A Simplified Method for Determining Reservoir Pressure from the Buildup of Bottomhole Pressure

L. A. Pukhlyakov

(Presented by Professor A. V. Akzharin)

Determination of reservoir pressure from the buildup of bottomhole pressure is usually carried out by means of complicated calculations and, as a rule, is performed by specialists in a narrow field. For this reason, in the fields, in order to determine reservoir pressures, wells are very often shut in for such a period during which the bottomhole pressure reaches the reservoir pressure. In this case, intermediate measurements are generally not used. Meanwhile, by using these measurements, reservoir pressure can be determined with sufficient accuracy by means of comparatively simple calculations, without waiting until it stabilizes.

To determine the reservoir pressure \(P_{\text{pl}}\) by the simplified method, the following parameters are used: \(P_{\text{zab}}\) is the bottomhole pressure at the moment the well is shut in; \(A\) is the total drawdown on the formation, or the deficiency of bottomhole pressure relative to reservoir pressure at the moment the well is shut in; \(\Delta P\) is the difference between the current and initial bottomhole pressures; \(\Delta P_0\) is the initial difference of bottomhole pressures, that is, between the first and second measurements; and \(d\Delta P\) is the increment of bottomhole pressure between the moments of any adjacent measurements. It is assumed that measurements are made at equal time intervals \(dt\).

The simplified method for determining reservoir pressure from the buildup of bottomhole pressure is based on the relation

\[ \frac{d\Delta P}{dt}=\Delta P_0 e^{-\alpha t}, \tag{1} \]

where \(e\) is the base of the natural logarithm.

In order to obtain the value \(\Delta P\) from this expression, it must be put in the form

\[ d\Delta P=\Delta P_0 e^{-\alpha t}\,dt. \tag{2} \]

and integrated within the limits \(0<t<t_1\), whence we obtain

\[ \Delta P=\Delta P_0\int_0^t e^{-\alpha t}\,dt \]

\[ \Delta P=\frac{\Delta P_0}{-\alpha}e^{-\alpha t}\bigg|_0^{t_1} \]

\[ \Delta P=\frac{\Delta P_0}{-\alpha}\left(e^{-\alpha t}-e^0\right) \]

or, carrying out the corresponding transformations and taking \(t_1\) as the current value \(t\),

\[ \Delta P=\frac{\Delta P_0}{\alpha}\left(1-e^{-\alpha t}\right) \tag{3} \]

If, in the expression obtained, the current value \(t\) is allowed to tend to infinity, then the second term of this expression, in parentheses, will become zero, and the value \(\Delta P\) will tend to the value \(A\), that is, to the total drawdown on the reservoir at the moment the well is shut in. As a result, this expression will take the form

\[ A=\frac{\Delta P_0}{\alpha}. \tag{4} \]

Thus, the problem of determining the total drawdown is reduced primarily to determining the value \(\alpha\). As initial quantities for determining it, let us take, as was recommended in its time by B. A. Bogachev, two values \(\Delta P\), for which the values of \(t\) will be related by the ratio

\[ t_1:t_2=1:2. \]

and divide them one by the other. As a result, we obtain

\[ \psi=\frac{\Delta P_2}{\Delta P_1} = \frac{\dfrac{\Delta P_0}{\alpha}\left(1-e^{-\alpha 2t}\right)} {\dfrac{\Delta P_0}{\alpha}\left(1-e^{-\alpha t}\right)} = \frac{1-e^{-\alpha 2t}}{1-e^{-\alpha t}}. \]

Expanding the numerator of the expression obtained by the rule for the difference of squares, we obtain

\[ \psi= \frac{\left(1-e^{-\alpha t}\right)\left(1+e^{-\alpha t}\right)} {1-e^{-\alpha t}} = 1+e^{-\alpha t}, \]

whence

\[ \frac{\Delta P_2}{\Delta P_1}-1=e^{-\alpha t} \tag{5} \]

or

\[ \ln\left(\frac{\Delta P_2}{\Delta P_1}-1\right)=-\alpha t \tag{6} \]

Suppose that in a certain well the bottomhole pressure increased in the manner shown in Table 1.

