Regular Article
Investigation on the pressure behavior of injectors influenced by waterfloodinduced fractures: field cases in Huaqing reservoir, Changqing Oilfield, China
^{1}
State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, 102249 Beijing, China
^{2}
Lusheng Petroleum Development Company Limited, SINOPEC Shengli Oilfield Company, 257000 Dongying, China
^{3}
PetroChina Changqing Oilfield Company, 710018 Xi’an, China
^{*} Corresponding author: petroyang@163.com
Received:
20
July
2019
Accepted:
20
February
2020
Pressure transient behaviors of water injectors with the influence of WaterfloodInduced Fractures (WIFs) in unconventional tight oil reservoirs are much more complicated than hydraulic fracturing producers due to the variation of the WIF properties in the fieldtesting period. In this paper, we present the interpretation on five different types of BottomHole Pressure (BHP) responses for water injectors by combining a commercial simulator tool and our proposed models, where the Shrinking Fracture Length (SFL) and Decreasing Fracture Conductivity (DFC) effects are captured for single long WIF while the multiple fracture closures effect is characterized for multiple WIFs. Results show that the commonplace phenomena of prolonged storage effect, bistorage effect, and interpreted considerably large storage coefficient suggest that WIF(s) may initiate by longtime water injection. Based on this interpreted large storage coefficient, the fracture halflength can be estimated. The upward buckling of pressure derivative curve may not necessarily be caused by closed outer boundary condition, but the Decreasing of Fracture Conductivity (DFC). As for multiple WIFs, they would close successively after shutting in the well due to the different stress conditions perpendicular to fracture walls, which behaves as several unit slopes on the pressure derivative curves in log–log plot. Aiming at different representative types of BHP responses cases in Huaqing reservoir, Changqing Oilfield, we innovatively analyze them from a different perspective and get a new understanding of water injector behaviors, which provides a guideline for the interpretation of water injectors in tight oil reservoirs.
© Y. Wang et al., published by IFP Energies nouvelles, 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Nomenclature
A: Fracture surface area, m^{2}
A_{fb1}, A_{fb2}, A_{fb3}: Surface area of three secondary fractures, m^{2}
A_{frb1}, A_{frb2}, A_{frb3}: Relative surface area of three secondary fractures, dimensionless
A_{fm} : Surface area of main fracture at the end of injection, m^{2}
C_{w} : Wellborestorage coefficient, m^{3}/Pa
c_{f} : Fracture compliance, m/MPa
C_{f} : Fracturestorage coefficient, m^{3}/Pa
c_{fm} : Leakoff coefficient of main fracture, m/h^{0.5}
C_{Lb1}, C_{Lb2}, C_{Lb3}: Leakoff coefficient of three secondary fractures, m/h^{0.5}
C_{Lm} : Leakoff coefficient of main fracture, m/h^{0.5}
C_{Lm1}, C_{Lm2}, C_{Lm3}, C_{Lm4}: Equivalent leakoff coefficient in the fractured falloff period, m/h^{0.5}
c_{t} : Total compressibility, MPa^{−1}
k: Permeability, 10^{−3}μm^{2}
p_{ki} : Closure pressure in any direction, MPa
p_{net} : Net pressure in the fracture, MPa
p_{shutin} : Bottomhole pressure in the shutin moment, MPa
: Dimensionless bottomhole pressure when fracture begins to shrink
q: Water injection rate, m^{3}/d
r_{p} : Ratio of permeable fracture surface area to the gross fracture area, dimensionless
R_{F} : Fracture radius in radius fracture model, m
S_{choke} : Chokedfracture skin
t_{p} : Fractured injection time at the end of injection, h
x_{F} : Fracture halflength, m
x_{F0} : Fracture halflength in the shutin moment, m
V_{frac} : Volume of the fracture, m^{3}
V_{frac,mf} : Volume of main fracture, m^{3}
V_{frac,b1}, V_{frac,b2}, V_{frac,b3}: Volume of three secondary fractures, m^{3}
V_{l,bf1}, V_{l,bf2}, V_{l,bf3}: Flow volume from three secondary fractures into matrix, m^{3}
V_{l,mf} : Flow volume from main fracture into matrix, m^{3}
V_{p} : Accumulated volume of water injected in fractured injection period, m^{3}
Δt_{fc1}, Δt_{fc2}, Δt_{fc3}: Fracture closure moment of three secondary fractures, h
Δt_{fcD1}, Δt_{fcD2}, Δt_{fcD3}: Dimensionless fracture closure moment of three secondary fractures, dimensionless
α: Area exponent, dimensionless
σ_{max} : Maximum principal stress, MPa
σ_{min} : Minimum principal stress, MPa
θ: Included angle between the fracture direction and maximum principal stress, radian
Subscript
Superscript
1 Introduction
Waterflooding is one of the most commonly used development methods in the field of petroleum industry and there are growing evidences showing that water injection may induce formation fracturing [1–5]. The long WIFs may be caused by plugging effects, depletion (poroelastic), thermal effects, and so on [6, 7]. As shown by Wright et al. [8] in West Texas by tiltmeter data, the halflength of the WIFs can reach a surprising 450–650 ft (70–100% of well spacing) as seen in Figure 1.
