Abstract
Full Text
Reports of the Academy of Sciences of the USSR
- Volume 116, No. 1
Hydraulics
S. N. Buzinov
On the Problem of Determining Residual Oil Saturation
(Presented by Academician L. I. Sedov, 23 IV 1957)
When oil is displaced by water, a zone is formed in which two liquids—oil and water—exist simultaneously. During the displacement of oil from a reservoir by water, the oil production rate continually decreases and eventually becomes equal to zero. However, some quantity of oil still remains in the reservoir. This oil is in an immobile, capillary-retained state. In the present work, the problem of determining this capillary-retained (“residual”) oil saturation is considered.
We assume that in a porous medium in which two liquids are simultaneously present, the filtration-velocity vector of the oil is determined by Darcy’s equation
[
\mathbf{w}{\mathrm{н}}=-\frac{k .}}(\sigma)}{\mu_{\mathrm{н}}}\operatorname{grad} P_{\mathrm{н}
\tag{1}
]
For the filtration-velocity vector of water, an analogous equation holds:
[
\mathbf{w}{\mathrm{в}}=-\frac{k,}}(\sigma)}{\mu_{\mathrm{в}}}\operatorname{grad} P_{\mathrm{в}
\tag{2}
]
where (P_{\mathrm{н}}) and (P_{\mathrm{в}}) are the reduced pressures:
[
P_{\mathrm{н}}=p_{\mathrm{н}}+\gamma_{\mathrm{н}} z,
\tag{3}
]
[
P_{\mathrm{в}}=p_{\mathrm{в}}+\gamma_{\mathrm{в}} z;
\tag{4}
]
(p_{\mathrm{н}}, p_{\mathrm{в}}) are, respectively, the pressures in the oil and water phases; (\gamma_{\mathrm{н}}, \gamma_{\mathrm{в}}) are the specific weights of oil and water, respectively; (z) is the vertical axis directed upward; (\mu_{\mathrm{н}}, \mu_{\mathrm{в}}) are the viscosities of oil and water, respectively; (k_{\mathrm{н}}(\sigma), k_{\mathrm{в}}(\sigma)) are the phase-permeability coefficients for oil and water, respectively; (\sigma) is the oil saturation.
It is assumed that the phase-permeability coefficients are experimental functions of saturation and do not depend on the filtration velocity.
We assume that the relation between the pressures in the oil and water phases is determined by the relation
[
p_{\mathrm{в}}-p_{\mathrm{н}}=p_{\mathrm{к}}(\sigma),
\tag{5}
]
where (p_{\mathrm{к}}(\sigma)), the difference between the pressures in the water and oil phases at a given point, is called the capillary-pressure function.
As established experimentally ((^{1})), the capillary-pressure function (p_{\mathrm{к}}(\sigma)) is a single-valued function of saturation and has approximately the form of the curve shown in Fig. 1.
Substituting relations (3), (4), and (5) into equations (1) and (2), we obtain
[
\mathbf{w}{\mathrm{н}}=-\frac{k}}(\sigma)}{\mu_{\mathrm{н}}
\left[\operatorname{grad} p_{\mathrm{н}}+\operatorname{grad}\gamma_{\mathrm{н}}z\right];
\tag{6a}
]
[
\mathbf{w}{\mathrm{в}}=-\frac{k}}(\sigma)}{\mu_{\mathrm{в}}
\left[\operatorname{grad} p_{\mathrm{в}}+\operatorname{grad}\gamma_{\mathrm{в}}z\right].
\tag{6б}
]
For (\mathbf{w}_{\mathrm{н}}=0) there may be two cases:
[
k_{\mathrm{н}}(\sigma)=0;
\tag{7}
]
[
\operatorname{grad} p_{\mathrm{н}}+\operatorname{grad}\gamma_{\mathrm{н}}z=0.
\tag{8}
]
Equation (7) establishes that the residual oil saturation in the formation does not change, which contradicts experimental data; therefore equation (7) is excluded.
From equation (5) it follows that
[
-\operatorname{grad} p_{\mathrm{н}}
=
-\operatorname{grad} p_{\mathrm{в}}
+
\frac{d p_{\mathrm{к}}(\sigma)}{d\sigma}\operatorname{grad}\sigma.
