Petroleum well

An oil well produces in an area of low permeability, with a bottom pressure flowing of 800 lb/(in ² ) _abs (55.15 Bar). Measured data for the decline in the well discharge are listed in Table 2. Meanwhile Table 3 shows well data and results of a pressure increase test.

In a complete well production study, the following characteristics are determined:

Initially, in order to have a complete characterization of the test, a flow rate as a function of time graph is useful (Figure 4), which can be used as an auxiliary for the general diagnosis. After doing a general conceptualization of the test, the flow declination model is identified, for which two graphic methods are used:

Figure 4 General behavior of flow rate as time function.  

Figure 5 shows the log-log graph of production data as a function of time, which is useful for acquiring a previous idea of declination behavior. Figure 6 is a graph of the flow logarithm as a function of time, elaborated using data in Table 2. Through data adjustment, the equation of a straight line corresponding to equation (3) (q=qie-Dt) is obtained.

Figure 5 Log-log graphic of flow rate versus time, using data of the analyzed well (^{Golan & Whitson, 2003}). 

Figure 6 Graphic showing flow rate in logarithmic scale versus times, whose data corresponds to the analyzed well and are shown in Table 2 (^{Golan & Whitson, 2003}). 

The adjustment to a straight line allows us to obtain its characteristic parameters (the ordinate to the origin and its slope). The ordinate to the origin corresponds to the initial flow q _i , while with the slope the decline rhythm (D) is obtained. In this way the values are:

In the absence of historical production measurements, parameters of equation (3) can also be determined from reservoir data. With equations (1) and (2), q _i and D are determined using data obtained from transient pressure tests. Table 3 shows the values determined from a pressure test carried out in the well.

Applying equation (1) and replacing pressure test values, we get:

Applying equation (2) and substituting the values provided in Table 3, we obtained:

Values of q _i and D calculated with ^{Arps equation (1945)}, differ slightly from values obtained using the equations proposed by Fetkovich. This is due to particular considerations with which both methods were developed, however what should be highlighted from the methodology is that it provides a criterion on the range of the parameters of the decline in the analyzed well. Likewise, the high value in the well decline rate can be identified in conjunction with its low initial flow rate. Due to previous reason to illustrate the use of type curves, Table 4 shows time data, volumetric flow and accumulated production of another oil well, with which an analysis of the decline of its production is developed.

In order to have a general diagnosis for the well behavior analysis, with the data provided, the graph of Figure 7 is developed, which illustrates the behavior of the flow with respect to time in Cartesian coordinates. Figure 8 shows the graph of the same variables in double logarithmic coordinates.

Figure 7 Data production behavior as time function in order to carry out a general diagnosis on the parameters behavior.  

Figure 8 Log-log graphic of the volumetric flow rate as time function. 

Figure 9 shows the behavior of the volumetric flow (q) with respect to the cumulative produced volume (Np). Figure 10 shows the overlap of production values (time, flow rate) on a double logarithmic scale on the type curve of Figure 1. It is convenient to emphasize the special care that must be taken in that the scales of both graphs are of the same magnitude to identify the type curve that best fits with the measured data.

Figure 9 Graph showing volumetric flow rate (q) behavior as function of cumulative produced volume (Np).  

Figure 10 Production data fit of analyzed well on the type curve for production decline analysis. 

Using the graph in Figure 8, and the production data in Table 4, it is assumed that the well has declined approximately 97.5% in 2 880 days. From the graph it can be inferred that at this rate of decline the critical flow limit (10 t/h) would be reached approximately a little less than 5 000 days. When using the analytical method, values closest to the extrapolation are selected from the well production history and are adjusted to an equation whose resulting expression is:

The combination of the accumulated produced volume (Np) and the volumetric flow (q) is used to determine the maximum recoverable volume in the well, that is, the total reserve. Assuming an output of 10 b/d as the economic limit of the well, the maximum producible volume can be determined by extrapolating the graph to this limit value, as shown in Figure 9. The graphical determination for this case turns out to be slightly higher than 500(10³) b, which, could be assumed as the total reserve of the well. The adjustment equation obtained for this well is given by:

Data in Table 4 show that the cumulative production in the well is 499 650 (b), this value represents 97.8% depletion in the reserve, therefore the remaining reserve is 2.2%. However, the approximate time in which the economic limit of the well will be reached is 4 854 days. That is, the 2 880 days that the well has been producing represent 59.3% of the total estimated useful life. It would seem that it is an excellent well, because it still has a 40.7% useful life. However, it must be considered that the initial flow rate in the well is 940 (b/d) and at 2 880 days the flow is 23.9 (b/d). This means that the accumulated produced mass represents 97.9% of the total exploitable fluid. Therefore, in the last 1 974 days of useful life (4 854 - 2 880), the remaining reserve of 10 750 (b) (510 400 - 499 650) will be extracted at a flow lower than 23.9 (b/d), until reaching to the economic limit.

