SPE 114991 Tight Gas Production Performance Using Decline Curves
C.L. Kupchenko, SPE, B.W. Gault, SPE, and L. Mattar, SPE, Fekete Associates Inc.
Copyright 2008, Society of Pe
troleum Engineers This paper was prepared for presentation at the CIPC/SPE Gas Technology Symposium 2008 Joint Conference held in Calgary, Alberta, Canada, 16–19 June 2008. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract Traditional (“Arps”) decline analysis is the most common reservoir engineering tool used for production performance forecasting. It has several advantages over other techniques in that it is simple to use, requires minimal data and is well understood by the industry. Currently, however, these methods are being misused in unconventional applications, such as tight gas. Production perfomance from tight gas reservoirs is characterized by steep initial decline rates and long periods of transient flow. If decline analysis is performed using this transient production data, the main assumption of boundary dominated flow (BDF) is violated and inaccurate forecasts may result. The goal of this work is to understand the behaviour of tight gas reservoirs during transient flow so that the familiar Arps method may be applied. The effects of different tight gas production responses (bilinear, linear, pseudo-radial, boundary dominated) are investigated. Finally, a methodology for applying traditional decline curve analysis to tight gas, with reference to long term transient flow, is presented. Introduction The goal of this work is to outline a method of production forecasting for tight gas reservoirs without the use of complicated tools. The most accepted and well understood forecasting tool is traditional decline curve analysis and for this reason is the focus of this paper. The general, hyperbolic form of the Arps decline equation is shown as Equation 1.
qi (1 + bDi t )
This equation is used to predict the gas flowrate (q) as a function of time (t). The hyperbolic decline exponent (b) can be determined by matching past production performance to Equation 1. For gas wells, hyperbolic decline exponents (0<b<0.5) are expected during BDF [Fetkovich (1) (1996), Okuszko (2007)]. However, decline exponents much greater than one (“superbolic”) have been used to forecast tight gas production with limited success. Many authors [Maley (1985), Cox (2002), Cheng (2007), Rushing (2007)] have investigated the use of traditional decline curve analysis to forecast production from tight gas reservoirs. Some authors [Maley (1985), Robertson (1988)] suggest limiting a hyperbolic or superbolic decline curve to an exponential decline curve at a certain time or at a specific decline rate. Other authors suggest that decline curve analysis be avoided altogether during transient flow and instead recommend using modern decline analysis methods [Fetkovich (2) (1980), Palacio (1993), Agarwal (1998), Anderson (2005)] as they are applicable to both transient and BDF flow behaviour. However, these methods are complicated, dependent on a complete (pressure/rate) dataset and not always available to the practising engineer. Figure 1 shows the production history of a typical tight gas well in transient flow and outlines the common problems associated with tight gas decline analysis. The blue line is an exponential (b = 0) fit of the final portion of the production history. This extrapolation is likely conservative and underestimates recovery. Conversely, the red line is a superbolic best fit of the production history with a decline exponent of 2.0 and likely overestimates recovery from this well.
In this paper, we suggest that traditional decline curve analysis can provide acceptable forecasts for tight gas reservoirs provided that the behaviour of the Arps equation under transient and BDF is well understood. Spivey et al (2001) introduced a method of calculating the hyperbolic decline exponent instantaneously on a rate-time production curve. They showed that the slope of the instantaneous decline rate, shown in Equation 2, can be used to calculate the instantaneous decline exponent as shown in Equation 3. Appendix A contains a complete derivation of Equation 3.
1 dq q dt
d ?1? ? ? dt ? D ?