These WIFs may be beneficial in improving injectivity or injection rates. However, if they are not managed adequately, it could be of great disadvantage (e.g., prematured water breakthrough in producers, steep rise in oilfield overall watercut profiles, unpredictable flood pattern, poor sweep), which would lead to undesired oilfield performance [9]. Figure 2 shows an example in a certain oilfield. The inverted ninespot pattern is adopted in this field (Fig. 2a). Influenced by WIFs in West–East direction, the performances of production wells in different positions are totally different. Quick water breakthrough, accompanied with low oil rate and high water cut, occurs in producers along the WIF direction (Fig. 2b) while longterm low production rate is sustained for producers that perpendicular to WIF (Fig. 2c). Such dynamic behaviors of WIFs need to be timely and carefully evaluated in the waterflooding program. Since it is difficult to detect the features of the reservoir and fractures underground, PressureTransient Analysis (PTA) becomes a desirable choice to evaluate well performances and reservoir characteristics (i.e., average formation permeability, wellborestorage effects, skin factor, fracture properties) on the basis of BottomHole Pressure (BHP) data in a more economical and environmentally friendly way.
Fig. 2 Well pattern and production dynamics in a certain oilfield. 
This paper aims at presenting several representative BHP responses of water injectors with WIF(s). Firstly, the dynamic evolution of WIF(s) in water injector is presented. After that, five common and typical pressure responses of water injectors in Huaqing reservoir, Changqing Oilfield, are interpreted by combining a commercial simulator (KAPPA Workstation 5.12) and models we proposed [10, 11] together to give us a new and comprehensive understanding of the BHP behaviors in water injectors.
2 The dynamic evolution of WIF(s) in water injector
We divide the whole water injection process into four stages as following (Fig. 3).
Fig. 3 Schematic of dynamic evolution of WIF in water injector. 
2.1 Reservoir injection
Generally, there are lots of closed minifractures under reservoir condition in tight formation. Due to the limited absorption capacity, the BHP of water injector increases rapidly as the process of water injection. When the formation pressure exceeds fracture opening pressure, the original closed minifractures begin to expand (Fig. 3a). The opening of minifractures increases the effective permeability of inner region. If the formation pressure exceeds fracture propagation pressure, the expanded minifractures extend continuously and communicate with each other and finally, a connected long WIF forms near the water injector (Fig. 3b).
2.2 Fractured injection
As the continuity of water injection, if the water injection rate into the fracture exceeds the rate flow from fracture to the matrix, the pressure in the fracture increases and then the fracture would continue to grow until the injection and leakoff rate balance again [12]. According to monitoring and analysis results in oilfields, two scenarios are given:
Single WIF: Generally, the WIF seems to follow the maximum stress direction [13, 14], which can be easily understood because the stress perpendicular to the fracture walls for maximum principal stress direction is least. This long fracture continues to grow as the process of water injection, which can even reach several times of well spacing. The waterfloods in Western Canada [12] and Ordos basin in China validate this phenomena. This dynamic process is sketched in Figures 3cI and 3dI.
