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Pressure-buildup analysis method for a post-treatment evaluation of hydraulically fractured tight gas wells
Journal of Natural Gas Science and Engineering 35 (2016) 753e760
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Pressure-buildup analysis method for a post-treatment evaluation of hydraulically fractured tight gas wells Fei Wang*, Shicheng Zhang College of Petroleum Engineering, China University of Petroleum, 102249 Beijing, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 March 2016 Received in revised form 8 September 2016 Accepted 9 September 2016 Available online 10 September 2016
Tight gas reservoirs have a very low permeability, usually approximately <0.1 md. Consequently, a fractured horizontal well drawn from these reservoirs will encounter difficulty reaching a radial flow regime after completion. The effective reservoir permeability and the effective fracture half-length cannot be determined using short-term well test data. In addition, the wellbore storage effect tends to obscure early flow regimes in hydraulic fractures, thereby hampering the calculation of fracture conductivity. Fitting well test curves in the absence of early flow regimes in the fracture and middle radial flow regime is not sufficient. In this paper, a deconvolution-based pressure buildup analysis method amended with a modified Schroeter deconvolution model is proposed. The proposed method utilizes short-term pressure buildup and long-term flow rate data to recover the true reservoir pressure response. A synthetic case is presented to demonstrate that the proposed deconvolution model and algorithm can eliminate the wellbore storage effect and recover the fracture linear flow and formation radial flow regimes. A field case in the East China Sea is further presented to demonstrate the feasibility of the proposed method. This study demonstrates that a short-term pressure buildup test data can still be used to calculate the fracture and reservoir dynamic parameters of tight gas wells. Thus, the hydraulic fracturing treatment can be quantitatively evaluated. © 2016 Elsevier B.V. All rights reserved.
Keywords: Tight gas Hydraulic fracturing Pressure-buildup Post-treatment evaluation Deconvolution
1. Introduction Completion of a horizontal well with a multistage hydraulic fracturing treatment has become an effective means to produce gas from tight gas reservoirs (Kang and Luo, 2007; Dai et al., 2012). These reservoirs exhibit a low permeability, usually no more than 0.1 md, which results in a long period of transient flow in fractured horizontal wells. In the last ten years, many studies have focused on mathematical models and the pressure transient response of fractured horizontal wells. Ozkan et al. (2009), Denney (2010), Yao et al. (2013), and Wang et al. (2014) proposed mathematical models and analytical solutions for multistage fractured horizontal wells. AlKobaisi et al. (2006) focused on the pressure transient response during the early flow stage of a horizontal well intercepted by multiple transverse fractures. The early flow stage is defined as a transient flow period prior to fracture interference, which indicates a flow state under the control of fracture storage, including fracture
* Corresponding author. China University of Petroleum, No. 18, Fuxue Road, Changping, Beijing, China. E-mail address: [email protected] (F. Wang). http://dx.doi.org/10.1016/j.jngse.2016.09.026 1875-5100/© 2016 Elsevier B.V. All rights reserved.
radial flow, radial linear flow, and bilinear flow. Zerzar et al. (2004), Luo et al. (2010), and Cheng (2011) studied the pressure transient response in subsequent stages (after the fracture storage dominated stage), including the formation linear flow, pseudo-radial flow, composite linear flow, infinite-acting radial flow, and boundary-dominated pseudosteady state flow or steady state flow. Song et al. (2011) introduced the concept of pseudo-pseudosteady state flow to describe a flow regime between pseudo-linear flow and composite linear flow, which is also called exhaustion flow in a stimulated reservoir volume. Wang et al. (2013) utilized numerical simulations to present the six flow regimes of a hydraulically fractured horizontal well and the corresponding pressuretransient-response type curve (Fig. 1). The flow regimes follow the sequence: fracture flow obscured or masked by wellbore storage effect, formation linear or bilinear flow, pseudo-pseudosteady state flow, composite linear flow, formation radial flow, and pseudosteady state flow. For a tight gas reservoir with 0.01 md permeability, a fractured horizontal well usually takes a decade to reach the formation radial flow regime. Well test analysis is an effective method to define flow regimes, as well as quantify reservoir and well-completion parameters
754
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
Fig. 1. Flow regimes of a simulated hydraulically fractured horizontal well and corresponding pressure-transient-response type curve (Wang et al., 2013).
