- Abstract:
- 1 Introduction▲
- 2 Reservoir mathemat...▲
- 3 Numerical solution▲
- 4 Calculation instan...▲
- 5 Conclusions▲
- References
- Figure
- Fig. 1 Nonlinear flow curve of low-permeability reservoir
- Fig. 2 Nonlinear flow curves with different parameters of a and b
- Fig. 3 Oil production under different flow models
- Fig. 4 Water cut under different flow models
- Fig. 5 Reservoir pressure ((a)-(c)) and pressure gradient ((a′)-(c′)) distributions of different flow models in ten years: (a), (a′) Darcy flow; (b), (b′) Nonlinear flow; (c), (c′) Quasi-linear flow
- Fig. 6 Oil saturation distribution of different flow models in ten years: (a) Darcy flow; (b) Nonlinear flow; (c) Quasi-linear flow
- Fig. 7 Dimensionless permeability coefficient distribution of nonlinear flow model in ten years: (a) ξx; (b) ξy
J. Cent. South Univ. (2012) 19: 1980-1987
DOI: 10.1007/s11771-012-1235-7
Nonlinear flow numerical simulation of low-permeability reservoir
YU Rong-ze(于荣泽)1, BIAN Ya-nan(卞亚南)1, ZHOU Shu(周舒)2, WANG Kai-jun(王楷军)3,
L? Qi(吕琦)4, CHEN Zhao-hui(陈朝辉)5
1. PetroChina Research Institute of Petroleum Exploration & Development-Langfang, Langfang 065007, China;
2. Special Oil Development Company of Liaohe Oilfield Company, Panjin 124010, China;
3. Down-hole Service Company of Xinjiang Oilfield Company, Kelamayi 834000, China;
4. Geological Scientific Research Institute of Shengli Oilfield Company, Dongying 257015, China;
5. CNOOC Energy Technology & Service-Oilfield Engineering Research Institute, Tianjin 300452, China
? Central South University Press and Springer-Verlag Berlin Heidelberg 2012
Abstract:
A nonlinear flow reservoir mathematical model was established based on the flow characteristic of low-permeability reservoir. The well-grid equations were deduced and the dimensionless permeability coefficient was introduced to describe the permeability variation of nonlinear flow. The nonlinear flow numerical simulation program was compiled based on black-oil model. A quarter of five-spot well unit was simulated to study the effect of nonlinear flow on the exploitation of low-permeability reservoir. The comprehensive comparison and analysis of the simulation results of Darcy flow, quasi-linear flow and nonlinear flow were provided. The dimensionless permeability coefficient distribution was gained to describe the nonlinear flow degree. The result shows that compared with the results of Darcy flow, when considering nonlinear flow, the oil production is low, and production decline is rapid. The fluid flow in reservoir consumes more driving energy, which reduces the water flooding efficiency. Darcy flow model overstates the reservoir flow capability, and quasi-linear flow model overstates the reservoir flow resistance. The flow ability of the formation near the well and artificial fracture is strong while the flow ability of the formation far away from the main streamline is weak. The nonlinear flow area is much larger than that of quasi-linear flow during the fluid flow in low-permeability reservoir. The water propelling speed of nonlinear flow is greatly slower than that of Darcy flow in the vertical direction of artificial fracture, and the nonlinear flow should be taken into account in the well pattern arrangement of low-permeability reservoir.
Key words:
low-permeability reservoir; nonlinear flow; mathematical model; numerical simulation;
1 Introduction
For many years, Darcy’s law has been considered a fundamental equation that governs fluid flow in porous media. In 1869, KING [1] firstly detected the nonlinear fluid flow characteristic in low-permeability porous media. In the chemical engineering, some authors [2-3] have also noted that there may be departures from Darcy’s law at low-velocity fluid flow. BASAK [4] identified the low pressure gradient and low-velocity fluid flow as “pre-Darcy flow”, where the increase of fluid flow velocity can be greater than that proportional to the increase of fluid pressure gradient. Since then, many scholars [5-8] have confirmed the nonlinear fluid flow characteristic in low-permeability porous media. In recent years, the petroleum industry has shown a renovated interest in nonlinear flow [9-13], in order to better understand reservoir performance. Nonlinear flow is typically observed in low-permeability reservoir. In a consequence, pressure drop cannot be estimated from the classic Darcy equation, where the pressure gradient is a linear function of the flow velocity. In that case, in fact, the use of conventional reservoir numerical simulation software based on Darcy equation will lead to inaccurate production performances evaluation. Figure 1 presents the typical nonlinear flow curve [14-15], where the flow velocity increases nonlinearly with the driving pressure gradient. There are two specific points on the non-Darcy flow curve. Point A is the intersection point between the non-Darcy flow curve and x axis, and the corresponding pressure gradient is defined as the minimum starting pressure gradient. The fluid flow only happens when the driving pressure gradient exceeds the minimum pressure gradient. Point C is the critical point where the non-Darcy flow turns into quasi-linear flow, and the corresponding pressure gradient is known as critical pressure gradient. OA is the inert segment; AC is the nonlinear flow segment; CD is the quasi-linear segment. The fluid flow curve of low-permeability consists of nonlinear flow segment and quasi-linear flow segment. Point B is a hypothetical point between the extended line of DC and x axis, and the corresponding pressure gradient is defined as quasi-linear starting pressure gradient. At present, there are many flow models to describe the nonlinear flow law in low-permeability reservoir, which can be easily divided into quasi-linear flow and nonlinear flow models.
