was then extended and combined with the trilinear flow model for the vertical well, which defined fractal porosity and permeability as follows (eqs and ). Here, ϕ r is the reference porosity (dimensionless). k r is the reference permeability (mD).…”