\tag{9}
]
Then equation (8) will take the form
[
\operatorname{grad} p_{\mathrm{в}}
-
\frac{d p_{\mathrm{к}}(\sigma)}{d\sigma}\operatorname{grad}\sigma
+
\operatorname{grad}\gamma_{\mathrm{н}}z
=0.
\tag{10}
]
Substituting (10) into (6б), we obtain
[
\mathbf{w}{\mathrm{в}}
=
-\frac{k}}(\sigma)}{\mu_{\mathrm{в}}
\left[
\frac{d p_{\mathrm{к}}(\sigma)}{d\sigma}\operatorname{grad}\sigma
-
\operatorname{grad}(\gamma_{\mathrm{н}}-\gamma_{\mathrm{в}})z
\right].
\tag{11}
]
Fig. 1
Let us assume that (\operatorname{grad}(\gamma_{\mathrm{н}}-\gamma_{\mathrm{в}})z=0), which is valid for two cases: for filtration of liquids in a horizontal plane and when the volumetric weights of the liquids are equal.
Then equation (11) will take the form
[
\mathbf{w}{\mathrm{в}}
=
-\frac{k}}(\sigma)}{\mu_{\mathrm{в}}
\frac{d p_{\mathrm{к}}(\sigma)}{d\sigma}\operatorname{grad}\sigma.
\tag{12}
]
Introduce the function
[
\Phi=\int \frac{k_{\mathrm{в}}(\sigma)}{\mu_{\mathrm{в}}}\,d p_{\mathrm{к}}(\sigma)+\mathrm{const}.
\tag{13}
]
Obviously,
[
-\operatorname{grad}\Phi
=
-\frac{k_{\mathrm{в}}(\sigma)}{\mu_{\mathrm{в}}}
\frac{d p_{\mathrm{к}}(\sigma)}{d\sigma}\operatorname{grad}\sigma,
]
whence we conclude
[
\mathbf{w}_{\mathrm{в}}=-\operatorname{grad}\Phi.
\tag{14}
]
Equation (14) indicates that the motion of one liquid, when the other is in a capillary-retained state, is potential motion, with the potential being the function (\Phi) defined by relation (13). This makes it possible to use the formulas of potential motion of liquids to calculate the liquid flow rate for known values of the potential on the boundaries of the region, as well as formulas for determining the potential at any point of the region.
Since the potential function (\Phi) is related to the saturation (\sigma), knowing the value of the potential at one or another point of the reservoir makes it possible to determine the saturation at that point. Thus the residual saturation of the reservoir is determined.
Let us consider, as an example, the determination of residual oil saturation during the displacement of oil by water from a rectilinear reservoir. We shall not take into account the difference in the specific weights of the liquids.
Water is injected into the reservoir from one side, while fluid is withdrawn from the other side of the reservoir. We assume that the filtering fluids are incompressible. Let us direct the (x)-axis along the reservoir, placing the origin at the point where the fluid enters the reservoir. The length of the reservoir is (L), and the cross-sectional area is (F).
For the fluid production rate we obtain the equation
[
Q=\frac{\Phi_1-\Phi_2}{L}F,
\tag{15}
]
where (\Phi_1) is the value of the potential in the section (x=0); (\Phi_2) is the value of the potential in the section (x=L); (Q) is the fluid production rate.
The distribution of the potential function is determined by the equation
[
\Phi=\Phi_2+\frac{Q}{F}x.
\tag{16}
]
It follows from equation (16) that
[
\frac{d\Phi}{d\sigma}\,d\sigma=\frac{Q}{F}\,dx.
\tag{17}
]
Let us determine the mean-weighted residual oil saturation of the reservoir (\bar{\sigma}):
[
\bar{\sigma}=\frac{1}{L}\int_{\sigma_1}^{\sigma_2}\sigma\,dx,
\tag{18}
]
where (\sigma_1) and (\sigma_2) are, respectively, the saturations in the sections (x=0) and (x=L).
