2019 International Conference on Artificial Intelligence and Computing Science (ICAICS 2019) ISBN: 978-1-60595-615-2
Establishment and Application of a Water-drive Curve Based on Relative
Permeability Following Power Function
Xiao-jiao QIU
PetroChina Liaohe Oilfield Company, Panjin 124010, China
Keywords: Non-piston type, Fractional flow equation, Power function, Water-free oil production period, Water drive curve.
Abstract. Based on Buckley-Leverett theory of water flooding in non-piston way, fractional flow equation, Welge average water saturation equation and the relative permeability curve, a new relationship between water cut and recovery efficiency and the corresponding water drive characteristic curve are derived making use of the general solution of ordinary differential equation formula with the initial condition. The ratio of oil-water relative permeability follows the power function in the process of derivation. It is different from the previous derived methods of water displacing curves to consider the effect of water-free oil production period on water drive type curve in this paper. The practical application in K oilfield indicates that the method is rational and effective.
Introduction
Water drive curve is one of the most important methods for development evaluation and recovery prediction of waterflooding oilfields. It is also an important basic research topic in reservoir engineering. It has been widely used in dynamic analysis of waterflooding oilfields, demonstration of development technology policy and development adjustment plans [1-4]. Many scholars in China and abroad have done lots of researches on waterflooding, and have proposed dozens of waterflooding characteristic curves [5-9], the most commonly used of which are the four curves of A, B, C, and D. In reference [10], Tong Xianzhang discussed the accuracy of the four water drive curves in calculating reserves, judging the identification of waterflooding status, evaluating the effect of measures and predicting the ultimate recovery. Chen Yuanqian [11-12] deduced the relationship of the above waterflooding characteristic curve in detail and put forward a new water drive characteristic curve. Based on Buckley-Leverett theory [13] of waterflooding in non-piston way, fractional flow equation, Welge average water saturation equation [14] and the relative permeability curve [11], the basic relationship between cumulative water production and cumulative oil production is presented, and a new relationship between water cut and cumulative oil production is obtained. Compared with previous methods of waterflooding characteristic curves, this method takes into account the influence of waterless recovery period on waterflooding curve in water drive process. The application for the development adjustment program in an actual oilfield shows that the effect is good.
Theoretical Basis and Model Establishment
Based on Buckley-Leverett waterflooding theory, the following assumptions are made: the direction of oil-water two-phase movement in formation is the same, one end is the water injection source, the other end is the liquid extraction port, and the influence of gravity and capillary force is not taken into account. According to the principle of material balance, the diversion equation of the liquid at the outlet is as follows:
w
ro w rw o 1
1 f
K K
= +
where Kro, Krw are oil and water relative permeability respectively, f; fw is water cut at outlet, f;μo, μw
are the oil and water viscosity, mPa·s.
Under the condition of two-phase steady seepage, there are many mathematical expressions for the change of oil-water phase permeability with water saturation. Yu Qitai has counted the experimental curves of oil-water relative permeability of many groups of waterflooding reservoirs at home and abroad in reference [15]. It is found that although there are many expressions of oil-water phase permeability curves, the expressions of oil-water relative permeability ratio curves can be divided into two types: exponential type and power function type. In this paper, a new relationship between water cut and recovery degree is deduced based on the power function form that the oil-water relative permeability ratio conforms to formula (2).
-ro w wc
rw 1 wc or n
K S S
C
K S S
−
= − −
(2)
where C and n is constant related to reservoir and fluid properties, dimensionless; Sw is water
saturation, f; Swc is irreducible water saturation, f; Sor is residual oil saturation, f.
Combining the relative permeability curve of oil and water and water saturation value measured by oil field core experiment and through regression fitting, the constant C and n can be obtained by comparative analysis. According to the experimental theory[11], when the viscosity ratio of oil to water is between 1 and 10, the water cut at the outlet of oil-water two-phase flow is as follows:
(
)
3w
w or we
o 50
1 1
f = − −S −S
(3)
where Swe is water saturation at outlet, f.