Multiple WIFs: The secondary fractures may also initiate when the pressure in the fracture exceeds the stress perpendicular to fracture walls in specific direction. As the continuously process of water injection, both main fracture and secondary fractures continue to propagate, which shown in Figures 3cII and 3dII.
2.3 Fracture falloff
Different with the propped (static) hydraulic fractures around producers, the WIF(s) around water injectors would close gradually after shutting in the well because there are no proppants inside (full of water). Similarly, two scenarios are shown for single WIF and multiple WIFs, respectively:
Single WIF: One may easily imagine that the fracture width reduces as fracture closing, which results in the DFC effect. In the meantime, experiments from van Dam et al. [15] showed that the fracture surfaces make mechanical contact at the tip first, and then subsequently towards the wellbore, which is called Shrinking Fracture Length (SFL) in this study. This dynamic process is shown in Figures 3eI and 3fI.
Multiple WIFs: As for multiple fractures, due to the different stress conditions in the WIFs, they may not close at the same time, but successively. This can be explained as the equation (1): the closure/opening pressure (p_{ki}) of WIFs in a specific direction, which depends on the maximum principal stress, σ_{max}, minimum principal stress, σ_{min} and included angle between the fracture direction and maximum principal stress, θ. Obviously, the closure pressure (p_{ki}) increases as the increase of included angel (θ), and equal to σ_{min} and σ_{max} in maximum and minimum principal stress directions, respectively. Thus, the WIF with larger included angle closes first, and lastly to the maximum principal direction. This dynamic process is shown in Figures 3eII and 3fII.
2.4 Reservoir falloff
At this stage, the WIFs have already closed completely, which means that the pressure drop of the injector is reservoir dominated. However, one should note that the permeability of closed fracture is still larger than matrix permeability due to the fracture roughness, especially for tight reservoirs.
3 Field cases
Huaqing reservoir is located in the central south of the Yishan slope in the Ordos Basin (Fig. 4). The Chang6 formation is the most developed and best oilbearing in this field, which is dominated by sandy debris flow and turbidity deposits. With fine rock particle size, high clay content, and complex rock composition, Huaqing reservoir is characterized as strong heterogeneity, low porosity and low permeability. The porosity for Chang6 formation in this area ranges from 6% to 16% while permeability ranges from 0.04 mD to 0.6 mD. Based on the statistics data of cores, thin section observation, and well logging with imaging, we find that minifractures are well developed in this field, which largely increases the difficulty of waterflooding management. The average interpreted permeability for water injectors and producers in this field are 2.7 mD and 0.3 mD, respectively, which reflects that longtime water injection induces formation fracturing. In this part, five representative wells that influenced by induced fractures in this field are selected (see Tab. 1). Among all the 329 water injectors we analyzed, the numbers of pressure responses that behave like Well A, B, C, D, and E are 94, 19, 43, 31, and 11. Aiming at these five cases, we try to analyze them by combining a commercial simulator and our models, and provide us a novel understanding of the performance of water injectors in tight oil reservoirs.
Summary of field cases.
3.1 Case #1: Well A
For PTA in well injectors, Morse and Ott [17] developed an unsteadystate pressure analysis method to locate the position of outer radius of water bank. Kazemi et al. [18] proposed an idealized illustration of two or three zones in liquidfilled reservoir and described the mathematical model in detail. Abbaszadeh and Kamal [19] improved previous models by taking the variation of fluid mobility and diffusivity in the reservoir into consideration. Then, Kong and Lu [20] presented an interpretation method for water injector and developed a solving process for twobank systems that possessing sharp change in interface. The pressure and its derivative curves are plotted in log–log coordinate, which has been widely used nowadays.
