holistic fracture diagnostics

Copyright 2007, Society of Petroleum Engineers This paper was prepared for presentation at the 2007 SPE Rocky Mountain Oil & Gas Technology Symposium held in Denver, Colorado, U.S.A., 16–18 April 2007. 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, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes 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 where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435. Abstract Since the introduction of the G-function derivative analysis, pre-frac diagnostic injection tests have become a valuable and commonly used technique. Unfortunately, the technique is frequently misapplied or misinterpreted leading to confusion and misdiagnosis of fracturing parameters. This paper presents a consistent method of analysis of the G-function, its derivatives, and its relationship to other diagnostic techniques including square-root(time) and log(∆p wf )-log(∆t) plots and their appropriate diagnostic derivatives. Actual field test examples are given for the most common diagnostic curve signatures. Introduction Pre-frac diagnostic injection test analysis provides critical input data for fracture design models, and reservoir characterization data used to predict post-fracture production. An accurate post-stimulation production forecast is necessary for economic optimization of the fracture treatment design. Reliable results require an accurate and consistent interpretation of the test data. In many cases closure is mistakenly identified through misapplication of one or more analysis techniques. In general, a single unique closure event will satisfy all diagnostic plots or methods. All available analysis methods should be used in concert to arrive at a consistent interpretation of fracture closure. Relationship of the pre-closure analysis to after-closure analysis results must also be consistent. To correctly perform the after-closure analysis the transient flow regime must be correctly identified. Flow regime identification has been a consistent problem in many analyses. There remains no consensus regarding methods to identify reservoir transient flow regimes after fracture closure. The method presented here is not universally accepted but appears to fit the generally assumed model for leakoff used in most fracture simulators. Four examples are presented to show the application of multiple diagnostic analysis methods. The first illustrates the expected behavior of normal fracture closure dominated by matrix leakoff with a constant fracture surface area after shut- in. The second example shows pressure dependent leakoff (PDL) in a reservoir with pressure-variable permeability or flow capacity, usually caused by natural or induced secondary fractures or fissures. The third example shows fracture tip extension after shut-in. These cases generally show definable fracture closure. The fourth example shows what has been commonly identified as fracture height recession during closure, but which can also indicate variable storage in a transverse fracture system. For each example the analysis will be demonstrated using the G-function and its diagnostic derivatives, the sqrt(time) and its derivatives, and the log-log plot of pressure change after shut-in and its derivatives. 1-4 When appropriate, the after- closure analysis is presented for each case, as is an empirical correlation for permeability from the identified G-function closure time. 5 A critical part of the analysis is the realization that there is a common event indicating closure that should be consistently identified by all diagnostic methods. To reach a conclusion all analyses must give consistent results. The goal of this paper is to provide a method for consistent identification of after-closure flow regimes, an unambiguous fracture closure time and stress, and a reasonable engineering estimate of reservoir flow capacity from the pressure falloff data, without requiring assumptions such as a known reservoir pressure. Other methods, based on sound transient test theory, require pressure difference curves based on the observed bottomhole pressure during falloff minus the “known” reservoir pressure. 5,8 While these methods are technically correct they can lead to confusing results at times, especially in low permeability reservoirs when pore pressure is difficult to determine accurately prior to stimulation. This is not a transient test analysis paper but is intended to present a practical approach to analysis of real, and frequently marginal-quality, pre-fracture field test data. The techniques applied are based on some transient test theory. Some of the results presented here are still under debate and development. The methods shown have been tested and, we believe, proven in the analysis of hundreds of tests. Application of these methods provides consistent analysis that helps to avoid misinterpretation of falloff data, and give the most useful information available from diagnostic injection tests. Step-rate injection tests and their analysis are not included in the scope of this paper. Determination of the pressure- dependent leakoff coefficient is also not described here, as it SPE 107877 Holistic Fracture Diagnostics R.D. Barree, SPE, and V.L. Barree, Barree & Assocs. LLC, and D.P. Craig, SPE, Halliburton 2 SPE 107877 has been previously reported. 3,4 Only the analysis of pressure decline following shut-in of a fracture-rate injection test is considered. Transient Flow Regimes During and After Fracture Closure Several transient flow regimes may occur during a falloff test after injection at fracture rate. The major flow regimes are graphically illustrated in the classic paper by Cinco-Ley and Samaniego. 