Using equation (17), we obtain
[
\bar{\sigma}=\frac{F}{LQ}\int_{\sigma_1}^{\sigma_2}\sigma\,\frac{d\Phi}{d\sigma}\,d\sigma.
\tag{19}
]
It follows from equation (13) that
[
\frac{d\Phi}{d\sigma}=\frac{k_{\mathrm{в}}(\sigma)}{\mu_{\mathrm{в}}}\frac{dp_{\mathrm{k}}}{d\sigma}.
]
Then equation (19) will take the form
[
\bar{\sigma}=\frac{F}{LQ}\int_{\sigma_1}^{\sigma_2}\sigma\,\frac{k_{\mathrm{в}}(\sigma)}{\mu_{\mathrm{в}}}\,\frac{dp_{\mathrm{k}}}{d\sigma}\,d\sigma.
\tag{20}
]
Introducing the notation
[
k_{\mathrm{в}}^{*}(\sigma)=\frac{k_{\mathrm{в}}(\sigma)}{k},\qquad
\varphi(\sigma)=\frac{\mu_{\mathrm{в}}}{k p_{\mathrm{k}}(1)}\,\Phi,
]
[
p_{\mathrm{k}}^{*}(\sigma)=\frac{p_{\mathrm{k}}}{p_{\mathrm{k}}(1)},\qquad
A=\frac{Q\mu_{\mathrm{в}}L}{Fkp_{\mathrm{k}}(1)},
\tag{21}
]
where (p_{\mathrm{k}}(1)) is the capillary pressure at (\sigma=1), and substituting into formulas (15) and (20), we obtain
[
\varphi(\sigma_1)-\varphi(\sigma_2)=A;
\tag{22}
]
[
\bar{\sigma}=\frac{1}{A}\int_{\sigma_1}^{\sigma_2}\sigma k_{\mathrm{в}}^{}\frac{dp_{\mathrm{k}}^{}(\sigma)}{d\sigma}\,d\sigma.
\tag{23}
]
For illustration, a calculation was carried out that makes it possible to estimate the influence of the parameter (A) (in other words, the filtration velocity) on the residual oil saturation (\bar{\sigma}). The relative permeability to water (k_{\mathrm{w}}^) and the dimensionless capillary pressure (p_{\mathrm{k}}^(\sigma)) were determined from the approximation formulas
[
k_{\mathrm{w}}^*=\frac{(0.9-\sigma)^3}{0.9^3},
]
[
p_{\mathrm{k}}^*(\sigma)=\frac{0.92}{\sigma-0.08}.
]
Fig. 2
The dependence of the dimensionless function (\varphi) on (\sigma) is shown in Fig. 2. Considering the curve of the dependence (\varphi(\sigma)), we find that at large values of oil saturation (\sigma) the function (\varphi(\sigma)) changes only slightly, while at small values of (\sigma) the function (\varphi(\sigma)) decreases sharply as (\sigma) increases. This circumstance is very important for practical calculations, since even a considerable error in determining the residual oil saturation at the outlet will have little effect on the results of the calculations.
Fig. 3
The graph of the dependence of the residual oil saturation (\bar{\sigma}) on the parameter (A) for (\sigma_2=0.65) is shown in Fig. 3.
The following conclusions follow from the solution presented: a) the residual oil saturation in the formation does not depend on the viscosity of the oil; this proposition is confirmed by experiments ((^2)); b) for small values of the parameter (A), the residual oil saturation decreases sharply with increasing parameter (A); for large values of the parameter (A), the oil saturation is almost independent of (A), i.e., of the velocity; this proposition is also confirmed by experiments ((^3)).
Thus, beginning with a certain displacement velocity, a further increase in velocity practically does not lead to an increase in the oil production rate; i.e., increasing the velocity becomes ineffective.
Moscow Petroleum Institute
named after I. M. Gubkin
Received
10 IV 1957
REFERENCES
- M. Muskat, Physical Foundations of Oil Production Technology, 1953.
- V. M. Baryshev, N. D. Markhasin, Neft. khoz., No. 10 (1951).
- V. M. Baryshev, E. I. Ibragimov, Al. N. Adonin, Tr. AzNII DN, issue 3, Baku (1956).