Two derivatives are required to calculate the instantaneous decline exponent from production data using Equations 2 and 3. Noisy data and production upsets may prohibit a straightforward calculation of these derivatives. However, data smoothing and modern differentiation algorithms will improve the derivative calculations. Data and Results The general behaviour of the decline exponent for typical tight gas wells was investigated using several simulated examples. Factors affecting tight gas performance were investigated, including: ? ? ? ? Hydraulic Fracturing Linear/Bilinear Flow Permeability Anisotropy Drawdown
General Behaviour of the Decline Exponent In theory a tight gas well that has been hydraulically fractured should experience the following production responses: ? ? ? ? Linear flow Pseudo-radial flow (transient) Transitional flow from transient to BDF BDF
The decline exponent during each of these production responses has been investigated using several simulated examples. To generate these examples, a square, homogeneous and isotropic reservoir model containing an infinite conductivity hydraulic fracture was used, as shown in Figure 2. Figure 3a shows the decline exponent, calculated using Equation 2 and 3, as function of time for two tight gas examples: an infinite-acting reservoir (i.e. no boundaries) and a bounded reservoir. Several corresponding flow periods, labeled A through D, are identified on the instantaneous decline exponent plot in Figure 3a: A: Hydraulic Fracture Linear Flow B: Transition from Linear to Infinite-Acting Pseudo-Radial Flow C: Transition from Infinite-Acting Pseudo-Radial Flow to Boundary Dominated Flow D: Boundary Dominated Flow The same flow periods have also been identified in Figures 3b and 3c, which contain plots of the reciprocal decline rate (1/D) and the welltesting pressure derivative (dp/dln(t)), respectively. The characteristics of each flow period are listed below: A: Hydraulic Fracture Linear Flow From the welltesting literature [ERCB (1979)], a flowrate proportional to t-1/2 is expected during linear flow. An examination of Equation 1 reveals that when the decline exponent is equal to 2 the production rate is proportional to t-1/2. Therefore, a decline exponent of 2 is expected during linear flow.
The hydraulic fracture linear flow regime is expected in tight gas wells [Stotts (2007), Arevalo-Villagran (2003)] as they are fracture-stimulated to establish deliverability. Relative to the time scale in Figure 3a (100 years) this flow period is very short in duration and plots as a ‘point’ on the decline exponent (b = 2). The log scale in time on the pressure derivative plot (Figure 3c) allows the characteristic half slope of linear flow to be visible. B: Transition from Linear to Infinite-Acting, Pseudo-Radial Flow This flow regime is characterized by: ? An increasing instantaneous decline exponent for approximately 4 years. ? A concave upward or constant slope reciprocal decline rate. ? A pressure derivative approaching a constant. C: Transition from Infinite-Acting Pseudo-Radial Flow to Boundary Dominated Flow The transition from infinite-acting flow to BDF occurs once all the boundaries begin to influence the flow. This response is characterized by: ? A decreasing instantaneous decline exponent. ? A concave downward reciprocal decline rate. ? An increasing pressure derivative. D: Boundary Dominated Flow Boundary dominated flow is characterized by: ? An instantaneous decline exponent equal to or less than 0.5. ? A reciprocal decline rate with a slope equal to or less than 0.5. ? A pressure derivative with a unit slope.
Effect of Hydraulic Fracturing The effect of the hydraulic fracture length on the decline exponent in an infinite-acting and bounded reservoir was investigated. (i) Infinite-Acting Reservoir Figure 4 plots the instantaneous decline exponent for wells with varying hydraulic fracture lengths (infinite conductivity) in an infinite reservoir. Two flow regimes (A and B as described in Figure 3a) are identified in these plots. The following observations can be made from Figure 4: ? ? During transient flow, the decline exponent decreases with increasing fracture length. In other words, as the fracture becomes more “effective” the decline exponent decreases. The decline exponent slowly increases during infinite-acting radial flow.
For comparison purposes, a vertical well with a skin factor of zero (‘Radial s = 0’) was plotted in Figure 4. This shows that the decreasing trend of the decline exponent with stimulation size is not related to fracture flow but to fracture “effectiveness”. A positive skin will produce a decline exponent curve falling above the ‘Radial s = 0’ curve in Figure 4; a negative skin will plot below the ‘Radial s = 0’ curve. (ii) Bounded Reservoir—Infinite Conductivity Hydraulic Fracture Similar cases as those shown in Figure 4 were generated for a bounded reservoir. In these cases, the length of the fracture (xf) was varied relative to the width of the reservoir (Xe), to create separate plots for various Dimensionless Fracture Lengths, xfD.
x fD =
The decline exponents for the various cases are shown in Figure 5. The same flow periods, A through D, as described in Figure 3a were identified on this plot. The following observations can be made: ? During transient flow (flow regimes A through C), the decline exponent decreases with increasing fracture length
(i.e. fracture “effectiveness”). Transient linear flow is evident during two periods: ? Fracture linear flow (all curves start at a decline exponent of 2). ? Pure linear flow for the xfD = 1.0 (constant b = 2 during transient flow). During boundary dominated flow, the decline exponent decreases with increasing fracture length. For the largest fractures (xfD = 1.0) the decline exponent stabilizes at approximately 0.25 during BDF.
