臺灣博碩士論文加值系統

本研究的目的是要利用迴旋積分及反迴旋積分方法分析井壓測試資料。模擬井壓測試資料及推求流體在不同邊界條件地層之源函數。並研究各種不同邊界條件之源函數，以了解地層特性。
壓力、流率及源函數關係中，若已知源函數及流率，可利用迴旋積分求得壓力，若已知壓力及流率，可利用反迴旋積分求得源函數。亦即已知壓力、流率及源函數之中的二個函數，就可利用迴旋積分或反迴旋積分求得另外一個函數。
本研究結果顯示，源函數與油層的邊界條件、完井狀況、地層中有無水及水驅有關。具有井筒效應的源函數（隨時間變化）之特性是在開始時為水平直線，井筒效應結束後，會與一般的無限表面圓柱源函數或無限線源函數相同。正值（大於零）的源函數不受膚表因子之變化而變化。但具有負值（小於零）之膚表因子的源函數在井壓測試初期也為水平直線，而且，不同的負值膚表因子，會有不同值的水平源函數，在膚表效應結束時也與無限表面圓柱源函數或無限線源函數一致。部份貫穿井的源函數則是開始時與無限表面圓柱源函數不同，貫穿比例越小越偏離無限表面圓柱源函數，但後來也會趨近無限表面圓柱源函數。
有限圓形邊界地層的源函數在初始的時間之特性與無限表面圓柱源函數或無限線源函數相同，但受到地層外邊界影響之後，其源函數會變成水平直線。部份貫穿井具有水錐效應的源函數開始時會與具有部分貫穿效應的源函數相同，並偏離無限表面圓柱源函數。具有邊際水驅的源函數則是開始時與無限表面圓柱源函數相同，當水驅效應開始影響時，源函數會向下偏離無限表面圓柱源函數，最後出水時，源函數會急遽的大於無限表面圓柱源函數。具有底水驅的源函數則是開始時會稍微大於無限表面圓柱源函數，有水流入地層底部後，源函數也會向下偏離無限表面圓柱源函數，當出水時，源函數會急遽的大於無限表面圓柱源函數。

The purpose of this study is to utilize the convolution and deconvolution to analyze the pressure testing data, including obtaining source functions with various boundary conditions from simulations and using source functions to study the characteristics of source functions of a producing well in the different types of reservoirs.
The solution of flowing fluid in the reservoir (or solutions of the diffusivity equation) can be represented by the relationship among the pressure, flow rate, and source function. With such relationship, in applying either convolution or deconvolution, the third unknown function can be obtained if the other two are known. The pressure drop can be obtained by convoluting flow rates and source functions. If the pressure drop and flow rates are known, the source functions can be derived by deconvoluting the other two.
The source function is dependent on the boundary conditions of the reservoir, including the inner and outer boundary conditions. The inner boundary conditions investigated in this study include wellbore storage effects, skin effects, and partial penetrations. The outer boundary conditions studied are no flow boundary for a closed boundary reservoir, such as a finite reservoir, and water influx to a reservoir from an aquifer, such as water drive. The source function, changing with time, for wellbore storage effects is characterized by a horizontal line at a very early time stage. Then this coincides to the source function of the infinite surface cylinder, for the well without wellbore storage in an infinite reservoir, after the end of wellbore storage effects. The source functions with damage effects, i.e. positive skin factors, are almost the same. For a negative skin factor, the source function is a horizontal line at an early time that subsequently coincides with the infinite line source or infinite surface cylinder source functions. The source functions from wells with various penetration ratios are higher than the infinite surface cylinder source function, and then coincide to the infinite surface cylinder source function at a later time.
The source functions for different external reservoir radii become horizontal lines at a late time or while the outer boundary is affected. The source function of the partially-penetrated well in the center of a reservoir with water coning is consistent with the source function of the partial penetration at an early time. Later, the source function for water coning is lower than the infinite surface cylinder source function. At a late time, when the water breaks through the wellbore, the source function is dependent on the reservoir permeability. The value of the source function for edgewater drive is lower than the infinite surface cylinder source function at a later time. Then the source function for a well producing water has higher value than it does for the infinite surface cylinder source function at a late time stage. The value of the source function for bottomwater drive coincides to the partially-penetrated source function at an early time, and then decreases dramatically at a late time. Finally, when the water breaks through the well, the source function suddenly becomes higher than the infinite surface cylinder source function.

Abstract --------------------------------------------------i
中文摘要--------------------------------------------------iv
Acknowledgements------------------------------------------vi
Table of Contents --------------------------------------viii
List of Tables -------------------------------------------xi
List of Figures ----------------------------------------xiii
Nomenclature -------------------------------------------xxii
Chapter 1 Introduction ------------------------------------1
1.1 Research background -----------------------------------1
1.2 Study purposes ----------------------------------------3
Chapter 2 Literature review -------------------------------4
2.1 Conventional solutions of the diffusivity equation ----4
2.2 Source function solutions of the diffusivity equation -6
2.3 Interpretation of pressure data affected by boundary conditions ---------10
2.4 Evaluation of completion of a well -------------------12
2.5 Interpretation of pressure data affected by water drives ------------------------13
2.6 Interpretation of water coning -----------------------15
Chapter 3 Basic theory and calculations ------------------18
3.1 Diffusivity equation ---------------------------------18
3.2 Pressure solutions from source functions -------------20
3.2.1 Convolution using the matrix method ----------------23
3.2.2 Convolution using the polynomial multiplication ----24
3.3 Source functions using the deconvolution method ------24
3.3.1 Deconvolution using the matrix method --------------25
3.3.2 Deconvolution using the polynomial multiplication --26
3.4 Fourier transformation -------------------------------27
Chapter 4 Results ----------------------------------------29
4.1 Analytical source functions and their characteristics 30
4.2 Pressure solutions from analytical source functions --34
4.3 Source functions from simulated pressure data --------36
4.3.1 Source functions from an infinite reservoir --------37
4.3.1.1 Pressure drawdown test ---------------------------37
4.3.1.2 Pressure buildup test ----------------------------38
4.3.1.3 Wellbore storage effects -------------------------38
4.3.1.4 Skin effects -------------------------------------40
4.3.1.5 Partial penetrations -----------------------------42
4.3.2 Source functions from the closed boundary reservoir 43
4.3.2.1 Circular reservoir -------------------------------43
4.3.2.2 Rectangular reservoir ----------------------------45
4.3.3 Source functions from a reservoir with an aquifer --48
4.3.3.1 Edgewater drive ----------------------------------48
4.3.3.2 Bottomwater drive --------------------------------50
4.3.3.3 Water coning -------------------------------------53
Chapter 5 Discussions ------------------------------------56
Chapter 6 Conclusions ------------------------------------60
6.1 Conclusions ------------------------------------------60
6.2 Suggestions for the future work ----------------------64
References -----------------------------------------------66

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