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1 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs Application of Recently Developed Time-Rate Relations V. Okouma, Shell Canada Energy, D. Symmons, Consultant, N. Hosseinpour-Zonoozi, D. Ilk, DeGolyer and MacNaughton, and T.A. Blasingame, Texas A&M University Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Hydrocarbon, Economics, and Evaluation Symposium held in Calgary, Alberta, Canada, September This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract The application of "Decline Curve Analysis" (DCA) in unconventional reservoirs is almost always problematic. The Arps relations (hyperbolic and exponential relations) have been the standard for evaluating estimated ultimate recovery (EUR) in petroleum engineering applications for more than 80 years. However; with the pursuit of low and ultra-low permeability plays, these relations often yield ambiguous results due to invalid assumptions (e.g., existence of the boundary-dominated flow regime, presumption of a constant bottomhole pressure, etc.). Misapplications of the Arps' relations to production data exhibiting long-term, transient flow generally results in significant overestimates of reserves specifically when the hyperbolic relation is extrapolated unconstrained, using an Arps b-value greater than 1. We note that the "modified hyperbolic" relation one with an initial (unconstrained) hyperbolic trend used during early times, coupled with an exponential decline trend using a standard terminal decline can be used effectively (with proper care) for predicting EUR and production extrapolations. However; we note that this approach is "non-unique" in the hands of most users, and often yields widely varying estimates of reserves with time, and/or "consistent" estimates of reserves, which are highly biased. In short, the modified hyperbolic relation can be effectively applied to production data from low/ultra-low permeability reservoirs systems, these analyses must be based on diagnostic interpretations of the data (as we have proposed earlier [Ilk et al. (2008)]), where multiple data functions are used to define the analyses. The use of diagnostics is a necessary, not a sufficient condition the underlying models must be able to characterize the selected flow regimes, and there must also be constraints applied to production extrapolations and EUR predictions. The issues related to the use of Arps' rate decline relations have led various authors [Ilk et al. (Power Law Exponential, 2008), Valkó (Stretched Exponential, 2009), Clark et al. (Logistic Growth Model, 2011), and Duong (2011)] to propose various rate decline relations which attempt to properly model the time-rate behavior specifically early transient and transitional flow behavior. However, none of these equations can be considered sufficient to forecast production for all unconventional plays, due to the characteristics and operational conditions of each play and the behavior of the time-rate equation. In other words, one equation could work very well in a specific play, but could possibly perform poorly in another play. Under these circumstances, it is critical to understand the behavior of each equation, and to apply these relations appropriately for production forecasts. This work presents guidelines for the application of the various time-rate relations currently being deployed in the petroleum industry. The results of time-rate analyses of wells from three different plays are presented, and the advantages/ disadvantages of each time-rate relation are discussed. Ultimately, our goal in this work is to define and demonstrate a process for the proper application of the time-rate analyses typically performed for production forecasting and EUR prediction.

