On the Determination of Physical Parameters of an Aquifer

B. A. Bogachev

(Presented by the Department of Combustible Minerals, TPI)

It is known that the determination of certain physical parameters of a productive formation (oil, gas, and water-bearing) from observations of unsteady regimes is of considerable value in comparison with determining them from cores and productivity coefficients [3]. In recent years, a large number of published works have appeared on this question, dealing chiefly with methods for determining the indicated parameters from pressure build-up curves in shut-in oil and gas wells.

Without considering the essence of each of the existing methods, and referring the reader to the cited work [3], we shall set forth the possibilities of determining certain physical parameters of a water-bearing productive formation by means of the integral method of E. B. Chekalyuk [2], which, in our opinion, has a number of advantages in comparison with other methods of processing pressure build-up curves. In addition, on the basis of the indicated method we shall present a procedure we have developed for determining reservoir pressure, which is characterized by the fact that it does not require a prolonged shut-in of a water well until complete restoration of the static level [1].

Having data on the rise of the water level in a well over time after the cessation of pumping from it, which had continued for a long period under steady-state conditions, we determine the physical parameters of the water-bearing productive formation by means of the basic relation given by E. B. Chekalyuk [2]:

\[ \frac{D \cdot \ln f}{f\left[Q_0-\frac{v(t)}{t}\right]} = \frac{\mu}{2\pi kh} \left[ \ln \frac{\chi}{r^2}+\ln t \right], \tag{1} \]

where \(f\) is the dimensionless time parameter; \(Q_0\) is the water rate before pumping from the well was stopped, in \(\mathrm{cm^3/sec}\); \(t\) is the time after cessation of pumping, in sec; \(v(t)\) is the inflow of water into the well after cessation of pumping, in \(\mathrm{cm^3}\); \(\mu\) is the viscosity of water under reservoir conditions, in centipoises; \(k\) is the permeability of the productive formation, in darcys; \(h\) is the thickness of the productive formation, in cm; \(\chi\) is the piezoconductivity of the formation, in \(\mathrm{cm^2/sec}\); \(r\) is the radius of a perfect well or the reduced radius of an imperfect one, in cm; \(D\) is the Duhamel integral.

The Duhamel integral is the area enclosed between the bottom-hole pressure build-up curve in a shut-in well—

and special coordinate axes \(\Delta P(t-\tau), G(f)\), where \(\Delta P(t)\) is the increase of bottom-hole pressure with time in \(\mathrm{kg}/\mathrm{cm}^2\); \(\tau\) is the integration parameter in sec; \(G(f)\) is the total dimensionless inflow of water into the well at unit depression, \(\Delta P = 1\ \mathrm{kg}/\mathrm{cm}^2\).

It is recommended that the quantity \(G(f)\) be determined from the simplified dependence proposed by us [1]

\[ G(f)=\frac{f}{0.5015\cdot \ln f}. \tag{2} \]

To determine the physical parameters of the formation it is necessary to construct, at a minimum, two curves of increase of bottom-hole pressure for two values of the time elapsed from the moment pumping was stopped, in the coordinates \(\Delta P(t-\tau), G(f)\), and, for purposes of self-checking, three or more curves. After the construction, the Duhamel integrals are determined by known methods of numerical integration or with the aid of a planimeter. Substituting the values of the Duhamel integrals into expression (1), the coordinates of the computed points in the system \(\ln t, y_0\) are found, where

\[ y_0=\frac{D\cdot \ln f}{f\left[Q_0-\frac{v(t)}{t}\right]}. \tag{3} \]

The computed points obtained in the named coordinate system must lie on a straight line. The slope of this straight line \(i\), and the intercepts cut off by it on the abscissa and ordinate axes, \(B\) and \(A\), are auxiliary data by means of which the physical parameters of the productive formation are determined.

1. Formation conductivity

\[ \frac{kh}{\mu}=\frac{1}{2\pi\cdot i},\qquad \frac{\mathrm{darcy}\cdot \mathrm{cm}}{\mathrm{centipoise}}. \tag{4} \]

1. Relative piezoconductivity of the formation

\[ \frac{\chi}{r^2}=e^{A/i},\qquad \mathrm{sec}^{-1}. \tag{5} \]

1. Elastic capacity of the formation

\[ \beta^*=\frac{e^B}{2\pi\cdot r^2\cdot h\cdot i},\qquad \mathrm{cm}^2/\mathrm{kg}. \tag{6} \]

To determine the static depression on the formation \(\Delta P_0\) at the moment of shut-in, assume that, up to the time pumping ceased, the well had been operating under steady-state conditions for \(T\) sec. Then, for a shut-in duration equal to the same time \(T\), the curve of bottom-hole pressure recovery in the special coordinate system \(\Delta P(t-\tau), G(f)\) will tend to form a rectangle with the named axes. The sides of this rectangle will be equal to: \(\Delta P_0, G(f_T)\), where \(G(f_T)\) is the total dimensionless inflow of water into the well at \(\Delta P=1\ \mathrm{kg}/\mathrm{cm}^2\), as a function of the dimensionless time parameter elapsed after pumping ceased, \(f_T\).