(Bourdet, 2002; Gringarten, 2005; Du, 2007; Ehlig-Economides et al., 2009; Clarkson, 2009). Short-term well tests (pressure buildups or drawdowns) are not practical or useful for the quantitative evaluation of a hydraulic fracturing treatment for fractured horizontal tight gas wells because the extraction of the effective fracture half-length, which is a key well-completion parameter, from the recorded transient pressure requires an estimate of the reservoir's effective permeability. Effective permeability can only be precisely estimated directly from a buildup or drawdown test if data from the formation radial flow regime reflecting this parameter were recorded. However, this flow regime occurs only after long shut-in or flowing times because of the extremely low permeability of the tight gas reservoir. Several months are required to test the well, and allowing the well to reach the radial flow regime is not economically feasible to producers. Meanwhile, the early flow regime in hydraulic fractures is usually obscured by the wellbore storage effect. This leads to a failure calculating the fracture conductivity, which is another key well-completion parameter, if early fracture flow is not recovered. Well test deconvolution is a new breakthrough in the well test field (Gringarten, 2010). A calculation using the deconvolution can yield the true reservoir pressure transient response during whole flow periods. Thus, the superposition effect caused by variable flow rates is avoided, and additional information about the reservoir is obtained and compared with data from a relatively short test duration. Well test deconvolution can be divided into two categories, i.e., deconvolution in the spectral domain and time domain. The former (Scott et al., 1991; Iseger, 2006; Al-Ajmi et al., 2008) transforms the convolution operation in the time domain into multiplication in the spectral domain by applying the Laplace or Fourier transform, and it converts the obtained transient pressure response to the time domain by the Stehfest numerical inversion. The latter (Baygu et al., 1997; von Schroeter et al., 2004; Levitan, 2005; Llk et al., 2006; Onur and Kuchuk, 2010; Onur et al., 2011) can be further divided into linear, restrained linear, and nonlinear solutions, in which the deconvolution considers the majority of errors and possesses a stable solution. However, Schroeter's decovolution process has not eliminated the wellbore storage effect. Therefore, the early flow regime in the fractures still cannot be recovered using their decovolution models. In this study, we propose a modified Schroeter deconvolution model specially designed to analyze the short-term pressurebuildup test data from hydraulically fractured horizontal wells in tight gas reservoirs. The wellbore storage effect can be eliminated, and the fracture linear flow and formation radial flow regimes can be recovered using the proposed analysis method. Consequently,
several well-reservoir parameters, such as reservoir effective permeability, hydraulic fracture half-length, and conductivity, can be determined. A numerical well testing synthetic study was performed to demonstrate the analysis procedures. A field application was then conducted to prove the feasibility of the proposed method.
2. Modified Schroeter deconvolution model and algorithm 2.1. Modified Schroeter deconvolution model Duhamel's principle in a single-well system states that the pressure drop is the convolution product of flow rate and reservoir response as a function of time (van Everdingen and Hurst, 1949).
DpðtÞ ¼ pi pðtÞ ¼
Zt qðtÞgðt tÞdt
(1)
0
The first step in well test analysis is to solve the Bourdet logarithm derivative. For the case of a single-flow period with a constant flow rate, the numerical differentiation of the pressure data in Eq. (1) with respect to the logarithm of time is equal to the product of the reservoir impulse response g and time t as follows (Bourdet et al., 1989):
dDp ¼ tgðtÞ d ln t
(2)
To maintain physical significance, Eq. (1) should meet the following constraint requirements to ensure that the flow rate and impulse response are greater than or equal to 0:
qðtÞ ¼
0; t 0 1; t > 0
(3)
The following equation can be obtained according to the definition by von Schroeter et al. (2004):
uðsÞ ¼ lnftgðtÞg; s ¼ ln t; t2½0; T
(4)
Thus, Eq. (1) can be transformed as follows:
Dpðt Þ ¼
Zln t
euðsÞ qsc ðt es Þdt
(5)
∞
Eq. (5) is the Schroeter deconvolution model, in which qsc is the
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
flow rate measured on the surface without considering the wellbore storage effect. Given the single-phase gas flow and according to the material balance equation, the flow rate underground can be calculated by the surface flow rate and wellbore after-flow rate (Lee et al., 2003):
qsf
. Tsc Vwb d pwf z ¼ qsc þ dt Tr psc
Zln t
analysis method using the modified deconvolution model to recover the true reservoir pressure response with a short-term pressure buildup and long-term flow rate data. The deconvolved pressure response and its derivative can be further applied to normal well test analysis. The buildup test analysis based on our modified deconvolution model is composed of the following four parts:
(6)
A deconvolution of the flow rate and pressure data can be performed for any flow period provided that the wellbore storage coefficient remains constant. The time interval corresponding to the calculated pressure response is from t ¼ 0 to the current time. Therefore, the reservoir pressure response without the wellbore storage effect can be obtained by substituting the sandface flow rate from Eq. (6) for the surface flow rate from Eq. (5). Hence, the modified deconvolution model is devised as follows.
Dpwf ðt Þ ¼
755
euðsÞ qsf ðt es Þdt
(7)
(1) Data preparation ❖ Pressure history: Transform pressure data of tight gas reservoirs into pseudo-pressure data to satisfy the linear requirement of the convolution formula. ❖ Flow rate history: Adjust flow rate history in the same time step as the pressure history. ❖ Initial reservoir pressure: Determine the original value of pi through an independent information source, such as a Repeat Formation Test (RFT). The original value can also be estimated by the extrapolation of the Horner curve during the pressure buildup period. (2) Algorithm adjustment
∞
Previous data suggest that the weights for pressure match, rate match, and smoothness should be adjusted to a reasonable range to avoid the distortion resulting from over smoothing.
