SPE Copyright 2012, Society of Petroleum Engineers

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1 SPE Rearaisal of the G Time Concet in Mini-Frac Analysis R.C. Bachman, Taurus Reservoir Solutions Ltd., D.A. Walters, Taurus Reservoir Solutions Ltd., R.A. Hawkes, Pure Energy Services Ltd., Fabrice Toussaint, Dinova Petroleum Ltd., A. Settari, University of Calgary Coyright 2012, Society of Petroleum Engineers This aer was reared for resentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 8-10 October This aer was selected for resentation by an SPE rogram committee following review of information contained in an abstract submitted by the author(s). Contents of the aer 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 osition of the Society of Petroleum Engineers, its officers, or members. Electronic reroduction, distribution, or storage of any art of this aer without the written consent of the Society of Petroleum Engineers is rohibited. Permission to reroduce in rint is restricted to an abstract of not more than 300 words; illustrations may not be coied. The abstract must contain consicuous acknowledgment of SPE coyright. Abstract Industry is currently using mini-frac analysis for the determination of fracture closure stress and after-closure reservoir roerties. The foundation of all mini-frac analysis is the one dimensional Carter leak-off model, which leads directly to the concet of G Time. For 30+ years, G Time (or the G function) has layed the dominant role for the determination of closure stress. The current norm uses combination G function and combination square root t lots for closure ressure determination. Each combination lot has three lotting functions associated with it. These combination lots also allow the identification of non-ideal behavior. Additionally, various log-log derivative techniques based on ressure transient analysis concets have been develoed to act as a guide for determining flow regimes and closure ressure. These PTA based techniques also allow the determination of after-closure flow regimes and roerties. Concurrently, various secialized afterclosure lotting techniques have been develoed for fracture/reservoir roerty determination. Desite all these techniques, there remains ambiguity in erforming mini-frac analysis. Part of the roblem is that the recommended lots do not rigorously identify the various flow regimes that occur during a mini-frac fall-off. Mini-frac analysis requires a general theory that accounts for all of the actual observed flow regimes. A systematic aroach based on ressure transient analysis (PTA) concets has been develoed to identify the various flow regimes (Carter leak-off being only one of them). The starting oint is the Bourdet log-log derivative lot, accomanied by the rimary ressure derivative (PPD) function. It will be shown that the PPD on its own has indeendent flow regime identification caabilities. Once secific flow regimes have been identified, secialized log-log lots can be constructed for further flow regime verification. New combination lots are then develoed for each flow regime to further assist in closure ressure determination. The theory will first be develoed and illustrated with various examle roblems. Introduction Nolte (1979) introduced the first rigorous technique for determining closure ressure using the Carter leak-off assumtion couled with material balance within the fracture. This led to a secial time function called G Time. Nolte (1986) further extended this work to account for different fracture geometries. These analysis techniques were the beginning of what is now referred to as before-closure analysis. Closure ressure was determined by linear lots of versus G. Deviation from straight line behavior indicated the closure ressure. Practical difficulties of where to draw the straight line often occurred. This is the analogous situation that occurred in welltest PTA for determining the correct straight line on the Horner lot. For PTA the situation was resolved by Bourdet et al. (1983) with the Bourdet log-log ressure derivative lot. This lot allowed for flow regime identification, reservoir roerties determination and defining the time range over which straight lines could be drawn on other secialized lots to comlete the analysis. Returning to mini-frac analysis, Mukherjee et al. (1991) and Barree and Mukherjee (1996) added secial derivative lots to more accurately ick deviations from straight line behavior on the versus G lot. Barree and Mukherjee (1996) recommended that three functions, d/dg and Gd/dG be lotted against G. This is the combination G function lot, which is the basis of all modern interretation methods. Imlicit in this methodology was that G Time with the associated Carter leak-off assumtion was the dominant flow regime, although allowances for ressure deendent leak-off (PDL) were accounted for. Further deviations from non-ideal behavior were addressed in Barree et al. (2007) where the additional cases of fracture ti extension and height recession or transverse storage were identified and analyzed. They also develoed combination square root t lots as well as a log-log derivative lot which while bearing similarities to the Bourdet log-log