(iii) Bounded Reservoir—Finite Conductivity Hydraulic Fracture To investigate the effects of bilinear flow on the decline exponent, simulations with fractures of varying conductivity were generated using the following definition for Dimensionless Fracture Conductivity, FCD:
kf w kx f
The decline exponents for these cases are shown in Figure 6. The same flow regimes as those shown in Figure 3a (A through D) are identified. The notable difference between Figures 5 and 6 is that the early time decline exponent is 4 rather than 2 for the lowest conductivity fracture (FCD = 1.0). From the welltesting literature, a flowrate proportional to t-1/4 is expected during bilinear flow. An examination of Equation 1 reveals that when the decline exponent is equal to 4, the production rate is proportional to t-1/4. Therefore, a decline exponent of 4 is expected during bilinear flow. The following observations can be made from Figure 6: ? During transient flow (flow regimes A through C), the decline exponent decreases with increasing fracture conductivity (i.e. fracture “effectiveness”). ? Bilinear flow (b=4) is only recognizable for very low fracture conductivity. The decline exponent for moderate to high fracture conductivities behaves similarly to an infinite conductivity fracture (b=2). ? During boundary dominated flow, the decline exponent slightly decreases with increasing fracture conductivity.
Effect of Linear Flow As outlined in Figures 3a, 4, and 5, the linear flow regime caused by hydraulic fractures is relatively short. Only in the case where xfD approaches 1.0 does long term (i.e several years) linear flow actually occur (Figure 5). To investigate the effects of long term linear flow on the decline exponent, channel shaped reservoir models as shown in Figure 7 were generated. The models were built such that the bi-wing hydraulic fractures extend the entire width of the reservoir (xfD = 1.0), so that the flow is perpendicular to the fracture. Figure 8 shows the decline exponents for three different channel lengths. Similar flow periods as those identified in Figure 3a are shown on these plots. However, the fracture geometry with respect to the reservoir dimensions prevents pseudo-radial flow. The decline exponent during linear flow remains constant and is equal to 2 as shown in Figure 8.
Effect of Permeability Anisotropy Linear flow has been observed in anisotropic tight gas reservoirs [Arevalo-Villagran (2003)]. The effect of permeability anisotropy on the decline exponent in an infinite-acting and bounded reservoir was investigated. (i) Infinite-Acting Anisotropic Reservoir To investigate the effect of anistropy on the decline exponent, five simulations with consistent for the following anisotropy ratios, using the infinite-acting reservoir model shown in Figure 9: ? ? ? kx: ky = 1 : 1 (isotropic) kx: ky = 10 : 1 kx: ky = 100 : 1
k x k y and xf were generated
kx: ky = 1 : 10 kx: ky = 1 : 100
Figure 10 shows the decline exponent for these cases. Similar flow regimes as those identified in Figure 3a have been identified in Figure 10, excluding the boundary dominated flow regime. The following observations can be made from Figure 10: ? A long term linear flow regime does not occur in an infinite-acting anisotropic reservoir. ? A lower decline exponent (a more “effective” stimulation) is observed for fractures oriented perpendicular to the maximum permeability. (ii) Bounded Anisotropic Reservoir The decline exponents for equivalent bounded cases as those shown in Figure 10 (excluding the isotropic case) are plotted in Figure 11. Similar flow regimes as those identified in Figure 8 are shown in Figure 11 for the 100:1 anisotropy ratio case. The following observations can be made from this Figure: ? Long term linear flow (b=2) does occur with a combination of no flow boundaries, a large anisotropy ratio and a fracture oriented parallel to the maximum permeability. ? A short term pseudo-radial flow regime is apparent for all anisotropy ratios prior to boundary influenced flow. ? Decline exponents less than 2 may occur prior to BDF.