Having the ordinates of two, three, etc., computed points in the system \(y_0, \ln t\), we construct a straight line on the plane in other coordinates, namely: \(y_0, \ln \dfrac{t}{T+t}\).

The intersection of the constructed straight line with the ordinate axis gives the value \(y_0^T\), for which \(\ln \dfrac{t}{T+t}=0\).

According to the adopted assumption, the Theis integral for the duration of well shutdown \(T\) sec will be equal to

\[ D_T=\Delta P_0\cdot G(f_T). \tag{7} \]

On the basis of relations (3) and (7), one may write

\[ y_0^T=\frac{\Delta P_0\cdot G(f_T)\cdot \ln f_T}{f_T\left[Q_0-\dfrac{v(T)}{T}\right]}. \tag{8} \]

In view of the smallness of the quantity \(\dfrac{v(T)}{T}\), it may be neglected.

Then, on the basis of expressions (2) and (8), the static depression on the formation at the moment of cessation of water pumping from the well will be equal to

\[ \Delta P_0=0.5015\cdot y_0^T\cdot Q_0. \tag{9} \]

Below is given a concrete example of determining the physical parameters of a water-bearing productive formation from data on the rise of the water level with time after cessation of pumping in one exploratory well.

Initial data for the exploratory well

Before pumping was stopped, the well operated under two regimes: the first regime—\(Q_1=770\ \text{m}^3/\text{day}\), for \(T_1=15\) days; the second regime—\(Q_2=1150\ \text{m}^3/\text{day}\), \(T_2=25\) days, through lowered \(2\ 1/2''\) tubing. The production casing is \(12\frac{3}{4}''\), with pipe wall thickness \(\Delta=12\ \text{mm}\). At the moment pumping was stopped, the well discharge was \(Q_0=1150\ \text{m}^3/\text{day}=13300\ \text{cm}^3/\text{sec}\), with a dynamic level \(h_g=58.03\ \text{m}\). The well, by degree and character of penetration of the productive formation, is complete; bottom-hole depth \(H=300\ \text{m}\), well diameter by bit \(15\frac{3}{4}''\), well radius \(r=20\ \text{cm}\), thickness of the productive part \(h=20\ \text{m}\), bottom-hole temperature \(t_z=22^\circ\text{C}\); viscosity of the water under reservoir conditions \(\mu=0.97\) centipoise; the water is weakly mineralized; its specific gravity is \(\gamma=1\ \text{g}/\text{cm}^3\).

The data of measurements of the rise of the water level in the well, made with an electric level gauge, and some processing of them are placed in Table 1.

Using the data of column 5 of Table 1, curves were constructed in the coordinates \(\Delta P(t-\tau)\), \(G(f)\) for four different values of time elapsed after cessation of water pumping from the well (Fig. 1). Curves 1, 2, 3, and 4 were constructed respectively for \(t_1=9000\ \text{sec}\), \(t_2=12600\ \text{sec}\), \(t_3=16200\ \text{sec}\), and \(t_4=19800\ \text{sec}\). In this case the time \(t_4=19800\ \text{sec}\) was distributed into \(f=19800\) units of the dimensionless-time parameter. After numerical integration, the Theis integrals were found to be: \(D_1=3240\); \(D_2=4520\); \(D_3=5730\); \(D_4=7010\ \text{kg}/\text{cm}^2\).

The water inflows into the well over the corresponding time intervals were determined from the expression

\[ v(t)=\frac{\pi}{4}\cdot \Delta h\,(D_e^2-d_1^2-d_2^2), \]

where \(D_e\) is the inner diameter of the production casing, in cm; \(d_1\) and \(d_2\) are, respectively, the inner and outer diameters of the tubing, in cm. Then the coordinates of four computed points in the system \(y_0,\ \ln t\) were determined.

1. First point

\[ y_0^{I}=\frac{3240\cdot \ln 9000}{9000\left[13300-\frac{1369000}{9000}\right]}=0{,}000249. \]

1. Second point

\[ y_0^{II}=\frac{4520\cdot \ln 12600}{12600\left[13300-\frac{1382500}{12600}\right]}=0{,}0002565. \]

1. Third point

\[ y_0^{III}=\frac{5730\cdot \ln 16200}{16200\left[13300-\frac{1402000}{16200}\right]}=0{,}00026. \]

1. Fourth point

\[ y_0^{IV}=\frac{7010\cdot \ln 19800}{19800\left[13300-\frac{1421000}{19800}\right]}=0{,}0002645. \]

From the known coordinates of the four computed points we construct a straight line on the plane in the system \(y_0,\ \ln t\). The slope of the straight line is \(i=0{,}0000196\);

Table

the intercepts cut off on the coordinate axes are \(A=0{,}000073,\ B=-3{,}72\) (Fig. 2).

