SPE-187191-MS New Model for DFIT Fracture Injection and Falloff Pressure Match G. Liu and C. Ehlig-Economides, University of Houston Copyright 2017, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 9-11 October 2017. 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 Analysts regard the fracture injection and falloff test known as the diagnostic fracture injection test (DFIT) as a reliable tool to quantify formation closure stress, leakoff coefficient, formation permeability, and pressure. The recent analytical DFIT model for before and after closure enables matching the pressure falloff for abnormal leakoff behaviors, and quantification of many more formation parameters than traditional DFIT models. However, the model design addressed only the falloff data after shut-in, and many express concerns that net pressure implied by the falloff is inconsistent with the injection pressure behavior. This paper provides a model capable of matching both injection and falloff pressure behavior. The pressure falloff model is capable of quantifying essential pressure values including, in order of occurrence, instantaneous shut-in pressure (ISIP), minimum fracture propagation pressure, one or more closure stress values, minimum stress, and pore pressure. The early pressure response represents the dissipation of three kinds of friction, wellbore, perforation, and near-wellbore friction, each of which are quantified, and which together comprise the difference between the pressure at the end of injection and the ISIP. Presence of tip extension enables quantification of the minimum fracture propagation pressure. The minimum stress is consistent with the final closure stress. Subtracting the closure pressure and friction pressure losses from the recorded or calculated bottomhole pressure provides the fracture net pressure. The model match for injection pressure behavior incorporates the same pressures and consistent values for 2D fracture geometry and leakoff coefficient. The global match confirms not only the estimation of formation and fracture properties from the pressure falloff analysis, but also the pressure distribution along the wellbore, through the perforations and the near wellbore, and in the fracture during and after injection. In particular, by matching both injection and falloff, the model incorporates friction pressure losses that can explain apparent excessive net pressure. The match with both injection and falloff pressure variation addresses concerns that the net pressure implied by the falloff model match cannot be consistent with observed injection behavior. When the pressure difference between the final DFIT pick for closure stress and the pressure at the end of injection is very large, the reason may be large friction pressure losses and/or tip extension. Design for the main fracture treatments could consider an altered perforation strategy to reduce friction losses and/or reduce cluster spacing to address very low effective permeability likely consistent with tip extension and confirmed by the after closure match. 2 SPE-187191-MS Introduction For a DFIT test, a relatively small amount of fluid is pumped into formation to create a fracture in the target layer, and then the well is shut in to allow the injected fluid leak off into formation (Feng and Gray 2016; Feng et al. 2015). Liu and Ehlig-Economides 2015, 2016 described available models for DFIT analysis. The Liu and Ehlig-Economides 2015, 2016 model is designed to match the entire falloff, including such abnormal leakoff behaviors as perforation and near wellbore friction, tip extension, pressure dependent leakoff area, height recession, variable compliance, and multiple closure events. The focus of this paper is matching the pressure injection during pumping with the model that matches the falloff behavior. Three two-dimensional (2D) fracture models still widely used in hydraulic fracturing analysis and design are known as PKN (Perkins and Kern 1961; Nordgren 1972), KGD (Khristianovic and Zheltov 1955; Geertsma and de Klerk 1969) and radial (Geertsma and de Klerk 1969). All of these models assume that the formation rock is continuous and homogeneous, and that it follows the linear elastic relationship between stress (net pressure) and strain (half of fracture width). Fracture width is a key parameter because it is the bridge connecting pressure change and material balance. Mack and Warpinski 2000 explained that net pressure at the tip of all three models is assumed to be zero implies that the fracture shape is elliptical during propagation. Therefore, fracture width varies with a maximum value at wellbore and zero at tip. For convenience, the average width is used for material balance calculations. The geometric factor, γ is introduced to denote the relationship between the average ( ) and the maximum (wmax) fracture width for three fundamental models, and given as, (1) The geometric factor, γ, equals to 0.628 for PKN model, 1 for KGD and 0.8 for radial