In addition, Gao Wenjun [16] proposed that when the oil-water viscosity ratio of does not conform to the range of 1-10, the water drive curve derived from the above formula and the actual fitting effect are also very good, so the water cut derived from the above formula can be written as:
(
)
2' w
w or we
o 150
1
f = −S −S
(4)
where fw' is the derivative of water cut at the outlet, f.
Considering the non-piston water flooding, the expressions of water saturation and average water saturation at the outlet are obtained from B-L water displacement theory [8]and Welge equation[9].
w w
we '
w 1 f
S S
f −
= −
(5)
where Sw is the average water saturation, f.
According to the principle of material balance, the average water saturation in reservoir can be expressed as
(
)
w wc 1 wc
S =S +R −S
(6)
Combining the above equations (1) ~ (7) and taking consideration of R = Np / N, we can obtain the
following equation,
(
)
(
)
p w
wd p wc wd
p o
2 n 3 1 n
dN C
NS N S NS
dW − = − − (8)
The intermediate variable of water saturation is definedSwd = −1 Sor−Swc.The initial conditions of the equation (8) are as follows: when the anhydrous period of the reservoir ends (fw=0.02), the
cumulative water production is Wp0, and the cumulative oil production is Np0. Solving ordinary
differential equations, we can get
(
)
10 p 3 p 1 wc wd
n C
B
W N S NS
A A
+
= − − +
(9)
where N is reserves,104m3, A, B and C0 is constant, their expressions are as follows:
(
)
w wd o 2 n CA= NS
;
(
)(
wc)
1 3 1 1 B
n S
=
+ − ;
(
)
1 0 p0 3 p0 1 wc wd
n
C =AW −B N −S −NS + .
Equation (9) is a new relationship between cumulative water production and cumulative oil production. Taken derivation of formula (9) from time on both sides, it can be found as
(
)
p p
p wc wd
1
3 1 n
dW dN
N S NS
dt = A − − dt (10)
Submit Eq.(7) into Eq.(10), one can get the following expression as
(
)
w wdwc w
1
3 1 n1
f
R A NS
N S f
= +
− −
(11)
This above expression is a new relationship between water cut and recovery degree for waterflooding reservoirs, through which the corresponding recovery degree can be predicted for any different water cut.
Example Application
Taking an oilfield in Kazakhstan as an example, the viscosity of crude oil is 1.1 mPa·s, the one of formation water is 0.5 mPa·s, and the oil-water viscosity ratio is 2.2, which meets the experimental conditions. The oilfield has the reserves of 9.08 million tons with strong bottom water. The ultimate oil recovery is 45% by reservoir simulation, and the original formation pressure is 23.8 MPa. The formation pressure level has been kept good since the reservoir was put into development, which is close to maintaining the formation pressure. At present, the formation pressure is 22.6 MPa. The oil-water relative permeability curves (Fig. 1) were fitted according to Eq.(2). The constant C and n
Figure 1. Relative permeability curves of oil and water in actual oilfield.
According to the above parameters and formula (11), the relationship curve between recovery degree and water cut is calculated and compared with the actual production curve. It is found that the calculated curve in this paper is in good agreement with the actual production data (Fig.2). When the water cut is 95% [12], the predicted oil recovery is 44.7% close to the approved recovery. It indicated that the mothed presented in this paper can be used for dynamic analysis and waterflooding effect evaluation for waterflooding oilfields.
Figure 2. Theoretical and actual curves of water cut and recovery degree for actual oilfield.
Summary
There are some conclusions obtained as following:
[image:4.595.179.408.386.577.2](3) By means of the derivation method of waterflooding characteristic curves proposed in this paper, it can also be used to study the waterflooding characteristic law and development effect of low permeability reservoirs considering the influence of capillary force.
References
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