2.2. Algorithm The deconvolution algorithm is dependent on the constraints during solving, of which the nonlinear total least squares is the most stable. The measurement errors in the pressure and rate data are considered in the objective function of the nonlinear regression. Thus, the objective function (or error measure) is composed of three parts, namely pressure match, flow rate match, and smoothness constraints. 2
kεk22 kdk2 kDgk22 Minðn; lÞ ¼ 2 þ n 2 þ l kgk22 Dpwf qsf 2
(8)
2 N Dpwf 2 nd ¼ n 2 qsf
(9)
2
and the default weight of the smoothness coefficient:
ld ¼
2
n
A deconvolution can be applied to several build-up periods separately. The initial reservoir pressure pi should be adjusted until all deconvolved pressure derivatives coincide in their late stages, during which pi is the real initial reservoir pressure. Additionally, deconvolution can be applied to the entire data history in the absence of a distinct difference in the wellbore storage coefficient and skin factor among each buildup period. (4) Data analysis
2
The two default weights dominate because the weight of the pressure is equal to 0 after rate normalization during the actual solving process. These are the default weights of the rate error:
2 Dpwf
(3) Data processing
(10)
The curve of u(s) can be obtained in terms of s using the modified deconvolution model [Eq. (7)] and the nonlinear total least square method [Eq. (8)]. This curve can match the measured and theoretical transient pressure response. Fitting time can begin at the start of production to the last measured time. u(s) is unknown during the deconvolution process, while the initial reservoir pressure and flow rate allowable errors are variables. Optimization is conducted to minimize the objective function by adjusting these three variables. 3. Buildup test analysis based on the modified deconvolution model This study was performed to propose a pressure-buildup
The obtained deconvolved pressure derivative, u(s) , is used to select the well test interpretation model that matches and calculate the reservoir and well-completion parameters. The reservoir boundary type and boundary location can be determined as well. For hydraulically fractured horizontal wells in tight gas reservoirs, the effective reservoir permeability, k, can be obtained by a straight-line analysis of the formation radial flow regime, and the fracture half-length, xf, can be determined by analyzing the formation linear flow. Additionally, the fracture conductivity, kfw, can be estimated from the analysis of the early fracture linear flow regime. 4. Synthetic case study A numerical well test is conducted to examine the modified deconvolution model and its algorithm. The simulated tight gas reservoir is infinite with a 5000 psi initial pressure, 100 C temperature, 30 ft thickness, 0.3 md permeability, 0.08 porosity, and 3 106 psi1 rock compressibility coefficient. A multi-stage hydraulically fractured horizontal well with a lateral length of 2000 ft is considered. Nine transverse fractures with fracture half-lengths of 400 ft are evenly distributed along the horizontal wellbore. The hydraulic fracture conductivity is assumed to be infinite, and the wellbore storage coefficient is set to 0.15 bbl/psi. Fig. 2 shows a sketch of the simulated model, where the x and y axes represent the reservoir length in x and y directions, respectively. The center position along the horizontal lateral is assumed to be coordinates (0,0).
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F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
Fig. 2. Numerical model of the simulated well.
Fig. 3. Production history of the simulated well.
Fig. 3 shows a 680-day production history of the simulated well, including three shut-in periods with pressure buildup test data. The test durations in order are 13, 22, and 1 day(s). The logelog plots of the three sets of pressure buildup data are presented in Fig. 4. The third buildup response (denoted in blue), has the shortest duration of only one day. It depicts the features of the linear flow regime from formation to fracture because its pressure derivative presents a half slope after the “hump” as a marker of the wellbore storage transition. The second plot, which is also the longest buildup response, (denoted in red) coincides with the early stage of the third plot. However, its pressure derivative increases during the late stage, reflecting the fracture interference effect. The duration of the first pressure buildup period is the second longest. Therefore, its pressure derivative (denoted in green) overlaps in a region between the other two responses. The early flow regimes in the hydraulic fracture of all three buildups are obscured by the wellbore storage effect. This is verified by the iconic unit-slope straight line evident in the early stage. Radial flow is not exhibited on the derivative
curves of all three buildup periods due to time limitations of the buildup tests. Fig. 5 shows the log-log plot of the deconvolved pressures and derivatives using the third pressure buildup data and the whole production history. The deconvolved pressure derivatives (denoted in red and green) coincide with that from the third pressure buildup (denoted in blue) during their middle stages, reflecting the formation of linear flow. Likewise, the curves reveal the formation radial flow, which is represented by a zero-slope straight line during the later stage. The deconvolved pressure and derivative calculated by the conventional Schroeter model (denoted in green) cannot eliminate the wellbore storage effect in the early stage. By contrast, the deconvolved pressure and derivative calculated by our modified model (denoted in red) can reflect the flow state in the fractures. Instead of a wellbore storage unit-slope, a half-slope straight line in the early flow period both in pressure and its derivative represents the fracture linear flow regime. These results demonstrate that the proposed deconvolution
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
757
Fig. 4. Pressure buildups on log-log plot.
Fig. 5. Deconvolved pressure-transient response on log-log plot.
model and algorithm can eliminate the wellbore storage effect and recover the fracture linear flow and formation radial flow regimes in fractured tight gas wells. The results also further facilitate the estimation of reservoir and well-completion parameters.