fracture model. Besides the nonuniform pressure distribution along fracture, all three models assume constant rate injection. However, field data frequently show that the surface injection rate is not constant during the test. Some operators deliberately vary injection rate before the end of pumping with a step rate test (SRT) also known as a step-down test, designed to measure friction pressure loss under different injection rates (Barree et al. 2014, Naidu et al. 2015). The commonly used 2D fracture models also assume that fluid leakoff is negligible during pumping. The PKN model in Nordgren 1972 obtains analytical solutions for two limiting cases: one with large leakoff rate and the other with zero. Typically, the analytical PKN solution with zero leakoff is preferred, especially in the tight formations. For the other two models, KGD and radial fracture model, fluid leakoff is neglected. At the end, the width change, or correspondingly, the fracture propagation rate, can be expressed in a simple power law function. (2) where Af is fracture one-side surface area at any time t during injection; Af0 is the fracture one-side surface area at shut-in, or at t = tp, and tp is the injection time; α is the area exponent. According to Nolte et al. 1993, the area exponent is close to 1 for low leakoff, and ½ for high leakoff. In particular, for the Newtonian fluid with no leakoff, area exponents for PKN, KGD and radial fracture model are 4/5, 2/3 and 8/9, respectively. In this paper, we construct a consistent model that can match the injection profile using the parameters quantified from the falloff data. We identify inconsistencies between injection and falloff modelling and explain how to resolve them. Finally, the new model is used in two field cases to demonstrate that one model is capable to match both injection and falloff data. SPE-187191-MS 3 Inconsistencies between Injection and Falloff Modelling As two successive processes, the injection and falloff should be consistent in following parameters: fracture geometry at shut-in, formation leakoff coefficient, natural fracture surface area if its effect is present, closure stress and rate dependent friction. These parameters are quantified via falloff analysis, and should be used as input parameters in the model to fit the injection pressure. Although the injection and falloff occur in the same fracture and formation setting, the pressure response and fracture behavior during these two periods may not be exactly same. In this discussion, we are to investigate the gap between pressure falloff modelling and the fracturing propagation modelling, and then try to build a more general model working for both. Wellbore storage and variable bottomhole injection rate In modelling fracture initiation and propagation, the injection rate refers to bottomhole (BH) rate into the created fracture, not the recorded rate at wellhead. Estimation of the bottomhole rate requires consideration of the wellbore storage effect. The volume change of fluid storage in the wellbore (ΔVfiuid) with pressure can be written as, (3) where cw is water compressibility, and Vw is the total wellbore volume; Δpw(t) is the treating pressure change. As suggested by Barree et al. 2014, besides fluid volume expansion, the expansion of the injection string should also be considered, and its diameter change with pressure is described as, (4) where Δp is the internal pressure change; d is the string diameter and Δd is its change; δ is the string thickness, and E is the steel modulus. Then, the volume change with pressure due to string expansion, ΔVstring, can be derived as, (5) The total volume change due to WBS (ΔVWBS) then can be determined as, (6) For a casing with outer diameter (OD) between 4.5 inches to 7 inches, the ratio between ( ) to cw lies between 11% to 28%, which means the string volume change is relatively small comparing with fluid volume change due to the large magnitude of the steel modulus, E, which is around 30 ×106 psi. These two factors are combined in Eq. (7) as one with one single effective WBS compressibility (cWBS) that is 11% to 28% larger than the water compressibility for casings with OD ranging from 4.5 to 7 inches. (7) The bottomhole (BH) rate is calculated by considering the compressibility of fluid and injection string, (8) where qsurf,i and qBH,i are surface and bottomhole rate during a time interval, Δti; pw(ti) is the treating pressure, and pw,start is the pressure level at the start of injection, or pw,start = Pw(t = 0). 