2 2 SPE derivative lot has a different time function. The Barree et al (2007) aroach reresents the current state of the art interretation methodology alied across the industry. Leschyshyn et al. (1996) resented an early attemt at using log-log ressure derivative techniques for before-closure analysis. They analyzed a near wellbore shear fracture zone for oilsands alications. The most recent investigations of minifrac analysis using the PTA based Bourdet log-log derivative are by Mohamed et al. (2011) and Marongiu-Porcu et al. (2011). Mohamed et al (2011) have identified a characteristic 3 / 2 sloe on the derivative lot when late time Carter flow is evident. One asect of welltest PTA analysis that has not been alied to mini-fracturing analysis is Mattar and Zaoral s (1992) PPD concet. The remise behind the PPD is that due to the nature of the diffusivity equation, transients resulting from shutting in a well should result in a decreasing ressure change with time. If the PPD increases at some oint in time, it can be attributed to wellbore effects as oosed to reservoir effects. The Bourdet log-log derivative during this ortion of the test, which often looks quite normal, must be ignored. The alication of the PPD concet to standard PTA is aarent because the reservoir fabric is not changing with time. With mini-fracturing, if the fracture snas shuts sufficiently quickly, the revious statements may not hold and the PPD may increase at a later oint in time because of the reservoir/fracture interaction. For a roer interretation of mini-frac tests it is critically imortant to monitor the PPD. It will be shown that the PPD also indeendently allows flow regime identification. The emhasis on before-closure analysis has been dominated by the G Time concet, which in turn imlies Carter leak-off. There are three other fracture flow regimes that have been identified when static roed fractures exist. It will be demonstrated that some of these also occur during mini-fracturing during the before-closure eriod. In fact in a number of cases more than one fracture flow regime occurs during the before-closure eriod. This comlicates the interretation rocess when analysis techniques are based on Carter leak-off. The first ste in an analysis is to identify the flow regimes and based uon what is observed, utilize aroriate secondary lots to determine closure ressure. Flow Regime Identification Within the well testing community the current aradigm for PTA is to first identify flow regimes using the Bourdet log-log derivative lot and then use secondary secialized lots to comlete the analysis. In contrast, for before-closure mini-frac interretation equal weight is given to various combination lots (square root t and G Function) and the secial DT (or Delta Time) log-log derivative lot. The DT log-log derivative function, defined as tdp/dt, is significantly different than the Bourdet log-log derivative lot used in PTA. The Bourdet derivative function used in the PTA based log-log derivative aroach was based on work by Agarwal (1980) and accounts for rate variation rior to the analyzed shut-in eriod. This time function is known as Agarwal equivalent time. The urose of this suerosition function is to remove the as yet unknown initial ressure from the resulting equations and relace it with the flowing ressure immediately at shut-in. It requires an assumtion with regards to the flow regime. As a result there is a different function for comuting radial, linear or bilinear Agarwal equivalent time (t er, t el or t eb resectively). In the majority of commercial well test software radial flow is assumed. Therefore the default Bourdet log-log derivative is defined as t er dp/dt er. The great ower of the Bourdet log-log derivative technique is that PTA interretation and identification of flow regimes is not strongly affected by the tye of flow regime secific generating function. Whether the well is in bounded flow or any of the other non-radial flow regimes, only a minor distortion of results occurs. Once a secific flow regime has been identified, a change in the Bourdet log-log derivative to that secific flow regime is ossible. In PTA this is rarely done. In the interretation workflow of this aer, changing of the Agarwal generating function to reflect the identified flow regime will be an integral art of our roosed mini-frac interretation technique. A brief review of the various fracture flow regimes will begin with a discussion of Carter leak-off. It is a secial tye of linear one dimensional flow into the formation normal to the fracture lane. During uming in a static fracture formation linear flow occurs. The flow velocity u into the formation from the static fracture is u C / LO t (1) When a dynamic fracture occurs, during roagation the initiation time for linear flow at a given oint in the fracture is a function of t 0, the time at which the fracture reaches the oint in question. For this case u CLO / t t 0 (2)

3 SPE This is the Carter leak-off assumtion. The concet is illustrated in Figure 1. Nolte (1979) made the further assumtion that during uming the fracture length L increases linearly with time (the so called low leak-off case). Once uming stos the fracture is assumed to stay at the maximum length until closure. Combining this secific Carter leak-off assumtion and material balance for a closing fracture, ressure is related to G Time as follows: DP P DT 0) C G( t ) (3) G( t d ( 1 d 16 ) (1 t 3 d ) 1.5 t 1.5 d 1 (4) where C 1 is a constant including the aroriate reservoir and geomechanical roerties. Alternatively if one multilies through by t 1.5 one gets P C (5) c ( t t) t t where C c is the aroriately adjusted Carter leak-off constant. For a static fracture, Cinco-Ley et al. (1978) has identified three tyes of fracture flow regimes which occur during roduction or injection. These are: 1. fracture linear flow 2. bilinear flow 3. formation linear flow These flow regimes are illustrated in Figure 2. They may also be observed in mini-frac treatments before or after-closure times. Fracture linear flow occurs at very early times and lasts at most a few minutes. It is not seen in conventional well tests because its effect is masked by wellbore storage. In mini-fracs, the effective wellbore volume may be small and for these cases wellbore storage effects are minimal. As a result, observing fracture linear flow is ossible. Bilinear flow is observed when linear flow occurs simultaneously in the fracture and erendicular to the formation. Bilinear flow requires a significant ressure dro in the fracture. It is associated with a finite conductivity static fracture. Alternatively, in the after-closure eriod of a mini-frac, residual fracture conductivity may exist in the now closed un-roed frac. Formation linear flow is normally associated with an infinite conductivity static fracture. It is mutually exclusive to the finite conductivity frac and is commonly seen in mini-frac after-closure analysis. For a single injection eriod with a subsequent fall-off the PTA solutions for the three static fracture cases can be exressed in terms of either i or P as follows: i n n C f t t t i C f tsf n n n t t t t C f tef ) P C ) f ( (6) ( (7) The second art of Equation 6 is in terms of the flow regime secific suerosition time function originally determined by Odeh and Jones (1965) for radial flow, but subsequently generalized for all flow regimes. Similarly the second art of Equation 7 is in terms of the flow regime secific equivalent time function. For fracture and formation linear flow n=0.5 and for bilinear flow n=0.25. For cases where there are variable rates throughout the test generalized suerosition and equivalent time functions are used for t sf and t ef in the second art of Equation 6 and 7. Comaring the Carter leak-off exression Equation 5 to Equation 7, and seeing how Equation 6 is simly a reformulation of Equation 7 it is hyothesized that: i P C c Cc t t) t i C f tsc t t) t t Cctec ( (8) ( (9) which means Equations 6 and 7 are very general and include Nolte s G Function concet. It also extends Nolte analysis to the variable rate case. Generalized suerosition and equivalent time functions are easily comuted based on concets develoed for linear flow. The alicability of using suerosition rinciles for Nolte s Carter leak-off case initially aears tenuous. Subsequent injection cycles, or varying the injection rate within a cycle would aear to invalidate the reeatability of the flow regime due to the time deendency of the fracture length during uming. Field data in Examle 1 shows that suerosition is valid for Nolte s Carter leak-off case. As a result unification of mini-frac analysis with PTA has been achieved, and mini-frac analysis should no longer be viewed as an indeendent disciline.

