Transient Phenomena of Bottom Hole Pressure (Part 3)

Application of Theory 2. Change of Flowing Presure and Static Pressure: In order to convert the expression of the Eqs. (2.48 a, b, c) to Q, T experession, put b-1, b0, bn as Eqs. (5.1), and flowing pressure Pf, which is considered to be equal to the pressure Pw when transient terms, -Σbne-an2AT, disappeared, are expressed by Eqs. (5.3) according to the reservoir type (a), (b) or (c) respectively. The coefficients in these equations are given by Eqs. (5.4 a, b, c). How do the coefficients b-1 and b0 change with λ, are shown in Figs. 5.1, 5.2 and 5.3. It is seen that, if the well is λ<2, it is effective to do acidizing of the well.Eqs. (5.3 a, b, c) and (5.4 a, b, c) are rewritten as Eqs. (5.6 a, b, c). In the case of (a), Pf does not change, and in the case of (b) or (c ), it change and lowers with T or log T.If the reservoir constant changes, for example, the apparent compressibility changes owing to gas channeling, T/c0 must be considered as variable as shown in the Eq. (5.7 b, c), and this fact is explained in Fig. 5.4.When the production rate changes with time, the variable T for (b) type, or (log T) for (c) type instead of T or log T respectively, must be adopted as shown in Eqs. (5.8 b, c) Putting the time after the well was shut in Tb, (refer to Fig. 5.5), the pressure change after that are expressed by Eqs. (5.9 a, b, c). The terminal pressure, when Tb→∞, is the static pressure Ps and they become as Eqs. (5.10 a, b, c). Then the specific productivity index, S. P. I., is expressed by the Eqs. (5.11 a, b, c) in its reciprocal form.The difference between S. P. I. in the case (a) and that in the case (b), which is resulted from the difference of pressure distribution as shown in Fig. 5. 6, is given by the Eq. (5.12). Application of Theory 3. Analysis of Build Up Curve of Bottom Hole Pressure: As the pressure changes after the well was closed, is given by Eqs. (5.9 a, b, c), and also Pf and Ps are given by Eqs. (5.3 a, b, c) and (5.10 a, b, c), the Eqs. (6.1 a, b, c) are obtained. These Eqs. express the pressure change during the pressure build up, an2s in the case of (a) and (b) are the quantities proportional to 1/re2, and the exponential terms disappeared in several hours or days. And in the case of (c) the exponential terms disappeared in several seconds.If x is put as Eq. (6.2 a, b) in the cases of (a) and (b), Eqs. (6.1 a, b) become Eqs. (6.3 a, b). The value of coefficients bn/b0 in these eqations depend upon that of re, ri, λ and ω, as shown in Fig. 6.1 as an example. Plotting the data (Ps-P)/(Ps-Pf) on the chart drawn as Figs. 6.2 and 6.3, in which the time axes are graduated in x, the curve except for the initial part becomes a stright line and its inclination is proportional to b1/b0. If the curves of λ-b1/b0 for various combinations of the values of re, ri, λ, and ω, are given similarly as Fig. 6.1, the value of λ can be easily obtained.In the case of type (c), put the variable x as Eq. (6.4 c) and graduate the time axis in x, as shown in Figs. 6.4 and 6.5. Plotting the data of (Ps-P)/Q on it, the inclination of the straight line which is the last part of the plotted curve, gives the value of μ0/k0 (refer to Eq. (6.6c)).Application of Theory 4. Production Decline Curve: Production declines of pumping well are shown by Eqs. (3.7a), (3.8b), (3.9a) and (3.10b), but converting these equations to the expression used Q and bn by the aid of Eqs. (2.7) and (7.1), and putting x as Eq. (7.2), the above mentioned Eqs. become Eqs. (7.3 a, b).