According to the law of conversation of mass, the underground reserves can be determined by subtracting the produced oil and gas from the amounts of original reserves and dry gas injection combined, molar number is used to describe the relationship because it can’t be influenced by temperature and pressure, the equation can be expressed as Equation (1).

where, n_i is the mole number of initial reserves; n_j is the mole number of underground remaining reserves; n_ij is the mole number of cumulative injected gas; n_p is the mole number of cumulative production.

The equation of authentic gas is [10] . Using different parameters to describe the number of moles of hydrocarbons at the initial state:

where, p_i, Z_i, V_i are the formation pressure, deviation factor and pore volume under the original condition respectively, p_sc, Z_sc, T_sc are the pressure, deviation factor and absolute temperature under the standard condition respectively, G is the original condensate gas geological reserves, R is universal gas constant, T is the absolute temperature of reservoir.

Two phases, gas and liquid, exist in the formation when the bottom-hole pressure is lower than dew-point pressure, yet the two-phase deviation factor will be adopted to take place of single-phase deviation factor. The molar volume of underground hydrocarbon is:

The mole number of cumulative injected gas is:

where, V_j is the pore volume in the process of production; V_ij is the volume of injected gas; Z₂ is the two-phase deviation factor.

Produced condensate gas will become dry gas and condensate oil after the separator, translate the number of condensate oil as gas volume [11] , thus the amount of produced hydrocarbons as follows:

Plug the above equations in Equation (1), which is:

In the above equations, G_Lp is the cumulative production of condensate gas which includ dry gas and conversion volume of condensate oil, GE_o is conversion volume of condensate oil, n_o is the cumulative production of condensate oil.

As a non-ignorable factor in water drive gas reservoir, we must consider the influence of water invasion in the application of material balance method. With the exploitation of gas reservoir, water from water bodies flows into stratum, part will output and the rest still remain in the pore space. If we neglect the elastic expansion volume of rock and bound water, there is an equation about voids volume:

Assuming that hydrocarbons exist as vapor phase in the strata under the original condition, the original pore volume of gas can be written as. We define the volume coefficient of remanent water as the ratio of the water storage in to the cubage, which is:

Then,

where, W_e is the cumulative water influx rate, W_p is the cumulative production of water, B_w is the water volume factor, B_gi is the original oil volume factor.

Combine Equation (6) with Equation (9), then the expression is:

Reinjection ratio is volume ratio of injected dry gas vs. total produced gas. Research and practice show that with the increase of reinjection ratio, some medium weight hydrocarbons transfer from oil to gas, the recovery of condensate oil improved. The volume ratio of accumulative production of dry gas vs. condensate gas is R_v, and K is the reinjection ratio, therefore, we get Equation (11).

We define an apparent reinjection ratio as, that the equation can be simplified further:

With the growing water influx, the nonlinear degree of Equation (12) becomes higher. Change the equation as a linear relationship of and, which can be used to determine the OGIP. If we can obtain the accurate value of and make a diagram, when, the OGIP is equal to the size of the abscissa of the intercept.

There are many methods can be used to obtain water influx [12] , but the process is complicated and apt to have deviation. In order to make the solution easier, new parameter, apparent relative pressure, is introduced to avoid using water influx.

There is a statistical relationship between and [13] :

We define and, then the prediction equation of the apparent relative pressure can be expressed as:

Assuming a series of A, B values to seek an optimal fitting between the measured values and the predicted ones. When the deviation value is the minimum, we get the best fitting. Using the values of A and B can calculate the reserves and water influx.

Least squares method is widely used in curve fitting. To solve Equation (17) is equivalent to solve Equation (18).

The Newton iteration method and the steepest descent principle are conventional solutions for unconstrained problem, but their application ranges are different. To solve this problem, we introduce the levy berg algorithms into the solution, which is a combination of the two methods.

where, is the A value get from the k times iteration; is the increment of A value; D is the diagonal made up by derivative of E with respect to A.

Plug the A value get from each iteration into Equation (17) to calculate the deviation, when the deviation reach the minimum, we get the optimal result.

According to the method in this paper, the reserves can be obtained by parameter A, and the calculation formula is Equation (20). We use three different methods to determine the reserves, the proven geological reserve is provided as a counterpoint, the final results are shown in Table 3.

As can be seen from Table 3, the number 1 is the proven geological reserve, and the number 2 - 4 are reserves calculated by three different methods. Comparing the results in the Table 3, we can find that method 2 has largest error and the suggested method has smallest error. To analyze the three methods, method 2 assumes that the gas reservoir has closed boundary and constant volume, which is the primary reason of error. Although the method 3 has significant improvement than method 2, the computed result is on the large side because of the calculation error of water influx. In contrast to the previous approaches, the application of the suggest method is more convenient and the result is more accuracy.

According to the Equation (8) and the Equation (15), the water influx rate can be determined by parameter B:

Related parameters of water invasion can be calculated, which were shown in Table 4. In order to discuss the accuracy of prediction of water invasion, production data of 2010 are provided to contrast with the predicted value. As is shown in Figure 2 [14] , log interpretation indicates that gas-water contact of gas reservoir rose an

average of only three meters while single well water cone up seven meters. We can use volumetric balance formula to obtain the cumulative water influx rate, which is 1021.65 × 10⁴ m³, the prediction result of suggested method is 1104.42 × 10⁴ m³.