Effect of Drawdown Three cases with varying drawdowns (90, 50 and 10%) were generated to investigate the effects of flowing pressure on the instantaneous decline exponent. The results of these simulations are shown in Figure 12. The same flow periods as those identified in Figure 3a are identified on these plots. During the boundary influenced flow regimes (C and D), the decline exponent decreases with drawdown. This effect has been recognized previously in the literature [Okuszko (2007), Cheng (2007)].
Summary - Decline Exponent Behaviour: ? Typical Decline Exponent Behaviour: ? Decline Exponent = 4: Bilinear Flow (Figure 6) ? Decline Exponent = 2: Linear Flow (Figure 8) ? Decline Exponent ≤ 0.5: BDF (Figure 12) ? Decline Exponent Increases: Pseudo-Radial Flow (2 < b < 20) ? Decline Exponent Decreases: Boundary Influenced Flow A lower decline exponent occurs with increasing fracture effectiveness. More effective fractures can be obtained by: ? Increasing the fracture length ? Increasing the fracture conductivity ? Orienting the fracture perpendicular to the maximum permeability. During boundary influenced flow, the decline exponent decreases with drawdown. Generally, tight gas wells are operated with high drawdown; therefore, decline exponents of 0.5 are anticipated for a single layer reservoir. However, for wells with large hydraulic fractures, operated at high drawdown, a decline exponent as low as 0.25 can be expected (see Figure 5) during BDF. The decline exponent is not constant during the producing life of the well, except during linear flow and BDF. It may take several years to reach BDF in tight gas wells.
Practical Application Figure 1 outlines the limitations of traditional decline curve analysis for forecasting tight gas production. Yet, the method is commonly applied with varied success. The current authors believe reliable estimates may be generated, even through extended transient flow periods, using a two part production forecast. 1. 2. Transient Flow Period: A decline exponent greater than or equal to 2 (superbolic) is anticipated, based on the information outlined previously. Boundary Dominated Flow Period: At this stage, the transient superbolic decline exponent must be limited to a value appropriate for boundary dominated flow. A decline exponent less than or equal to 0.5 is anticipated based on the information outlined previously.
Decline Limit Plots—General Plots for estimating an appropriate time to limit the decline exponent during a tight gas production forecast are presented in Figures 13a and 13b. Simulation models were created for typical ranges of drainage area, permeability and dimensionless fracture lengths (xfD) for tight gas reservoirs. For each case, the time for the decline exponent to reach 0.5 (tb=0.5) was determined and plotted in Figures 13a and 13b. The negative unit slopes in Figures 13a and 13b are consistent with the radius of investigation (rinv) equation (field units):
Rearranging Equation 6,
2 log(tinv ) = ? log(k ) + log(948φμct rinv )
According to Equation 7, a plot of time versus permeability on log-log co-ordinates has a slope of -1 and a y-axis intercept of 948φμctr2inv. Based on these observations, the negative unit slope in Figures 13a and 13b is expected. Decline Limit Plot—Simplified Figures 13a and 13b present a method to calculate tb=0.5 for several values of xfD; however, in practice it is difficult to estimate this parameter for a tight gas well. To simplify, the xfD = 1.0 curves from Figures 13a and 13b are plotted in Figure 14 as this is consistent with the long term linear flow observed in many tight gas wells. The data in Figure 14 is represented by Equation 8:
t b =0.5 =
Where, tb=0.5 A k
= time to reach a decline exponent of 0.5, years = drainage area, acres = formation permeability, md
Fluid and Reservoir Property Correction Figures 13a, 13b and 14 were created assuming that φμct is equal to 2 E-7 cp/psi [(φμct)o]. Results for tb=0.5 obtained from these Figures [(tb=0.5)o] must be corrected for the actual φμct factor [(φμct)new] using Equation 9.
? (φμct ) new ? (tb=0.5 ) new = ? ? (tb=0.5 ) o ? (φμct ) o ?