6 Immediately after shut-in the pressure gradient along the length of the fracture dissipates in a short-duration linear flow period. In a long fracture in low permeability rock the initial fracture linear flow can be followed by a bi-linear flow period with the linear flow transient persisting in the fracture while reservoir linear flow occurs simultaneously. After the fracture transient dissipates the reservoir linear flow period can continue for some time, depending on the permeability of the reservoir and the volume of fluid stored in the fracture and subsequently leaked off during closure. After closure the pressure transient established around the fracture propagates into the reservoir and transitions into elliptical, then pseudoradial flow. Each of these flow regimes has a characteristic appearance on various diagnostic plots. Fluid leakoff from a propagating fracture is normally modeled assuming one-dimensional linear flow perpendicular to the fracture face. Settari has pointed out that in some cases of moderate reservoir permeability the linear flow regime may not occur, even during fracture extension and early leakoff. 7 During fracture extension and shut-in the transient may already be in transition to elliptical or pseudoradial flow. In this case analyses based on an assumed pseudolinear flow regime will give incorrect results. In all cases an understanding of the flow regime and its relation to the fracture geometry is critical to arriving at a consistent interpretation of the fracture falloff test. Diagnostic Derivative Examples For each analysis technique various curves are used to help define closure, leakoff mechanisms, and after-closure flow regimes. On each plot the curves are labeled as the primary (y vs. x), the first derivative (∂y/∂x), and the semilog derivative (∂y/∂(lnx) or x∂y/∂x). For convenience the primary curve is plotted on the left y-axis and all derivatives are plotted on the right y-axis for all Cartesian plots. For the log-log plot all curves are shown on the same y-axis. For pre-closure analysis, and consistent identification of fracture closure, three techniques are illustrated for each example: G-function, Square-root of shut-in time, and log-log plot of pressure change with shut-in time. All these analyses begin at shut-in. The instantaneous shut-in pressure (ISIP) is taken as the incipient fracture extension pressure for all cases. When there is significant wellbore afterflow (fluid expansion or continued low-rate injection), or severe near-well pressure drop, the ISIP can be difficult to interpret accurately and may be too high to represent actual fracture extension pressure. In all the examples in the paper the pressures have been offset to an approximate ISIP of 10,000 psi to remove any relation to the original field test data. The following sections detail the data and analysis for the four major leakoff type examples. Normal Leakoff Behavior Normal leakoff is observed when the composite reservoir system permeability is constant. The reservoir may exhibit only matrix permeability or have a secondary natural fracture or fissure overprint in which the flow capacity of the secondary fracture system does not change with pore pressure or net stress. After shut-in the fracture is assumed to stop propagating and the fracture surface area open to leakoff remains constant during closure. Normal Leakoff G-Function As noted in previous papers, the expected signature of the G-function semilog derivative is a straight-line through the origin (zero G-function and zero derivative). 4 In all cases the correct straight line tangent to the semilog derivative of the pressure vs. G-function curve must pass through the origin. Fracture closure is identified by the departure of the semi-log derivative of pressure with respect to G-function (G∂p w /∂G) from the straight line through the origin. During normal leakoff, with constant fracture surface area and constant permeability, the first derivative (∂p w /∂G) should also be constant. 2 The primary p w vs. G curve should follow a straight line. 1 The example in Figure 1 shows some slight deviation from the perfect constant leakoff but is a good example of the expected curve shapes with a clear indication of closure at G c =2.31. The closure event is marked by the dashed vertical line [1]. 0 5 10 15 20 25 G(Time) 7500 8000 8500 9000 9500 10000 10500 0 200 400 600 800 1000 1200 1400 1600 1800 2000 1 P vs. G GdP/dG vs. G dP/dG vs. G Fracture Closure Pressure Derivatives Figure 1: Normal leakoff G-function plot Normal Leakoff Sqrt(t) Analysis The sqrt(t) plot has frequently been misinterpreted when picking fracture closure, even for the simplest cases. The primary p w vs. sqrt(t) curve should form a straight line during fracture closure, as with the G-function plot. Some users suggest that the closure is identified by the departure of the data from the straight line trend, similar to the way the G- function closure is picked. This is incorrect and leads to a later closure and lower apparent closure pressure. The correct indication of closure is the inflection point on the p w vs. sqrt(t) plot. The best way to find the inflection point is to plot the first derivative of p w vs. sqrt(t) and find the point of maximum SPE 107877 3 amplitude of the derivative. Many fracture-pressure analysis software packages plot the inverse of the actual first derivative and show the inflection point as the minimum of the derivative. The plot in Figure 2, shows that the slope of the pressure curve starts low, then increases and reaches a maximum rate of decline at the inflection point, then decreases again after closure. The first derivative curve in Figure 2 is plotted with the proper sign. The dashed vertical line [1] is the G-function closure pick that is synchronized in time and pressure with the sqrt(t) plot. Clearly the consistent closure lies at the inflection point and not at the point of departure from the straight line tangent to the pressure curve. The semilog derivative of the pressure curve is also shown on the sqrt(t) plot. This curve is equivalent to the semilog derivative of the G-function for most low-perm cases. The closure pick falls at the departure from the straight line through the origin on the semilog derivative of the P vs. sqrt(t) curve. A single closure point must satisfy the requirement on both the G-function and sqrt(t) plots. 