4 SPE-187191-MS If there is no pre-existing fracture, initially all injected fluid is pressurizing the wellbore with little flowing out of wellbore until the formation breaks down. During this period, the pressure rises linearly with the injection volume, and time when the injection rate remains constant. This feature can be used to estimate the wellbore compressibility defined in Eq. (7). Frequently DFIT injection has a varying injection rate. Some operators include a step rate test (SRT), also called a step-down test, to estimate a relationship between friction pressure loss and injection rate. In such cases, the WBS effect distorts changes in bottomhole injection rate compared to step rate changes at the surface. There are two possible ways to determine the friction-rate relationship from an SRT: either maintain each rate step for a relatively long time to see a flat pressure level so that the bottomhole rate will be similar to the surface, or correct the bottomhole rate with Eq. (8) so that friction loss can be estimated using the corrected bottomhole rate values. Another impact of WBS on pressure response is after shut-in. The bottomhole pressure (BHP) after surface shut-in does not drop to zero immediately, some stored fluid continues flowing into the fracture with the declining pressure. Liu and Ehlig-Economides 2016 modelled this behavior by coupling WBS with friction dissipation. Two important parameters for the DFIT analysis are injection time and total injected volume. Injection volume is the basis for the material balance, which must be strictly obeyed for DFIT analysis. Variation in the injection rate has little explicit impact on the pressure falloff behavior, and it usually assumed to be constant. Although the injection rate is a trivial factor for falloff analysis, it has significant impact on the pressure response during pumping. Firstly, friction losses are rate dependent, and at high rate friction losses often account for much of the total pressure increases during pumping. As a result, pressure and rate behavior is highly correlated. Secondly, increasing the pumping rate increases the bottomhole rate into the created fracture, which increases the net pressure in the fracture and should accelerate fracture propagation. However, the commonly used fracture propagation model described in Eq. (2), which is derived by assuming constant injection rate, implies that the fracture propagation rate is independent of injection rate and only dependent on time. To honor the rate effect and material balance, we modify Eq. (2) by replacing time on the right-hand side with the injection volume, giving, (9) This equation uses the bottomhole rate to calculate the effective injection volume into the fracture, and VBH,inj is the total volume flowing into fracture before shut-in, which can be calculated as, (10) where, Vsurf,inj is the total volume injected at surface, and pws is the treating pressure at shut-in, which is not equal to the instantaneous shut-in pressure (ISIP). The relation between the pressure before the start of injection and the shut-in pressure depends on operating procedures. For example, after plug and perf, the wellbore pressure may be low before the start of injection. In such cases, there may be a sharp linear rise in pressure associated with pressurizing the wellbore, and the total fluid volume into fracture at shut- in is less than the surface injection volume because part of it is used to pressurize the wellbore. In contrast, we have seen field data in which the hydraulic pressure required to open a sleeve exceeded the shut-in pressure, so the total fluid flowing into fracture at shut-in may be larger than the surface designed volume due to fluid expansion. SPE-187191-MS 5 Fracture Internal Pressure Distribution The DFIT model commonly assumes that the fracture internal pressure is uniform. This has two implications. First, this assumption implies that there is no friction along fracture, and that the fracture width is also uniform, which further implies that the geometric factor in Eq. (1) is 1 for all mentioned 2D fracture models. Secondly, it implies that the net pressure is uniform and equals to the product of the fracture compliance and the fracture maximum width, which is equal to the average width. The assumption is acceptable for falloff analysis because rate dependent fracture internal friction is negligible for rate near zero. However, the assumption of uniform fracture internal pressure is incorrect for fracture propagation models, in which net pressure is assumed to be maximum close to the wellbore and zero at tip. The error in assuming uniform fracture internal pressure or neglecting internal fracture friction depends on the bottomhole flow rate into fracture. If the rate is maintained in a high level, flow friction magnifies the internal fracture pressure gradient and the net pressure estimate from the maximum fracture width close to wellbore will be greater than the pressure at any other location in the fracture. In contrast, for small bottomhole injection rate or after-flow rate, the internal fracture pressure gradient is small and there is less error in assuming uniform net pressure in the fracture. Thus, the geometry factor, γ, in Eq. (1) is not constant, but should be a function of injection rate. To address internal fracture friction, we assume γ to correspond to the known reference values for different 2D fracture models when the