Research On Some Differential Equations From Natural Gas Hydrate Production

As a kind of unconventional energy,natural gas hydrate,or hydrate for short,is widely distributed in permafrost and marine sediments.It is characterized as large scale storage,high energy density and clean energy etc.So it is one potential new energy source.More and more researchers pay attention to the exploitation of natural gas hydrate.In this paper,three mathematical models of heat injection and depressurization are studied,including the fractional thermal decomposition model,the radially symmetric thermal decomposition model with density difference and the steady radial symmetric depressurization model.This paper consists of five chapters.Chapter 1 summarizes the research background and significance of hydrate,the research history of mathematical model of hydrate exploitation,and the basic knowledge.Content and innovation of this paper are introduced here.The second to the fourth chapter is the main content of the paper.In the last chapter,the thesis is summarized and prospected.In chapter 2,it is discussed for fractional thermal decomposition model of hydrate.In order to describe the temperature distribution in the process of hydrate decomposition,a fractional partial differential equation with moving boundary was established.Then,by using Wright's function,the analytical solution is obtained.An example is given for a hydrate reservoir in a semi-infinite region,and the following conclusions are drawn: If fractional order is given,the temperature decreases continuously from the initial heat injection temperature to the reservoir temperature with the increasing decomposition distance.As time flies,the closer the fixed location is to the wellbore,the shorter time the area takes to approach to the initial heat injection temperature.The fractional model is an extension of the integral model.When the value of the fractional order approaches 1,fractional model describing the temperature change becomes the integral one.As the fractional order increases gradually,the decomposition distance also increases.When the initial heat injection temperature becomes higher or fractional order is larger,the decomposition rate also increases.In chapter 3,it is investigated for the radially symmetric thermal decomposition model with density difference.Based on the law of conservation of mass,the formula of velocity is derived which is caused by density difference.The radially symmetric temperature distribution equation is then derived with density difference.Combining with the moving boundary,the analytical solution is obtained by using the self-similarity transformation technique.An example is given to analyze the conclusion.When the time is fixed,the greater the density difference between water and hydrate,the longer the decomposition distance.When the density of hydrate is smaller,in other words,the density difference is larger between water and hydrate,the decomposition is more hydrate.The higher the initial heat injection temperature,the more hydrate decomposition.The decomposition rate decreases with the increasing of decomposition distance.In Chapter 4,it is analyzed for the steady-state radially symmetric depressurization model.Though the conservation of mass,energy and Darcy's law,the model is established for the change of pressure and saturation under the steady-state condition.The analytical expressions are derived in detail for the hydrate saturation and gas saturation.Then,a case analysis is provided and draw the following conclusions.The closer to the wellbore,the higher the saturation of water and natural gas.The lower the saturation of hydrate,the faster the pressure drops.