Fig. 1 Nonlinear flow curve of low-permeability reservoir
The quasi-linear flow model [16-17] takes the starting pressure gradient as a constant by extending the quasi-linear flow segment and neglecting the nonlinear flow segment. Due to the introduction of a constant in Darcy equation, the quasi-linear flow model is widely used in reservoir engineering, well test analysis and reservoir numerical simulation. CHENG and HAN [18-19] developed a two-dimensional and two-phase non-Darcy flow numerical simulator based on quasi-linear flow model. ZHAO [20] implemented the three-dimensional and three-phase quasi-linear flow numerical simulator which takes the starting pressure as a variable. YUAN and HAN [21] also contributed to the development of quasi-linear flow numerical simulation. However, the quasi-linear flow model does not take the nonlinear flow segment into consideration. The fluid flow only happens when the driving pressure gradient exceeds the quasi-linear starting pressure gradient, which shortens the flow range of low-permeability reservoir. There will be evident deviation between the field production and the theoretical analysis based on quasi-linear flow model, and a low economic and technical index will also be gained based on this flow model. Therefore, the quasi-linear flow model also has limitations in field application.
The nonlinear flow model [22] which takes both the nonlinear flow segment and the quasi-linear flow segment into consideration could describe the nonlinear flow law accurately. LONG [23] achieved the numerical simulation based on nonlinear flow model and developed the corresponding numerical simulator. This simulator which only establishes the relationship between the driving pressure gradient and permeability does not provide a smooth flow model and a complete reservoir mathematical model. Because of the function complexity and nonlinearity, there are not sufficient researches on the reservoir numerical simulation based on the nonlinear flow model. Therefore, it is of great necessity to include the nonlinear flow into low-permeability reservoir numerical simulation.
In this work, a nonlinear flow reservoir mathematical model is established based on the flow characteristic of low-permeability reservoir. The well-grid equations were deduced and the dimensionless permeability coefficient was introduced to describe the permeability variation. The nonlinear flow numerical simulation program was complied. A quarter of a five-spot well pattern unit geological model was simulated to study the effect of nonlinear flow on the exploitation of low-permeability reservoir. The oil production, water cut, reservoir pressure distribution and reservoir pressure gradient distribution under Darcy flow model, quasi-linear flow model and nonlinear flow model were provided. At last, the dimensionless permeability coefficient distribution was gained to describe the nonlinear flow degree.
2 Reservoir mathematical model
2.1 Basic assumptions
The assumptions for the nonlinear flow reservoir mathematical model are as follows. Firstly, the flow in low-permeability reservoir is isothermal flow. Secondly, there are at most three components in reservoir; the liquid component exists in the liquid phase completely; the gas component can exist in the gas phase as free gas, or exist in the liquid phase as dissolved gas; the liquid phase and the gas phase can achieve the phase equilibrium state instantaneously. Thirdly, the liquid phase and the gas phase are miscible. Fourthly, the liquid phase follows nonlinear flow law and the gas phase follows Darcy equation. Fifthly, the liquid permeability is the function of formation pressure gradient.
2.2 Mathematical equations
The motion equation:
(1)
The state equation:
(2)
(3)
The continuity equation:
(4)
(5)
The auxiliary equation:
(6)
where V is flow velocity, cm/s; K is absolute permeability, 10-3 μm2; Kr is relative permeability; μ is phase viscosity, mPa·s; ρ is density, g/cm3; P is pressure, MPa; S is the saturation; Pcgl is the capillary pressure of gas/liquid interface, MPa; φ is porosity; B is volume factor, m3/m3; q is mass rate of injection (or production, if negative), g/(cm3·s3); a is the dimensionless nonlinear parameter; b is nonlinear parameter, m/MPa. Subscripts “l” is liquid phase and “g” is gas phase.
Two nonlinear parameters are focused: a is an affecting factor that influences the nonlinear flow concave curve segment; b is equivalent to the reciprocal of the quasi-linear starting pressure gradient in the quasi-linear flow model. The second part of the permeability state equation reflects the loss of liquid permeability, resulting in nonlinear flow.
Abstract: A nonlinear flow reservoir mathematical model was established based on the flow characteristic of low-permeability reservoir. The well-grid equations were deduced and the dimensionless permeability coefficient was introduced to describe the permeability variation of nonlinear flow. The nonlinear flow numerical simulation program was compiled based on black-oil model. A quarter of five-spot well unit was simulated to study the effect of nonlinear flow on the exploitation of low-permeability reservoir. The comprehensive comparison and analysis of the simulation results of Darcy flow, quasi-linear flow and nonlinear flow were provided. The dimensionless permeability coefficient distribution was gained to describe the nonlinear flow degree. The result shows that compared with the results of Darcy flow, when considering nonlinear flow, the oil production is low, and production decline is rapid. The fluid flow in reservoir consumes more driving energy, which reduces the water flooding efficiency. Darcy flow model overstates the reservoir flow capability, and quasi-linear flow model overstates the reservoir flow resistance. The flow ability of the formation near the well and artificial fracture is strong while the flow ability of the formation far away from the main streamline is weak. The nonlinear flow area is much larger than that of quasi-linear flow during the fluid flow in low-permeability reservoir. The water propelling speed of nonlinear flow is greatly slower than that of Darcy flow in the vertical direction of artificial fracture, and the nonlinear flow should be taken into account in the well pattern arrangement of low-permeability reservoir.
- Nonlinear flow numerical simulation of low-permeability reservoir