Numerical Simulation of Nonlinear Seepage in Super-low Permeability Reservoir

Numerical Simulation of Nonlinear Seepage in Super-low
Permeability Reservoir
XU Qingyan1, YANG Zhengming1, 2
1. Institute of Flow and Fluid Mechanics, Chinese Academy of Science, China, 065007
2. Research Institute of Petroleum Exploration and Development-Langfang, China, 065007
Abstract: The fluid seepage in the super-low permeability reserves conforms to the nonlinear seepage
law, and according to this characteristic, the nonlinear seepage numerical simulation software has been
developed. In addition, a nonlinear seepage model considering the nonlinear bending section is
established in this paper. Moreover, with the combination of field application and laboratory experiment
data, an ideal model of diamond inverted nine spots well pattern is also built. Besides, the simulation
result of different well spacing and well array, different fracture penetration ratio and different fracture
conductivity is comparative analyzed. The result shows that: the smaller the well spacing and well array
is, the higher the oil recovery and water cut; the target block has an optimal fracture penetration ratio,
which is about 0.28; artificial fracture conductivity has little effect on the oilfield development.
Keywords: super-low permeability reservoir, nonlinear seepage, numerical simulation,
diamond inverted nine spots well pattern, well spacing and well array, fracture penetration ratio, fracture
conductivity
1 Introduction
In the super-low permeability reservoir, the fluid flow channel is very small and the liquid-solid
interaction is significant. Fluid boundary layer in the pore surface results in increased flow resistance
and the seepage velocity is not linear with the pressure gradient any longer. Only when the pressure
gradient exceeds a certain value, the seepage velocity just has a pseudo-linear relationship with the
pressure gradient. As for the nonlinear flow of super-low permeability reservoir, the permeability is not
a constant, which in fact is a parameter varying with pressure gradient due to characteristic of super-low
permeability reservoir. There are three zones in the super-low permeability reservoir: dead oil zone,
nonlinear seepage zone and pseudo-linear seepage zone. When the pressure gradient in a zone is smaller
than the real threshold pressure gradient (λa in Fig.1), the zone is called dead oil zone; when the pressure
gradient in a zone is smaller than the critical pressure gradient (λc in Fig.1)but bigger than the real
threshold pressure gradient, the zone is called nonlinear seepage zone; when the pressure gradient in a
zone is bigger than the critical pressure gradient, the zone is called pseudo-linear seepage zone.
According to the seepage characteristics of super-low permeability reservoir, the nonlinear flow
numerical simulation software is developed. An ideal model of diamond inverted nine spots well pattern
is also built with the real field data of Da 45 block of Jilin Oilfield. At last, the simulation result of
different well spacing and well array, different fracture penetration ratio and different fracture
conductivity is comparative analyzed.
2 The Establishment of Nonlinear Flow Numerical Model
2.1 Nonlinear flow model
285
Figure 1 The typical curve of the nonlinear flow in super-low permeability reservoir
Typical flow curves of super-low permeability reservoir include three parts: non-flow section, nonlinear
flow bend section and the pseudo-linear flow bend section. In order to describe the nonlinear flow curve
of super-low permeability reservoir in mathematical sense, experimental correction is used on the basis
of classical Darcy law. Hence the equation of the motion in the super-low permeability reservoir can be
formulated as follows:
Vi = − M
KK ri
∇Pi
(1)
M = 1−
1
a + b ∇Pi
(2)
Vg = −
µi
KK rg
µg
∇Pg
(3)
In the formulas above, Vi is seepage velocity for each phase (i=o,w);Vg is seepage velocity for gas; K is
absolute permeability; Kri is relative permeability for each phase; Krg is relative permeability for gas;
Pi is pressure gradient for each phase; Pg is pressure gradient for gas; ui is viscosity for each phase;
ug is viscosity for gas; M is nonlinear flow correction factor; a and b are nonlinear flow parameters,
which are both functions of permeability and can be obtained by laboratory experiments.
In the formula (2), b is equivalent to the reciprocal of pseudo threshold pressure gradient (λc in Figure1).
a is a factor that affect the nonlinear concave curve (a≥0)and is also a dimensionless parameter. When
a=0, the formulas above turn to pseudo threshold pressure gradient model; when b is infinite, reflecting
infinitesimal pseudo threshold pressure gradient, the interaction between fluid and solid is weak and M
approaches zero, which make the formulas above become Darcy linear flow model. In summary, when
a=0, appearing as pseudo threshold pressure gradient model, the nonlinear flow curve is a straight line
and the line intersects the x axis at 1/b; when 0 a 1, the curve intersects the x axis at (1-a)/b and the
value is minimum starting pressure gradient; when a≥1, the curve gets through the coordinate origin and
the minimum threshold pressure gradient is zero.
When a and b are given different values, according equation (1) and equation (2), seepage curves are
shown in Figure 2.
▽
▽
＜＜
286
Seepage velocity, m/d
0.018
a=0.0,b=15
0.016
a=0.5,b=15
0.014
a=1.0,b=15
a=2.0,b=15
0.012
a=2.0,b=25
0.010
0.008
0.006
0.004
0.002
0.000
0.00
0.05
0.10
0.15
0.20
0.25
Pressure gradient, MPa/m
Figure 2 Nonlinear seepage curves
The Figure 2 shows that, when the pressure gradient is small, the seepage curve is concave; when the
pressure gradient is large, the seepage is linear. At the minimum pressure gradient, the biggest pores and
throats participate in the flow; with the increase of the pressure gradient, more and more pores and
throats take part in the flow and the seepage curve presents a concave curved section; after the pressure
gradient has reached the critical value, the number of the pores and throats involved in the flow is stable
and the curve turns into straight line.
2.2 Fundamental assumption
Fundamental assumption of the model: (1) There are three phases (oil, gas, and water) in the reservoir (2)
Water and oil can not be soluble with each other (3) The gas can be dissolved in the oil phase (4) The
dissolved gas and the free gas can exchange freely (5) Oil and water reach the balance instantaneously
(6) The seepage in the reservoir is isothermal (7) The seepage of oil and water abides by the nonlinear
seepage law (8) The compressibility of rock and fluid is considered.
2.3 Mathematical model
By substituting the equation of motion into the equation of continuity of each phase, the mathematical
model of super-low permeability reservoir can be obtained.
Oil component equation:
 kk