bottomhole rate equals to the average bottomhole injection rate, and allow it to linearly increase to 1 when the rate drops to 0. The newly constructed geometric factor function, given in Eq. (11), can be applied in both injection with variable bottomhole flow rate, and falloff with dissipating after flow rate. (11) where, γ0 is the reference value for 2D fracture model, equal, as before, to 0.628 for PKN, 1 for KGD and 0.8 for radial fracture model; q is bottomhole flow rate, and qBH,inj is the average bottomhole injection rate. Leakoff during Fracture Propagation As discussed in the Introduction, leakoff is neglected in the three commonly used 2D fracture propagation models. Accordingly, the area exponent in Eq. (2) can be set to a value close to 1. However, this assumption is incomplete and questionable, even for tests in tight formations. It has been well known that the natural fractures in tight formations are likely reactivated during pumping, and a large part of injected fluid volume flows into these natural tissues and leakoff through their surface into matrix. Because the total leakoff area could be much larger than the hydraulic fracture, the leakoff rate may not be negligible, so Eq. (2) or (9) should be further accommodated by considering the possible fluid leakoff loss during injection. According to Nolte et al. 1993, the leakoff loss, or fluid efficiency, is represented by the area exponent, a, in Eq. (2), and its values are anchored to 4/5, 2/3 and 8/9 for PKN, KGD and radial fracture model by assuming zero leakoff. Depending on fluid efficiency, the area exponent could be ranging from ½ for high leakoff to 1 for low. Therefore, it is reasonable to vary the area exponent for each 2D fracture model shown in Eq. (2) and (9). Successfully matching injection pressure data in several field tests suggests that using the fluid efficiency at shut-in as the initial guess for the area exponent during injection results in error less than 10%. This finding is consistent with the conclusion of Nolte et al. 1993 that area exponent has positive correlation with the fluid efficiency. Case Studies In this paper, a DFIT field cases demonstrate the ability to match both injection and falloff pressure data with one consistent model. Liu and Ehlig-Economides 2015, 2016 provided before-closure model matches for this test. 6 SPE-187191-MS Well A The DFIT for Well A was performed in the toe stage of a horizontal well using a single perforation cluster. About 31.45 bbl fresh water was injected into the well in 7.13 minutes with an average rate at 4.41 bbl/min, as shown in Figure 1. It should be pointed out that the rate profile is missing, and we assume that the rate is constant during injection at the value given by the quotient of the injected volume and the injection rate. Figure 1—injection profile of Well A Fluid was pumped into the formation through casing with inner diameter (ID) of 121.4 mm (4.78 inch). The measure depth (MD) of perforation is at 4644 m (15,236 ft). From this information, the wellbore volume is estimated at about 338 bbl, which is much larger than the injected fluid volume and sufficiently large to note that the WBS effect cannot be neglected. Indeed, the WBS effect is apparent from the straight line in Fig. 15 marked by the dashed green line during the early injection. Assuming all injected volume is stored in the wellbore, Eq. (6) enables estimation of the WBS fluid and pipe compressibility defined by Eq. (7) at 2.92 × 10-6 psi, which corresponds to the water and pipe compressibility at the test temperature and pressure conditions, and the observed behavior also confirms that the flow rate is constant. Figure 2 shows the analysis presented previously by Liu and Economides 2015. The left graph in Fig. 2 is the Bourdet derivative log-log plot with markings indicating slope trends of interest, and the right shows the composite G-function plot displaying exactly the same closing trends. Briefly, the first behavior indicated by the green unit slope (marked as circle 1) in the Bourdet log-log plot is identified as WBS with friction dissipation. The next relatively flat derivative trend (circle 2) could be the tip extension after shutin coupled with near-wellbore friction dissipation. The minimum fracture propagation pressure at the end of tip-extension is 6640 psi. Then, the unit slope ending at around Δt = tp (circle 3) is identified as the first constant area closure trend. The following four 3/2-slopes (circle 4 – 7) are successive additional closure trends, giving five successive closures in total. Mohamed et al. 2011 showed a similar multiple-closure field case from a different formation. SPE-187191-MS 7 Figure 2—Bourdet derivative plot (Left) and composite G-function plot (Right) for Well A Liu and Ehlig-Economides 2016 previously quantified a different leakoff coefficient from each apparent closure trend. In our recent work which is still under peer review, we assume all leakoff trends relate to the same