DETERMINATION OF RESERVOIR PRESSURE IN OIL- AND GAS-BEARING FORMATIONS

Yu. A. MEDVEDEV

(Presented by Professor Doctor I. V. Lebedev)

This article proposes a new method for determining reservoir pressure, based on certain solutions of E. B. Chekalyuk [4] with subsequent simplifications by B. A. Bogachev [3].

E. B. Chekalyuk gave the basic analytical expression relating the pressure-buildup function in a shut-in well to the continuing inflow of fluid into it after shut-in:

\[ V(t)=Q\cdot t-D(t). \tag{1} \]

Because of the difficulty of graphical integration and a certain complexity of graphical constructions when determining, from expression (1), the Duhamel integral \(D(t)\), B. A. Bogachev proposed in [3] the simplified analytical expression

\[ D(t)\simeq G(t)\frac{\displaystyle\int_{0}^{t}\Delta P(t)\,dt}{t}. \tag{2} \]

Substituting expression (2) into (1) and using B. A. Bogachev’s formula for determining the total fluid inflow under unit drawdown \((\Delta P=1\ \text{kg}/\text{cm}^2)\) from [1],

\[ G(t)=\frac{4\pi kh}{\mu}\cdot\frac{t}{\ln\frac{\chi}{r^2}t}, \tag{3} \]

we shall obtain the final analytical expression for processing well-test data:

\[ Y(t)=\frac{\displaystyle\int_{0}^{t}\Delta P(t)\,dt}{Q_0\cdot t-V(t)} = \frac{\mu}{4\pi kh} \left[ \ln\frac{\chi}{r^2}+\ln t \right]. \tag{4} \]

Expression (4) can be used for monotonic pressure-buildup curves with an error, insignificant for practice, within \(\pm 5\%\).

To refine this expression, the UkrNIGRI staff members A. P. Kanyuk and Filias proposed determining the integral

Duhamel’s integral \(D(\bar t)=\int_0^t \Delta P(t-\tau)\,dG(\tau)\) is determined from a universal chart, which is constructed from pressure-build-up curves at the bottomhole of oil and gas wells (and also from arbitrarily chosen monotonically decaying curves at different scales) in the coordinates \(\varphi(P)\) and \(\alpha(d)\) [5].

On the abscissa axis \(\varphi(p)\) and the ordinate axis \(\alpha(d)\), the ratios of the areas of the instantaneous and momentary pressure recoveries are plotted, respectively:

for oil wells

\[ \varphi(p)=\frac{\int_0^t \Delta P(t)\,dt}{\Delta P(t)\cdot t}, \tag{5} \]

\[ \alpha(d)=\frac{\int_0^t \Delta P(t-\tau)\,dG(\tau)}{\Delta P(t)\cdot G(\tau)} \tag{6} \]

and for gas wells

\[ \varphi(p)=\frac{\int_0^t \Delta P^2(t)\,dt}{\Delta P^2(t)\cdot t}, \tag{7} \]

\[ \alpha(d)=\frac{\int_0^t \Delta P^2(t-\tau)\,dG(\tau)}{\Delta P^2(t)\cdot G(\tau)} \tag{8} \]

for various moments of time elapsed after shutting in the well. The dependence between the quantities is expressed by the equation of a straight line in the plane

\[ \alpha(d)=0.075+0.925\varphi(p). \tag{9} \]

The averaged depression on the inflow axis \(G(t)\) is determined by the expression

\[ \overline{\Delta P}(t)=\frac{\int_0^t \Delta P(t-\tau)\,dG(\tau)}{(Gt)}=\alpha(d)\cdot \Delta P(t). \tag{10} \]

The final calculation expressions have the form:

for oil wells

\[ Y(t)=\frac{\overline{\Delta P}(t)}{Q_0-\bar q(t)} =\frac{\mu}{4\pi kh}\left[\ln\frac{\chi}{r^2}+\ln t\right], \tag{11} \]

for gas wells

\[ Y(t)=\frac{\overline{\Delta P^2}(t)}{P_0\left[Q_0-\bar q(t)\right]} =\frac{\mu}{2\pi kh}\left[\ln\frac{\chi}{r^2}+\ln t\right], \tag{12} \]

where \(Q_0\) is the well flow rate before shut-in, cm\(^3\)/s;

\[ q(t)=\frac{V(t)}{t} \]

is the averaged flow rate of fluid from the formation into the well over the time after its closure, cm\(^3\)/s;

\(V(t)\) is the amount of liquid or gas accumulated in the wellbore after shutting it in, cm\(^3\).