Table 1

Then, comparing \(\Delta P_1\) and \(\Delta P_2\) at \(t=200\) sec and \(2t=400\) sec, we obtain

\[ \frac{\Delta P_2}{\Delta P_1} = \frac{1.813}{0.952} = 1.905 \]

whence

\[ \frac{\Delta P_2}{\Delta P_1} - 1 = 0.905 . \]

Next, in accordance with expression (5), one may use a table of the exponential function, for example the table of B. I. Segal and K. A. Semendyaev [1, p. 10], from which we find that, for

\[ \frac{\Delta P_2}{\Delta P_1} - 1 = 0.905, \]

the value \(-\alpha t\) will be equal to \(0.09982\), whence \(\alpha\) is expressed by the relation

\[ \alpha_1 = \frac{0.09982}{200} = 0.0004991 . \]

In the absence of tables of the exponential function, one may use tables of common logarithms or an ordinary logarithmic slide rule. In that case the calculation should be carried out according to formula (6). In particular, when solving the example considered above, with

\[ \frac{\Delta P_2}{\Delta P_1} - 1 = 0.905 \]

we obtain: the common logarithm of the quantity \(0.905\) will be equal to \(\overline{1}.95665\), or \(-0.4335\), and, when multiplied by \(2.3026\), it gives \(-0.09982\), whence \(\alpha\) is expressed by the relation

\[ -\alpha = \frac{-0.09982}{200}, \]

\[ \alpha_1 = 0.0004991 . \]

Having determined the value of \(\alpha\) for the first pair, we turn to the second pair, \(\Delta P_1=1.813\) and \(\Delta P_2=3.297\), that is, to the case \(t=400\) sec and \(2t=800\) sec; substituting these into expression (5), we obtain

\[ \frac{\Delta P_2}{\Delta P_1} - 1 = 0.8285, \]

whence, according to the table of the exponential function, \(\alpha t = 0.1881\) and

\[ \alpha_2 = \frac{0.1881}{400} = 0.00047 . \]

Next, we turn to the third pair \(\Delta P_1=2.592\) and \(\Delta P_2=4.512\), that is, we take \(t=600\) sec and \(2t=1200\) sec; substituting these into expression (5), we obtain

\[ \frac{\Delta P_2}{\Delta P_1}-1=0.74074, \]

whence, according to the table of the exponential function, \(\alpha t=0.30011\), and

\[ \alpha_3=\frac{0.30011}{600}=0.0005002. \]

We turn to the fourth pair \(\Delta P_1=3.297\) and \(\Delta P_2=5.507\), which correspond to \(t=800\) and \(2t=1600\) sec. Substituting them into expression (5), we obtain

\[ \frac{\Delta P_2}{\Delta P_1}-1=0.67031, \]

whence, according to the table of the exponential function, \(\alpha t=0.400\), and

\[ \alpha_4=\frac{0.400}{800}=0.00050. \]

From the obtained values of \(\alpha\), its average value is determined. In the present case it is expressed by the relation

\[ \alpha_{\mathrm{av}}= \frac{0.0004991+0.00047025+0.0005002+0.000500}{4} =0.0004924. \]

Next, it should be noted that formula (4) should not be used for the final determination of the total drawdown on the formation, since the value \(\Delta P_0\), determined from two measurements, is not sufficiently accurate. For more accurate calculations it is necessary to take all measurements of the increase in bottomhole pressure, which is achieved with the aid of the Duhamel integral.

The Duhamel integral is the area bounded by the pressure-buildup curve, a certain ordinate \(\Delta P\), and the time axis (Fig. 1-a). Usually this quantity is determined with a planimeter from the drawing. In this case, however accurately the drawing may have been plotted, certain errors will nevertheless occur. In this connection, it is proposed here to determine this quantity analytically as the sum of the products of the individual \(\Delta P\)’s by \(dt\), in accordance with the formula

\[ \mathfrak{D}=\Delta P_0\,dt+\Delta P_1\,dt+\ldots+\frac{1}{2}\Delta P_n\,dt . \tag{7} \]

Or, taking into account that the quantity \(dt\) remains constant within a single investigation, in accordance with the formula

\[ \mathfrak{D}=dt\left(\Delta P_0+\Delta P_1+\ldots+\frac{1}{2}\Delta P_n\right) \tag{8} \]

which is shown in Fig. 1-b.

Fig. 1. Areas corresponding to the Duhamel integral:
a) when it is determined by the graphical method and
b) by the analytical method. The dashed lines separate from one another the areas corresponding to the individual terms of formula (7). The points corresponding to the individual values of \(\Delta P\) given in Table 1 are circled and filled in with ink.