1/24/2007 04:00 08:00 12:00 1/24/2007 16:00 Time 7500 8000 8500 9000 9500 10000 10500 0 100 200 300 400 500 600 700 800 900 1000 1 Pressure Derivatives P vs. √t √tdP/d√tvs. √t dP/d√tvs. √t Fracture Closure Figure 2: Normal leakoff sqrt(t) plot Normal Leakoff Log-Log Pressure Derivative The log-log plot of pressure change from ISIP versus shut- in time for the normal leakoff example is shown in Figure 3. The heavy curve is the pressure difference and the dashed curve is its semilog derivative with respect to shut-in time. The vertical dashed line is the unique closure pick from the G- function and sqrt(t) plot. It is common for the pressure difference and derivative curves to be parallel immediately before closure. The slope of these parallel lines is diagnostic of the flow regime established during leakoff before closure. In many cases a near-perfect ½ slope is observed, strongly suggesting linear flow from the fracture. In this example the slope is greater than ½ suggesting possible linear flow coupled with changing fracture/wellbore storage (See Appendix B). The separation of the two parallel lines always marks fracture closure and is the final confirmation of a consistent closure identification. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 0.1 1 10 100 1000 Time (0 = 8.15) 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 10 100 1000 (m = -1) (m = 0.632) BH ISIP = 9998 psi 1 Delta-Pressure and Derivative ∆P vs. ∆t ∆td∆P/d∆t vs. ∆t Fracture Closure Radial Flow Figure 3: Normal leakoff log-log plot After closure the semilog derivative curve will show a slope of -1/2 in a fully developed reservoir pseudolinear flow regime and a slope of -1 in fully developed pseudoradial flow. In the example the derivative slope is -1 indicating that reservoir pseudoradial flow was observed. The late-time data shows a drop in the derivative probably caused by wellbore effects such as gas entry and phase segregation. The use of the semilog derivative of the log-log plot for after-closure flow regime identification, as well as closure confirmation, is a powerful new addition to fracture pressure decline diagnostics. After-Closure Analysis for Normal Leakoff Example The Talley-Nolte After-Closure Analysis (ACA) flow regime identification plot for the normal leakoff example is shown as Figure 4. 5 The heavy solid line is the observed bottomhole pressure during the falloff minus the initial reservoir pressure. The slope of the semi-log derivative of the pressure difference function (dashed line) is 1.0 during the identified pseudoradial flow period. If a linear-flow period existed in this data set a derivative slope of ½ would exist. It is critical to remember that the slope of the pressure difference curve on this plot is determined solely by the guess of reservoir pressure used to construct the plot. The slope of the derivative is not affected by the input reservoir pressure value. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 0.001 0.01 0.1 1 Square Linear Flow (FL^2) 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 10 100 1000 10000 (m = 1) Delta-Pressure and Derivative ∆P vs. F L 2 F L 2 d∆P/dF L 2 vs. F L 2 ∆P=(p w -p r ) Start of Radial Flow Figure 4: Normal leakoff ACA log plot 4 SPE 107877 If a pseudoradial flow regime is identified, then the Cartesian Radial Flow plot (Figure 5) can be used to determine reservoir far-field transmissibility, kh/µ. The viscosity used is the far-field mobile fluid viscosity and h is the estimated net pay height. For the analysis of the example data kh/µ = 299 md-ft/cp. For gas viscosity at reservoir temperature, kh=7.9 md-ft. For the assumed net pay, the effective reservoir permeability is 0.097 md. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Radial Flow Time Function 7200 7400 7600 7800 8000 8200 8400 8600 8800 9000 9200 (m = 4814.2) Results Reservoir Pressure = 7475.68 psi Transmissibility, kh/µ = 298.94991 md*f t kh = 7.94014 md*ft Permeability, k = 0.0968 md Start of Pseudo Radial Time = 2.15 hours 1 Pressure Figure 5: Normal leakoff ACA radial flow plot Horner Analysis for Normal Leakoff Example If a pseudoradial flow period is identified, then a conventional Horner plot can also be used to determine reservoir transmissibility. In Figure 6 the Horner slope through the radial flow data is 14411 psi. Using an average pump rate of 18.4 bpm, kh/µ = 298 md-ft/cp. For the assumed gas viscosity kh=7.9 md-ft. Using the same assumed net gives k=0.097 md. This result is consistent with the ACA results. 2 3 1 Horner Time 7250 7500 7750 8000 8250 8500 8750 9000 9250 9500 9750 (m = 14411) (Reservoir = 7476) 1 Pressure Figure 6: Normal leakoff Horner Plot G-Function Permeability Estimate An empirical correlation has also been developed to estimate formation permeability from the G-function closure time when after-closure data is not available. The correlation is described in detail in the Appendix. Figure 7 shows the G- function correlation permeability estimate for the observed closure time and other input parameters. The permeability estimate of 0.097 md is consistent with the Horner and ACA results. 0. 001 0. 01 0. 1 1 10 100 0 2 4 6 8 101214161820 G c Data Input r p 1 φ 0.09 V/V c t 7.50E-05 psi - 1 E3.5Mpsi µ 1cp Gc 2.44 Pz 966.0 psi Estimated Permeability = 0.0974 md Permeability, m d Figure 7: Normal leakoff permeability estimate Pressure Dependent Leakoff Pressure dependent leakoff (PDL) occurs when the fluid loss rate changes with pore pressure or net effective stress in the rock surrounding the fracture. PDL is not caused by the normal change in transient pressure gradient during leakoff. This is part of the normal leakoff mode and is handled by the one-dimensional linear flow solution of the diffusivity equation used to model fracture leakoff in a constant permeability system. The pressure dependence referred to here is a change in the transmissibility of the reservoir fissure or fracture system that dominates the fluid loss rate. PDL is only apparent when there is substantial stress dependent permeability in a composite dual-permeability