ON THE HYDRODYNAMIC CHARACTERIZATION OF RESERVOIR B₂ OF THE UST-BALYK OIL FIELD

Yu. A. MEDVEDEV

(Presented by the scientific seminar of the Department of Fossil Fuels)

The article presents preliminary data on the hydrodynamic characterization of the productive oil-bearing reservoir B₂ of the Ust-Balyk oil field in Western Siberia. Lithologically, reservoir B₂ is represented by fine- to medium-grained polymictic, weakly cemented sandstones of yellowish-gray to dark-brown color and is confined to the Vasyugan-Barremian strata. According to laboratory analyses, the open porosity of the sandstones ranges from 20 to 23.7%, and permeability from 74.2 to 388.4 millidarcies.

For calculating the hydrodynamic parameters of the reservoir, a method was adopted for processing pressure-buildup curves of wellhead and bottomhole pressures by an integral procedure, which, with a reduced volume of computational work, is not inferior in accuracy to other methods. This latter circumstance was pointed out by a number of authors [1, 2, 3, 6]. The theoretical substantiation of E. B. Chekalyuk’s integral method and its further simplification were given in works [4, 5, 2, 3].

The field data that we processed were obtained by the field-research group of VNII under the direction of I. D. Umrikhin during the trial operation of the Ust-Balyk oil field in 1964. Studies on flowing wells Nos. 67, 74, 75, 80, and 214 were carried out using DGM-4 differential pressure gauges and standard pressure gauges according to the usual procedure.

According to the adopted method [2, 4], in order to calculate the physical parameters of the reservoir, two curves are required as initial data: the bottomhole-pressure buildup curve \(\Delta P(t)\) and the curve of accumulation of fluid in the wellbore after its shut-in, \(V(t)\). The first of these is recorded by a downhole pressure gauge; the second is found from measurements of wellhead pressures according to the following relation:

\[ V(t)=\frac{1}{\gamma}\left[(f_k+f_b)\Delta P_c(t)-f_k\cdot \Delta P_k-f_b\cdot \Delta P_b\right], \tag{1} \]

where \(f_k\) and \(f_b\) are the cross-sectional areas of the oil columns in the annular space and in the flow tubing, cm²; \(\Delta P_c(t)\), \(\Delta P_k\), \(\Delta P_b\) are the increments of bottomhole, annular, and buffer pressures, kg/cm²; \(\gamma\) is the specific weight of the reservoir oil, kg/cm³.

In the absence of well interference, according to [5], the relation between the drawdown at the well bottom and the total withdrawal of elastic fluid is expressed by the Duhamel integral.

\[ V_0(T)=\int_0^T \Delta P_0(T-\tau)\,dG(\tau), \tag{2} \]

where \(T\) is the time from the beginning of well operation; \(V_0(T)\) is the volume of produced reservoir fluid; \(\Delta P_0(T)\) is the drawdown of bottom-hole pressure; \(G(\tau)\) is a function determining the volume of fluid produced from the reservoir at a constant drawdown equal to unity.

The volume of fluid \(V(t)\) accumulated after the well is shut in, taking expression (2) into account, is determined as

\[ V(t)=V_0(T_0+t)-V_0(T_0), \tag{3} \]

where \(T_0\) is the time of well operation before shut-in; \(t\) is the investigation time.

After prolonged operation of the well at a constant production rate \(Q_0\), or a constant drawdown \(\Delta P_0\), for \(t \ll T_0\),

\[ Q_0\cdot t - V(t)=\int_0^t \Delta P(t-\tau)\,dG(\tau). \tag{4} \]

In dimensionless form, the right-hand side of equation (4) gives the value of the Duhamel integral for dimensionless \(\bar t\)

\[ D(\bar t)=\int_0^t \Delta P(t-\tau)\,dG(\bar\tau). \tag{5} \]

To construct the function \(\Delta P(t-\tau)=f[G(\tau)]\), E. B. Chekalyuk [4] proposed using type curves. The processing of pressure-build-up data in a shut-in oil well is then carried out using the basic computational relationship

\[ \frac{D(\bar t)\cdot \ln \bar t}{m\,[Q_0\cdot t - V(t)]} = \frac{\mu}{2\pi k h} \left[ \ln \frac{\chi}{r^2}+\ln t \right], \tag{6} \]

where \(m\) is the scale of the type curve; \(\bar t\) is a dimensionless time parameter \((\bar t=m\cdot t)\); \(\mu\) is the oil viscosity under reservoir conditions, centipoises; \(k\) is the permeability, darcies; \(h\) is the formation thickness, cm; \(\chi\) is the hydraulic diffusivity of the formation, cm\(^2\)/sec; \(r\) is the reduced well radius, cm; \(V(t)\) is the amount of fluid accumulated in the wellbore by time \(t\), determined from expression (1).

After determining the Duhamel integrals for three or more time moments and the corresponding values of \(V(t)\) by formula (1), according to expression (6) one finds the position of the calculated points in the coordinate system \(Y(t), \ln t\), where

\[ Y(t)=\frac{D(\bar t)\cdot \ln \bar t}{m\,[Q_0\cdot t - V(t)]}, \tag{7} \]

and a straight line is drawn through the obtained points. The slope \(i\) of the calculated straight line and the value of the segment \(x_0\) cut off on the abscissa axis are used to determine certain reservoir parameters.