4 4 SPE A comarison is made of the DT log-log derivative lots currently used in mini-frac interretation and the Bourdet log-log derivative lot used in conventional PTA. The first thing to note is that there is no suerosition built into the DT log-log derivative function. Nolte s Carter leak-off case will be examined first. By definition - for all times the fracture is oen. For simlicity a uming time of 1 day is assumed. While an unreasonable value, in no way does it detract from the solution. The advantage is that numerical values of t (which is measured in days) are equivalent to t D in all lots. Figure 3 shows the DT log-log derivative lot for Carter leak-off (the fracture is oen for all times). There are two issues with this lot that make it non-ideal for identifying the Carter leak-off flow regime. First, there is a different asymtotic sloe at early time (unit sloe) and late time (½ sloe). In ractice with real data; how would the analyst distinguish a single flow regime which has a changing sloe versus a transition to another flow regime? Second, the late time asymtotic sloe is ½. The PTA community is used to ½ sloes being associated with linear flow. For this case the ½ sloe reresents an entirely different flow regime (late time Carter leak-off). Figure 4 shows the corresonding Bourdet log-log derivative lot. The early time asymtotic sloe is unit sloe and the late time value is 3 / 2 as has been reorted by Mohamed et al. (2011). There is no ½ sloe evident in this lot. It can be concluded that Carter leak-off never looks like linear flow. Early time unit sloe is exected considering that Nolte s solution accounts for fracture storage effects. Sloes greater than 1 are not common in traditional PTA analysis, so a reliminary ick of the late time Carter leak-off flow regime is straight forward. This is the first flow regime the authors are aware of where the early and late time Bourdet log-log derivative sloes are not identical. For interretation it is strongly recommended that the PPD function introduced by Mattar and Zaoral (1992) should also be included on the Bourdet log-log derivative lot. It is noted that the reviously mentioned DT log-log derivative function tdp/dt is equivalent to t*ppd. The PPD has no suerosition effects built into its calculation, but suffers from early and late time sloe differences within a single flow regime. The PPD function therefore has ractical indeendent diagnostic caabilities. For Carter leak-off the early time PPD sloe on a log-log lot is 0 and the late time sloe is -½. The significance of this will be demonstrated in the examle roblems. To remove the objection of changing sloes during one flow regime, an alternative derivative lot is roosed. Take the derivative of Equation 7 with resect to t ec, which is the derivative with resect to Carter equivalent time, and then multily by t. For all times for which Carter leak-off is aroriate the log-log sloe is unity. Alternatively one could have chosen to multily by t 1.5 in which case the log-log sloe would be 3 / 2 for all times. The idea of taking the derivative with resect to a secific flow regime time function and multilying by another time function so that a desired sloe is obtained on a articular lot will be a recurring theme in this aer. Figure 5 shows the resulting lot for Carter leak-off. Deviation from the unit sloe line for real data will indicate that this flow regime is no longer aroriate. This can be cross checked by concurrently checking the PPD curve. For the linear flow regime the log-log derivative lots are given in Figure 6, Figure 7 and Figure 8. Figure 6 shows the DT log-log derivative lot. At early time the asymtotic sloe is ½ while at late time it is -½. The time to the establishment of late time -½ sloe should corresond to when the Soliman et al. (2005) late time after-closure analysis technique would work. Unfortunately the early time linear flow asymtotic ½ sloe is the same as the late time asymtotic DT log-log derivative Carter leak-off sloe. This has created significant confusion within the industry; what flow regime is identified by ½ sloe on the DT log-log derivative lot? We recommend that industry discontinue using the DT log-log derivative lot. The inherent advantages of the Bourdet log-log derivative lot are aarent from Figure 7. Even though radial flow was used for the base equivalent time function, early and late time asymtotic ½ sloes are evident. Only minor distortion occurs at middle time. The robustness of the Bourdet log-log derivative lot is one reason for it being one of the outstanding achievements in the field of reservoir engineering over the last 30 years. Once linear flow is identified, a new time derivative function defined as t 0.5 dp /dt el could be used. It has a ½ sloe for all times when in linear flow. This is shown in Figure 8. Analogous lots could also be constructed for the bilinear flow case. Combination Plots For Carter leak-off, the combination G function lot is shown in Figure 9. Since there is only one flow regime, the various curves show the exected straight line behavior. Figure 10 shows the corresonding combination square root t lot. Only at late time does one get straight line behavior. The lot is comletely unreliable in icking any flow regime deviation from Carter leak-off. A new combination lot is roosed. A highly desirable feature of an alternative combination lot would be that the x-axis be linear in t instead of a transformed time function such as G Time or square root t. This is beneficial in the case where the t range of a secific flow regime has been identified on another lot, i.e. the Bourdet log-log derivative lot. A straight