In the case under consideration, the Duhamel integral is expressed by the relation

\[ \mathfrak{D}=200\left(0.952+1.813+2.592+3.297+3.935+4.512+5.034+\frac{1}{2}\,5.507\right)= \]

\[ =200\cdot 24.889=4977.8 \]

On the other hand, the Duhamel integral can be expressed from formula (3), if in it, instead of \(\dfrac{\Delta P_0}{\alpha}\), its value from formula (4) is substituted, that is, \(A\), and then the resulting expression is integrated over the limits \(0<t<t_1\)

\[ \mathfrak{D}=A\int_{0}^{t_1}\left(1-e^{-\alpha t}\right)\,dt \]

\[ \mathfrak{D}=A\left(t-\frac{1}{-\alpha}e^{-\alpha t}\right)\bigg|_{0}^{t_1} \]

\[ \mathfrak{D}=A\left(t+\frac{1}{\alpha}e^{-\alpha t}\right)\bigg|_{0}^{t_1} \]

\[ \mathfrak{D}=A\left(t_1+\frac{e^{-\alpha t_1}-e^{0}}{\alpha}\right) \]

\[ \mathfrak{D}=A\left(t_1-\frac{1-e^{-\alpha t_1}}{\alpha}\right) \tag{9} \]

Substituting into the expression obtained \(t_1 = 1600\) sec and \(\alpha = 0.0004924\), we obtain

\[ 4977.8 = A\left(1600 - \frac{1 - 0.45475}{0.0004924}\right) \]

\[ 4977.8 = A(1600 - 1107.33); \]

whence

\[ A = 10.1 \text{ atm}. \]

Thus, the reservoir pressure in this deposit at the moment the well is shut in proves to be equal to \(60.1\) ati.

As pressure is restored near the well bottom, a decrease in reservoir pressure should occur in zones farther away from it; however, this decrease will be so insignificant that it may be neglected and the reservoir pressure at the moment the well is shut in may be taken as the calculated value.

The relationship obtained in the process of determining reservoir pressure, (4), if it is brought to the form

\[ \alpha = \frac{\Delta P_0}{A}, \tag{10} \]

can be used to determine the permeability of the reservoir. For this purpose, the pressure increase over the first time interval \(\Delta t_1\), which we expressed through \(\Delta P_0\), must be expressed through the inflow into the well under reservoir conditions \(Q_{\text{pl}}\), the cross-sectional area of the well \(F\), and the density of oil under reservoir conditions \(\gamma_{\text{pl}}\). This expression takes the form

\[ 1000 \Delta P_0 = \frac{Q_{\text{pl}}\Delta t}{F\gamma_{\text{pl}}}. \tag{11} \]

The coefficient 1000 in the expression obtained appears because a column of liquid with specific weight \(1.0\ \text{g/cm}^3\) and height \(1000\ \text{cm}\) creates a pressure of \(1\) ati. The parameter \(\gamma_{\text{pl}}\)—the density of oil under reservoir conditions—requires some explanation. At present it is usually calculated by the formula

\[ \gamma_{\text{pl}} = \frac{\gamma_{\text{s.n.}}}{b} + \frac{f_v \cdot \rho_g}{1000b}, \tag{12} \]

where \(\gamma_{\text{s.n.}}\) is the density of separated oil in \(\text{g/cm}^3\), \(f_v\) is the gas content referred to a unit volume of oil, \(\text{cm}^3/\text{cm}^3\), \(\rho_g\) is the density of gas in \(\text{g/liter}\), 1000 is the coefficient for converting gas density from \(\text{g/liter}\) to \(\text{g/cm}^3\), and \(b\) is the oil formation volume factor, i.e., the ratio of its volume under reservoir conditions to its volume under surface conditions (always greater than unity).

Next, the inflow into the well under reservoir conditions must be expressed from the formula derived by the author [1] for inflow into a hydrodynamically imperfect well,

\[ A = \frac{Q_{\text{pl}}\mu}{n \cdot 2\pi k}(G_l + G_k), \tag{13} \]

where \(A\) is the total drawdown on the reservoir, averaged over the first time interval or, equivalently, at the moment the well is shut in, in ati; \(\mu\)—

the viscosity of oil under reservoir conditions in centipoises, \(n\) is the number of holes in the filter, \(k\) is the reservoir permeability in darcies, \(G_1\) is the geometric characteristic of the zone of influence of the holes, and \(G_{k2}\) is the geometric characteristic of the zone of plane-radial flow. The first of these quantities is determined differently for different wells. Namely, for wells with a high perforation density, that is, if the reservoir thickness per one hole does not exceed 2.55 times the well radius before the casing is run, or 1.27 times the bit diameter, this quantity is determined by the formula