5. Field application Well A is a representative tight gas well in the Chunxiao gas field located in East China Sea. The reservoir is primarily composed of fine sandstone and does not exhibit gas condensate. The reservoir fluid is mainly dry gas. Table 1 shows the formation parameters obtained by well logging and core experiments. Among them, the permeability is a static measurement obtained by nuclear magnetic resonance (NMR) data. Due to its limited investigation depth and state of the rock, the permeability of NMR is occasionally two orders of magnitude higher than the dynamic effective permeability of well tests (Haddad et al., 2000; Cantini et al., 2013), and that will be proved true by the effective permeability derived from the deconvolution-based pressure-buildup analysis in a later section of this paper. Well A is a development well in that reservoir, with a
depth of 13,530 ft. The well was put into production on July 11, 2010. Hydraulic fracturing treatment was implemented on well A on September 11, 2011 to increase the yield. The red curve in Fig. 6 shows the production history after treatment. Well A was shut in for a pressure buildup test on April 10, 2012. The blue part of Fig. 6 shows the pressure buildup history. The logelog plot in Fig. 7 reveals that the 7.5-day pressure buildup data only reflects the wellbore storage dominated flow and the subsequent formation linear flow because the pressure and its derivative both appear as a unit-slope straight line followed by a
Table 1 Basic formation and fluid parameters. Parameter
Value
Formation thickness/ft Porosity/% Permeability/md Total compressibility/psi1 Natural gas viscosity/cp
75.9e100.3 11.4e12.5 2.7e4.9 2.34 106 0.0248
758
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
Fig. 6. Post-fracturing production history of well A.
Fig. 7. Pressure buildup on log-log plot.
Fig. 8. Deconvolved pressure-transient response on log-log plot.
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
759
Table 2 Determined key parameters of well A. Wellbore storage coefficient C/bbl/psi
Skin factor s
Permeability k/md
Fracture half-length xf/ft
Fracture conductivity kfw/md$ft
0.0121
0.05
0.0238
267.7
626.3
Fig. 9. Production history matching of well A.
half slope straight line. The derivative curve finally goes slightly upwards, but the reservoir model cannot be determined yet. Fig. 8 shows that the deconvolved pressure derivative reflects the feature signaling formation radial flow, i.e., a zero-slope straight line period, using the deconvolution of pressure build-up data and production history data. Accordingly, a radial composite reservoir model, in which well A is located in the inner zone with higher mobility and storability, can be confirmed. Moreover, the deconvolved pressure and derivative obtained from the modified deconvolution model reflect the early linear flow regime in hydraulic fractures because they are parallel and both appear as a half-slope straight line section during the early flow period. The key parameters interpreted by straight-line analysis of the well-tests according to the recovered four flow regimes, i.e., wellbore storage, fracture linear flow, formation linear flow and formation radial flow, are obtained and listed in Table 2. The effective reservoir permeability, 0.0238 md, indicates the reservoir is typically tight. The fracture half-length and conductivity values are good, indicating a successful hydraulic fracturing treatment. Finally, the parameters obtained from the proposed deconvolution-based pressure-buildup analysis method were used as input values to build a numerical model that yields the production rate and casing pressure by history matching well A. Fig. 9 shows a satisfactory matching result.
(2) A deconvolution-based pressure-buildup analysis method supplemented with a modified deconvolution model is proposed. A numerical well testing synthetic study is performed to demonstrate the pressure buildup analysis procedures using the developed model. The results demonstrate that the proposed method works well by eliminating the wellbore storage effect and recovering the flow regime for fractured horizontal tight gas wells. (3) A field application from the Chunxiao gas field in the East China Sea is further presented to demonstrate the feasibility of the proposed method. The result proves that the proposed method can be used to interpret well tests of fracture and reservoir dynamic parameters. Thus, the hydraulic fracturing treatment of horizontal wells in tight gas reservoirs can be quantitatively evaluated. Acknowledgments The authors would like to acknowledge the Beijing Natural Science Foundation(No.3154038), the National Natural Science Foundation of China(No.51504266) and the Science Foundation of China University of Petroleum, Beijing (No.2462015YQ0212) for their financial support. Nomenclature
6. Conclusions (1) This study provides a modified Schroeter deconvolution model. In this model the sandface flow rate is calculated based on surface flow rate and wellbore after-flow rate, by which the deconvolved pressure response and its derivative can avoid the wellbore storage effect and recover the flow regimes of a fractured horizontal well during the early period. Therefore, the subsequent well-test analysis can be finished.