2 2 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Introduction The starting point for any discussion of decline curve analysis (DCA) for unconventional reservoirs (e.g., tight gas, shale gas, liquid-rich shales, and coalbed methane) must be an understanding that no simplified time-rate model can accurately capture all elements of the performance behavior. In addition, no time-rate model can be expected to provide a completely unique forecast of future performance or prediction of EUR. We must be both realistic and practical when attempting to characterize production performance from systems where the permeability is on the order of nd (or 10x10-6 to 500x10-6 md) the reservoir flow system is complex (we are not actually certain what is flowing from where) and although the induced (i.e., created) hydraulic fracture system enables (and dominates) the production performance, we have only the most rudimentary understanding of the flow structure in the hydraulic (and natural) fracture systems. It is essential that we establish these conditions as a starting point to not do so will inevitably lead the analyst to interpretations based on incorrect assumptions (e.g., boundary-dominated flow) as well as significant bias (e.g., the "expectation" that a particular regime has a given character, regardless of the flow conditions in the reservoir-well system such as large volumes of flowback water and backpressures being induced in the horizontal and vertical portions of the well). These issues are not insurmountable, we believe that reasonable production forecasts and predictions of EUR can be made, but not in isolation, not solely looking at the data and the selected time-rate model the analyst must consider the nature of the resource and the significant uncertainty in our ability to apply simple time-rate relations to a very complex reservoir system. As background, we need to look at the historical development of time-rate relations, obviously focusing on modern developments, but also the rationale for such developments. In the studies provided by Rushing et al. [2007] and Lee and Sidle [2010] these authors showed that the unconstrained use Arps' hyperbolic rate relation (particularly for cases where the b-values are greater than 1) can and almost always does yield significant overestimates of reserves. The b>1 point requires discussion and clarification the possible flow regimes encountered at early times for a horizontal well with multiple hydraulic fracture stages (as many as 30) are summarized as:! Linear Flow: (1:2 slope) a q( = LF t (very high conductivity vertical fractures)! Bilinear Flow: (1:4 slope) a q( = BLF 4 t (low/very low conductivity vertical fractures)! Multi-Fracture Flow: (1:3 slope) a q( = MFF 3 t (observed occasionally in practice and from simulations with multiple sets of vertical and horizontal fractures) We note that all of these are "power law" flow regimes, where rate is related to time raised to an exponent. This is an important distinction (and perhaps even coincidence) as the Arps hyperbolic time-rate relation can be reduced to a power law form under certain circumstances. This ability of the Arps hyperbolic time-rate relation to represent a power-law flow regime has led to this relation being used (often incorrectly) to analyze and extrapolate early-time production data. Recalling the Arps hyperbolic time-rate relation, we have: [Arps (1945), Johnson and Bollens (1927)] qi q( =... (1) 1/ (1 + bd b i By inspection, if we substitute:! b=2 into Eq. 1 (and assume that bd i t >> 1); then we obtain the square-root time relation (linear flow)! b=4 into Eq. 1 (and assume that bd i t >> 1); then we obtain the fourth-root time relation (bilinear flow)! b=3 into Eq. 1 (and assume that bd i t >> 1); then we obtain the third-root time relation (multi-fracture flow) In simple terms, if these power-law flow regimes are represented by the Arps hyperbolic time-rate relation, then extrapolation of the Arps hyperbolic relation will almost always tend to significant overestimations of EUR and performance extrapolations. As the Arps hyperbolic relation does often represent the early-time flow behavior, the industry has adopted a protocol to "constrain" the ultimate extrapolation by "splicing" a terminal exponential decline trend hence, the "modified hyperbolic" designation. Again, we recognize that this is practice-based approach and we appreciate that the modified hyperbolic time-rate relation can be used effectively, we simply have concerns about bias and about the significant potential for overestimation using this approach.