reservoir. G-Function for Pressure-Dependent Leakoff Figure 9 shows the G-function behavior expected for PDL. The primary p w vs. G curve is concave upward and curved while PDL persists. The semilog derivative exhibits the characteristic “hump” above the straight line extrapolated to the derivative origin. The end of PDL and the critical fissure opening pressure corresponds to the end of the “hump” and the beginning of the straight line representing matrix dominated leakoff. Fracture closure is still shown by the departure of the semilog derivative from the straight line through the origin. SPE 107877 5 0 2 4 6 8 10 12 14 16 18 20 G(Time) 8250 8500 8750 9000 9250 9500 9750 10000 10250 10500 0 100 200 300 400 500 600 700 800 900 1000 21 P vs. G GdP/dG vs. G dP/dG vs. G Fracture Closure Pressure Derivatives Figure 9: PDL G-function plot Sqrt(t) Analysis for PDL Interpretation of the sqrt(t) plot in PDL cases has often led to incorrect closure picks. Figure 10 shows an expanded view of the sqrt(t) plot for the example with the curves scaled for better visibility. Note that the semilog derivative is nearly identical in shape and information content to the G-function semilog derivative. It clearly shows the PDL “hump” and closure, which has been synchronized to the G-function result. Incorrect closure picks on the sqrt(t) plot will not occur if the semilog derivative is used. Problems arise when the first derivative is used exclusively to pick closure. In PDL cases the obvious derivative maximum, or most prominent inflection point, is caused by the changing leakoff associated with PDL and does not indicate fracture closure. The false closure indication is shown on the plot. Many fracture diagnostic tests have been badly misdiagnosed because the early and incorrect closure was picked because of dependence on only the sqrt(t) plot. This example clearly illustrates why all available diagnostic plots must be used in concert to arrive at a single consistent closure event. 1/24/2007 00:20 00:40 01:00 01:20 01:40 1/24/2007 02:00 Time 8250 8500 8750 9000 9250 9500 9750 10000 102 5 0 0 100 200 300 400 5 00 1 False Closure Pressure Derivatives P vs. √t √tdP/d√t vs. √t dP/d√t vs. √t Fracture Closure Figure 10: PDL Sqrt plot Log-Log Pressure Derivative for PDL Example Figure 11 shows the log-log plot for the PDL example. The normal matrix leakoff period, following the end of PDL, appears as a perfect ½ slope of the semilog derivative with a parallel pressure difference curve exactly 2-times the magnitude of the derivative. The parallel trend ends at the identified closure time and pressure difference. In this example a well-defined −½ slope, or reservoir pseudolinear flow period, is shown shortly after closure. The later data approach a slope of −1, which indicates pseudoradial flow has been established. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 0.1 1 10 100 1000 Time (0 = 9.133333) 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 10 100 1000 (m = 0.5) (m = -1) (m = -0.5) BH ISIP = 10000 psi 1 Pressure Difference and Derivative ∆P vs. ∆t ∆td∆P/d∆tvs. ∆t Fracture closure Linear Flow Radial Flow Figure 11: PDL log-log plot After-Closure Analysis for PDL Example The ACA log-log plot (Figure 12) shows both the reservoir linear and radial flow periods in their expected locations. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 0.001 0.01 0.1 1 Square Linear Flow (FL^2) 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 10 100 1000 10000 (m = 1) (m = 0.5) 123 Delta-Pressure and Derivative ∆P vs. F L 2 F L 2 d∆P/dF L 2 vs. F L 2 ∆P=(p w -p r ) Start Linear Flow End Linear Flow Start Radial Flow Figure 12: PDL ACA log plot Figures 13 and 14 show the ACA Cartesian plots for the linear and radial flow analyses. Both give consistent estimates of reservoir pore pressure. The pseudoradial flow analysis gives a transmissibility of 37.2 md-ft/cp and estimated permeability of 0.047 md. 6 SPE 107877 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Linear Flow Time Function 8000 8200 8400 8600 8800 9000 9200 (m = 1438.5) Results Reservoir Pressure = 8056.66 psi Start of Pseudo Linear Time = 15. 9 End of Pseudo Linear Time = 54.3 9 12 Pressure Start Linear Flow End Linear Flow Figure 13: PDL ACA linear flow plot 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Radial Flow Time Function 8000 8200 8400 8600 8800 9000 9200 (m = 11373) Results Reservoir Pressure = 8068.81 psi Transmissibility, kh/µ = 37.21984 m kh = 0.93764 md*ft Permeability, k = 0.0469 md Start of Pseudo Radial Time = 11.2 6 1 Pressure Figure 14: PDL ACA radial flow plot Horner Analysis for PDL Example For an average pump rate of 6.7 bpm the Horner plot gives kh/µ=35.72 md-ft/cp. The Horner estimated permeability is 0.046 md compared to 0.047 md from the ACA Radial Flow analysis. Pore pressure estimated from the Horner plot is also consistent with both the linear and radial analyses because a well-developed pseudoradial flow period does exist in this case. The vertical dotted line in Figure 15 shows the start of pseudoradial flow. If a pseudoradial flow period does not exist, extrapolation of an apparent straight-line on the Horner plot can give extremely inaccurate estimates of pressure and flow capacity. 1 Horner Time 8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 (m = 43920) (Reservoir = 8064) 1 Pressure Figure 15: PDL Horner plot G-Function Permeability Estimate for PDL Example The G-function permeability correlation for the PDL example is shown in figure 16. It also gives a consistent permeability of 0.045 md. The impact of the accelerated leakoff during PDL gives an estimate of the composite reservoir effective permeability. Note that the injected fluid viscosity is used for the permeability estimate based on closure time. 0. 001 0. 01 0. 