\[ G_{\lambda} = \frac{1}{\lambda} + \frac{1}{S} - \frac{0.25}{r}\ln\frac{S}{\lambda} + 0.0625\,\frac{S-\lambda}{r^{2}} - 0.01172\,\frac{S^{2}-\lambda^{2}}{r^{3}} + \]

\[ + 0.00260\,\frac{S^{3}}{r^{4}} - 0.00079\,\frac{S^{4}}{r^{5}} + 0.00021\,\frac{S^{5}}{r^{6}} - 0.00006\,\frac{S^{6}}{r^{7}}, \tag{14} \]

where \(\lambda\) is the radius of the hole in the casing, in cm; \(r\) is the well radius before the casing is run (one-half the bit diameter), in cm; \(S\) is the radius of influence of a hole (one-half the average distance between holes), in cm. In this case this parameter is determined by the formula

\[ S = 0.5\sqrt{\frac{2\pi r h}{n}}, \tag{15} \]

where \(h\) is the total reservoir thickness, in cm, and \(n\) is the number of holes in the casing in the filter zone.

For wells with a low perforation density, that is, if the reservoir thickness per one hole exceeds 2.55 initial radii, or 1.27 bit diameters, the geometric characteristic of the zone of influence of the holes is calculated by the formula

\[ G_{\lambda}^{*} = \frac{1}{\lambda} - 0.14362\,\frac{1}{r} - \frac{0.25}{r} \left( \ln\frac{2r}{\lambda} - \frac{0.5}{S} - 0.125\,\frac{r^{2}}{S^{3}} + 0.05625\,\frac{r^{4}}{S^{5}} \right). \tag{16} \]

In this case the radius of influence of the holes is calculated not by formula (15), but by the formula

\[ S = 0.5\sqrt{\frac{h^{2}}{n^{2}} + 0.96\pi^{2}r^{2}}. \tag{17} \]

The remaining parameters of expression (16) are the same as in expression (14).

The geometric characteristic of the zone of plane-radial flow is calculated in all cases by one and the same formula

\[ G_{R}=\frac{n}{h}\ln\frac{R}{r+S}, \tag{18} \]

where \(R\) is the radius of influence of the well, in cm (taken equal to one-half the average distance between neighboring wells); the remaining notation is the same as in expressions (14) and (16).

From formula (13), the inflow into the well can be expressed by the relation

\[ Q_{\mathrm{pl}}=\frac{A n\,2\pi k}{\mu\left(G_{\lambda}+G_{R}\right)}, \tag{19} \]

substituting which into expression (11), we obtain

\[ 1000\,\Delta P_0 = \frac{A n\,2\pi k}{\mu\left(G_{\ell}+G_R\right)} \cdot \frac{\Delta t}{F\gamma_{\text{pl}}} \tag{20} \]

or, expressing the cross-sectional area of the well through the internal radius of the casing \(r_k\) and carrying out the corresponding transformations,

\[ \Delta P_0 = \frac{A n\,2\pi k}{1000\,\mu\left(G_{\ell}+G_R\right)} \cdot \frac{\Delta t}{\pi r_k^2\gamma_{\text{pl}}} \tag{21} \]

\[ \Delta P_0 = \frac{2 A n k}{1000\,\mu\left(G_{\ell}+G_R\right)} \cdot \frac{\Delta t}{r_k^2\gamma_{\text{pl}}} \tag{22} \]

Substituting the obtained value of the pressure increase into expression (10), we obtain

\[ \alpha = \frac{2 A n k\,\Delta t} {1000\,\mu r_k^2\gamma_{\text{pl}}\cdot A\left(G_{\ell}+G_R\right)} \tag{23} \]

whence the permeability is expressed by the relation

\[ k = \frac{1000\,\alpha\,\mu r_k^2\gamma_{\text{pl}}} {2 n\,\Delta t} \cdot \left(G_{\ell}+G_R\right) \tag{24} \]

It should be noted here that determining the permeability of formations by the method described is possible only when processing data for shut-in deep-pump wells, where the pressure at the bottomhole is governed only by the pressure of the liquid column in the well. If the pressure at the well bottomhole depends mainly on gas liberated from the oil (shut-in flowing wells), then the data for them should be processed by other methods.

REFERENCES

1. L. A. Pukhlyakov. Derivation of the inflow formula for a hydrodynamically imperfect well. Proceedings of the Tomsk Polytechnic Institute, vol. 201, 1972.

2. B. I. Setal and K. A. Semendyaev. Five-Digit Mathematical Tables. 3rd ed. State Publishing House of Physico-Mathematical Literature, 1962.