In order to simplify the computational operations and reduce their volume, it was proposed in [2, 3] to use a universal type curve for determining the Duhamel integral. On the type curve, points calculated for various time moments for oil and gas wells satisfy the equation of a straight line of the form

\[ z(d)=0.075+0.925\,\varphi(p), \tag{8} \]

where $\alpha(d)$ and $\varphi(p)$ are the ratios, respectively, of the areas of the Duhamel integral and of the area bounded by the curve $\Delta P=f(t)$ and the coordinate axes, to the areas of instantaneous pressure buildup in the corresponding coordinates.

Knowing the value of $\alpha(d)$ from expression (8), the averaged drawdown on the influx axis $G(t)$ is determined as

\[ \Delta \overline{P}(t)=\alpha(d)\cdot \Delta P(t). \tag{9} \]

The principal calculation expression of the proposed method for oil wells then takes the form

\[ Y(t)=\frac{\Delta \overline{P}(t)}{Q_0-q(\overline{t})} = \frac{\mu}{4\pi kh} \left[ \ln \frac{\chi}{r^2}+\ln t \right], \tag{10} \]

where $q(\overline{t})=\dfrac{V(t)}{t}$ is the average production rate from the reservoir over the time $t$ after the well is shut in, cm³/sec; $V(t)$ is determined from formula (1).

Processing field data by the simplified procedure is carried out by planimetering the pressure-buildup curves and subsequently determining, by means of a chart or from formula (8), the coefficient $\alpha(d)$. Then, using expression (10), in the coordinates $Y(t), \ln t$ a straight line is constructed and, from its slope $i$ and intercept $x_0$, the hydrodynamic conductivity of the reservoir is determined:

\[ \frac{kh}{\mu}=\frac{1}{4\pi i} \tag{11} \]

and the relative piezoconductivity

\[ \chi/r^2=e^{-x_0}. \]

In addition to these parameters, it is possible to determine the static drawdown in the reservoir at the moment of shut-in, and, consequently, the reservoir pressure, from the same test data [2].

For clarity, an example is given below of calculating the physical parameters of oil-bearing reservoir B$_2$ for well No. 75 of the above-mentioned field.

Example. Data on the well: bottomhole—2104 m; filter—5″ in the interval 2095–2103 m; initial reservoir pressure at a depth of 2098 m—207.62 kg/cm²; stabilized oil production rate through a 6-mm choke—60 m³/day; bottomhole pressure before shut-in at a depth of 2098 m—183.7 (by MGT-2U) and 188.23 kg/cm² (by DGM-4); operating drawdown—19.39 kg/cm²; pressure at the buffer—24.87 kg/cm²; pressure in the annulus—18.24 kg/cm²; saturation pressure about 90.0 kg/cm²; specific weight of reservoir oil—0.810 t/m³; specific weight of separated oil—0.879 t/m³; shrinkage coefficient—1.16; date of well shut-in for observation of the pressure-buildup process—9.03.64; cross-sectional area of the flow tubing $f=21.2$ cm² and of the annular space $f_k=135.7$ cm².

The observation results are given in Table 1.

From the bottomhole-pressure-increase data given in Table 1, the curves $\Delta P(t-\tau)=f[G(\overline{t})]$ in Fig. 1 were constructed for three time values: $t=160$, 220, and 280 min. For constructing the curves, $\overline{t}=1000$ units was adopted, i.e., $m=1/2$. The results of the calculations by the two above-indicated methods for all wells are summarized in Table 2. In Fig. 2 a calculation chart is given for determining the reservoir parameters, the values of which are also given in the corresponding columns of Table 2.

Fig. 1. Curves of the increase in bottomhole pressure in well No. 75 of the Ust-Balyk field.

Fig. 2. Characteristic of the pressure increase in well No. 75 of the Ust-Balyk field:
1 — calculated points by the method of E. B. Chekalyuk,
2 — by the simplified method.

Table 1

Conclusions

1. For calculating the physical parameters of productive formations in the Middle Ob region, the method mentioned above is quite acceptable.

2. The convergence of the results obtained by the one and the other method is quite satisfactory.

3. As can be seen from Table 2, the scatter of the obtained parameter values for different parts of the formation is considerable: the formation is heterogeneous.

4. It should be noted that the inflow of fluid into the well after the latter was shut in has a peculiar character. In well No. 74, in the final moments of the bottomhole-pressure recovery period, an outflow of fluid into the formation was observed. A similar phenomenon was noted in well No. 236, the data for which are not presented in the article owing to the large-

Table 2

Note. $^{*}$ In the numerator—the values of the parameters by the method of E. B. Chekalyuk; in the denominator—by the simplified method.

this scatter of the calculated points. Apparently, the outflow of fluid was associated with the accumulation of gas liberated from the oil in the wellbore region and with the creation, owing to this, of an additional pressure, sometimes compensated by a considerable outflow into the formation.

1. Investigation of wells according to the adopted procedure makes it possible to reduce the observation time and, consequently, to obtain an additional thousand tons of oil.

LITERATURE

1. B. A. Bogachev. On the analysis of hydrodynamic methods of well investigation. Izv. VUZov, Neft’ i gaz, No. 1, 1963.

1. A. P. Kanyuga, Yu. A. Medvedev. A simplified integral method for processing bottom-hole pressure buildup curves. Tr. VNII, issue 24, 1964.

2. A. P. Kanyuga. Filtration properties of oil reservoirs of the Pre-Carpathian region. Avtoreferat dissertatsii, Lvov, 1964.

3. E. B. Chekalyuk. A method for determining the physical parameters of a formation. Neftyanoe khozyaistvo, No. 11, 1958.

4. E. B. Chekalyuk. Fundamentals of the piezometry of oil and gas deposits. Kiev, 1961.

5. R. G. Shagiev. Some features of processing field pressure buildup curves. Izv. VUZ, neft i gaz, No. 6, 1962.