Basic data of water injector Well A is shown in Table 2 and the corresponding BHP responses in 2006 and 2008 are presented in Figure 5, which behave as composite radial reservoir. We assume that the formation is homogeneous and isotropic with constant thickness, and the interface location of the twobank system is immobile during the fieldtesting period. In addition, thermal and gravitational effects are ignored. The commercial simulator (KAPPA Workstation 5.12) is adopted to match these BHP responses. The matching and interpretation results are shown in Figure 5 and Table 3, respectively. Traditionally, the composite radial model in water injector is explained as the fluid mobility contrast in each bank [19]. Based on the discussion of “The Dynamic Evolution of WIF(s) in Water Injector”, we think this mobility difference can be reflected in two aspects:
Water viscosity is generally smaller than oil under reservoir condition.
Permeability in the inner region is larger than outer region because of the long time water erosion and the reopening of original closed fractures (see Fig. 3a).
Fig. 5 Pressure matching results of Well A in (a) 2006 and (b) 2008. 
Basic data of Well A.
Well testing interpretation results for Well A in 2006 and 2008.
The increasing of inner radius from 42.9 m in 2006 to 88 m in 2008 indicates the moving of waterflooding front. At the same time, the increase of permeability from 1.06 mD to 4.48 mD illustrates that the reactivation of original closed fractures improves the effective permeability in the inner region.
Since the fieldtesting pressure of Well A after 2008 is unavailable, we would like to analyze the performance of this well from dynamic production data to verify the variation of flow pattern during water injection. There are six producers named B1, B2, B3, B4, B5, and B6 around injector Well A (Fig. 6). The production histories of Well B1, B4, and B6 are given in Figure 7. It is shown that the water cut of Well B1 surged from 14.18% to 53.44% in December 2009 (Fig. 7a) while the water cut of Well B4 soared from 0% to 92.67% in October 2009 (Fig. 7b). The unexpected and abrupt water breakthrough in B1 and B4 suggests that the WIF may be induced in B1–A–B4 direction, which is line with the maximum principle stress direction. The performances of the producers in the two sides of B1–A–B4 are quite different with Well B1 and B4. Taking B6 as a representative, the production history is plotted in Figure 7c, which shows stable oil production and water cut. The distinct behaviors between these wells confirm the existence of WIF in B1–A–B4 direction again. These performances validate the evolution of WIF during water injection process in Figure 3.
Fig. 6 Well pattern of A. 
Fig. 7 Production history curves. (a) B1, (b) B4, and (c) B6. 
One should note that well testing analysis is an inverse question, which suffers from nonuniqueness. This nonuniqueness may come from the reservoir uncertainties itself, pressure behavior matching, or the multiple solutions of mathematical models. In this case, one of the disputes may be the model selected for Figure 5b. Before applying radial composite model, the vertical well + one fault and fractureinfinite conductivity models are tried, but unreasonable results are obtained. Therefore, we still use radial composite model to interpret it. In the actual field, different methods are strongly suggested together to lower the uncertainties and increase the confidence of the results.
3.2 Case #2: Well B
There are growing evidences, including step rate test, production history of water injector and producer, well testing analysis, interconnectivity analysis, indicate the initiation of WIF in water injection process [21]. The fundamental differences between propped hydraulic fractures around producer and WIFs around water injector are summarized in Table 4. Recently, we proposed a new “Fracture – SFL & DFC” model to characterize the BHP behaviors of water injector with single long WIF considering the fracture properties variation in tight reservoir [10]. The flow between WIF and reservoir is characterized by fracturestorage effect and fractureface skin while the flow between wellbore and WIF is represented by wellbore storage effect and chokedfracture skin. Compared to the current models [22–27], the novelty of this model is that we take the dynamic variation effects of WIF (e.g., fracture storage effect, SFL, and DFC) after shutting in the well into consideration, which makes our methodology more practical and reliable. A brief introduction of this model is presented in Appendix A.
Case #2 examines a field case to demonstrate the characteristic of bistorage (wellbore + variable fracture storage) effect. The basic parameters for Well B is provided in Table 5 and its BHP response is shown in Figure 8. The bistorage phenomenon, which characterized by two unit slopes on the pressure derivative curve, dominates the whole falloff period and masks the reservoir response. Similar pressure responses have also been found in the Arabian Peninsula [29]. Using our “Fracture – SFL & DFC” model, we can obtain the matching result of this well, shown in Figure 8.