3 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 3 As an attempt to better represent the general character of time-rate production data for a multi-stages-fractured horizontal well in an ultra-low permeability reservoir, numerous authors have developed time-rate relations using certain specific bases in order to best represent a particular scenario. These developments include the following time-rate relations:! Power-Law Exponential Model [Ilk et al. (2008, 2009) a similar form was proposed independently by Jones (1942)]! Stretched Exponential Model [Valkó (2009) also Kisslinger (1993) and Kohlrausch (1854)]! Logistic Growth Model [Clark et al. (2011)]! Duong Model [Duong (2011)] Each relation has its own strengths, and in absolute fairness, at this time, each/all of these models can only be described as empirical, there is no direct link with reservoir engineering theory, other than via analogy. For example, the Stretched Exponential model is essentially an infinite sum of exponentials, so in as an analog, the concept of adding the rigorous exponential decline to some limit could be thought to "define" this model. The Power-Law Exponential model is essentially the same as the Stretched Exponential model (except for a constraining variable [D! ]), and this relation was derived exclusively from the observed behavior of D( and b( however; there may be a basis for this model from theory (e.g., for the variable-compressibility case), but this has not been proven, only inferred. At this point, we must assume that the proposed models are essentially empirical in nature, and generally center on a particular flow regime and/or characteristic behavior. Our workflow for this paper is as follow:! To apply the "Db," "!-derivative," and "q/g p " diagnostic plots to each data.! To apply each model to a given data set and provide: EUR predictions Production projections! To compare EUR predictions obtained from each model by investigating model behavior Orientation to Field Case Data In this work, we focus mainly on three different shale gas plays in North America. The first play (Field A) is a formation composed of siltstone and dark grey shale, with dolomitic siltstone in the base and fine grained sandstone towards the top. Particularly the formation of interest is a highly unusual " ft thick package of continuous, gas charged siltstone with a very small clay content. The formation is slightly over pressured with pressure gradients " psi/ft. Field B is a black, organic rich shale of Upper Jurassic age, which is deposited with mainly heavier clay minerals, silica, and calcite. The depth of the Field B ranges from approximately 10,000 ft in the northwest part to 14,000 ft in the southeast. It is overpressured with pressure gradients higher than 0.9 psi/ft. Field C is a Middle Devonian age black, low density, organic rich shale at an approximate average depth of 5,000-6,000 ft. We also evaluate performance from a tight-gas well in East Texas which has a very well-defined production character. These cases are provided largely as examples which have very long-term performance, and should provide unique diagnostics, analyses, interpretations, and predictions. Time-Rate Analysis Relations The basic definitions and diagnostic functions for time-rate analyses are given as: 1 dq( D( "! (Definition of the decline parameter)... (1) q( dt 1 q( "! (Definition of the loss-ratio)... (2) D ( dq( / dt d & 1 # d & q( # b( ( $! ( ' $! (Derivative of the loss-ratio)... (3) dt % D( " dt % dq( / dt " 1 dq( " ( t )! t t D( q( dt! ("Beta" function relates rate and derivative function)... (4) A complete summary of the time-rate analysis relations is provided in Appendix A, the fundamental "rate" relations for each case considered in this work are provided below for reference and orientation.

4 4 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE qi q( = [Modified-Hyperbolic Relation valid for t<t exp ]... (5) 1/ (1 + bd b i q( = qi exp[! Di t] [Exponential Relation valid for t> t exp ; D i = D lim ]... (6) ˆ exp[ ˆ n q( = q " D t " D t] [Power-Law Exponential Model (PLE)]... (7) i i! n q ( = qˆ i exp[ "( t /! ) ] [Stretched Exponential Model (SE)]... (8) ' & ' # ( ) = mdng adng (1 ) q t q1 t exp$ [ t mdng ' 1]! [Duong Model (DNG)]... (9) $ % (1 ' mdng )!" q( = dg ( ) ( n 1) p t LGM! K nlgm algm t = dt nlgm [ a ] 2 LGM + t [Logistical Growth Model (LGM)]... (10) These relations are formulated as diagnostic relations (primarily using qdb-plots) and are used to make long-term rate projections as well as predictions of estimated-ultimate-recovery (EUR). As a matter of process, any given relation is calibrated against the historical rate and cumulative data using a diagnostic approach