\(P_0\) — atmospheric pressure, kg/cm\(^2\);
\(k\) — formation permeability, darcies;
\(h\) — formation thickness, cm;
\(\mu\) — viscosity of the reservoir oil or gas, centipoise;
\(\chi\) — piezoconductivity of the formation, cm\(^2\)/sec;
\(r\) — reduced radius of an imperfect well, cm.

We have used the simplified relations given above to determine the static depression on the formation at the moment the well is shut in and, consequently, the reservoir pressure from the same test data. To process these data, the following premise is used:

First, that before shut-in the well had been operating under steady-state conditions for \(T\) sec; second, that, for a shut-in duration equal to the same time \(T\), the pressure-buildup curve in the coordinates \(\Delta P(t-\tau), G(\tau)\) for oil wells and \(\Delta P^2(t-\tau), G(\tau)\) for gas wells will tend to form a rectangle with the named axes. For the time \(T\) sec the quantity \(\alpha(d)=1\), and the quantity

\[ \overline{q}(T)=\frac{V(T)}{T}\to 0. \]

Hence, from expressions (11) and (12), we shall have formulas for determining the static depression on the formation:

for oil wells

\[ \Delta P_0 = Q_0 \cdot Y(T) \tag{13} \]

and for gas wells

\[ \Delta P_0^2 = P_{\mathrm{pl}}^2 - P_z^2(0) = Q_0 \cdot Y(T)\cdot P_0, \tag{14} \]

Table 1

Fig. 1. Graph for determining reservoir pressure

where \(P_{\mathrm{pl}}\) is the reservoir pressure to be determined, kg/cm\(^2\);
\(P_z(0)\) — bottom-hole pressure at the moment the well is shut in, kg/cm\(^2\);

\(Y(T)\) is the magnitude of the segment cut off by the extension of the straight line on the ordinate axis.

In order to determine the accuracy of the proposed method, a number of theoretical examples for hypothetical wells were solved. Their results agree satisfactorily with the adopted standards. Below is the calculation for one of the theoretical examples for a hypothetical well, taken from the work of B. A. Bogachev [2]. The necessary calculations from expressions (9) and (11) for \(t = 600, 1200, 1800\), and \(2400\) sec are entered in Table 1 and are presented in Fig. 1.

It is seen from Fig. 1 that the calculated points in the coordinate system \(y(t), \ln t/T + t\) lie strictly on a straight line. The value \(Y(T)\) is determined either graphically—by extending the straight line to its intersection with the \(y(t)\) axis—or analytically, from the two extreme points on the straight line. In the present case, the static drawdown on the formation is equal to

\[ \Delta P_0 = 500 \cdot 0.01988 = 9.94 \text{ at.} \]

As is seen from the calculation presented, the error in determining the static drawdown on the formation was \(0.6\%\).

REFERENCES

1. B. A. Bogachev. Analytical method for determining formation parameters. NK, No. 3, 1962.

2. B. A. Bogachev. On the analysis of hydrodynamic methods for investigating wells. Izv. VUZ, Oil and Gas, No. 1, 1963.

3. B. A. Bogachev. A simplified integral method for determining formation parameters. Izv. VUZ, Oil and Gas, No. 2, 1964.

4. E. B. Chekalyuk. Method for determining the physical parameters of a formation. NK, No. 11, 1958.

5. E. B. Chekalyuk. A universal method for determining formation parameters. NK, No. 2, 1964.