The stochastic-continuum model developed by Schwartz and Smith [1988] is a new approach to the traditional continuum methods for solute transport in fractured media. Instead of trying to determine dispersion coefficients and an effective porosity for the hydraulic system, statistics on particle motion (direction, velocity and fracture length) collected from a discretely modeled sub-domain network are used to recreate particle motion in a full-domain continuum model. The discrete sub-domain must be large enough that representative statistics can be collected, yet small enough to be modeled with available resources. Statistics are collected in the discrete sub-domain model as the solute, represented by discrete particles, is moved through the network of fractures. The domain of interest, which is typically too large to be modeled discretely is represented by a continuum distribution of the hydraulic head. A particle tracking method is used to move the solute through the continuum model, sampling from the distributions for direction, velocity and fracture length. This thesis documents extensions and further testing of the stochastic-continuum two-dimensional model and initial work on a three-dimensional stochastic-continuum model. Testing of the model was done by comparing the mass distribution from the stochastic-continuum model to the mass distribution from the same domain modeled discretely. Analysis of the velocity statistics collected in the two-dimensional model suggested changes in the form of the fitted velocity distribution from a gaussian distribution to a gamma distribution, and the addition of a velocity correlation function. By adding these changes to the statistics collected, an improvement in the match of the spatial mass distribution moments between the stochastic-continuum and discrete models was effected. This extended two-dimensional model is then tested under a wide range of network conditions. The differences in the first spatial moments of the discrete and stochastic-continuum models were less than 10%, while the differences in the second spatial moments ranged from 6% to 30%. Initial results from the three-dimensional stochastic-continuum model showed that similar statistics to those used in the two-dimensional stochastic-continuum model can be used to recreate the nature of three-dimensional discrete particle motion. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

A research report submitted to the Faculty of Engineering and Built Environment, University of the Witwatersrand, Johannesburg, South Africa, in partial fulfilment of the requirements for the Degree of Master of Science in Engineering, November 2018 / The main aim of the study was to develop mathematical flow model of the shale gas at the Karoo Basin of South Africa (SA). The model development incorporates three systems (phases) to form a triple continuum flow model, the phases include matrix (m), natural (NF) and hydraulic fracture (HF). The model was developed from the continuity equation, and the general equations were formed. (0.05������ ���� = 3.90087 × 10−15 ��2���� ����2 + 3.90087 × 10−15 ��2���� ����2 − 1.95043 × 10−16(20 × 106 − ������), 0.01 �������� ���� = 2.00 × 10−15(20 × 106 − ������) − 2.00 × 10−9(20 × 106 − ������) + �� ���� [7.80 × 10−5 �������� ���� ] + �� ���� [7.80 × 10−5 �������� ���� ] �� ���� [0.1248269 �������� ���� ] + 0.1248269(20 × 106 − ������)− 4.98 × 10−4 = �������� ���� The model was solved using numerical method technique known as Finite Difference Method (FDM). For each phase a computer program MATLAB was used to plot the pressure gradient. Hydraulic pressure gradient fractures propagate between the distance of 100m and 500m. The model was verified using the data of Barnett Shale. Sensitivity analysis was also performed on the hydraulic permeability, drainage radius and the initial pressure of the reservoir. / XL2019

Proppant flowback in fractures under confining pressures is not well understood and difficult to reproduce in a laboratory setting. Improper management of proppant flowback leads to flow restrictions near the well bore, poor fracture conductivity and costly production equipment damage. A simple, scalable model is developed using a discrete element method (DEM) particle simulator, to simulate representative cubic volumes consisting of fracture openings, fracture walls and the confining formation. The effects of fracture width, confining stress, fluid flow velocity and proppant cohesion are studied for a variety of conditions. Fracture width is found to be dependent on confining stress and fluid flow velocity while proppant production is also dependent on cohesion. Three regimes are observed, with complete fracture evacuation occurring at high flow rates and low confining stresses, fully packed fractures occurring at high confining stresses and open but mostly evacuated fractures occurring in-between. From these observations, a recommended flowback rate can be estimated for a given set of conditions. A slow and controlled well flowback is recommended to improve proppant pack stability. The rate ramp-up time is dependent on the leak-off coefficient. / text