5 SPE forward comarison then is ossible between various lots at the same t range. An additional requirement of such a lot is that it exhibit straight line behavior for critical curves in a manner analogous to the combination G function lot. The new combination lots will be flow regime deendent. For an arbitrary but fixed flow regime, can be differentiated with resect to t sf..(in Equation 6). t sf reresents suerosition time with resect to the identified flow regime. d/dt sf = -C f is a constant when the aroriate flow regime occurs. Then one takes the absolute value of d/dt sf. This term is analogous to d/dg. Then one multilies the absolute value of d/dt sf by t. This term is roortional to t as long as that flow regime is aroriate. This term is analogous to Gd/dG. These two new functions are lotted along with as a function of t. For Carter leak-off, the lot is called the combination suerosition Carter lot. It is shown in Figure 11. The only disadvantage of this lot versus the combination G function lot is that the curve is not a straight line. Since the emhasis is on the derivative curves the lot is accetable. This new combination lot is not restricted to mini-frac analysis and is comletely general with resect to any PTA determined flow regime. One advantage they have is that one can ick deviations from a given flow regime on linear lots, as oosed to log-log derivative lots. Therefore the resolution of the ick is imroved. Exerience indicates that the absolute value of d/dt sf function should also be lotted as a third curve on the reviously discussed flow regime secific equivalent time log-log derivative lot. This curve is analogous to the recommended PPD curve on the Bourdet log-log derivative lot. This will be shown in the Examles section. For linear flow, the combination G function lot, the combination square root t lot and the combination suerosition linear lot are shown in Figure 12, Figure 13 and Figure 14 resectively. Neither the combination G function lot nor the square lot give meaningful results. Only the combination suerosition linear lot in Figure 14 gives meaningful results. Prior to generating the combination suerosition linear lot it is imortant to have identified the linear flow t time range on the log-log derivative lot. A review of Figure 10 and Figure 13 would lead to the conclusion that square root t lots should never work. Exerience has shown that the combination square root t lot can give reasonable closure ressures for certain roblems. Barree et al (2007) are cautious in recommending the lot unless it corroborates icks from other lots. Fundamentally the question is: what flow regimes are contained within the square root t function? Figure 15 shows the log-log lot of three functions; square root t, G and t el versus t D. When t D is below 0.03 square root t is equivalent to t el. When t D is greater than 10 the sloe of square root t on log-log co-ordinates is equivalent to the sloe of G. Between these values square root t function is transitioning from an aarent early time linear flow to late time Carter leak-off. In ractice closure rarely occurs before t D = 1. Therefore the square root t function has a transitional feature that can look like Carter leak-off at large t D. The transition along with its sloe change through this time interval makes its interretative caabilities unreliable on its own. The square root t does not carry any information that is not available from other lots. Desite it being comlex to interret, industry is highly unlikely to give u on this lot. Analysis Workflow When erforming a mini-frac analysis a systematic workflow is required. One should always start with the Bourdet log-log derivative lot with the accomanying PPD curve. Flow regimes are then icked, and secialized log-log derivative lots are constructed, followed by the aroriate combination lots. The work flow and the required lotting variables for each lot are shown in Figure 16. Closure stress is icked by the following hierarchal rocedure: 1. Any PPD increase or jum may indicate either raid closure (snaing shut) or wellbore effects. Interretation must be made on a case by case basis. 2. If Carter leak-off is resent, the end of the Carter leak-off flow regime indicates closure. 3. If Carter leak-off does not end then the fracture did not close. 4. If radial, linear or bilinear flow regimes occur after Carter leak-off ends, these are after-closure flow eriods. For these after-closure flow eriods, traditional after-closure analysis techniques can be used for roerty determination. Alternatively, PTA techniques based on drawing the aroriate straight lines are accetable. 5. If Carter leak-off is not seen, the end of any linear flow regime will be considered a closure event. Subsequent well defined flow regimes will be considered after-closure flow eriods. 6. If linear flow does not end (similarly for bilinear flow) then the fracture did not close. 7. Picking a ressure or time range where closure may occur is accetable. 8. Picking multile closure events may be ossible in excetional cases. 9. If no Carter leak-off or linear flow is observed, then the test cannot be interreted. Table 1 and Table 2 have been develoed to aid the analyst in determining the various sloes for each recommended curve for each flow eriod and time range.

6 6 SPE Derivative Function Equivalent Time Derivative Function Early Time Sloe Ends t D =0.08 Log-Log Flow Regime Carter Linear Bilinear Late Early Late Early Time Time Time Time Sloe- Sloe Sloe Sloe Starts Ends Starts Ends t D =5.0 t D =0.04 t D =4.0 t D =0.13 Late Time Sloe Starts t D =8.0 t er dp /dt er 1/1 3/2 1/2 1/2 1/4 1/4 (Radial) tdp /dt ec 1/1 1/1 1/2 0 1/4-1/4 (Carter) t 0.5 dp /dt el 1/1 3/2 1/2 1/2 1/4 1/4 (Linear) t 0.25 dp /dt eb (Bilinear) 1/1 3/2 1/2 1/2 1/4 1/4 Table 1: Secialized Equivalent Time Plotting Functions and their Sloes Early Time Sloe Log-Log Flow Regime Radial Carter Linear Bilinear Late Early Late Early Late Early Time Time Time Time Time Time Sloe Sloe Sloe Sloe Sloe Sloe Late Time Sloe tdp /dt 0-1/1 1/1 1/2 1/2-1/2 1/4-3/4 PPD = dp /dt -1/1-2/1 0-1/2-1/2-3/2-3/4-7/4 Table 2: DT Derivative and PPD Sloes for Various Flow Regimes Examle 1 - Suerosition Effects The urose of this examle is to show that suerosition of Carter leak-off is a reasonable assumtion based uon field evidence. In normal mini-frac alications very few tests are run over multile cycles within the same zone. An excetion is in the shallow Athabasca oilsands in north-eastern Alberta. Measured carock closure ressure directly affects the regulated maximum oerating ressure of these shallow thermal rojects. Reeatability of tests is a major concern. As a result, multicycle mini-fracs are routinely conducted on all zones. The test in this examle used a MDT tool in an oen hole wellbore over a shale interval. The MDT tool injects relatively small volumes of fluid during each cycle. The tool and the details of its oeration are described in Mishra et al (2011). Exerience has shown that in the Athabasca oilsands minimum effective stress gradients are often equal to the overburden gradient; imlying that a horizontal fracture is being generated. Oerationally it is desirable that an initial vertical fracture exists. To ensure this a sleeve frac is tyically erformed rior to the main test. This involves setting a acker over the interval to be tested and inflating it to high ressures. High hoo stresses are generated in the near wellbore area and a vertical fracture is mechanically created. Prior to injecting above the estimated fracture gradient, a number of low rate injection/fall-off cycles are run. The ressure is carefully monitored to ensure that ressures do not exceed fracturing ressure. Fluid enters the re-existing artially oen vertical fracture. Figure 17 shows the rate normalized derivative for three successive flow eriods. It is clear that the late time behavior of these tests is reeatable and that a transition from an early flow eriod within the shale/re-existing vertical fracture into a storage flow eriod is occurring. Storage is the first art