and the model extrapolations are made only from the end of the data (not the body of the data). This approach ensures that all extrapolations/projections are based on the actual (not model-based) cumulative production. Diagnostics and Characteristic Time-Rate Behavior In this section we present the characteristic time-rate performance from each play. We present six wells from Field A and Field B, and five wells from Field C. The primary objective of this effort is to demonstrate time-rate behavior of the wells using diagnostic plots without performing analysis corresponding to each play. As shown earlier utilized diagnostic plots are D and t, b and t,! and t, q/g p and t. Diagnostic plots have significant importance in our applications as these plots provide direct insight into our understanding of decline behavior. For example, a straight line trend of the continuously evaluated D-parameter [i.e., D(] versus t on log-log scale could indicate power-law behavior which would yield the "power-law exponential" (or "stretched exponential") function when the ordinary differential equation (Eq. 1) is solved for the rate function. Furthermore, from the continuous evaluation of the b- parameter, it is possible to verify the hyperbolic behavior. A constant b-parameter trend [i.e., b( = constant] suggests hyperbolic rate decline behavior; and as such, it is possible to establish the value of b-parameter in the hyperbolic equation. In addition, a constant!-derivative trend verifies "power-law" flow regimes such as linear and/or bi-linear flow. These diagnostic functions involve differentiation of time-rate data, and therefore errors and inconsistencies associated with the data are amplified in the derivative functions, which may prevent the analyst from establishing a unique interpretation. The diagnostic plot of q/g p and t [as suggested by Duong (2011)] provides significant diagnostic value as it does not include any numerical differentiation and it serves as a complementary diagnostic tool to the previously mentioned diagnostic plots. As a matter of process, we prefer to use all of the diagnostic plots in an effort to understand data characteristics prior to performing time-rate analysis. We believe this procedure decreases the uncertainty associated with forecasts and offers confidence in time-rate analysis results. From another point of view, diagnostic plots are particularly useful while performing time-rate analysis. Each time-rate relation has more than two model parameters and it is generally difficult to establish the values directly from production rate data. In particular, the log[d(] versus log[t] plot is used to establish the power-law exponential and stretched exponential model parameters (i.e., Dˆ i, n) as these parameters are related with the slope and intercept values on this log-log plot. Also, the model parameters for Duong's equation (i.e., a Dng and m Dng ) can be estimated using the log[q/g p ] versus log[t]. The carrying capacity parameter (K) of the logistic growth model is eliminated from the model if the derivative functions (and plots) are used (as indicated in Appendix A).

5 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 5 Our general procedure for time-rate analysis is to simultaneously use the diagnostic plots and calibrate the parameters of each model until an optimum (visual) match is achieved. This procedure ensures consistency in the analysis and prevents the nonuniqueness associated with simply matching a single variable (e.g., only matching the model with the production rate). We observe in industry the increasing use of regression as a primary tool for matching data we believe that sole use of regression (without any diagnostic guidance) for time-rate analysis will eventually result in non-uniqueness and any forecast/ results will be prone to errors. We only suggest the use of regression as a final step to further refine model parameters once a diagnostic interpretation is complete and model parameters are calibrated using the diagnostic plots. As mentioned earlier, we utilize time-rate data from three different plays for our purposes. We demonstrate the characteristic time-rate behavior associated with each play using multi-well cross-plots. We begin with Field A. In Fig. 1 we present the time-rate behavior of six wells producing in Field A. Shallower decline behavior and dominantly "power-law" type flow regimes (e.g., 1/2 slope, 1/3 slope) are observed throughout production history. When data are plotted on the log[q/g p ] versus log[t] plot (see Fig. 2), almost all wells exhibit almost identical behavior (i.e., we observe an apparent linear trend on log-log scale). The log[d(] versus log[t] data are presented in Fig. 3 (left axis), and we observe that certain (but not major) differences exist in the slope values of these wells which could be related with production characteristics (e.g., permeability, fracture half-length, etc.). The log[b(] versus log[t] data are presented in Fig. 3 (right axis) and these data suggest that the hyperbolic relation could be applicable to model time-rate data as the b( trend exhibits a very gradual decrease with time and we could reasonably assume a constant b-value, in the 2-3 range. In Fig. 4 we present the log[!