C g k kfw m N n p(t) pi psc
wellbore storage coefficient, bbl/psi reservoir impulse response, psi/(h$mscf/D) permeability, md fracture conductivity, md$ft slope on log-log plot number of flow rate data number of pressure data flowing pressure, psi initial reservoir pressure, psi pressure on surface condition, psi
760
pwf Dp(t) Dpwf q(t) qsf qsc s t T Tsc Tr Vwb xf z
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
bottom-hole flowing pressure, psi pressure drop, psi bottom-hole pressure drop, psi flow rate, mscf/D sandface flow rate, ft3/h surface flow rate, ft3/h skin factor time, h test duration temperature on surface condition, R reservoir temperature, R wellbore volume, ft3 fracture half-length, ft gas deviation factor
Greek ε
pressure measurement error, psi flow rate error between the calculated and true sandface flow rate, ft3/h g relative smoothness coefficient Dg derivative sequence of u(s) n flow rate error weight l smoothness coefficient weight nd default weight of flow rate error ld default weight of smoothness coefficient Min(n,l) error measure s logarithmic function of time u(s) logarithmic response function t integration variable
d
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Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Pressure-buildup analysis method for a post-treatment evaluation of hydraulically fractured tight gas wells Fei Wang*, Shicheng Zhang College of Petroleum Engineering, China University of Petroleum, 102249 Beijing, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 March 2016 Received in revised form 8 September 2016 Accepted 9 September 2016 Available online 10 September 2016
Tight gas reservoirs have a very low permeability, usually approximately <0.1 md. Consequently, a fractured horizontal well drawn from these reservoirs will encounter difficulty reaching a radial flow regime after completion. The effective reservoir permeability and the effective fracture half-length cannot be determined using short-term well test data. In addition, the wellbore storage effect tends to obscure early flow regimes in hydraulic fractures, thereby hampering the calculation of fracture conductivity. Fitting well test curves in the absence of early flow regimes in the fracture and middle radial flow regime is not sufficient. In this paper, a deconvolution-based pressure buildup analysis method amended with a modified Schroeter deconvolution model is proposed. The proposed method utilizes short-term pressure buildup and long-term flow rate data to recover the true reservoir pressure response. A synthetic case is presented to demonstrate that the proposed deconvolution model and algorithm can eliminate the wellbore storage effect and recover the fracture linear flow and formation radial flow regimes. A field case in the East China Sea is further presented to demonstrate the feasibility of the proposed method. This study demonstrates that a short-term pressure buildup test data can still be used to calculate the fracture and reservoir dynamic parameters of tight gas wells. Thus, the hydraulic fracturing treatment can be quantitatively evaluated. © 2016 Elsevier B.V. All rights reserved.
Keywords: Tight gas Hydraulic fracturing Pressure-buildup Post-treatment evaluation Deconvolution
1. Introduction Completion of a horizontal well with a multistage hydraulic fracturing treatment has become an effective means to produce gas from tight gas reservoirs (Kang and Luo, 2007; Dai et al., 2012). These reservoirs exhibit a low permeability, usually no more than 0.1 md, which results in a long period of transient flow in fractured horizontal wells. In the last ten years, many studies have focused on mathematical models and the pressure transient response of fractured horizontal wells. Ozkan et al. (2009), Denney (2010), Yao et al. (2013), and Wang et al. (2014) proposed mathematical models and analytical solutions for multistage fractured horizontal wells. AlKobaisi et al. (2006) focused on the pressure transient response during the early flow stage of a horizontal well intercepted by multiple transverse fractures. The early flow stage is defined as a transient flow period prior to fracture interference, which indicates a flow state under the control of fracture storage, including fracture
* Corresponding author. China University of Petroleum, No. 18, Fuxue Road, Changping, Beijing, China. E-mail address: [email protected] (F. Wang). http://dx.doi.org/10.1016/j.jngse.2016.09.026 1875-5100/© 2016 Elsevier B.V. All rights reserved.
radial flow, radial linear flow, and bilinear flow. Zerzar et al. (2004), Luo et al. (2010), and Cheng (2011) studied the pressure transient response in subsequent stages (after the fracture storage dominated stage), including the formation linear flow, pseudo-radial flow, composite linear flow, infinite-acting radial flow, and boundary-dominated pseudosteady state flow or steady state flow. Song et al. (2011) introduced the concept of pseudo-pseudosteady state flow to describe a flow regime between pseudo-linear flow and composite linear flow, which is also called exhaustion flow in a stimulated reservoir volume. Wang et al. (2013) utilized numerical simulations to present the six flow regimes of a hydraulically fractured horizontal well and the corresponding pressuretransient-response type curve (Fig. 1). The flow regimes follow the sequence: fracture flow obscured or masked by wellbore storage effect, formation linear or bilinear flow, pseudo-pseudosteady state flow, composite linear flow, formation radial flow, and pseudosteady state flow. For a tight gas reservoir with 0.01 md permeability, a fractured horizontal well usually takes a decade to reach the formation radial flow regime. Well test analysis is an effective method to define flow regimes, as well as quantify reservoir and well-completion parameters
754
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Fig. 1. Flow regimes of a simulated hydraulically fractured horizontal well and corresponding pressure-transient-response type curve (Wang et al., 2013).