7 SPE of Carter leak-off. It is concluded that Carter leak-off is being re-initiated for each injection cycle and that full rate suerosition is valid. Examle 2 Tight Oil This examle was chosen to comare traditional interretation techniques to the new aroach for a well having classic behavior. The test is from a low ermeability oil well in the Uer Devonian Redknife formation, which occurs throughout a widesread area of northwestern Alberta, northeastern British Columbia, and the southern Northwest Territories shale lays. This is a redominantly limestone unit with minor amount of dolomite. The mini-frac is into the toe stage of a multi-stage frac ort horizontal well. Puming rate during the test was 0.65 m3/min of oil with a total of 3.25 m3 of oil being injected during the test. Pressure was measured from subsurface gauges in the vertical section of the well. Figure 18 shows the ressure versus time lot. Wellbore friction effects are aarent immediately after shut-in. Figure 19, Figure 20 and Figure 21 show the traditional combination G function, combination square root t and the DT log-log derivative lot resectively. A review of the combination G function lot indicates that the well has ressure deendent leakoff (PDL) according to Barree et al. (2007). The DT log-log derivative lot shows two ½ sloes followed by a -½ sloe. All three lots give the same closure ressure. The new technique starts with the Bourdet log-log derivative lot shown in Figure 22. The Bourdet log-log derivative curve has an early ½ sloe flow eriod which is interreted as being fracture linear flow. Continuing along this curve there is a transition into late time Carter leak-off ( 3 / 2 sloe) after which fracture closure occurs. The final flow regime is a second linear flow eriod, which is an after-closure flow regime. The associated PPD curve indicates smooth downward behavior. The associated sloes for the identified flow regimes are labeled on the figure. There is consistency for the flow regime icks between the PPD curve and the Bourdet log-log derivative curve. Since PPD function does not rely on suerosition as does the Bourdet log-log derivative, it is an indeendent means for flow regime verification. Taken as a whole, Figure 22 rovides the analyst with significantly more information than the conventional DT log-log derivative lot of Figure 21. The DT loglog derivative has ambiguity as to the meaning of the two ½ sloes. Once Carter leak-off has been identified; the Equivalent Carter log-log derivative lot is constructed as shown in Figure 23. In this lot the entire Carter flow regime (early and late time) would aear as a unit sloe line on the tdp /dt ec curve. The d/dt sc curve shows when Carter flow ends and also verifies the late after-closure linear flow regime (late time linear flow has a zero sloe on this curve as shown in Table 1). Finally, the combination suerosition Carter lot is given in Figure 24. The ractice of drawing lines through the origin, which is aroriate when there is one flow regime, is not justified. The time range of Carter flow should be identified and a straight line is drawn through the aroriate oints. The result is a non-zero y intercet. In ractice this would rarely change the closure ick from the traditional rocedure of going through the origin. In this examle, the closure icks are identical for both interretation methodologies. The roosed Bourdet log-log derivative lot with the accomanying PPD curve is shown to be very useful for both before- and after-closure flow regime identification. The one interretation difference is that the Bourdet log-log derivative lot does not show evidence of PDL (an issue to be exlored in the next examle). Examle 3 Tight Gas Well 1 This gas well is comleted in the Montney formation in British Columbia Canada. The Montney formation is areally extensive and is comosed of siltstones interbedded with shales. The mini-frac was conducted in the toe stage of a 1800 meter long lateral, multi-stage frac-ort horizontal well at a deth of m CF TVD. Puming rate during the test was 0.48 m3/min of water. A total of 10.0 m3 of water was injected over aroximately days (23 minutes). Pressure was measured at the surface and converted to bottomhole deth using a fresh water gradient. During uming BHP was between 84,000 kpa and 88,000 kpa, values significantly above the vertical stress gradient. Clearly there are wellbore related effects during uming which are not accounted for. Figure 25 and Figure 26 show early time BHP versus time after shut-in. Figure 25 shows that BHP dros below the vertical stress value at days (2.9 minutes). There is a more raid dro in ressure at DT=0.03 days in Figure 26. This corresonds to a PPD increase and is indicative of a wellbore event or closure. Figure 27 and Figure 28 show the late and middle time combination G Function lots resectively. An earlier time combination G Function lot (not shown) dislayed similar character but would result in a foc greater than the vertical stress gradient. The late time combination G Function in Figure 27, if interreted as shown, would be an indication of height recession or transverse storage according to current ractice, giving foc =46,900 kpa. The middle time combination G Function lot in Figure 28 would also indicate the same behavior, although a different closure ressure would be icked ( foc =49,510 kpa). Figure 29 shows the middle time Combination square root t lot which indicates the same closure ressure ( foc =49,510 kpa) as the middle time combination G Function lot. The DT log-log derivative is shown in Figure 30. If one were to ick closure based uon a roll-over of the derivative curve, one might ick a