(] versus log[t] trend and we note a stabilization of data with time, which suggests that "power-law" type flow regimes are being established. In conclusion, we complete our diagnostic interpretation of time-rate behavior of wells in Field A with the remark that time-rate behavior is being dominated by "power-law" type flow regimes. Our next example presents six wells from Field B. Contrary to Field A; the time-rate data for the wells in Field B exhibit steep decreases that is, the log[q(] versus log[t] trends (Fig. 5) are close to unit slope after 100 days, indicating "depletion"-type behavior. The log[q/g p ] versus log[t] trends are shown in Fig. 6; and it is not clear that a linear trend ever fully exists for these cases (as is required by the Duong method [Doung (2011)]). This behavior is discussed in more detail in the next section where the performance for each well is analyzed using each of the time-rate models. Due to somewhat random oscillations in the rate data, the resolution of the log[d(] and log[b(] versus log[t] data trends (Fig. 7) yields only qualitative character the data trends are quite erratic, but the D( trend does suggest linearity, hence the power-law exponential time-rate model may be the most appropriate model for these cases. The b( data trends are more difficult to interpret; but at least we do observe that on average, the b( trends are not as high as those observed in Field A we note that most of the b( trends are in vicinity of unity (or less). In Fig. 8 we present the log[!(] versus log[t] trends, and although the trends shown on Fig. 7 were quite erratic, the trends in Fig. 8 appear well-behaved, and this observation suggests that transient "power-law" type flow regimes likely do not exist for the wells in Field B, except to note that most (or all) trends shown on Fig. 8 appear to be converging to unity at long times, which validates the "depletion" (i.e., unit slope) observation. Our last example includes 5 wells from Field C, and from the time-rate behavior shown in Fig. 9, we can consider Field C to be a possible analog of Field A however; the time-rate trends for Field C do not achieve a "half-slope" character as expected for high conductivity vertical fractures. It is possible that the "power-law"-type flow regimes exist but are not dominant. In Fig 10 we present the log[q/g p ] versus log[t] data, and we observe almost identical character to Field A (i.e., strongly linear trends). As shown in Fig. 11 (left axis), the log[d(] versus log[t] trends are strongly linear and quite consistent, verifying the likelihood of the viability of the power-law exponential time-rate model. Also on Fig. 11 (right axis) we present the log[b(] versus log[t] data trends, and in these cases we note consistently declining (apparently linear) trends of b( which also validates the power-law exponential concept. On the other hand, we could suggest the possibility of using an average b-value if hyperbolic relation where to be forced on these data (it is improbable that a constant b-value would represent the observed behavior, but we are simply suggesting that this could be done given the uncertainty in some of the b( data). In Fig. 12 we present the log[!(] versus log[t] data and we observe generally increasing data trends, although there does appear to be convergence of several trends to!( = 0.5, suggesting at least some portion of the data are governed by linear flow. It is also worth noting that one particular case exhibits!( # 1, suggesting progression towards the "depletion" flow regime. It is vitally important that the analyst realize that the diagnostic analysis of production data is a necessary step. Although some of our conclusions are qualitative, the diagnostic analysis of multiple data functions ensures a degree of impartiality in the data analysis, and helps us to at least qualify the uncertainty in the data, which will likely ensure that we isolate the relevant time-rate models and that we do not attempt analyses/interpretations which are not justified by the quality and/or nature of the given production data. It is critical that data diagnostics always be performed as part of the data analysis.