1 1 10 100 0 2 4 6 8 101214161820 G c Data Input r p 1 φ 0.08 V/V c t 6.00E-05 psi - 1 E5Mpsi µ 1cp Gc 2.9 Pz 841.0 psi Estimated Permeability = 0.0453 md Permeability, md Figure 16: PDL permeability estimate Fracture Tip Extension In very low permeability reservoirs the decline in wellbore pressure observed after shut-in may be caused by the dissipation of the pressure transient established in the fracture during pumping. The near-well pressure decreases as the fracture closes, which results in a decrease of fracture width at the well. The closing of the fracture volumetrically displaces fluid to the tip of the fracture, causing continued extension of the fracture length. Much of the pressure decline is therefore not related to leakoff but to the dissipation of the linear transient along the fracture length. SPE 107877 7 G-Function Analysis for Tip Extension During fracture tip extension the G-function derivatives fail to develop any straight-line trends. The primary P vs. G curve is concave upward, as is the first derivative. The semilog derivative starts with a large positive slope and the slope continues to decrease with shut-in time, giving a concave-down curvature. 3,4 Figure 17 shows a typical case of fracture tip extension with minimal leakoff. This is another case that is frequently misdiagnosed. 0 5 10 15 20 25 30 G(Time) 8400 8600 8800 9000 9200 9400 9600 9800 10000 10200 0 25 50 75 100 125 150 1 P vs. G GdP/dG vs. G dP/dG vs. G Pressure Derivatives Figure 17: Tip extension G-function plot Sqrt(t) Analysis with Fracture Tip Extension Many times the first break in the semilog derivative curve has been misinterpreted as a closure event. The mistake is often compounded by the use of the sqrt(t) plot. Figure 18 shows the sqrt(t) plot for the same data. The first derivative shows a large maximum very shortly after shut-in. This is often mistaken for closure. The semilog derivative on the sqrt(t) plot helps to avoid this mistaken closure pick, and shows the same continuously increasing trend as seen on the G-function semilog derivative plot. In low permeability systems it is generally safe to assume that as long as the semilog derivative is still rising, the fracture has not yet closed. This is not true in very high permeability reservoirs and should always be checked using the log-log pressure difference plot. 1/25/2007 04:00 08:00 12:00 16:00 1/25/2007 20:00 Time 8400 8600 8800 9000 9200 9400 9600 9800 10000 10200 0 10 20 30 40 50 60 70 80 90 100 1 Incorrect Closure Pressure Derivatives P vs. √t xd P/d x vs. √t d P/d x vs. √t Figure 18: Tip extension sqrt(t) plot Log-Log Pressure Derivative Analysis with Tip Extension The log-log plot of pressure change after shut-in is particularly useful for diagnosing fracture tip extension. Figure 19 shows the pressure difference and pressure derivative (semilog) for the tip extension example. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 1 10 100 1000 Time (0 = 33.7) 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 10 100 1000 (m = 0.25) BH ISIP = 10000 psi 1 Delta-Pressure and Derivative ∆P vs. ∆t ∆td∆P/d∆t vs. ∆t Figure 19: Tip extension log-log plot In Figure 19 the pressure derivative departs from the early unit-slope (storage) and establishes a ¼ slope during fracture tip extension. The pressure difference curve falls on a parallel ¼ slope line separated by 4-times the magnitude of the derivative. The ¼ slope signature is diagnostic of bilinear flow representing a continued dissipation of the linear pressure transient along the fracture length (extension and concomitant fluid flow) and some linear flow driving minimal leakoff. For tip extension to occur the leakoff rate to the formation must be low. As long as the parallel ¼ slope trend continues, the fracture has not closed and is still in the process of extending. Closure cannot be determined and no after-closure analysis can be conducted. Height Recession or Transverse Storage There are two different mechanisms that can generate a similar diagnostic derivative signature during fracture closure. Both are caused by an excess stored volume of fluid in the fracture at shut-in relative to the expected surface area of the fracture for a planar, constant-height geometry model. Traditionally this signature has been called “fracture height recession”. The usual model assumes that leakoff occurs only through a thin permeable bed and that the fracture extends in height to cover impermeable strata with no leakoff. At shut-in there is a large volume of fluid stored in the fracture and the leakoff rate relative to the stored volume is small, hence the rate of pressure decline is likewise small. As the fracture empties, the rate of leakoff relative to the remaining stored fluid accelerates and the pressure declines more rapidly. If the fracture height changes during leakoff, the fracture compliance may also decrease, adding to the rate of pressure loss. However, the same signature is observed in many cases where fracture height growth out of zone is not observed by tracers, inclinometer, or micro-seismic mapping. Some of these cases show treating behavior similar to PDL cases, with 8 SPE 107877 a tendency for rapid screenout and difficulty placing high proppant concentration slurries. These observations suggest that another mechanism may be responsible for the same diagnostic derivative signature. The alternate mechanism is called “transverse fracture storage”. In transverse fracture storage a secondary fracture set is opened when the fluid pressure exceeds the critical fissure- opening pressure, just as in PDL. As the secondary fractures dilate they create a storage volume for fluid which is taken from the primary hydraulic fracture. While the fracture storage volume increases, leakoff can also be accelerated so PDL and storage are aspects of the same coupled mechanism of fissure dilation. The relative magnitude of the enhanced leakoff and storage mechanisms determines whether the G-function derivatives show PDL or storage. Numerical modeling studies indicate that the storage mechanism can easily dominate even large PDL. At shut-in the secondary fractures will close before the primary fracture because they are held open against a stress higher than the minimum in-situ horizontal stress. As they close fluid will be expelled from