formation and, therefore, are governed by one constant leakoff coefficient. Different leakoff rates, indicated by unit and 3/2-slope in the Bourdet derivative or extrapolated Gdp/dG slopes in the G-function plot, are caused by different leakoff areas. The leakoff area decreases from its highest value associated with the first closure trend to the lowest with the last final closure. A possible interpretation for the closure trends is that each represents a different secondary fracture set, and a possible explanation for the distinct closing pressures may relate to the angle of the fracture planes in a given fracture set. Table 1 lists the interpretation parameter values for the final model match. Figure 3 shows the leakoff area for each closure trend associated with the pressure at the end of the constant area leakoff. By subtraction, each closure trend has an associated leakoff area. Figure 3 starts from the second closure event because the area associated with the first closure is extremely large and of very short duration. This might be related to near wellbore fracture initiations under local drilling-modified stress, and the higher leakoff rate may be partly explained by a higher leakoff coefficient through a smaller apparent leakoff area. Table 1—Interpretation result of Well A 8 SPE-187191-MS Figure 3—The decline of total (one side) fracture surface area with pressure during falloff. The apparent ½-slope (circle 8) observed after the final closure is usually interpreted as formation linear flow. From this behavior, we estimate formation permeability as 336 nano-Darcy using the method introduced by Liu et al. 2016 and estimate the formation pressure at 4,065 psi. The derivative curve seems to bend to flat at the very end (circle 9). Permeability estimated by identifying this trend as pseudo-radial flow results in an unlikely pore pressure estimate. Figures 4 and 5 show matches for both falloff and injection models achieved through slight adjustments in the previously published friction and area exponent estimates based only on the falloff data. Several interesting points merit discussions. First, the total fluid volume flowing into formation before shut-in is 25.6 bbl, which is almost 6 bbl less than the surface injected volume because of WBS. Further, about 9.6 bbl out of the injected 25.6 bbl leaks off into the formation during injection. It means that the fluid efficiency during injection is relatively low (about 0.625) even the well is drilled in shale. As mentioned above, initial injection may be initiating multiple fractures under drilling-modified stress near the wellbore that close at high leakoff rate opposite high stress at the start of the falloff. Figure 4—Model match for falloff pressure behavior of the Well A. SPE-187191-MS 9 Figure 5—Model match for injection and falloff pressure of the Well A. Figure 6 (A) presents the comparison between surface injection rate and the bottomhole flow rate at the formation face. At the very early injection time, injected fluid only pressurizes the wellbore with no fracture initiation. After breakdown, the bottomhole rate rises quickly to the wellhead rate level. The surface rate drops to zero immediately after shut-in, while the bottomhole rate continues flowing at a low rate for some time as the after-flow effect. The rate dependent geometric factor, γ, defined in Eq. (11) varies between its reference value (0.8 for radial fracture model) and 1. Figure 6 (B) shows the fracture internal pressure, bottomhole shale face pressure and its components: friction occurring in wellbore & perforation and near wellbore tortuosity, net pressure at the wellbore which is the maximum net pressure along the fracture due to fracture internal friction, and the average net pressure. The gap between net pressure at the wellbore and the average net pressure can be taken as the fracture internal friction, which is great during injection, but dissipates soon after shut-in. The fracture internal pressure is calculated by adding average net pressure to the closure stress quantified from closure stress of the last closure. It is clear that the fracture internal pressure (red line) has much less pressure drop after shut-in compared with the bottomhole pressure, so the Carter 1957 leakoff model is justified. 