6 6 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Application of the Time-rate Models to Long Term Production Data We present a (relatively) long term production data example in order to investigate the model behavior of the rate decline equations considered in this work. As mentioned earlier, this field example consists of a tight gas well from East Texas (permeability values are estimated to be around 7.0 $D) and we have more than 7 years of production. For this case, we demonstrate our diagnostic interpretation procedure for matching data and performing forecasts. In Figs we present the matches of given production data with each of the time-rate models. We note that all of the matches are performed simultaneously by calibrating model parameters as we are guided by the diagnostic plots. As observed in Figs. 13 and 14, each of the models matches the data throughout entire production history. In particular, when the log[q/g p ] versus log[t] plot is used, we note that the models can approximate the data trend to a considerable extent (as mentioned, this rendering tends to force the impression of a linear relationship, which may not be the case). Therefore, the differences in EUR will be dictated by the long term model behavior and this is where the differences between the timerate models begins to emerge. Duong's model is based on the linear behavior of q/g p -t data trend (on a log-log scale); whereas the power-law exponential, the stretched exponential, and the logistic growth models exhibit non-linear behavior (although this is somewhat difficult to distinguish on Fig. 14 essentially one should note that the other models fall below the Duong model). This difference in behavior dictates that the EUR estimates from Duong's model should (almost always) be higher than that for the other models. On the other hand, when a terminal decline is imposed on the modified-hyperbolic relation, deviations from the linear trend are readily evident (as seen in Fig. 14). We note that the modified-hyperbolic and power-law exponential have specific terms which limit the over-estimation of EUR, the Duong model does not. We would note as a suggestion that the Duong model could be similarly constrained by adding a terminal exponential decline, but this is not the intent of the Duong model and we do not make a specific recommendation that this be done. It is worth noting that for methods which utilize a terminal decline, the prescribed value of the terminal decline is generally an arbitrary number and is often based on a company's policies and/or the analyst's experience. In Fig. 15 we present the log[d(] and log[b(] versus log[t] data trends for this case. Very strong linear behavior of the computed log[d(] versus log[t] trend is observed confirming the applicability of the power-law exponential time-rate model. It can be argued that the latest-time data are affected by the numerical differentiation algorithm and therefore can be considered as artifacts (note the late-time behavior in the D( and b( functions). Nevertheless, each of the models matches the data trends in their own fashions. The b( trend does appears to be decreasing with time (with the noted artifact near the endpoin; and an average b-value can be inferred from the data behavior. Lastly, we review the log[!(] versus log[t] data in Fig. 16 and we note that each of the models matches the!( data trends to some degree and we note that there are differences are observed at early and late times. In particular, the late-time behavior of a given model plays a dominant role in the prediction of EUR. In Table 1 we summarize the EUR values computed at 30 years as obtained from the models. For this case, Duong's model yields the highest EUR and the logistic growth model yields the lowest EUR values. We use five percent (5%) terminal decline value for the modified hyperbolic equation, we note that increasing this value will yield a decrease in the predicted EUR. Table 1 Time-rate analyses results for the East TX gas well (long term production data) (All models) EUR PLE EUR SE EUR DNG EUR LGM EUR MHYP Well Name (BSCF) (BSCF) (BSCF) (BSCF) (BSCF) East TX gas well Time-Rate Analyses In this section we present the "Time-Rate" analyses for each well from a given shale play using each of the models specified in this study. Before starting, we believe it is worthwhile to note that each of the matches produced in this study are based uniquely on our own interpretation of the model behavior and we realize that different matches with different EUR values can be obtained with similar probability. As we have mentioned many times earlier in this work, our matches are the direct result of a systematic interpretation process which is focused primarily on the use of specialized diagnostic plots. We again note that we do not simply perform regression analysis, each analysis is based on diagnostic behavior coupled with an intrinsic understanding of each time-rate model and the diagnostic character of the given time-rate model.