8 8 SPE closure time at DT=16.6 days ( foc =46,900 kpa). This closure ressure would be consistent with the late time combination G Function lot in Figure 27. Alternatively one could ick the end of ½ sloe on the derivative curve at DT=0.9 days ( foc =49,290 kpa). The Bourdet log-log derivative lot with the PPD is shown in Figure 31. It shows some comlex behavior. Early time Carter leak-off (ending at days or 5.8 minutes) seems to be indicated by the unit sloe on the log-log derivative curve and the zero sloe on the PPD curve. The ressure is above the vertical stress, suggesting an incorrect interretation. This early flow eriod must be wellbore related. The PPD curve dros raidly and has a sloe < -1. U to at least DT=0.002 days when the ressure dros below the vertical stress gradient, there must be significant wellbore effects. The time around DT=0.01 days (0.6 times the uming time) still has a PPD sloe <-1. This is interreted to be PDL by the first author, although it could still be wellbore related effects. A review of Table 2 shows that the steeest PPD derivative sloe for traditional flow regimes at early time is that associated with radial flow (-1 sloe). For PDL to occur, leak-off must be more raid than these traditional flow regimes. Therefore it is hyothesized that a PPD sloe steeer than -1 is an indication of PDL if wellbore effects are not resent. At DT=0.03 a PPD violation occurs. This is interreted as a closure event, which would be in the secondary natural fractures associated with the earlier in time PDL behavior. The well then goes into late time Carter leak-off which ends at DT 1.0 days ( foc =49,280 kpa). Both the derivative lot (sloe = 3 / 2) and the PPD (sloe = -½) verify this flow regime. Since late time Carter flow follows the PPD violation, the main fracture is still oen and is so until DT 1.0 days. Subsequently there is another PPD violation occurring at DT=5.0 days. This gives an absolute latest time of closure. No distinctive after closure flow regimes are identified. The Equivalent Carter log-log derivative with the d/dt sc curve is shown in Figure 32. This lot shows the same flow regimes as the Bourdet log-log derivative lot. A later time has been icked for the closure time DT=1.3 days ( foc =49,240 kpa). The Combination Suerosition Carter lot in Figure 33 was used to make the final closure time DT=1.05 days ( foc =49,270 kpa). The d/dt sc is not erfectly flat during Carter leak-off. It is increasing at a more raid rate after DT=1.05 days indicating the end of Carter leak-off. The ability of the Bourdet log-log derivative with the PPD lot greatly enhances our ability to interret mini-fracs. It also leads to different interretations than traditional techniques (PDL versus height recession or transverse storage in this case). Examle 4 Tight Gas Well 2 This gas well was comleted in the Lower Montney formation in British Columbia by a different oerator than the revious well. The mini-frac was conducted in the toe stage of a 1300 meter long multi-stage frac ort horizontal well. Puming rate during the test was 0.75 m3/min of water with a total of 5.0 m3 of water being injected over aroximately 7 minutes. Pressure was measured at the surface and converted to bottomhole deth. Figure 34, Figure 35, Figure 36 and Figure 37 show the ressure versus time, combination G function, combination square root t and the DT log-log derivative lots resectively. The ressure dros off linearly (storage event) at a very raid rate at early time. There is then a slight ressure rebound at days, when the ressure decline resumes at a much lower rate. The rebound is not interreted to be a closure event. This early ressure decline is robably an artifact of surface oerational issues related to shutting the um down, and is not a reservoir effect. Both the combination G function and square root t lots show a large uward curvature, which is traditionally considered an indication of height recession/transverse storage. Hyothetical closure ressures are icked from both of these lots. Taking the average value closure ressure would be 51,750 kpa. The DT log-log derivative lot shows two clearly identifiable flow regimes, the first is a ½ sloe flow regime; the second is late time after closure bilinear flow as a result of the -¾ sloe. Following the ½ sloe flow regime, there is a di followed by a significant sloe increase. Picking the end of the ½ sloe flow as closure, results in a closure ressure of 55,300 kpa. This is significantly different than the value determined from the combination lots. The Bourdet log-log derivative lot with the PPD is shown in Figure 38. An aarent radial flow eriod occurs from days to days. During this time the PPD has a sloe of -1, which is consistent with the zero sloe on the derivative curve. It is not reasonable that radial flow is occurring at this time. It is ossible that this is a PDL effect as discussed in the revious examle. At days a unit sloe occurs on the derivative curve lasting until 0.04 days (58 minutes or a t D =8.3). This would normally be interreted as early time Carter leak-off. From Table 2 early time Carter leak-off should be over by t D =0.08 and the late time Carter leak-off should have started by t D =5.0. The lot should have had a 3 / 2 sloe instead of unit sloe (there are no 3 / 2 sloes on this lot and in fact the derivative curve after 0.1 days has a sloe that is > than 3 / 2). Therefore this unit sloe

9 SPE is not early time Carter leak-off and deviations from this sloe should not be used for closure icks. Additionally the PPD curve does not show early time Carter leak-off (which should have a zero sloe). This illustrates the advantages of two indeendent flow regime identification curves. For this reason closure icks were based on when the PPD increases. The earliest ossible closure would be at 0.07 days where foc = 55,700 kpa, but we feel that this is wellbore related. A more reasonable value is at 0.35 days where a smooth increase in the PPD starts and continues over ½ a log cycle. This gives a foc = 55,000 kpa. These two values bound the closure ressure ick. The advantages of using the Bourdet log-log derivative along with the PPD are aarent. It is extremely imortant to erform consistency checks to ensure false signals are identified. Conclusions A thorough review of mini-frac analysis techniques and how they relate to conventional PTA concets has been erformed. It is concluded: 1. Nolte analysis has been extended to account for the variable rate case using suerosition rinciles. 2. Nolte s Carter leak-off flow regime associated with a dynamically generated fracture is just one more flow regime that can be handled within the PTA aradigm. 3. The ambiguities of the DT log-log derivative lot have been revealed. 4. The starting oint for any mini-frac study should be the standard Bourdet log-log derivative lot with the PPD curve for consistency checking. 5. The PPD curve has been shown to contain flow regime identification roerties indeendent of the Bourdet log-log derivative. 6. Once various flow regimes have been identified, additional flow regime secific log-log derivative lots have been develoed so as to further clarify the interretation rocess. 7. New derivative functions have been defined for the log-log derivative lots which ensure invariant sloes at both early and late time within a given flow regime. These functions have been constructed in a way that they give sloes consistent with the associated flow regimes in PTA. 8. New combination lots and associated derivative functions have been determined to generalize the combination G function lot. They have been built so that they can handle all flow regimes and can mimic the functionally of the combination G function lot when lotted against t. This allows the easy comarison of the flow regimes across many lots, as they all now have t as the x axis. These combination lots are very general and could also be used in traditional PTA.