the transverse storage volume back into the main fracture decreasing the normal rate of pressure decline and, in effect, supporting the observed shut-in pressure by re-injection of stored fluid. Accelerated leakoff can still occur at the same time but if the storage and expulsion mechanism exceeds the enhanced leakoff rate then the only signature observed during falloff will be storage. In many cases a period of linear, constant area, constant matrix permeability dominated leakoff will occur after the end of storage. G-Function Analysis with Storage The characteristic G-function derivative signature is a “belly” below the straight line through the origin and tangent to the semilog derivative of p w vs. G at the point of fracture closure. Figure 20 shows an example of slight to moderate storage. In Figure 20 fracture closure, indicated by the same departure of the tangent line from the semilog derivative, occurs just after the end of the storage effect. 5 10 15 20 G(Time) 7500 8000 8500 9000 9500 10000 10500 0 200 400 600 800 1000 1200 1400 1600 1800 2000 1 P vs. G GdP/dG vs. G dP/dG vs. G Fracture Closure Pressure Derivatives Figure 20: Storage G-function plot Sqrt(t) Analysis with Storage or Height Recession The sqrt(t) plot (Figure 21) shows a clear indication of closure based on both the first-derivative inflection point and the semilog derivative curve. Picking closure in the case of storage is not generally a problem. 1/24/2007 02:00 04:00 06:00 08:00 1/24/2007 10:00 Time 7500 8000 8500 9000 9500 10000 10500 0 100 200 300 400 1 Pressure Derivatives P vs. √t xdP/dx vs. √t dP/dx vs. √t Fracture Closure Figure 21: Storage sqrt(t) plot The storage model, whether caused by height recession or transverse fractures, requires that a larger volume of fluid must be leaked-off to reach fracture closure than is expected for a single planar constant-height fracture. In either case the time to reach fracture closure is delayed by the excess fluid volume that must be lost. Any estimation of reservoir permeability will give an incorrect result if the uncorrected closure time (either G c or time-to closure in minutes, t c ) is used. The observed closure time must be corrected by multiplying by the storage ratio, r p . The magnitude of r p can be determined by taking the ratio of the area under the G-function semilog derivative up to the closure time, divided by the area of the right-triangle formed by the tangent line through the origin at closure. For normal leakoff and PDL the value of r p is set to 1.0 even though the ratio of the areas will be greater than 1 for the PDL case. It is possible that the closure time for PDL leakoff is proportional to the composite system permeability including both the matrix and fractures. For severe cases of storage r p can be as low as 0.5 or less. Log-Log Pressure Derivative with Storage Figure 22 shows the log-log plot of pressure difference and semilog derivative for the storage case. Prior to closure, and while transverse storage is dominant, the semilog derivative approaches a unit slope, with the pressure difference curve nearly parallel. In some cases the two curves lie together on a single unit-slope line. In this case the curves are separated slightly and the slope is not exactly 1.0. After closure the reservoir transient signature is defined as in the previously presented cases. All fracture storage effects are eliminated and the reservoir pseudolinear flow period is shown by a -1/2 slope with a pseudoradial flow period indicated by a -1 slope of the semi-log derivative. SPE 107877 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 0.1 1 10 100 Time (0 = 9.416667) 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 10 100 1000 (m = 0.5) (m = -0.5) BH ISIP = 10000 psi 1 Delta-Pressure and Derivative ∆P vs. ∆t ∆td∆P/d∆t vs. ∆t Figure 22: Storage log-log plot Conclusions The use of pre-frac injection/falloff diagnostic tests has become commonplace. Many important decisions regarding fracture treatment designs and expectations of post-frac production are based on the results of these tests. In too many cases individual diagnostic plots and analysis techniques are misapplied, leading to incorrect interpretations. The analyses presented here lead to the following conclusions: 1. With consistent application of all available pressure decline diagnostics, a single unambiguous determination of fracture closure time and pressure can be made. 2. A single, unique closure event can be identified on all diagnostic plots. 3. The conventional analysis of the sqrt(t) plot, using the inflection point identified by the first derivative, gives incorrect indications of closure for cases of PDL and tip extension and should not be relied upon. 4. A modified sqrt(t) analysis, using the semilog derivative, is equivalent to the G-function analysis and helps avoid incorrect closure picks in cases of PDL and tip extension. 5. Flow regimes can be identified using the semilog pressure derivative on the log-log plot of ∆p wf − ∆t during the shut- in period following the fracture injection test. 6. As in conventional transient test analysis, a pseudolinear flow period is identified by parallel ½ slope lines, separated by 2x, on the log-log ∆p wf − ∆t plot up until fracture closure. 7. Bilinear flow can be identified by parallel ¼ slope lines separated by 4x on the log-log ∆p wf − ∆t plot prior to fracture closure. 8. After closure the pseudolinear reservoir flow period is identified by a -1/2 slope of the semilog derivative of the pressure difference on the log-log ∆p wf − ∆t plot, and a −3/2 slope of the first derivative of the pressure difference with shut-in time on the same plot. 9. Pseudoradial flow is identified by a -1 slope of the semilog derivative on the log-log plot. 10. When a stable pseudolinear flow period exists, the after- closure Cartesian plot of the linear flow function can be used to estimate reservoir pressure. 