As can be seen in Figure 10, the flow as a function of time data plotted on log-log paper fits the standard curve whose value of n equals 0.5. According to the adjustment of production data of the analyzed well, it can be inferred that its type of decline is hyperbolic.

Values of the set point resulting from the comparison between the values plotted (t, q) in log-log scale and the type curve of Figure 1 are:

For the graph of measured values in the well, we have: t = 180 and q = 1000.

While the values corresponding to the same point in the type curve are: (t _D ) = 1.4 y; (q _D ) = 0.38.

The determination of the initial volumetric flow (q _i ) and the initial declination rate (D _i ) are obtained by means of the substitution of the previous values in equations (5) and (6), as shown below:

and

Furthermore, from the values of the dimensionless variables t _D and q _D , obtained in the adjustment with the type curve, it is possible to determine the characteristics of the formation by applying equations (8) and (11). To satisfy the conditions of the reservoir variables that these equations require, values obtained through the use of other techniques are used, such as pressure tests, laboratory measurements and/or geological or geophysical studies.

With equation (8) is obtained, according to Table 1, the permeability (k) or depending on the variables (kh) or (k/() is obtained or (kh/() of the formation with the equation (9). Whereas with equation (11) the porosity ( is determined, or in its case the storage of the formation (C _T h with equation (12). This depends on the knowledge of the thickness of the production formation (h), the viscosity of the fluid (() and the total compressibility of the system (C _T ). The volume factor (B) is obtained from laboratory analysis on the behavior of the fluid starting from background conditions and ending in surface conditions. The initial pressure (p _i ) is determined from the measurements doing in the well. It is possible to determine the background pressure flowing (p _wf ) either with the transient pressure tests, or through the use of simulators or, if possible, also from measurements with recording tools.

Case of a geothermal well

Throughout the development of the geothermal industry many of the techniques applied to oil fields have been adapted for their application in geothermal systems. The only condition for the application of this methodology is to take into account the fluid characteristics of each system, maintain consistency in the units and be careful in the use of the corresponding conversion factors.

Ordinarily in geothermal wells the production is expressed in mass flow (W) unlike how it has been expressed for oil wells that the volumetric flow (q) is used. In order to demonstrate the use of decline analysis techniques, the graph of Figure 11 is shown, which corresponds to production data from a well of a Mexican geothermal field (^{Arellano, Torres & Barragán, 2005}; ^{Aragón-Aguilar et al., 2013}). Well production history covers a period of 340 months and in this graph it is possible to observe the behavior of its mass flow, its head pressure and its discharge orifice diameters as a function of time.

Figure 11 Plot for general diagnosis of behavior of well production its pressure at head conditions and variation of discharge orifice as function time. 

The utility of the graph for the general diagnosis of the data and its application to the analysis of declination is oriented to obtain a vision of the general behavior of the test. As more data correlated with each other is available, an analysis with greater reliability can be doing. This is because in field operations it is not always possible to keep the flow constant, which influences the pressure.

Due to fluid mobility and boiling processes to maintain the discharge pressure in geothermal wells, variations in the orifice diameter are applied; however, the declining behavior in production prevails. One of the diagnostic plots is shown in Figure 11, in which the declination behavior in flow measurements can be identified under different opening conditions of the well discharge orifice.

In this figure it is possible to identify a closure in the well, in the period of time between months 150 and 200, which causes the head pressure to increase abruptly, reaching values close to its static condition. However, when opening the well again to a discharge hole similar to the one that was operating, the flow is slightly higher (by approximately 5%) than it existed at the time of closure. The above is evidence that the closure caused a slight recovery in the well, however the trend in the decline remained with a slope similar to the one that had been showing before the closure.

Figure 12 shows the graph of production data as a function of time. This same graph, but in log-log scale, is the one used within the adjustment methodology by comparison with the type curve.

Figure 12 Graph showing economic limit determination of well production. 

Another interesting application of this graph is its usefulness to determine the time in which the well, according to the trend in its decline, could reach the economic production limit. In the case of geothermal fields, the economic limit is governed by the plant capacity index to generate at least 1 MWe, with 10 (t/h) of steam at 7 bars of pressure. In accordance with the above, the graph shows the extrapolation of the last slope of the declination trend until the value for mass flow of 10 (t/h) is cut. Graphically, it can be seen that the time at which the flow reaches the economic limit is approximately 495 months. Using data adjustment of this last declination section with constant slope, an equation with the following form is obtained:

Being able to determine the value of t = 492 months, for W = 10 (t/h); which is a value close to that obtained graphically. According to the results obtained by one part, from graph 11 it is assumed that the well already has approximately 340 months of production. Therefore, the remaining time of production could be estimated at approximately 152 (492-340) months. That is, the well has already consumed, approximately (340/492), 69% of its productive life.