Fig. 8 History matching result of Well B by use of the model we proposed. 
Basic data of Well B.
The interpreted effective permeability is 2.91 mD and the calculated storage coefficient from the long unitslope line in the pressure derivative curve is considerably large as 8.92 m^{3}/MPa. The significant large storage coefficient suggests that the WIF(s) that extend to some distances in the reservoir. After shutting the fractured well, the water that stored in the fracture is unloaded and squeezed into reservoir, which is similar to the phenomenon of wellbore storage effect. We call it as fracture storage effect in this paper. The relationship between fracture storage coefficient and fracture halflength is comprehensively discussed based on three 2D models [10], which are,
where C_{f} is fracture storage coefficient, x_{f} is fracture halflength, h is fracture height, E is the planestrain modulus, R_{F} is the fracture radius of a radial fracture. PKN model is assumed by Perkins and Kern [30] and Nordgren [31], which denotes the fracture length is larger than the fracture height while KGD is abbreviation of Khristianovich et al. [32] and Geertsma and de Klerk [33], which assumes that fracture height is more than the fracture length. Radial model occurs for fracture propagation from a point source of injection, as an intermediate condition between the two limiting cases of the PKN and KGD models.
From equation (2), we find that the fracture storage coefficient decreases with the SFL. Therefore, a slightly larger than unit slope in the fracture storage is identified, which is a new flow regime – variable fracture storage  in water injectors. In this field, the planestrain modulus is approximately 10 000 MPa, reservoir thickness is 23.9 m. Therefore, the fracture halflength is estimated as 49.7 m by PKN model.
3.3 Case #3: Well C
This case examines a field case of Well C that seems quite common, but extremely easy to be misinterpreted if the WIF is not well realized or understood. The basic information of this well is given in Table 6 and its BHP behavior is shown in Figure 9. The pressure and its derivative curves coincide as unit slope and last about three log cycles, which signifies the severe storage effect. Due to this long storage period, the true reservoir response is masked completely. A commercial simulator (KAPPA Workstation 5.12) is used to match this pressure behavior (see Fig. 9). The interpreted reservoir permeability is 0.38 mD and the matched storage coefficient from this long unitslope line in the pressure and its derivative curves is considerably large as 22.7 m^{3}/MPa. Similarly, we use the method in Case #2, the fracture halflength can also be estimated as 121.4 m.
Fig. 9 History matching result of Well C by use of the Saphir. 
Basic data of Well C.
Different with bistorage effect in Case #2, the wellbore and fracture storage curve merge into single long unit slope in this case. This is because there is no extra pressure drop between wellbore and WIF, that is, the wellbore and WIF can be considered as the whole system. Since the WIF is not hold by proppant, but full of water, it would close gradually with the decreasing of BHP after shutting in the well. As a result, the water that stored in the WIF is squeezed into the formation, behaving like storage effect. If there is no pressure drop between the wellbore and WIF, it had been validated that the total storage coefficient is attributed by four parts: compressibility of water in the wellbore, change rate of wellbore volume, compressibility of water in the fracture, and change rate of fracture volume. Generally, the change rate of wellbore volume and compressibility of water in the fracture are negligible compared to another two terms [10]. Thus, the interpreted storage coefficient of 22.7 m^{3}/MPa is actually the total storage coefficient, including wellbore and fracture. That is [10, 34],
where C_{s} is total storage coefficient, C_{w} is wellbore storage coefficient, C_{f} is fracture storage coefficient, c_{fluid} is the fluid (water) compressibility, V_{w} is wellbore volume.
In most cases, fracture storage coefficient is several orders of magnitude larger than wellbore storage [9]. Therefore, the wellbore storage coefficient can be ignored when calculating the fracture halflength by total storage coefficient.