(Bourdet, 2002; Gringarten, 2005; Du, 2007; Ehlig-Economides et al., 2009; Clarkson, 2009). Short-term well tests (pressure buildups or drawdowns) are not practical or useful for the quantitative evaluation of a hydraulic fracturing treatment for fractured horizontal tight gas wells because the extraction of the effective fracture half-length, which is a key well-completion parameter, from the recorded transient pressure requires an estimate of the reservoir's effective permeability. Effective permeability can only be precisely estimated directly from a buildup or drawdown test if data from the formation radial flow regime reflecting this parameter were recorded. However, this flow regime occurs only after long shut-in or flowing times because of the extremely low permeability of the tight gas reservoir. Several months are required to test the well, and allowing the well to reach the radial flow regime is not economically feasible to producers. Meanwhile, the early flow regime in hydraulic fractures is usually obscured by the wellbore storage effect. This leads to a failure calculating the fracture conductivity, which is another key well-completion parameter, if early fracture flow is not recovered. Well test deconvolution is a new breakthrough in the well test field (Gringarten, 2010). A calculation using the deconvolution can yield the true reservoir pressure transient response during whole flow periods. Thus, the superposition effect caused by variable flow rates is avoided, and additional information about the reservoir is obtained and compared with data from a relatively short test duration. Well test deconvolution can be divided into two categories, i.e., deconvolution in the spectral domain and time domain. The former (Scott et al., 1991; Iseger, 2006; Al-Ajmi et al., 2008) transforms the convolution operation in the time domain into multiplication in the spectral domain by applying the Laplace or Fourier transform, and it converts the obtained transient pressure response to the time domain by the Stehfest numerical inversion. The latter (Baygu et al., 1997; von Schroeter et al., 2004; Levitan, 2005; Llk et al., 2006; Onur and Kuchuk, 2010; Onur et al., 2011) can be further divided into linear, restrained linear, and nonlinear solutions, in which the deconvolution considers the majority of errors and possesses a stable solution. However, Schroeter's decovolution process has not eliminated the wellbore storage effect. Therefore, the early flow regime in the fractures still cannot be recovered using their decovolution models. In this study, we propose a modified Schroeter deconvolution model specially designed to analyze the short-term pressurebuildup test data from hydraulically fractured horizontal wells in tight gas reservoirs. The wellbore storage effect can be eliminated, and the fracture linear flow and formation radial flow regimes can be recovered using the proposed analysis method. Consequently,
several well-reservoir parameters, such as reservoir effective permeability, hydraulic fracture half-length, and conductivity, can be determined. A numerical well testing synthetic study was performed to demonstrate the analysis procedures. A field application was then conducted to prove the feasibility of the proposed method.
2. Modified Schroeter deconvolution model and algorithm 2.1. Modified Schroeter deconvolution model Duhamel's principle in a single-well system states that the pressure drop is the convolution product of flow rate and reservoir response as a function of time (van Everdingen and Hurst, 1949).
DpðtÞ ¼ pi pðtÞ ¼
Zt qðtÞgðt tÞdt
(1)
0
The first step in well test analysis is to solve the Bourdet logarithm derivative. For the case of a single-flow period with a constant flow rate, the numerical differentiation of the pressure data in Eq. (1) with respect to the logarithm of time is equal to the product of the reservoir impulse response g and time t as follows (Bourdet et al., 1989):
dDp ¼ tgðtÞ d ln t
(2)
To maintain physical significance, Eq. (1) should meet the following constraint requirements to ensure that the flow rate and impulse response are greater than or equal to 0:
qðtÞ ¼
0; t 0 1; t > 0
(3)
The following equation can be obtained according to the definition by von Schroeter et al. (2004):
uðsÞ ¼ lnftgðtÞg; s ¼ ln t; t2½0; T
(4)
Thus, Eq. (1) can be transformed as follows:
Dpðt Þ ¼
Zln t
euðsÞ qsc ðt es Þdt
(5)
∞
Eq. (5) is the Schroeter deconvolution model, in which qsc is the
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
flow rate measured on the surface without considering the wellbore storage effect. Given the single-phase gas flow and according to the material balance equation, the flow rate underground can be calculated by the surface flow rate and wellbore after-flow rate (Lee et al., 2003):
qsf
. Tsc Vwb d pwf z ¼ qsc þ dt Tr psc
Zln t
analysis method using the modified deconvolution model to recover the true reservoir pressure response with a short-term pressure buildup and long-term flow rate data. The deconvolved pressure response and its derivative can be further applied to normal well test analysis. The buildup test analysis based on our modified deconvolution model is composed of the following four parts:
(6)
A deconvolution of the flow rate and pressure data can be performed for any flow period provided that the wellbore storage coefficient remains constant. The time interval corresponding to the calculated pressure response is from t ¼ 0 to the current time. Therefore, the reservoir pressure response without the wellbore storage effect can be obtained by substituting the sandface flow rate from Eq. (6) for the surface flow rate from Eq. (5). Hence, the modified deconvolution model is devised as follows.
Dpwf ðt Þ ¼
755
euðsÞ qsf ðt es Þdt
(7)
(1) Data preparation ❖ Pressure history: Transform pressure data of tight gas reservoirs into pseudo-pressure data to satisfy the linear requirement of the convolution formula. ❖ Flow rate history: Adjust flow rate history in the same time step as the pressure history. ❖ Initial reservoir pressure: Determine the original value of pi through an independent information source, such as a Repeat Formation Test (RFT). The original value can also be estimated by the extrapolation of the Horner curve during the pressure buildup period. (2) Algorithm adjustment
∞
Previous data suggest that the weights for pressure match, rate match, and smoothness should be adjusted to a reasonable range to avoid the distortion resulting from over smoothing.