D.Sc. / Please refer to full text to view abstract

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for the degree of Master of Science. 21 March 2016. / The problem of turbulent hydraulic fracturing is considered. Despite it being a known phenomenon, limited mathematical literature exists in this field. Prandtl’s mixing length model is utilised to describe the eddy viscosity and a mathematical model is developed for two distinct cases: turbulence where the kinematic viscosity is sufficiently small to be neglected and the case where it is not. These models allow for the examination of the fluid’s behaviour and its effect on the fracture’s evolution through time. The Lie point symmetries of both cases are obtained, and a wide range of analytical and numerical solutions are explored. Solutions of physical significance are calculated and discussed, and approximate solutions are constructed for ease of fracture estimation. The non-classical symmetries of these equations are also investigated. It was found that the incorporation of the kinematic viscosity within the modelling process was important and necessary. / MT2016

No description available.

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains xii, 99 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 53-54).

A sizable amount of literature exists analyzing the effect of confining stress on fracture conductivity in sandstones. This thesis attempts to answer similar questions with regard to shale formations. The low Young’s Moduli and Brinell hardness values characteristic of many prospective shale formations may lead to a great deal of embedment and fines production which can drastically reduce fracture conductivity. Furthermore, shales exhibit sensitivity to aqueous fluids which may cause them to be weakened in the presence of certain fracturing fluids. Previous work analyzing shale fluid sensitivity has failed to preserve the shales’ formation properties by allowing the shale to dry out. This paper presents a study of propped fracture conductivity experiments at reservoir temperature and pressure using various North American shale reservoir cores. Exposure to the atmosphere can alter the mechanical properties of the shale by either drying or hydrating the samples, so care was taken to preserve these shales in their native state by maintaining constant water activity (relative humidity). Variations in applied closure stress and aqueous fluid exposure were analyzed and in certain cases altered the propped fracture conductivity by crushing proppant, embedding the proppant into the fracture face, and producing fines. The damage to fracture conductivity is correlated to mineralogy for the various shale samples. These findings show that a one-size-fits-all frac design will not work in every shale formation, rather a tailored approach to each shale is necessary. In the future, the results of this work will be analyzed alongside easier to perform Brinell hardness tests, swelling tests, and other characterization techniques incorporated into the UT Shale Characterization Protocol. Correlations were developed to relate the simpler tests to the fracture conductivity experiments which yield a straight forward method to determine the role embedment and fluid sensitivity have on post treatment fracture conductivity in shales. The UT Shale characterization Protocol can then be used to optimize the design and execution of fracing treatments. / text

In this work, we present methodologies for optimization of hydraulic fracturing design under uncertainty specifically with reference to the thick and anisotropic reservoirs in the Lower Tertiary Gulf of Mexico. In this analysis we apply a stochastic programming framework for optimization under uncertainty and apply a utility framework for risk analysis. For a vertical well, we developed a methodology for making the strategic decisions regarding number and dimensions of hydraulic fractures in a high-cost, high-risk offshore development. Uncertainty is associated with the characteristics of the reservoir, the economics of the fracturing cost, and the fracture height growth. The method developed is applicable to vertical wells with multiple, partially penetrating fractures in an anisotropic formation. The method applies the utility framework to account for financial risk. For a horizontal well, we developed a methodology for making the strategic decisions regarding lateral length, number and dimensions of transverse hydraulic fractures in a high-cost, high-risk offshore development, under uncertainty associated with the characteristics of the reservoir. The problem is formulated as a mixed-integer, nonlinear, stochastic program and solved by a tailored Branch and Bound algorithm. The method developed is applicable to partially penetrating horizontal wells with multiple, partially penetrating fractures in an anisotropic formation.

Thesis (M.S. - Hydrology and Water Resources)--University of Arizona. / Includes bibliographical references (leaves 201-204).