1
∂ φS
)∇( Po − ρ o gD ) + qov = ( 0 )
∇ ⋅  ro (1 −
a + b ∇( Po − ρ 0 gD )
∂t Bo

 Bo µ o
(4)
Water component equation:
 kk

1
∂ φS
)∇( Pw − ρ w gD) + q wv = ( w )
∇ ⋅  rw (1 −
a + b ∇( Pw − ρ w gD)
∂t Bw
 Bw µ w

Gas component equation:
287
(5)
 kk rg

 R kk

∂  S g Rso S o 
∇⋅
∇( Pg − ρ g gD ) + ∇ ⋅  so ro ∇( Po − ρ o gD ) + q gv = φ (
) (6)
+
Bo 
∂t  B g
 Bo µ o

 Bg µ g

3 Establishment of Field Model
3.1 Brief introduction to the field
Diamond inverted nine spots well pattern is adopted in Da 45 block of Jilin Oilfield. At present, the well
pattern is 500m×150m. The porosity is 10.7%, and the permeability is 0.4md. The bubble point pressure
is 9.67MPa, and the effective thickness is 9.4m. Then the effect of well spacing and well array, fracture
penetration ratio and fracture conductivity on the oilfield development is researched.
3.2 Well spacing and array
According to the physical parameter of Da 45 block, the geological model of diamond inverted nine
spots well pattern is established. The well spacing is 400m, 450m and 500m respectively, while the well
array is 80m, 120m and 150m. In other words, nine different diamond inverted nine spots well pattern
models are built for numerical simulation.
.
Figure 3 Three dimensional geological model diagram of Da 45 block
As shown in Figure 3, there are five water injection wells and eight oil production wells. The simulation
continues for 20 years and the result is also comparative analyzed. As shown in Fig.4, the oil recovery is
at a low level. The oil recovery and water cut curves of different schemes indicate that the smaller the
well spacing and well array, the higher the oil recovery and water cut.
288
Figure 4 The curves of oil recovery versus time for different well patterns
Figure 5 The curves of water cut versus time for different well patterns
Figure 6 The curves of water cut versus oil recovery for different well patterns
3.3 Fracture penetration ratio
According to the physical parameter of Da 45 block, the geological model of diamond inverted nine
spots well pattern can be established. Fracture penetration ratio is defined as the ratio of half-fracture
length to the well spacing. Under present 500 × 150m well pattern, six geological models are established
to study the influence caused by different fracture penetration ratios. The fracture penetration ratio is
designed to be 0.00, 0.18, 0.28, 0.38 and 0.48 respectively.
289
Figure 7 The curves of oil recovery versus time for different fracture penetration ratios
Figure 8 The curves of water cut versus time for different fracture penetration ratios
Figure 9 The curves of water cut versus oil recovery for different fracture penetration ratios
It can be seen from the figures above that the oil recovery and water cut increase with the fracture
penetration ratio. When the penetration ratio increases to a certain value, recovery rate of increase is
relatively small, so for the inverted nine spots well pattern there is an optimal fracture penetration ratio.
As shown in the curves of oil recovery and water cut, when the fracture penetration ratio is 0.28, a
higher degree of recovery can be achieved and the water cut can be controlled effectively, so the optimal
fracture penetration ratio is 0.28.
290
3.4 Fracture conductivity
Under present 500 × 150m well pattern, five geological models are established to study the influence
caused by different fracture conductivities. The fracture conductivity is designed to be 300, 350, 400,
450 and 500 ×10-3µm2·m respectively.
Figure 10 The curves of oil recovery versus time for different fracture conductivities
Figure 11 The curves of water cut versus time for different fracture conductivities
Figure 12 The curves of water cut versus oil recovery for different fracture conductivities
It can be concluded from the figures that artificial fracture conductivity has little effect on the oilfield
development. For the development, the artificial fracture conductivity is an insensitive parameter.
291
4 Conclusion
(1) The result of nonlinear flow numerical simulation shows that the oil recovery of Da 45 block under
present well pattern for developing 20 years is at a low level and the smaller the well spacing and well
array, the higher the oil recovery and water cut.
(2) The target block of Jilin Oilfield has an optimal fracture penetration ratio, which is about 0.28.
(3) As to Da 45 block, the artificial fracture conductivity has little effect on the oilfield development and
the artificial fracture conductivity is an insensitive parameter.
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