Tight Gas Production Performance Using Decline Curves—Method The decline exponent limit plots presented in Figures 13a, 13b and 14 may be used to forecast production from tight gas wells in transient flow. The purpose of the Figures is to determine when to limit a superbolic decline exponent to a decline exponent typical of boundary dominated flow (b ≤ 0.5). A three step method to perform a decline forecast for a tight gas well in transient flow is proposed. Step 1: Fit a Decline Exponent to Actual Production History ? Use a best fit regression or a predetermined decline exponent to fit the production history. The authors suggest using a decline exponent equal to 2 for the transient portion of the decline history. This is consistent with hydraulic fracture and long term linear flow observed in tight gas wells. Step 2: Determine tb=0.5 ? Estimate the well’s drainage area and reservoir permeability o Drainage area may be determined from a detailed analysis of analog wells. A review of older wells in the area using volumetric or rate transient analysis may provide a range of appropriate drainage areas to apply to the subject well. Well spacing in higher density developments may also be used to determine an appropriate drainage area. o Permeability may be estimated from offset core data, pressure transient tests or rate transient analysis. ? Determine tb = 0.5 o Figure 14 or Equation 8 is used to determine tb=0.5 for the case of xfD = 1.0 where the hydraulic fracture spans the width of the reservoir. The assumption of xfD = 1.0 is consistent with the long term linear flow observed in many tight gas wells. o Figures 13a and 13b are used to determine tb=0.5 for other values of xfD. Step 3: Limit the Decline Exponent at tb=0.5 ? Hold the decline exponent from Step 1 until tb = 0.5 (from Step 2). ? Limit the decline exponent to b ≤ 0.5 at tb = 0.5 and continue the production forecast. The appropriate value of the decline exponent after tb = 0.5 depends on drawdown (flowing pressure) and completion effectiveness. If we assume xfd =1.0 and a high drawdown then a decline exponent of 0.25 may be appropriate (see Figure 5). Tight Gas Production Performance Using Decline Curves—Simulated Example Figures 15a and 15b show the first six months of production history (black points) for a simulated tight gas well in transient flow. The simulation parameters are as follows: ? Drainage Area = 40 acres ? k = 0.01 md ? φμct at initial conditions = 2E-7 cp/psi ? OGIP = 1300 MMSCF The preceding method for forecasting production performance using decline curves may be applied to this well. Step 1: Fit a Decline Exponent to Actual Production History ? A superbolic decline exponent of 2.0 (red curve) is fit to the end of the production history in Figures 15a and 15b. This decline exponent was chosen as linear flow is expected in this well. Step 2: Determine tb=0.5 ? The drainage area of the well is 40 acres, an input of the simulation. ? The permeability of the reservoir is 0.01 md, an input of the simulation. ? tb = 0.5 is determined to be approximately 2 years using Figure 14. No correction for the φμct product (Equation 9) is required in this case. Step 3: Limit the Decline Exponent at tb=0.5 ? The superbolic decline exponent of 2.0 was limited to a value of 0.25 at tb=0.5 = 2 years. This decline exponent is appropriate for this well as it has a large hydraulic fracture and is being produced at a high drawdown. The recovery factor predicted by the decline forecast in Figures 15a and 15b is approximately 80%. The actual simulation output is plotted as the yellow line in the figure. There is good agreement between the decline forecast (red – transient/green – boundary dominated) and the simulation output (yellow line).
Tight Gas Production Performance Using Decline Curves—Field Example Figures 16a and 16b show the first seven months (0.58 years) of production history for a tight gas well in transient flow. A production forecast is needed for this well; however, decline curve analysis is the only tool available. The preceding method for forecasting production performance using decline curves may be applied to this well. Step 1: Fit a Decline Exponent to Actual Production History ? A superbolic decline exponent of 2.0 (red curve) is fit to the end of the production history in Figures 16a and 16b. This decline exponent was chosen as linear flow is expected in this well. Step 2: Determine tb=0.5 ? The field is drilled to 10 acre spacing. ? Reservoir permeability is estimated to range from 0.005 md to 0.010 md based on testing on some of the initial wells in the field. ? Using Figure 14, tb = 0.5 is determined to be 0.53 and 1.05 years for the k = 0.010 md and k = 0.005 md cases, respectively. No correction for the φμct product (Equation 9) is required in this case. Step 3: Limit the Decline Exponent at tb=0.5 ? The superbolic decline exponent of 2.0 was limited to a value of 0.25 at tb=0.5 = 0.53 years and 1.05 years. The boundary dominated decline exponent of 0.25 is appropriate for this case as we are assuming xfD = 1.0. The results of the production forecasts for the well in Figures 16a and 16b are shown in Table 1.