3.4 Case #4: Well D
This case shows another “common and conventional” water injector Well D and provide us a new understanding on it. The basic information of this well is presented in Table 7 and the BHP response is shown in Figure 10. We find that the pressure and its derivative curve coincide in the early time and then separate, followed by the upward buckling of pressure derivative curve in the late time, which is generally considered as the existence of closed boundary near the well. This closed might be caused by sedimentary facies, lithology change, the drainage areas of several other wells, and so on. Meanwhile, we found that the interpreted storage coefficient is very large, which is similar with Case #3. One of the biggest differences of WIF and propped hydraulic fracture is that it closes gradually as the decrease of BHP during the fieldtesting period. As a result, the fracture conductivity is decreasing during the fracture closing process, which has a big influence on BHP response. In our previous study [10], we have validated that the DFC effect would lead to the upward buckling of pressure derivative curve in the late time. Fracture linear and bilinear flow characteristics corresponding to 0.5 and 0.25 slopes for different conductivities may not be observed in water injectors with WIF because of the change of fracture geometry, which would not occur in producers with propped hydraulic fractures during well testing period. Using “Fracture – SFL & DFC” model, we interpreted the Well D, shown in Figure 10. The results are given in Table 8.
Fig. 10 History matching result of Well D by use of the model we proposed. 
Basic data of Well D.
Well testing interpretation results for Well D.
The interpreted permeability is 0.37 mD, suggesting the poor flow condition around this well. The fractureface skin is 1. This means that there exists pressure drop in the fracture face, which may be the reason of “filter cake”, caused by long time water injection. As for fracture halflength, we recommend combine two methods to decrease the uncertainty of the analysis and double check the results. One is the method presented in Case 2 by the matched considerably large storage coefficient value. The other one is conventional well testing analysis, that is, matching the typical curve with measured pressure data. In theory, the fracture halflength values estimated by these two methods should be very close to each. However, due to the uncertainty of formation parameters and the nonuniqueness results of pressuretransient analysis, neither of these methods are that reliable for the field cases. For Well D, the matched storage coefficient is 43.3 m^{3}/MPa, which should not be wellbore storage coefficient only that presented in Case #3. Anand and Subrahmanyam [9] demonstrated that the fracture storage coefficient ≫ wellbore storage coefficient. Therefore, C_{f} ≈ 43.3 m^{3}/MPa in this case, which is used to calculate fracture halflength as 73.5 m via equation (2). This is consistent with the fracture halflength (90 m) matched by “Fracture – SFL & DFC” model.
The meaning of this example is that it reminds us the upward buckling of pressure derivative of water injector in the late time may not necessarily related to closed boundary, but the decreasing of fracture conductivity caused by fracture closing process.
3.5 Case #5: Well E
Some evidences show that waterflooding may also induce multiple fractures during long time water injection [35]. Compared with single fracture closure, the characterization of the closure of multiple fractures is much more complicated. According to equation (1), we conclude that the fracture open/closure pressure varies in different directions, which increases with the increase of θ. Specially, fracture closure pressure is equal to σ_{min} and σ_{max} in maximum and minimum principal stress directions, respectively. Because the different stress condition perpendicular to the fracture walls, the WIFs would not close simultaneously, but successively. Assuming the same pressure system in the fractures, Liu and Economides [36, 37] proposed a mathematical model to describe the multiple closures in MiniFrac analysis based on the classical Carter’s model, which indicates that the rate of fluid loss decreases with time at any point along the fracture. Leakoff coefficient is used to characterize the fluid loss ability from fracture into matrix and the rate of fluid flow behavior can be represented as , among which C_{L} is leakoff coefficient, q_{L} is fluidloss rate, A is the element of the fracture area, t is the time measured from the start of pumping, τ is the time when fracture is created or opened. He assumed that each fracture (main and secondary fractures) has own leakoff coefficient and area, which makes his model very limited due to too many unknown parameters need to be solved. Applying Liu’s research in MiniFrac analysis to waterflooding field, we propose a new “Multiple Fracture Closures” model, in which the closure of WIFs are equivalent to the decrease of leakoff rate (the rate at which a fracturing fluid flows from the fracture into the surrounding formation) in the main fracture (Appendix B) [11]. We find that 3/2 slope recognized by Mohamed et al. [38] in MiniFrac analysis would not appear in waterflooding, but unit slope instead, which can be interpreted as fracture storage effect.