11. When a pseudoradial flow period exists, both the conventional Horner analysis and after-closure radial- flow analysis can be used to determine reservoir transmissibility and pore pressure. Acknowledgements The authors would like to thank Kumar Ramurthy, Halliburton, Mike Conway, Stim-Lab, and Stuart Cox, Marathon, for their discussion and contribution to the procedures described. Sincere thanks are also due to the many operators whose diligence in pre-frac testing has allowed these diagnostic analysis procedures to be developed and tested. Nomenclature A f = fracture area, L 2 , ft 2 B = formation volume factor, L 3 /L 3 , RB/STB c t = total compressibility, Lt 2 /m, psi -1 C ac = after-closure storage, L 4 t 2 /m, bbl/psi C bl = bilinear flow constant, m/Lt 5/4 , psihr 3/4 C pl = pseudolinear flow constant, m/Lt 3/2 , psihr 1/2 C pr = pseudoradial flow constant, m/Lt, psihr C fbc = before-closure fracture storage, L 4 t 2 /m, bbl/psi F L = linear flow time function, dimensionless F R = radial flow time function, dimensionless g = loss-volume function, dimensionless G = G-function, dimensionless h = height, L, ft k = permeability, L 2 , md L f = fracture half-length, L, ft m H = slope of data on Horner plot, m/Lt 2 , psia m L = slope of data on pseudolinear flow graph, m/Lt 2 , psia m R = slope of data on pseudoradial flow graph, m/Lt 2 , psia p = pressure, m/Lt 2 , psia p wf = fracture pressure measured at wellbore, m/Lt 2 , psia q = flow rate, L 3 /t, bbl/D Q t = total injection volume, L 3 , bbl r p = storage ratio, dimensionless S f = fracture stiffness, m/L 2 t 2 , psi/ft t = time, hr t a = adjusted pseudotime, hr Greek  = constant, dimensionless  = difference, dimensionless  = constant, dimensionless µ = viscosity, m/Lt, cp φ = porosity, dimensionless Subscripts a = adjusted c = closure D = dimensionless e = end of injection f = filtrate p = pumping 0 = end of injection w = wellbore z = process zone 10 SPE 107877 References 1. Nolte, K. G.: “Determination of Fracture Parameters from Fracturing Pressure Decline, paper SPE 3841, presented at the Annual Technical Conference and Exhibition, Las Vegas, NV, Sept. 23-26, 1979. 2. Castillo, J. L.: “Modified Fracture Pressure Decline Analysis Including Pressure-Dependent Leakoff,” paper SPE 16417, presented at the SPE/DOE Low Permeability Reservoirs Joint Symposium, Denver, CO, May 18-19, 1987. 3. Barree, R. D., and Mukherjee, H.: “Determination of Pressure Dependent Leakoff and Its Effect on Fracture Geometry,” paper SPE 36424, presented at the 71st Technical Conference and Exhibition, Denver, CO, Oct. 6-9, 1996. 4. Barree, R.D.: "Applications of Pre-Frac Injection/Falloff Tests in Fissured Reservoirs—Field Examples," paper SPE 39932 presented at the SPE Rocky Mountain Regional/Low- Permeability Reservoirs Symposium, Denver, Apr. 5-8, 1998. 5. Talley, G. R., Swindell, T. M., Waters, G. A. and Nolte, K. G.: “Field Application of After-Closure Analysis of Fracture Calibration Tests,” paper SPE 52220, presented at the 1999 SPE Mid-Continent Operations Symposium, Oklahoma City, OK, March 28–31, 1999. 6. Cinco-Ley, H., and Samaniego-V., F.: “Transient Pressure Analysis for Fractured Wells,” JPT (September 1981) 1749. 7. Settari, A.: “Coupled Fracture and Reservoir Modeling,” presented at the Workshop on Three Dimensional and Advanced Hydraulic Fracture Modeling, held in conjunction with the Fourth North American Rock Mechanics Symposium, July 29, 2000, Seattle, WA. 8. Craig, D. P. and Blasingame, T. A.: “Application of a New Fracture-Injection/Falloff Model Accounting for Propagating, Dilated, and Closing Hydraulic Fractures,” paper SPE 1005778 presented at the SPE Gas Technology Symposium, Calgary, Alberta, Canada, May, 15-17, 2006. 9. Hagoort, J.: "Waterflood-Induced Hydraulic Facturing," PhD Thesis, Delft Technical University, 1981. 10. Koning, E.J.L. and Niko, H.: "Fractured Water-Injection Wells: A Pressure Falloff Test for Determining Fracture Dimensions," paper SPE 14458 presented at the 1985 Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Las Vegas, NV, September, 22-25, 1985. 11. Cinco-Ley, H., Kuchuk, F., Ayoub, J., Samaniego-V, F., and Ayestaran, L.: "Analysis of Pressure Tests Through the Use of Instantaneous Source Response Concepts," paper SPE 15476 presented at the 61 st Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, New Orleans, LA, October, 5-8, 1986. Appendix A - Definition of diagnostic functions The G-Function The G-function is a representation of the elapsed time after shut-in normalized to the duration of fracture extension. Corrections are made for the superposition of variable leakoff times while the fracture is growing. The form of the G-function used in this paper assumes high fluid efficiency in low-permeability formations. Under that assumption the surface area of the fracture is assumed to vary linearly with time during fracture propagation. The dimensionless pumping time used in the G-function is defined as: ( ) / Dpp tttt∆=− . (A-l) The elapsed total time from the start of fracture initiation (not start of pumping) is t and the total pumping time (elapsed time from fracture initiation to shut-in) in consistent time units is t P . For the assumption of low leakoff the dimensionless time (∆t D ) is used to compute an intermediate function: () () [ ] 5.15.1 1 3 4 DDD tttg ∆−∆+=∆ . (A-2) The G-function used in the diagnostic plots is derived from the intermediate function as follows: () () 0 4 DD Gt gt g π ∆ =⎡∆ − ⎤ ⎣ ⎦ , (A-3) where g 0 is the dimensionless loss-volume function at shut-in (t = t p or ∆t D = 0). All derivatives are calculated using a central difference function of pressure and G-function (normalized shut-in time). After-Closure Analysis and Flow Regime Identification After-closure pressure decline analysis requires the identification of fully-developed reservoir pseudolinear and pseudoradial transient flow regimes. The flow regimes can be identified by characteristic slopes on a log-log plot of observed falloff pressure minus reservoir pressure, (p w (t) − p i ), versus the square of the linear-flow time function (F L 2 ) and the semilog derivative, (X*dY/dX), of the pressure difference curve. 