10 10 SPE Nomenclature C b = Constant for bilinear flow regime C c = Constant for Carter leak-off C f = Generic constant for a flow regime C l = Constant for linear flow regime C LO = leak-off coefficient C 1 = constant including reservoir and geomechanical roerties to give Nolte s G Function relationshi d = derivative DP = Delta Bottomhole Pressure ((DT=0) ) (same as P) DP Deriv = t er ddp/dt er Bourdet log log derivative, si or kpa DT = Delta time from current time to end of last injection eriod (same as t) G = G Time, dimensionless L = Fracture half-length at an instant in time during uming, ft or m n = exonent associated with one of the fracture flow regimes = Bottomhole Pressure, si or kpa foc = Oening and closure ressure (closure stress), si or kpa i = Initial Pressure, si or kpa QDT= DT 0.25 = t 0.25, days 0.25 SDT= DT 0.5 = t 0.5, days 0.5 T = Time, days t eb = Equivalent Time bilinear, days t ec = Equivalent Time Carter, days t ef = Equivalent Time for an arbitrary flow regime, days t el = Equivalent Time linear, days t er = Equivalent Time radial, days t = Puming Time t sb, = Suerosition Time bilinear, days t sc = Suerosition Time Carter, days t sf = Suerosition Time for an arbitrary flow regime, days t sl = Suerosition Time linear, days t rl = Suerosition Time radial, days t 0 = Time at which leak-off begins at a secific location in the roagating fracture, days u = velocity normal to the fracture face, ft/day or m/day T2 = T 2, days 2 P = Delta Bottomhole Pressure ((DT=0) ) (same as DP) t = Delta time from current time to end of last injection eriod (same as DT) t D = dimensionless shut-in time equal to t/t References Agarwal, R.G A New Method for Accounting for Producing Time Effects when Drawdown Tye Curves are Used to Analyze Pressure Build-us and other Test Data, Paer SPE resented at the SPE Annual Technical Conference and Exhibition, Dallas, Setember. Barree, R.D. and Mukherjee, H Determination of Pressure Deendent Leakoff and its Effects on Fracture Geometry. Paer SPE resented at the SPE Annual Technical Conference and Exhibition, Las Vegas, 6-9 October. Barree, R.D., Barree, V.L. and Craig, D.P Holistic Fracture Diagnostics. Paer SPE resented at the SPE Rocky Mountain Technical Symosium, Denver, Aril. Bourdet, D.P, Whittle, T.M., Douglas, A.A and Pirard, Y.M A New Set of Tye Curves Simlifies Well Test Analysis, World Oil, May. Cinco-Ley, H., Samaniego-V, F. and Dominguez, N Transient Pressure Behavior for a Well with a Finite Conductivity Vertical Fracture, SPEJ, August. Leschyshyn T., Farouq Ali S.M. and Settari A Mini-frac Analysis of Shear Parting in Alberta Reservoirs and Its Imact towards On-site Fracture Design, Paer No , 47th Annual Technical Meeting of Petroleum Society. of CIM,

11 SPE Calgary, June. Mattar, L. and Zaoral,, K The Primary Pressure Derivative (PPD) A new Diagnostic Tool in Well Test Interretation, JCPT, Aril. Marongiu-Porcu, M., Ehlig-Economides, C.A. and Economides, M.J Global Model for Fracture Falloff Analysis. Paer SPE resented at the North American Unconventional Gas Conference and Exhibition, The Woodlands, June. Mishra, M.K., Lywood, P. and Ayan, C Alication of Wireline Stress Testing for SAGD Carock Integrity, Paer SPE resented at the Canadian Unconventional Resource Conference, Calgary, November. Mohamed, I.M., Nasralla, R.A., Sayad, M.A. Marongiu-Porcu M. and, Ehlig-Economides, C.A Evaluation of Afterclosure Analysis Techniques for Tight and Shale Gas Formations, Paer SPE resented at the Hydraulic Fracturing Technology Conference and Exhibition, The Woodlands, January. Mukherjee, H., Larkin, S. and Kordziel, W Extension of Fractured Decline Curve Analysis to Fissured Formation, Paer SPE resented at the Low-Permeability Reservoir Symosium, Denver, Aril. Nolte, K.G Determination of Fracture Parameters from Fracturing Pressure Decline, Paer SPE 8341 resented at the SPE Annual Technical Conference and Exhibition, Las Vegas, Setember. Nolte, K.G A General Analysis of Fracturing Pressure Decline with Alications to Three Models, SPEFE, December. Odeh, A.S. and Jones, L.G Pressure Drawdown Analysis, Variable Rate Case, JPT, August. Soliman, M.Y., Craig, D., Bartko, K. and Rahim, Z After-Closure Analysis to Determine Formation Permeability, Reservoir Pressure, and Residual Fracture Proerties, Paer SPE resented at the SPE Middle East Oil Show and Exhibition, Bahrain, March.

12 12 SPE Figure 1: Reresentation of Carter Leak-off Figure 2: Static Fracture Flow Regimes from Cinco-Ley (1978)

13 SPE Early Time Sloe = 1 Late Time Sloe = 0.5 t = 1 day d( P) t dt Figure 3: Carter Leak-Off Solutions using the DT log-log derivative Early Time Sloe = 1 Late Time Sloe = 1.5 t = 1 day t er d( P) DP _ Deriv dt er t er t t t t Figure 4: Carter Leak-Off Solutions using the Bourdet log-log derivative

14 14 SPE All Sloes = 1 t = 1 day P C ( t d( P) C dt ec c t) d( P) t Cct dt ec c 1.5 t 1.5 t 1.5 C t c ec Figure 5: Carter Leak-Off Solutions using the Equivalent Carter log-log derivative Early Time Sloe = 0.5 Late Time Sloe = -0.5 t = 1 day d( P) t dt Figure 6: Linear Fall-off Solutions using the DT log-log derivative

15 SPE Early Time Sloe = 0.5 Late Time Sloe = 0.5 t er d( P) DP _ Deriv dt er t er t t t t Figure 7: Linear Fall-off Solutions using the Bourdet log-log derivative All Sloes = 0.5 t = 1 day P C d( P) dt t el 0.5 l ( t C d( P) dt el l t) 0.5 C t l t t 0.5 Ct el Figure 8: Linear Fall-off Solutions using the Equivalent Linear log-log derivative