We determine the physical parameters of the water-bearing productive formation according to relations (4), (5), and (6).

1. Formation conductivity

\[ \frac{kh}{\mu}=\frac{10^6}{2\pi\cdot 19{,}6}=8120\ \text{D}\cdot\text{cm}/\text{cP}. \]

\[ \Delta P(t-\hat c),\ \mathrm{kg/cm^2} \]

\[ D_1 = 3240\ \mathrm{kg/cm^2} \]

\[ D_2 = 4520\ \mathrm{kg/cm^2} \]

\[ D_3 = 5730\ \mathrm{kg/cm^2} \]

\[ D_4 = 7010\ \mathrm{kg/cm^2} \]

\[ C(t) \]

Fig. 1.

\[ y_0(t)\cdot 10^{-4} \]

\[ A = 0.000023 \]

\[ \ln t \]

Fig. 2.

1. Aquifer permeability

\[ k=\frac{8120\cdot 0.97}{2000}=3.94\ \text{darcies}. \]

1. Relative piezoconductivity

\[ \frac{\chi}{r^2}=e^{3.72}\simeq 41.5\ \text{sec}^{-1}. \]

1. Aquifer piezoconductivity

\[ \chi=41.5\cdot (20)^2=16600\ \text{cm}^2/\text{sec}. \]

1. Aquifer elastic capacity

\[ \beta^*=\frac{10^6\cdot e^{-3.72}}{2\pi\cdot (20)^2\cdot 19.6\cdot 2000}\simeq 2.44\cdot 10^{-4}\ \text{cm}^2/\text{kg}. \]

1. Radius of influence of the well during operation in two regimes

In the figure:

\[ \chi_0(t)\cdot 10^{-4}, \qquad \chi_0(t)=\varphi\!\left(\ln\frac{t}{T+t}\right), \qquad \chi_0=0.000372, \qquad \ln\frac{t}{T+t}. \]

Fig. 3.

Before pumping is stopped, it is determined by taking into account the reduced time, which will be equal to

\[ T_{\text{pr.}}=\frac{Q_1\cdot T_1+Q_2\cdot T_2}{Q_2}\simeq 35\ \text{days}. \]

Then the radius of influence of the well is

\[ R=r\sqrt{\frac{\chi}{\pi\cdot r^2}\cdot T_{\text{pr.}}}\simeq 3.97\cdot 10^5\ \text{cm}. \]

To determine the static drawdown on the aquifer at the moment pumping is stopped, we construct a straight line on the plane in the coordinates

\[ y_0,\ \ln \frac{t}{T+t}. \]

For

\[ \ln \frac{t}{T+t}=0,\quad y_0^{T}=0.000372. \]

According to expression (9), the static drawdown will be equal to:

\[ \Delta P_0=0.5015\cdot 13300\cdot 3.72^{-4}\simeq 2.5\ \text{kg}/\text{cm}^2. \]

Then the reservoir pressure is determined from the expression

\[ P_{\text{pl.}}=\frac{(H-h_{\text{d}})}{10}\cdot \gamma+\Delta P_0= \]

\[ =\frac{300-58.03}{10}\cdot 1.0+2.5\simeq 26.7\ \text{kg}/\text{cm}^2. \]

It is interesting to note that the productivity coefficients of the well for the near-wellbore zone and averaged over the productive aquifer, determined respectively from the test data and from the flow rate and drawdown at the moment pumping is stopped, are almost equal to each other.

The productivity of the near-wellbore zone is

\[ k_3=\frac{2\pi kh}{\mu\cdot \ln \frac{R}{r}} =\frac{2\pi\cdot 8120}{\ln \frac{3.97\cdot 10^5}{20}} =5160\ \text{cm}^3/\text{kg}\cdot\text{s}. \]

The productivity averaged over the aquifer is

\[ k_c=\frac{Q_0}{\Delta P_0}=\frac{13300}{2.5}=5320\ \text{cm}^3/\text{kg}\cdot\text{s}. \]

The last calculations show that the productive water-bearing aquifer is homogeneous.

REFERENCES

1. Bogachev B. A. Izvestiya VUZov, Neft’ i gaz, No. 10, 1961.
2. Chekalyuk E. B. Neftyanoe khozyaistvo, No. 11, 1958.
3. Shchelkachev V. N. Development of Oil-Bearing Formations under Elastic-Drive Conditions. Gostoptekhizdat, 1960.