5 It is important to note that the guess of reservoir pressure, p i , used in construction of the flow regime plot severely impacts the slope and magnitude of the pressure difference curve. The pressure derivative, because of the difference function used to generate it, is not affected by the initial guess of reservoir pressure. The linear-flow time function is defined by: () c c cL ttfor t t ttF ≥= −1 sin 2 , π . (A-4) The linear-flow function also requires an accurate determination of the time required after shut-in to reach fracture closure, t c . In the pseudolinear flow period the slope of the derivative curve on the log-log plot should be ½. For the correct estimate of reservoir pore pressure, the pressure difference curve should also have a slope of ½ and should be exactly twice the magnitude of the derivative. If a stable pseudolinear flow period is identified then a Cartesian plot of observed pressure during the falloff, p w (t), versus F L should yield a straight line with intercept equal to the reservoir pore pressure, p i , and with a slope of m L . ( ) ( ) , wiLLc p tpmFtt−= . (A-5) If a pseudoradial flow period exists, the slope of the derivative and correct pressure difference curves on the log- log flow regime plot should both be 1.0 and the two curves should coincide. In the pseudoradial flow period, a Cartesian plot of pressure versus F R should also yield a straight line with intercept equal to p i and slope of m R . [...]... = te + t and pw(te) is the instantaneous shut-in pressure Define a fracture- pressure difference as pwf 141.2(24) Qt 2 kh C pr , (B-6) where Af is the fracture area (one wing) and Sf is the fracture stiffness Hagoort's solution predicts that the before-closure falloff is a combination of fracture storage and linear flow As fracture storage becomes small, CfbcD 0, linear flow will dominate the... empirical function to approximate formation permeability has been derived from numerous numerical simulations of fracture closure The correlation is based on the observed G-function time at fracture closure: 0.0086 f k ct 0.01 pz 1.96 Gc E rp (A-10) 0.038 The rate of fluid loss from the fracture before closure is dominated by the mobility of the injected fluid instead of the far-field viscosity... relative permeability in the invaded zone Note that the correlation gives permeability and not kh The time to closure is related to fracture volume versus created area so both the fracture height and length do not appear in the equation The process zone stress, Pz, is the net fracture extension pressure above closure pressure, pc, or pz = pISIP pc The net extension pressure and Young’s Modulus provide... parameters in Equation A-10 are defined, except the storage ratio, rp This parameter represents the amount of excess fluid that must be leaked-off to reach fracture closure when the fracture geometry deviates from the normally assumed constant-height planar fracture The storage ratio, rp, is the ratio of the area under the G-function semilog derivative up until closure divided by the area of the triangle... Pseudoradial, Pseudolinear, and Bilinear Flow AfterClosure Craig & Blasingame8 developed an analytical solution for a fracture- injection/falloff sequence with a propagating and closing hydraulic fracture, and they derived the "complete" after-closure impulse solutions accounting for fracture storage The after-closure impulse solution for pseudoradial flow is written as pw (te t) pi 141.2(24) Qt kh 2... time to fracture closure with time zero set as the beginning of fracture extension, pi is the initial reservoir pressure, and mR is the Cartesian slope of the “correct” straight line The radial-flow function (FR) is given by5 FR t , t c tc 1 ln 1 , t tc 4 16 2 1.6 (A-7) In the properly identified pseudoradial flow period the reservoir far-field transmissibility can be determined from the slope, fracture. .. rate in Equation A-9 is assumed to be in barrels per minute and is the average rate for the time the fracture was extending In both Equations A-8 and A-9, the viscosity is the far-field mobile fluid viscosity The propagation of the transient in pseudoradial flow occurs at a great distance from the fracture and is not affected by the injected fluid viscosity The major problem with the Horner analysis... hL2 f (B-5) ct hL2 f Here, Cfbc is the before-closure fracture storage constant, which is defined as8 C fbc 2 Af 5.615 S f pw (te t) t , (B-7) pw (te ) pw (te t) , (B-8) and a log-log graph of pwf vs t will exhibit a ½ slope during linear flow before closure An analytical before-closure bilinear flow solution accounting for fracture tip extension during shut-in does not exist, but... is therefore 1.0 For the transverse storage and height recession signature, rp is some value less than 1.0 The observed G-function closure time (Gc) is always delayed by the excess fluid volume in the fracture for either height recession or transverse storage Likewise the apparent time to closure used in the after-closure analysis will be delayed and will cause errors in the reservoir transmissibility... t2, respectively Table B-1 shows the derivative terms for each after-closure flow regime Log-Log Diagnostic Graph Before-closure linear or bilinear flow are identified by the relationship between the fracture- pressure difference, pwf, and total time, t, but the after-closure flow regimes are identified by the relationship between the reservoir-pressure difference, pw, and t, which requires knowing . established in the fracture during pumping. The near-well pressure decreases as the fracture closes, which results in a decrease of fracture width at the well. The closing of the fracture volumetrically. by natural or induced secondary fractures or fissures. The third example shows fracture tip extension after shut-in. These cases generally show definable fracture closure. The fourth example. fracture-rate injection test is considered. Transient Flow Regimes During and After Fracture Closure Several transient flow regimes may occur during a falloff test after injection at fracture