10 SPE-187191-MS Figure 6—(A) Surface injection rate, bottomhole flow rate and the rate dependent geometric factor of the Well A; (B) Bottomhole pressure and fracture internal pressure, friction, net pressure at wellbore and average net pressure during injection and falloff of the Well A. Discussion 1. Reduced or eliminated need for step-down test. To measure directly the relationship between flowing pressure and injection rate related functionally to friction pressure loss, some operators perform a step-down test just before stopping pumping for the DFIT injection falloff. As a consequence of wellbore storage, there is a delay between a change in the surface injection rate and the corresponding change in bottomhole flow rate. Therefore, it is essential to use the measured or calculated bottomhole rate to estimate rate-dependent friction, not the wellhead surface rate. A down side to the step-down test is that it compromises either the injection rate or the injection volume. To ensure fracture closure in a reasonable time frame, the DFIT design calls for minimizing the injection volume. If some of the volume is used to perform the step-down test, the remaining injection must be done at lower rate. This, in turn, may fail to activate main or secondary fracture behaviors that occur at higher rate. The Liu and Ehlig-Economides 2016 model for rate-dependent friction enables quantifying friction loss. When a step-down test has been performed, it should confirm the falloff model. Otherwise, the injection and falloff pressure match should eliminate the need for a step-down test. 2. Net pressure magnitude Large pressure drop observed after shut-in could be caused by friction in the wellbore, perforation, near wellbore and/or fracture internal flow friction. Usually, the net pressure in most parts of the fracture is relatively small, and sufficiently large in-situ net pressure will trigger tip extension, which serves to dissipate high net pressure. 3. Uniformity of the fracture internal pressure SPE-187191-MS 11 In tests with high closure stress and relatively small net pressure, the fracture internal pressure variation can be viewed as negligible or constant, and the Carter 1957 leakoff model remains valid. Hence, while it may be useful to considering modeling leakoff as flow into a permeable formation, there is no requirement to do so. 4. Implication of falloff behaviors on the injection pressure response In this paper, all leakoff behaviors are quantified from the falloff analysis. Their behaviors are simplified when they are used as input data in the injection modelling. For instance, the multiple closure case shown in Well A can be explained as four sets of secondary fracture with different directions, closure stresses and areas. Secondary fracture sets were opened and extended during pumping by a combination of shear and tensile behaviors. A simplifying assumption is that they propagate essentially in the same manner together with the hydraulic fracture. This assumption facilities leakoff volume computation and fracture area estimation, and results in pressure calculated at a stable level during injection, which fits the field data shown in this paper very well. However, if these fractures are not propagating in the same manner, the observed treating pressure may not remain at a constant level. Alternatively, if pressure during injection at constant rate shows an uneven pattern, this suggests abnormal behavior that should also be observed in the falloff pressure response. We anticipate that the matching both injection and falloff data will improve the model reliability and provide additional insights valuable for fracture design. 5. Meaning of the minimum propagation pressure at the end of tip extension after shut-in. The minimum propagation pressure picked at the end of tip extension after shut-in represents the minimum pressure to create a static fracture in the formation. Fracture propagation will stop when pressure drops below this pressure. The minimum propagation pressure should be less than the stable injection pressure during injection even after friction is deducted because extra momentum or pressure is required to maintain fracture growth at a high rate. Conclusions The paper shows a model capable of matching both injection and falloff data from a DFIT test. Understanding the inconsistency between classic 2D fracture models and models used for the falloff analysis reveals the way to construct a consistent model that is based on the falloff analysis and able to match the entire injection and falloff pressure behavior. Following are key conclusions: 1. The bottomhole rate must be calculated in the injection and falloff analysis. In particular, wellbore storage must always be considered when estimating rate dependent friction from a step-down test. 2. We use a rate dependent fracture geometric factor to estimate fracture internal pressure during injection that satisfies both the constant fracture internal pressure gradient during injection and zero internal fracture gradient during falloff. 