The basic information and the BHP responses of Well E are shown in Table 9 and Figure 11, respectively. Several characteristics are identified.
The pressure drop slows down repeatedly and the pressure derivative shows multiple fluctuations.
Several unit slopes in the pressure derivative are identified.
Storage coefficient interpreted by the commercial simulator is considerably large as 7.93 m^{3}/MPa, and the interpreted permeability is 3.14 mD.
Fig. 11 Comparison of the matching results of Well E between (a) Saphir and (b) Model we proposed. 
Fig. A1 Schematic of relationship between wellbore, WIF and reservoir. 
Basic data of Well E.
Firstly, a commercial simulator (KAPPA Workstation 5.12) is used to interpret this pressure behavior and the matching result is shown in Figure 11a. We find that the fluctuations of pressure derivative curve cannot match by current models. In our previous paper [11], we have comprehensively discussed that such pressure characteristics shown above may be cause by multiple fracture closures during the shutting in period. In this case, five fractures are assumed since there are five unit slopes in the pressure derivative curve. Interpreted by our “Multiple Fracture Closures” model, we match the pressure behavior, which is shown in Figure 11b. The interpretation results of leakoff coefficient for each fracture are C_{Lm1} = 0.4 m/h^{0.5}, C_{Lm2} = 0.3 m/h^{0.5}, C_{Lm3} = 0.16 m/h^{0.5}, C_{Lm4} = 0.055 m/h^{0.5}, C_{Lm5} = 0.015 m/h^{0.5}. By the equation [34], we can obtain the reservoir permeability as 0.16 mD, which is consistent with the formation permeability in this field.
4 Conclusion
This work intends to show a comprehensive analysis of pressure responses in water injection wells considering the influence of WIF(s). After interpreting several hundreds of water injection wells in Changqing Oilfield, China, we find that the BHP performances of water injectors with WIF(s) are much more complicated compared with fractured producers due to the variation of fracture properties. Aiming at five representative types of BHP responses, we innovatively analyze them by combining a commercial simulator (KAPPA Workstation 5.12) and our new models to get a new understanding of water injector behaviors. Results show that the closing of WIF(s) during the field testing period should be paid particular attentions when interpreting the BHP behaviors. If they are not well realized or characterized, misleading interpretation results may be obtained.
Appendix A
Derivation of the closure of single WIF considering properties variations
We assume that the reservoir is isothermal, homogenous, isotropic and of uniform thickness and porosity. Sing water phase flow in the WIF and reservoir. The dimensionless parameters are defined.
Dimensionless pressure,
Dimensionless pressure,
Dimensionless wellbore storage coefficient and fracture storage coefficient,
Dimensionless fracture conductivity in the shutin moment,
Diffusivity ratio,
Dimensionless stresssensitivity coefficient,
Compared with current methods, the key and novelty of our model is that SFL and DFC effects are considered. We divide the whole system into three sections: wellbore, WIF and reservoir. The relationship between wellbore, WIF and reservoir is shown in Figure A1. The wellbore and WIF are linked by chokedfracture skin (S_{choke}) while the WIF and reservoir are connected by fracture face skin (S_{fs}). The brief derivation of our model is presented below.
The decreasing of fracture conductivity/permeability is assumed as,
Considering the stress sensitivity in the WIF and the constant pressure out boundary condition, the BHP solution of bilinear flow can be written as [10],
where,
The reason we use is that the differential equation in the fracture region is strongnonlinear when DFC effect is considered. On the other hand, since the differential equation is nonlinear, the superposition principle is not applicable. Therefore, the Pedrosa substitution [39] is applied to eliminate this nonlinearity, which is given in equation (A11). Experiments show that the fracture shrinks from the tip first, then subsequently towards the wellbore [15], which can be expressed as [25],
where is the dimensionless pressure where lengthwise shrinkage starts, and the parameter “delpat” is fracture length shrinkage coefficient, which is used to measure how fast the WIF closes.