Table 1: Production Forecast Results
Forecast b = 0.0 b = 0.25 b = 0.25 b = 2.0 Exponential tb=0.5 = 0.53 yr tb=0.5 = 1.05 yr Superbolic
Expected Recovery (MMSCF) 3100 3720 5240 >>5240
The exponential (b = 0.0) forecast predicts an unreasonably low recovery while the superbolic (b = 2.0) forecast predicts an unreasonably high recovery. In this case, a range of recovery is defined by the uncertainty in formation permeability. The k = 0.010 md (tb=0.5 = 0.53 year) forecast has no remaining transient component; it likely represents a conservative recovery expectation. The k = 0.005 md (tb=0.5 = 1.05 year) forecast incorporates over five months of transient (b = 2.0) flow before the decline exponent is limited to 0.25. Its recovery is reasonable for this tight gas reservoir.
Discussion Success in decline curve forecasting for tight gas wells depends on defining a reasonable range in recovery. The extremes of this range—the conservative and optimistic recovery expectations—should be defined first and subsequent sensitivities be bound within this range. For a well in transient flow, the conservative recovery expectation may be defined by assuming that transient flow will end immediately. In other words, the entire forecast will have a decline exponent less than or equal to 0.5. The optimistic recovery expectation may be defined with the proposed method by using a reasonable drainage area and permeability. Decline curve forecasting for tight gas wells involves predicting the end of transient flow. Uncertainties in formation permeability, φμct product and drainage area make this prediction difficult. In the preceding field example, the drainage area was fixed by well spacing and the permeability was varied to place a range on recovery. In other fields with lower density development, optimum well spacing may not yet be defined and forecast sensitivities based on drainage area may be useful.
Advantages and Limitations The proposed method for performing decline curve analysis on tight gas wells must consider engineering and geological factors to greatly reduce the potential for obtaining non-unique and impractical results. Gas-in-place calculated from volumetrics, permeability ranges derived from well testing and a geological model derived from seismic or mapping are all important sources of information to incorporate in the analysis. The proposed method has the following advantages and limitations: Advantages ? Limiting superbolic decline curves is no longer completely arbitrary. Instead of limiting a decline curve based on a certain decline rate or forecast period, the limit can now be based on reservoir parameters such as drainage area and permeability. ? Sensitivities on the effects of fracture size may be incorporated into the production forecast. ? Figures 13a, 13b and 14 may be used to determine whether a well is still in transient flow. If the resulting tb = 0.5 occurs before the end of data, the well is interpreted to be in boundary dominated flow. Limitations ? Permeability information may not be available if no prior testing (pressure transient testing, rate transient analysis, core study) has been done in the field. ? Subjectivity exists in defining appropriate permeability and drainage area ranges for an analysis. It is up to the evaluator to define ‘reasonable’ ranges and decide if the results are acceptable. ? This method, like all modeling, involves non-uniqueness.
Conclusions Two advancements in tight gas production performance using decline curves are presented: ? The behaviour of the decline exponent in transient flow has been summarized for bilinear, linear and pseudo-radial flow. ? A method is introduced to forecast production throughout the entire life of a tight gas well using the Arp’s equation in its original form. The method uses simple plots to limit superbolic decline exponents based on drainage area, permeability and hydraulic fracture size.
Acknowledgement The authors would like to thank Gary Metcalfe for his thoughtful reviews and contributions to this work.