The declination constant of the well and the parameters of the formation are obtained through the graph of Figure 12, taking care that the logarithmic axes are of the same magnitude as those of the type curve. In order to obtain the parameters of the formation, this graph is compared with the type curve of Figure 1 and the curve to which the data fit best is searched. This procedure is shown in Figure 13.

Figure 13 Overlapping of production data on the type curve for determining decline type through fitting technique. 

Values of the match point that are determined in each graph are:

For the log-log graph (measured data): t=50 (months); q=50 (t/h)

While for the type curve, we have: t _D =0.22; q _D =0.78

The determination of the initial volumetric flow (q _i ) and the initial decline rate (D _i ) is carried out by means of the substitution of the previous values in equations (5) and (6). Therefore, we get:

And applying equation (6):

In relation to the initial expense (q _i ) obtained from the type curve, it is convenient to mention the graphs of Figures 11 and 12. In both it can be seen that even when there is dispersion in the values, an initial value of the mass flow (q _i ) can be estimated, close to 64 (t/h), which is congruent with that obtained using equation (5).

Using equation (7) a typical value of B was assumed for a geothermal system and a well flow simulator was used to determine (p _i -p _wf ). Special care was taken to use consistent units for the calculation of the transmissivity value of the formation:

On the other hand, using equation (8), and replacing the value of the transmissivity previously obtained, it is feasible to determine the storage of the formation. The apparent radius value r _wa of the well is obtained from pressure tests. However, in the absence of more information, a range between 1E² and 1E³ can be assumed, so values according to this variation are obtained. Substituting values and using consistent units we get:

Due to the uncertainty in the knowledge of r _wa , the result can be left indicated as a function of this variable. From this point of view, the storage value for this well is equivalent to :

In oil reservoirs, the compressibility of the formation (C _t ) and the thickness (h) are parameters that can be determined by measurements and for this reason the porosity value (() can be calculated directly. However, in geothermal systems, compressive experimental data are generally not available, due to the impossibility of preserving the reservoir temperature in the core test, in their transfer to laboratories. Likewise, it is not possible to determine with certainty the thickness (h), because most of the geothermal systems are nested in volcanic formations, whose characteristics cause uncertainty in the determination of the length of the production thicknesses. Therefore, it is an ordinary practice in geothermal systems to determine an index of transmissivity (kh/() and the storage (C _t h instead of the permeability and porosity that is calculated in oil systems.

The other graph that is used is the flow against cumulative production (Np), whose extrapolation to the economic limit allows the estimation of the total reserve of the well, as shown in Figure 14. The behavior of production, closer to the extrapolation is considered representative by the overcoming of the transitory effects in the analyzed well. For this reason, the last slope of the declination trend was selected, extrapolating it to the economic limit of the flow of 10 (t/h).

Figure 14 Behavior of cumulative produced volume as function of mas flow rate, for determining the feasible reserve. 

With the graphic interpolation it is obtained that when reaching the economic limit, the cumulative produced fluid (Np) would be 14.75 E⁶ (ton), as can be seen in Figure 14. By adjusting the data of this last period an equation is obtained as follows:

Writing the expression to obtain Np as a function of W, we obtain :

Substituting the value of W = 10 in the previous expression, we obtain the value of Np=14.794 (10⁶ (ton)), which represents the total reserve of the well. According to the measured data, the accumulated produced mass in the analyzed well is a little more than 10x10⁶ (ton), which represents an approximate recovery of 67.6% (10/14.794) of the total reserve.

Case of a water well

Data used in this work correspond to water well located in the zone of Oakwood of Montogomery, Ohio, USA (^{Dursch & Wenclewicz, 2017}). The well is completed in a thickness with high hardness water (CaCO3) in the rank of 200 and 250 ppm and Cl (with concentrations between 300 and 350 ppm). These conditions influence in salt depositions inside pipe, causing production diminishes. Obtained data (Dursch & Wenclewicz, 2017) correspond to 50 months of its productive life and are shown in the diagnosis general graph of Figure 15. It can be identified two decline trends: the first at early time (since the month nine) and the second, from month 32. Salt scale in the early decline had duration time of six months. After early decline, were carried out jobs for its productivity improve, which is manifested in production increase. Nevertheless, the well again downward in its productive trend and since month 31 started a new decline, even though with minor severity. Both decline periods in the well are shown in graph of Figure 16.