2.2. Algorithm The deconvolution algorithm is dependent on the constraints during solving, of which the nonlinear total least squares is the most stable. The measurement errors in the pressure and rate data are considered in the objective function of the nonlinear regression. Thus, the objective function (or error measure) is composed of three parts, namely pressure match, flow rate match, and smoothness constraints. 2
kεk22 kdk2 kDgk22 Minðn; lÞ ¼ 2 þ n 2 þ l kgk22 Dpwf qsf 2
(8)
2 N Dpwf 2 nd ¼ n 2 qsf
(9)
2
and the default weight of the smoothness coefficient:
ld ¼
2
n
A deconvolution can be applied to several build-up periods separately. The initial reservoir pressure pi should be adjusted until all deconvolved pressure derivatives coincide in their late stages, during which pi is the real initial reservoir pressure. Additionally, deconvolution can be applied to the entire data history in the absence of a distinct difference in the wellbore storage coefficient and skin factor among each buildup period. (4) Data analysis
2
The two default weights dominate because the weight of the pressure is equal to 0 after rate normalization during the actual solving process. These are the default weights of the rate error:
2 Dpwf
(3) Data processing
(10)
The curve of u(s) can be obtained in terms of s using the modified deconvolution model [Eq. (7)] and the nonlinear total least square method [Eq. (8)]. This curve can match the measured and theoretical transient pressure response. Fitting time can begin at the start of production to the last measured time. u(s) is unknown during the deconvolution process, while the initial reservoir pressure and flow rate allowable errors are variables. Optimization is conducted to minimize the objective function by adjusting these three variables. 3. Buildup test analysis based on the modified deconvolution model This study was performed to propose a pressure-buildup
The obtained deconvolved pressure derivative, u(s) , is used to select the well test interpretation model that matches and calculate the reservoir and well-completion parameters. The reservoir boundary type and boundary location can be determined as well. For hydraulically fractured horizontal wells in tight gas reservoirs, the effective reservoir permeability, k, can be obtained by a straight-line analysis of the formation radial flow regime, and the fracture half-length, xf, can be determined by analyzing the formation linear flow. Additionally, the fracture conductivity, kfw, can be estimated from the analysis of the early fracture linear flow regime. 4. Synthetic case study A numerical well test is conducted to examine the modified deconvolution model and its algorithm. The simulated tight gas reservoir is infinite with a 5000 psi initial pressure, 100 C temperature, 30 ft thickness, 0.3 md permeability, 0.08 porosity, and 3 106 psi1 rock compressibility coefficient. A multi-stage hydraulically fractured horizontal well with a lateral length of 2000 ft is considered. Nine transverse fractures with fracture half-lengths of 400 ft are evenly distributed along the horizontal wellbore. The hydraulic fracture conductivity is assumed to be infinite, and the wellbore storage coefficient is set to 0.15 bbl/psi. Fig. 2 shows a sketch of the simulated model, where the x and y axes represent the reservoir length in x and y directions, respectively. The center position along the horizontal lateral is assumed to be coordinates (0,0).
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Fig. 2. Numerical model of the simulated well.
Fig. 3. Production history of the simulated well.
Fig. 3 shows a 680-day production history of the simulated well, including three shut-in periods with pressure buildup test data. The test durations in order are 13, 22, and 1 day(s). The logelog plots of the three sets of pressure buildup data are presented in Fig. 4. The third buildup response (denoted in blue), has the shortest duration of only one day. It depicts the features of the linear flow regime from formation to fracture because its pressure derivative presents a half slope after the “hump” as a marker of the wellbore storage transition. The second plot, which is also the longest buildup response, (denoted in red) coincides with the early stage of the third plot. However, its pressure derivative increases during the late stage, reflecting the fracture interference effect. The duration of the first pressure buildup period is the second longest. Therefore, its pressure derivative (denoted in green) overlaps in a region between the other two responses. The early flow regimes in the hydraulic fracture of all three buildups are obscured by the wellbore storage effect. This is verified by the iconic unit-slope straight line evident in the early stage. Radial flow is not exhibited on the derivative
curves of all three buildup periods due to time limitations of the buildup tests. Fig. 5 shows the log-log plot of the deconvolved pressures and derivatives using the third pressure buildup data and the whole production history. The deconvolved pressure derivatives (denoted in red and green) coincide with that from the third pressure buildup (denoted in blue) during their middle stages, reflecting the formation of linear flow. Likewise, the curves reveal the formation radial flow, which is represented by a zero-slope straight line during the later stage. The deconvolved pressure and derivative calculated by the conventional Schroeter model (denoted in green) cannot eliminate the wellbore storage effect in the early stage. By contrast, the deconvolved pressure and derivative calculated by our modified model (denoted in red) can reflect the flow state in the fractures. Instead of a wellbore storage unit-slope, a half-slope straight line in the early flow period both in pressure and its derivative represents the fracture linear flow regime. These results demonstrate that the proposed deconvolution
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
757
Fig. 4. Pressure buildups on log-log plot.
Fig. 5. Deconvolved pressure-transient response on log-log plot.
model and algorithm can eliminate the wellbore storage effect and recover the fracture linear flow and formation radial flow regimes in fractured tight gas wells. The results also further facilitate the estimation of reservoir and well-completion parameters.