Nomenclature A= b= cf = cg = co = ct = cw = D= Di = FCD = k= kf = kx = ky = μ= φ= (φμct)o = drainage area, acres decline exponent, dimensionless formation compressibility, 1/psi gas compressibility, 1/psi oil compressibility, 1/psi total compressibility = cwSw + coSo + cgSg + cf, 1/psi water compressibility, 1/psi decline rate, 1/year initial decline rate, 1/year dimensionless fracture conductivity, dimensionless permeability of the reservoir, md permeability of the hydraulic fracture, md permeability in the x-direction, md permeability in the y-direction, md viscosity, cp porosity, fraction 2E-7 cp/psi
(φμct)new = q= qi = rinv = s= Sg = So = Sw = t= tinv = tb = 0.5 = w= Xe = xfD = xf = Ye = References
1. 2. 3. 4. 5.
adjusted fluid and formation property product, cp/psi production rate, MSCFD initial production rate, MSCFD radius of investigation, ft skin, dimensionless gas saturation, fraction oil saturation, fraction water saturation, fraction time, years time of investigation, hours time to reach a decline exponent of 0.5, years width of the hydraulic fracture, ft half of the external dimension of the reservoir in the x-direction, ft dimensionless fracture length, dimensionless fracture half length, ft half of the external dimension of the reservoir in the y-direction, ft
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SI Conversion Factors ac cp ft ft3 md psi x x x x x x 4.046873 1.0* 3.048* 2.831685 9.869233 6.894757 E+03 E-03 E-01 E-02 E-04 E+00 = m2 = Pas =m = m3 = μm2 = kPa
*The conversion factor is exact.
Appendix A: Derivation of Instantaneous Decline Exponent From the general form of the Arps Equation:
Differentiating with respect to time, t:
qi (1 + bDit )
1 ( ? ?1) dq ?1 = qi ( )(bDi )(1 + bDit ) b dt b
Which reduces to:
? ? ?1? dq = ?qi Di (1 + bDi t )? b ? dt ? 1 ?
From the definition of Decline Rate (D):
Combining Equations 1, A.2 and 2:
1 dq q dt
1 ? ? ? qi Di 1 dq (1 + bDit ) b ? ? D=? =? 1 ? ( +1) ? q dt qi ? (1 + bDit ) b ? ? ?
Which reduces to:
Di (1 + bDit )
1 1 = + bt D Di
Differentiating Equation A.5 with respect to time:
d?1? ? ? dt ? D ?
Figure 1: Typical Tight Gas Decline Curve Forecasts
Figure 2: Reservoir Model used to Create Tight Gas Simulations
Figure 3a: Instantaneous Decline Exponent for Infinite-Acting and Bounded Tight Gas Reservoirs
Figure 3b: Instantaneous Reciprocal Decline Rate for Infinite-Acting and Bounded Tight Gas Reservoirs
Figure 3c: Pressure Derivative for Infinite-Acting and Bounded Tight Gas Reservoirs
Figure 4: Decline Exponent for Varying Fracture Lengths in an Infinite Reservoir
Figure 5: Instantaneous Decline Exponent for Varying Fracture Length in a Bounded Reservoir
Figure 6: Instantaneous Decline Exponent for Varying Fracture Conductivity in a Bounded Reservoir
Figure 7: Channel Shaped Reservoir Model
Figure 8: Instantaneous Decline Exponent for Infinite-Acting and Bounded Channel Reservoirs
Figure 9: Infinite-Acting Anisotropic Reservoir
Figure 10: Instantaneous Decline Exponent for Infinite-Acting Anisotropic Reservoirs
Figure 11: Instantaneous Decline Exponent for Bounded Anisotropic Reservoirs
Figure 12: Instantaneous Decline Exponent for Varying Drawdown
Figure 13a: Decline Exponent Limit Plot A = 5/20/80 Acres
Figure 13b: Decline Exponent Limit Plot A = 10/40/160 Acres
Figure 14: Decline Limit Plot xfD = 1.0
Figure 15a: Tight Gas Production Performance Using Decline Curves—Simulated Example: Rate-Cumulative
Figure 15b: Tight Gas Production Performance Using Decline Curves—Simulated Example: Rate-Time
Figure 16a: Tight Gas Production Performance Using Decline Curves—Field Example: Rate-Cumulative
Figure 16b: Tight Gas Production Performance Using Decline Curves—Field Example: Rate-Time