Figure 15 General diagnosis graph, showing behavior of water well productivity (^{Dursch & Wenclewicz, 2017}), as exploitation time function. 

Figure 16 Productivity behavior of water well (^{Dursch & Wenclewicz, 2017}), showing two decline periods: the first at early time (between 9 and 16 months) and the second, since the month 28 of its productive life.  

Using the well data, is carry out a graph of log (flow rate) versus time as shown in Figure 17. A data fit to a line according to equation (3) q=qie-Dt is obtained and is determined the value qi=211 t/h. For estimating the useful life time of the well is used the graph log-log (flow rate versus time) shown in Figure 18. In order to achieve a realistic estimation, the decline data more recent are used and it is established an economic limit of well production. For this case, 78 months of useful life of the well is determined.

Figure 17 Graph of flow rate logarithmic versus time for determining the initial water flow rate (^{Dursch & Wenclewicz, 2017}) using fit to a line.  

Figure 18 Log-log graph (flow rate versus time) using the more recent production decline data of the water analyzed well (^{Dursch & Wenclewicz, 2017}), after established an economic productive limit. 

After had been established an economic productive limit of the well in10 (t/h) and using log-log methodology of cumulative produced mass (Np) as flow rate (q), function, the total extractable of the well is determined, as shown in Figure 19. Knowing the difference between cumulative produced mass, the maximum reserve, the useful life and economic limit, remaining reserve is determined, which in this case results in 264 MM (ton).

Figure 19 Produced mass (Np) as flow rate (q) function, for determining total extractable reserve from the well. 

After the first production decline in the well, jobs for improving its productivity were carried out (^{Dursch & Wenclewicz, 2017}) it, resulted in an production increase. However, a year later, the well again showed decline tendency (month 28). Under such circumstances, in this work, in Figure 20 and Figure 21, two analyzes using the type curves technique are shown.

Figure 20 Measured production data, in water well, at early time period, graphed on the decline type curve proposed by ^{Fetkovich, (1973)}.  

Figure 21 Measured production data, in water well, at late period, graphed on the decline type curve proposed by ^{Fetkovich, (1973)}.  

Figure 20 shows the fit with the type-curve, using data of the first decline trend. While Figure 21 shows the fit with the type-curve, using data of decline trend, from month 28 of operative well life.

According to the fit with the type-curve for the first period it can be determined a decline trend of “harmonic” type. This behavior is related with productive characteristics of the well taking into account its storage and feed.

For decline analysis in the first stage, through the use of data on the type curve and taking care for achieving the best match, following values are obtained :

For log-log graphic (measured data): t=10 (months); q=200 (t/h).[/p]

While, for the type-curve, are obtained tD=0.40; qD=0.85.

Determination of initial volumetric flow rate (qi) is done by substituting last values in equation (5). Therefore, it is obtained:

In relation with initial flow (qi) determined from values of the match with the type curve, it is appropriate to mention graph of Figure 17. Even though there is some dispersion in graphed values, it can be estimate an initial value of flow rate (qi) close to 211 (t/h), Which is consistent with that obtained using equation (5).

For period time of major system depletion, wich in this case is associated with decline trend of the second stage, match point of the well data on the type-curve, show different behavior. The achieved fit by comparing measured data with the type-curve, shows a decline of "Hyperbolic" type, with coefficient b=0.4.

For decline analysis at second stage, data of the well are graphed on the type-curve, treating to obtain the best match. In this case, it were obtained the “match” point values determined for each graph, as following:

For the log-log graph (measured data): t=30 (months); q=200 (t/h).

While for the type-curve are obtained tD=1.6; qD=0.80.

The initial volumetric flow rate determination (qi) is carried out by substituting last values in equations (5) and (6), for determining the flow rate and production decline at initial conditions. Therefore, it results following:

In relation with initial flow (qi) determined from values of the match with the type curve, it is appropriate to mention graph of Figure 17. Even though there is some dispersion in graphed values, it can be estimate an initial value of flow rate (qi) close to 250 t/h), Which is consistent with that obtained using equation (5).

Results of the developed analyzes have their practical application in obtaining technical criteria for the formulation of well exploitation designs and their future management. Table 5 shows a summary of results of the main parameters that support the arguments for decision making. Special care was taken to show the results of the analysis of the well data used in the examples with their respective work units. Each column of the table explains itself and shows the parameters with which it is feasible to configure decisions related to wells management.

Within results of decline analysis that have been exposed it is advisable to involve the petrophysical characteristics of the formation (kh/() and ((C _t h), to achieve an integral characterization of the well. In this research we have reviewed two analysis techniques: The type curves and the decline analysis and its application to petroleum and geothermal systems. The application of the analysis methods to hydraulic systems would not represent complications since it is a fluid in a single phase.