5. Field application Well A is a representative tight gas well in the Chunxiao gas field located in East China Sea. The reservoir is primarily composed of fine sandstone and does not exhibit gas condensate. The reservoir fluid is mainly dry gas. Table 1 shows the formation parameters obtained by well logging and core experiments. Among them, the permeability is a static measurement obtained by nuclear magnetic resonance (NMR) data. Due to its limited investigation depth and state of the rock, the permeability of NMR is occasionally two orders of magnitude higher than the dynamic effective permeability of well tests (Haddad et al., 2000; Cantini et al., 2013), and that will be proved true by the effective permeability derived from the deconvolution-based pressure-buildup analysis in a later section of this paper. Well A is a development well in that reservoir, with a
depth of 13,530 ft. The well was put into production on July 11, 2010. Hydraulic fracturing treatment was implemented on well A on September 11, 2011 to increase the yield. The red curve in Fig. 6 shows the production history after treatment. Well A was shut in for a pressure buildup test on April 10, 2012. The blue part of Fig. 6 shows the pressure buildup history. The logelog plot in Fig. 7 reveals that the 7.5-day pressure buildup data only reflects the wellbore storage dominated flow and the subsequent formation linear flow because the pressure and its derivative both appear as a unit-slope straight line followed by a
Table 1 Basic formation and fluid parameters. Parameter
Value
Formation thickness/ft Porosity/% Permeability/md Total compressibility/psi1 Natural gas viscosity/cp
75.9e100.3 11.4e12.5 2.7e4.9 2.34 106 0.0248
758
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Fig. 6. Post-fracturing production history of well A.
Fig. 7. Pressure buildup on log-log plot.
Fig. 8. Deconvolved pressure-transient response on log-log plot.
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759
Table 2 Determined key parameters of well A. Wellbore storage coefficient C/bbl/psi
Skin factor s
Permeability k/md
Fracture half-length xf/ft
Fracture conductivity kfw/md$ft
0.0121
0.05
0.0238
267.7
626.3
Fig. 9. Production history matching of well A.
half slope straight line. The derivative curve finally goes slightly upwards, but the reservoir model cannot be determined yet. Fig. 8 shows that the deconvolved pressure derivative reflects the feature signaling formation radial flow, i.e., a zero-slope straight line period, using the deconvolution of pressure build-up data and production history data. Accordingly, a radial composite reservoir model, in which well A is located in the inner zone with higher mobility and storability, can be confirmed. Moreover, the deconvolved pressure and derivative obtained from the modified deconvolution model reflect the early linear flow regime in hydraulic fractures because they are parallel and both appear as a half-slope straight line section during the early flow period. The key parameters interpreted by straight-line analysis of the well-tests according to the recovered four flow regimes, i.e., wellbore storage, fracture linear flow, formation linear flow and formation radial flow, are obtained and listed in Table 2. The effective reservoir permeability, 0.0238 md, indicates the reservoir is typically tight. The fracture half-length and conductivity values are good, indicating a successful hydraulic fracturing treatment. Finally, the parameters obtained from the proposed deconvolution-based pressure-buildup analysis method were used as input values to build a numerical model that yields the production rate and casing pressure by history matching well A. Fig. 9 shows a satisfactory matching result.
(2) A deconvolution-based pressure-buildup analysis method supplemented with a modified deconvolution model is proposed. A numerical well testing synthetic study is performed to demonstrate the pressure buildup analysis procedures using the developed model. The results demonstrate that the proposed method works well by eliminating the wellbore storage effect and recovering the flow regime for fractured horizontal tight gas wells. (3) A field application from the Chunxiao gas field in the East China Sea is further presented to demonstrate the feasibility of the proposed method. The result proves that the proposed method can be used to interpret well tests of fracture and reservoir dynamic parameters. Thus, the hydraulic fracturing treatment of horizontal wells in tight gas reservoirs can be quantitatively evaluated. Acknowledgments The authors would like to acknowledge the Beijing Natural Science Foundation(No.3154038), the National Natural Science Foundation of China(No.51504266) and the Science Foundation of China University of Petroleum, Beijing (No.2462015YQ0212) for their financial support. Nomenclature
6. Conclusions (1) This study provides a modified Schroeter deconvolution model. In this model the sandface flow rate is calculated based on surface flow rate and wellbore after-flow rate, by which the deconvolved pressure response and its derivative can avoid the wellbore storage effect and recover the flow regimes of a fractured horizontal well during the early period. Therefore, the subsequent well-test analysis can be finished.
C g k kfw m N n p(t) pi psc
wellbore storage coefficient, bbl/psi reservoir impulse response, psi/(h$mscf/D) permeability, md fracture conductivity, md$ft slope on log-log plot number of flow rate data number of pressure data flowing pressure, psi initial reservoir pressure, psi pressure on surface condition, psi
760
pwf Dp(t) Dpwf q(t) qsf qsc s t T Tsc Tr Vwb xf z
F. Wang, S. Zhang / Journal of Natural Gas Science and Engineering 35 (2016) 753e760
bottom-hole flowing pressure, psi pressure drop, psi bottom-hole pressure drop, psi flow rate, mscf/D sandface flow rate, ft3/h surface flow rate, ft3/h skin factor time, h test duration temperature on surface condition, R reservoir temperature, R wellbore volume, ft3 fracture half-length, ft gas deviation factor
Greek ε
pressure measurement error, psi flow rate error between the calculated and true sandface flow rate, ft3/h g relative smoothness coefficient Dg derivative sequence of u(s) n flow rate error weight l smoothness coefficient weight nd default weight of flow rate error ld default weight of smoothness coefficient Min(n,l) error measure s logarithmic function of time u(s) logarithmic response function t integration variable
d
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