16 16 SPE t = 1 day Pressure (si) Figure 9: Carter Leak-Off Combination G Function Plot t = 1 day Pressure (si) Combination Square Root Plot P Early Time Constant Late Time Sqrt(DT) dpdsdt Early Time Sqrt(DT) Late Time Constant SDTdPdSDT Early Time DT Late Time Sqrt(DT) Figure 10: Carter Leak-Off Combination Square Root t Plot

17 SPE t = 1 day Pressure (si) C d dt t sc i d dt C sc c c ( t C t c t) 1.5 t 1.5 C t i c sc Figure 11: Carter Leak-Off Combination Suerosition Carter Plot t = 1 day Pressure (si) Figure 12: Linear Flow Combination G Function Plot

18 18 SPE t = 1 day Pressure (si) Combination Square Root Plot P Early Time Constant Late Time 1/Sqrt(DT) dpdsdt Early Time Constant Late Time 1/DT SDTdPdSDT Early Time Linear Late Time 1/Sqrt(DT) Figure 13: Linear Flow Combination Square Root t Function Plot t = 1 day Pressure (si) d dt t d dt C C ( t sl i sl l l C t l t) 0.5 t 0.5 C t i l sl Figure 14: Linear Flow Combination Suerosition Linear Plot

19 SPE Figure 15: Comarison of square root t, G and t el Functions Start Log-Log Plot P, Bourdet deriv, PPD vs t Flow Regime Identification Carter Linear Bilinear Log-Log Plot P, td(p)/dt ec, d/dt sc vs t Log-Log Plot P, t 0. 5 d(p)/dt el, d/dt sl vs t Log-Log Plot P, t 0.25 d(p)/dt eb, d/dt sb vs t Linear-Linear Combination Plot, d/dt sc, td/dt sc vs t Linear-Linear Combination Plot, d/dt sl, td/dt sl vs t Linear-Linear Combination Plot, d/dt sb, td/dt sb vs t Note: when we write d/dt sf we mean d/dt sf Figure 16: Workflow associated with Mini-Frac Analysis

20 20 SPE Sloe = 1/1 Figure 17: Examle 1 - Rate Normalized Derivative over Multile Flow Periods Friction effects not Pressure Deendent Leak-off (PDL) Two searate injection cycles due to oerational issues in the field FP_001_Inj = 6.0 minutes FP_002_FO = 3.1 minutes FP_003_Inj = 5.3 minutes FP_004_FO = 5626 minutes or 1825 times as long as FP_003_Inj Figure 18: Examle 2 Pressure versus Time

21 SPE foc = 8400 kpa Indicator of ressure deendent leak-off Figure 19: Examle 2 Combination G Function Plot foc = 8400 kpa Figure 20: Examle 2 Combination Square Root t Plot

22 22 SPE ½ ½ -½ Cross over of Middle and Late ½ Sloe Delta Time = 0.28 Days foc = 8300 kpa Figure 21: Examle 2 DT log-log derivative Plot -½ 3/2 ½ -½ End of Carter Leak-off Fracture has closed foc = 8300 kpa ½ -3/2 Figure 22: Examle 2 Bourdet log-log derivative Plot

23 SPE End of Carter Leak-off Fracture has closed Linear flow signatures 0 1/1 foc = 8300 kpa Figure 23: Examle 2 Equivalent Carter log-log derivative Plot foc = 8300 kpa Figure 24: Examle 2 Combination Suerosition Carter Plot

24 24 SPE Figure 25: Examle 3 Pressure versus Delta Time to 0.01 days PPD Violation 1 Figure 26: Examle 3 Pressure versus Delta Time to 1 day

25 SPE Closure at G = 82? DT = 16.6 days foc = 46,900 kpa Figure 27: Examle 3 Combination G Function Plot (Late Time) Closure at G = 3.0? DT =.045 days foc = 49,510 kpa Figure 28: Examle 3 Combination G Function Plot (Middle Time)

26 26 SPE Closure at Sqrt(DT) = 0.215? DT = days foc = 49,510 kpa Figure 29: Examle 3 Combination Square Root t Plot (Middle Time) 1/1 Sloe = Gray ½ Sloe = Blue Closure at end of ½ sloe at DT = 0.9 days? foc = 49,290 kpa 1/1 ½ Closure near end? DT = 16.9 days foc = 36,950 kpa Figure 30: Examle 3 DT log-log derivative Plot

27 SPE /1 0 PPD Deriv sloe <-1 Pressure Deendent Leak-off (PDL) PPD Violation at DT=0.03 days foc =49,560 kpa? Possible Closure of secondary fractures 3/2 Late Time Carter Until 1.0 days foc =49,280 kpa? -½ PPD Violation at DT=5.0 days foc =48,850 kpa? Absolute latest time for closure Figure 31: Examle 3 Bourdet log-log derivative Plot 0 Since Carter leak-off is occurring after PPD violation at DT=0.03 days main fracture still oen 0 1/1 Late Carter Closure at 1.3 days foc = 49,240 kpa? 1/1 Figure 32: Examle 3 Equivalent Carter log-log derivative Plot

28 28 SPE End of Carter Leak-off? Yes DT=1.05 Days foc =49,270 kpa Final closure ick Figure 33: Examle 3 Combination Suerosition Carter Plot Figure 34: Examle 4 Pressure versus Time

29 SPE Closure at G=28.5? foc =51,600 kpa Figure 35: Examle 4 Combination G Function Plot Closure at SDT=0.93? foc =51,900 kpa Figure 36: Examle 4 Combination Square Root t Plot

30 30 SPE Closure at DT=1.000? foc =50,600 kpa -3/4 ½ Closure at DT=0.025? foc =55,300 kpa Figure 37: Examle 4 DT log-log derivative Plot 0 1/4-1 <0 0 1/1 False early Carter Leak-off As PPD sloe is not 0 No late Carter Leak-off resent (sloe is >3/2 after DT=0.1) -7/4 PPD Closure Picks Highest closure stress at DT=0.07 foc =55,700 kpa Most likely closure stress at DT=0.35 P foc =55,000 kpa Figure 38: Examle 4 Bourdet log-log derivative Plot