3. The leakoff volume may not neglected during injection even for tight formation, and the match with injection data must consider the resulting fluid volume loss. 4. Net pressure in the fracture could be much smaller than the observed pressure drop after shut-in. Extra net pressure could be the driver for the tip extension, and then the net pressure in a static fracture should be at a relatively low level. Therefore, the Carter leakoff model is justified for the fracture with high closure stress but limited net pressure. Nomenclature Af = fracture one side surface area, L2 Af0 = fracture one side surface area at shut-in, L2 BHP = bottomhole pressure, m/Lt2 12 SPE-187191-MS cw CL d G E ISIP k p pc pi pp, min pw pws pw, start q qaf qp Rf Rf,initial Rf,final t tp V Vl Vp Vw w Greek = water compressibility, Lt2/m = leakoff coefficient, L/√t = wellbore diameter, L = G-function, dimensionless = steel modulus, m/Lt2 = instantaneous shut-in pressure, m/Lt2 = permeability, L2 = pressure, m/Lt2 = closure pressure, m/Lt2 = formation initial pressure, m/Lt2 = minimum fracture propagation pressure, m/Lt2 = bottomhole pressure, m/Lt2 = shut-in pressure, m/Lt2 = bottomhole pressure before injection, m/Lt2 = flow rate, L3/t = after-flow rate into fracture due to fluid expansion in the wellbore, L3/t = pumping rate, L3/t = fracture radius in radial fracture model, L = fracture radius at shut-in, L = fracture radius at the end of tip extension, L = time, t = pumping time, t = volume, L3 = leakoff volume, L3 = pumping volume, L3 = injection string or wellbore volume, L3 = fracture width, L α Δ δ γ η = area exponent, dimensionless = difference, dimensionless = string thickness, L = geometric factor, dimensionless = fluid efficiency, percentage c eca f fric inj max perf tort w, wb WBS = closure = end of constant area = fracture = friction = injection = maximum = perforation = near-wellbore fracture tortuosity = wellbore = wellbore storage Subscripts SPE-187191-MS Reference 13 Barree, R.D., Miskimins, J.L., and Gilbert, J.V. 2014. Diagnostic Fracture Injection Tests: Common Mistakes, Misfires, and Misdiagnoses. Paper presented at the SPE Western North American and Rocky Mountain Joint Meeting, Denver, Colorado. Society of Petroleum Engineers. DOI: 10.2118/169539-MS. Carter, R.D. 1957. Derivation of the General Equation for Estimating the Extent of the Fractured Area. In Appendix of "Optimum Fluid Characteristics for Fracture Extension", ed. Howard, G.C. and Fast, C.R., Drilling and Production Practices, New York: API. Feng Yongcun and Gray, K. E. A Comparison Study of Extended Leak-off Tests in Permeable and Impermeable Formations In the 50th US Rock Mechanics / Geomechanics Symposium. Houston, Texas, USA, 2016. Feng Yongcun, Jones John F., and Gray, K. E. Pump-in and Flow-back Tests for Determination of Fracture Parameters and In-situ Stresses. In the 2015 AADE National Technical Conference and Exhibition, San Antonio, Texas, 2015. Geertsma, J. and De Klerk, F. 1969. A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures. Journal of Petroleum Technology 21 (12): 1,571 - 571,581. DOI: 10.2118/2458-PA Khristianovic, A. and Zheltov. 1955. 3. Formation of Vertical Fractures by Means of Highly Viscous Liquid. Paper presented at the 4th World Petroleum Congress, Rome, Italy. World Petroleum Congress. Liu, G. and Ehlig-Economides, C. 2015. Comprehensive Global Model for before-Closure Analysis of an Injection Falloff Fracture Calibration Test. Paper presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, USA. Society of Petroleum Engineers. DOI: 10.2118/174906-MS. Liu, G. and Ehlig-Economides, C. 2016. Interpretation Methodology for Fracture Calibration Test before-Closure Analysis of Normal and Abnormal Leakoff Mechanisms. Paper presented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, USA. Society of Petroleum Engineers. DOI: 10.2118/179176-MS. Liu, G., Ehlig-Economides, C., and Sun, J. 2016. Comprehensive Global Fracture Calibration Model. Paper presented at the SPE Asia Pacific Hydraulic Fracturing Conference, Beijing, China. Society of Petroleum Engineers. DOI: 10.2118/181856-MS. Mack, M.G. and Warpinski, N.R. 2000. Mechanics of Hydraulic Fracturing. Ed Economides, M.J. and Nolte, K.G. Reservoir Stimulation (Third Edition). Chichester, England: John Wiley & Sons Ltd,. Original edition. ISBN 0471491926. Mohamed, I.M., Azmy, R.M., Sayed, M.A.I. et al 2011. Evaluation of after-Closure Analysis Techniques for Tight and Shale Gas Formations. Paper presented at the North American Unconventional Gas Conference and Exhibition, The Woodlands, Texas. Society of Petroleum Engineers. DOI: 10.2118/140136-MS. Naidu, R.N., Guevara, E.A., Twynam, A.J. et al 2015. Understanding Unusual Diagnostic Fracture Injection Test Results in Tight Gas Fields - a Holistic Approach to Resolving the Data. Paper presented at the SPE Middle East Unconventional Resources Conference and Exhibition Muscat, Oman. Society of Petroleum Engineers. DOI: 10.2118/172956-MS. Nolte, K.G., Mack, M.G., and Lie, W.L. 1993. A Systematic Method for Applying Fracturing Pressure Decline: Part I. Paper presented at the SPE Rocky Mountain Regional Low Permeability Reservoirs Symposium, Denver, Colorado. Society of Petroleum Engineers. DOI: 10.2118/25845-MS. Nordgren, R.P. 1972. Propagation of a Vertical Hydraulic Fracture. Society of Petroleum Engineers Journal 12 (04): 306 - 314. DOI: 10.2118/3009-PA Perkins, T.K. and Kern, L.R. 1961. Widths of Hydraulic Fractures. Journal of Petroleum Technology 13 (09): 937 - 949. DOI: 10.2118/89-PA

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