The dimensionless pressure solution outside of wellbore in the fracture is obtained based on Duhamel’s principle,
where,
and S_{fs} is fractureface skin.
Finally, we can obtain the BHP by taking chokedfracture skin and wellbore storage into account,
where S_{choke} is the chokedfracture skin.
Applying Stehfest inversion [40], the can be translated into in time domain. Then, the inversion of Pedrosa substitution [39] is applied, the BHP response in water injection well with WIF can be achieved.
Appendix B
Derivation of multiple closures of WIFs modelling
We use the models presented by [41, 42], Liu and Economides [36, 37] as a starting point. MiniFrac analysis and material balance function are combined to derive the BHP response of water injector with multiple WIFs. We assume the main and secondary fractures share the same pressure system. The abbreviated derivation of the model is developed below.
The relationship between fracture width and net pressure is defined as,
where c_{f} is fracture compliance, m/MPa; is average fracture width, m; p_{net} is the net pressure on fracture surface, which can be calculated as,
and σ_{resistant} is the confining stress on the fracture face, MPa.
Average fracture width is,
For the falloff analysis, we define Δp_{w}(t) as,
where p_{ws} is the BHP in the shutin moment, MPa; p_{w} is BHP, MPa.
Combining equations (B1)–(B4) yields,
If a main fracture and three secondary fractures are assumed and each fracture has its flow coefficient and fracture area. The pressure drop in these fractures is ignored. According to material balance function, we have,
The flow volume from main fracture and secondary fractures to matrix is [36],
This g function is defined by [41] and later reformulated by [43]. It gives the ratio of the flow rate from fracture into the matrix to the quantity 4C_{L}A_{f}t_{p}^{0.5}, but now is valid for all times greater than the end of pumping while g_{0} gave the ratio only at one time point, namely exactly at the end of pumping (Valko and Economides, 1999),
where α is area exponent, which ranges from 1/2 to 1.
Combining equations (B1) through (B5) and integrating, yields,
where,
Even though above model is more accurate, which investigates the properties of main fracture and secondary fracture separately. It is very limited because there are too many unknown parameters need to be determined. Thus, a simplified model is proposed based on Liu and Economides [36]. The closure of fractures is considered as the decreasing of leakoff coefficient in the main fracture. Then, we can get
where C_{Lm1}, C_{Lm2}, C_{Lm3}, and C_{Lm4} are the equivalent leakoff coefficient in the fractured falloff period, m/h^{0.5}.
Acknowledgments
The authors thank the financial supports from National Natural Science Foundation of China (No. 11872073) and National Science and Technology Major Projects (No. 2017ZX05013002). We also would like to thank the PetroChina Ltd. in Changqing Oilfield for providing data for our studies. An earlier version of this work was presented as a conference paper at the SPE Improved Oil Recovery Conference held in Tulsa, Oklahoma, USA as SPE 190264 and the feedback obtained from colleagues in this forum is greatly appreciated.
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All Tables
All Figures
Fig. 1 Sideview for all tiltmeter mapped fractures in the waterflood project in West Texas [8]. 

In the text 
Fig. 2 Well pattern and production dynamics in a certain oilfield. 

In the text 
Fig. 3 Schematic of dynamic evolution of WIF in water injector. 

In the text 
Fig. 4 Location and basic parameters of Huaqing Oilfield (modified from Su et al. [16]). 

In the text 
Fig. 5 Pressure matching results of Well A in (a) 2006 and (b) 2008. 

In the text 
Fig. 6 Well pattern of A. 

In the text 
Fig. 7 Production history curves. (a) B1, (b) B4, and (c) B6. 

In the text 
Fig. 8 History matching result of Well B by use of the model we proposed. 

In the text 
Fig. 9 History matching result of Well C by use of the Saphir. 

In the text 
Fig. 10 History matching result of Well D by use of the model we proposed. 

In the text 
Fig. 11 Comparison of the matching results of Well E between (a) Saphir and (b) Model we proposed. 

In the text 
Fig. A1 Schematic of relationship between wellbore, WIF and reservoir. 

In the text 