The transient pressure behavior of a vertically fractured reservoir has been solved successfully based on numerical solutions. These solutions enhance the understanding of early time portion during the drawdown or the build-up test conducted in a vertically fractured reservoir. To completely understand the performance of a gas reservoir, solutions for later time are also necessary. However, stability considerations always complicate the design of the model. The main objective of this work is to develop a numerical model which could be stable for all times.
The model is formulated for a fracture of limited and radial extent completely penetrating the producing formation vertically. The model equations are transformed using conformal mapping. Appropriate stability criteria are developed and the resulting finite difference equations are solved using the Iterative ADIP method. The boundary conditions are handled using a special numerical approach resulting in stable solutions for all times. Another feature of the developed procedure is the ability to handle finite fracture procedure is the ability to handle finite fracture conductivity and stress sensitivity.
It has become a common practice to increase the flow capacity of gas wells by hydraulic fracturing. Although the shapes of these fractures are undoubtedly complicated, real fractures are often idealized as being either horizontal or vertical planes intersecting the wellbore. The orientation depends on the least principal stress and in deeper reservoirs the induced principal stress and in deeper reservoirs the induced fractures are usually vertical.
The purpose of this research was to develop a finite difference model by using the theory which has been developed by Wattenbarger and Ramey to interpret the pressure drawdown and pressure build-up tests in a vertically fractured reservoir and to study the pressure performance of the reservoir with both the pressure performance of the reservoir with both the fractures and the reservoir having pressure dependent properties. properties. Build-up and drawdown tests are usually performed on gas wells to determine the flow capacity (kh) of the formation, the condition of the region near the wellbore and the average reservoir pressure. These conditions can also be used to make long range economic projections or evaluate the need for remedial work.
Because of the relatively high permeability in most fractures compared to that of the formation, the first step was to assume an infinite flow capacity in the fracture, and to check the numerical model on this basis. In actual cases, however, fractures tend to heal after fracturing because of the effective overburden pressure. Sand, aluminum, glass beads etc., are used to prop the fractures and these proppants will reduce the flow capacity. The permeability of both the fracture and the formation may be stress sensitive which implies that it is a function of proppant concentration, depth and effective overburden proppant concentration, depth and effective overburden pressure. Hence the second step of this study was to pressure. Hence the second step of this study was to model these effects.
Differential Equations of Gas Flow
The equation for the gas flow in a porous medium with uniform porosity and thickness is,
(1)
where 'm(p)' is real gas pseudo-pressure, or real gas potential, defined as:potential, defined as: (2)
P is an arbitrary low pressure. P is an arbitrary low pressure.
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