7 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 7 Field A: Time-rate analyses for six wells are presented for Field A in Figs The summary table for the predicted EUR values is given below (Table 2): Table 2 Time-rate analyses results for Field A (All models) Field A EUR PLE EUR SE EUR DNG EUR LGM EUR MHYP Well Name (BSCF) (BSCF) (BSCF) (BSCF) (BSCF) Well A Well A Well A Well A Well A Well A It can be suggested that wells in Field A exhibit "power-law" type flow regimes. The basis for this observation is mainly the signature on time-rate plot and the near-constant character (at intermediate and late times) exhibited by data on the log[!(] versus log[t] plot. And almost all of the models match the entire production history. Differences in model behavior are observed at late times in the forecasts. Generally, EUR values from the power-law exponential and stretched exponential relations are very similar (as should be expected) and these models, along with the results from the logistic growth model, tend to provide conservative estimates across all wells. The Duong model and the modified-hyperbolic model always yield the highest EUR predictions. We note that a five percent terminal decline rate is used for the modified hyperbolic relation. Field B: Wells in Field B exhibit higher initial rates and steeper decline trends due to the unique characteristics of Field B. This behavior is completely different than the behavior observed in Fields A and C. In general, we had difficulties in matching the entire production history for a given well in Field B; primarily due to early-time behavior. In particular, we suggest that the Duong model may be the least appropriate model for this specific play due to observation that the Duong model and the observed data do not seem to agree in general (we note that disagreement is most evident in the log[d(] and log[!(] versus log[t] plots). The model matches are shown in Figs and the EUR results are presented in Table 3. Table 3 Time-rate analyses results for Field B (All models) Field B EUR PLE EUR SE EUR DNG EUR LGM EUR MHYP Well Name (BSCF) (BSCF) (BSCF) (BSCF) (BSCF) Well B Well B Well B Well B Well B Well B As with Field A, the highest EUR values are obtained with the Duong and modified hyperbolic models. The log[d(] versus log[t] trends exhibit more or less linear behavior and we note that it is difficult to interpret the log[b(] versus log[t] trends due to the quality of the production data for these wells. We note that the stretched exponential, power-law exponential and the logistic growth models again provide (in general) the most conservative estimates of predicted EUR. Field C: We apply the same procedures to analyze and forecast the 5 wells provided for Field C. In terms of performance behavior, Field C is different than Fields A and B, but there is some commonality of character with Field A (i.e., the trends have similarity of character, particularly for the log[!(] versus log[t] plots). The analysis and forecast plots for Field C are presented in Figs , and the EUR prediction results are summarized in Table 4. As was the case for Fields A and B, we note that the Duong model yields the highest EUR estimates due to the nature of this model. We again note that most cases are matched reasonably well by each time-rate model, but we also note that the Duong model is always the most aggressive extrapolation model.

8 8 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Table 4 Time-rate analyses results for Field C (All models) Field C EUR PLE EUR SE EUR DNG EUR LGM EURM HYP Well Name (BSCF) (BSCF) (BSCF) (BSCF) (BSCF) Well C Well C Well C Well C Well C In this particular case, the logistic growth model appears to provide the most conservative EUR values (quite comparable to both the power-law exponential and stretched-exponential time-rate models). Interpretation of Results We provide our interpretation of results by presenting comparison plots of the EUR values predicted by each model. For our purposes, we choose the power-law exponential model results as the reference results and we compare EUR values from different models with respect to the power-law exponential model. This approach should identify any correlations and/or inconsistencies that might exist between models. For Field A, comparison plots are presented through Figs Fig. 85 presents the comparison between the power-law exponential model and the Duong model and we observe from this plot that the EUR values from the Duong model are consistently higher than the results for the power-law exponential model. In Fig. 86 we present a comparison between the power-law exponential model and the logistic growth model, and with the exception of a single case (Well A.5), the EUR predictions are very similar. Our next comparison considers the power-law exponential model and the modified hyperbolic model and is presented in Fig. 87 we note some consistency in predicted EUR values, but we also note a couple of outliers that would suggest that the modified hyperbolic relation will always predict higher EUR values compared to the power-law exponential model where we also note that the terminal decline (for the modified hyperbolic model) is probably the controlling variable in this case. Lastly, in Fig. 88 we present a comparison of the power-law exponential model and the stretched exponential model and essentially identical results are obtained, somewhat as expected as these relations have essentially the same mathematical formulation (the power-law exponential does have a constraint parameter (i.e., the D! - term). Almost identical results are seen, due to the fact that these two equations are essentially the same relations, and wells in Field A are not (ye in boundary-dominated flow regime after a few years of production (2-4 years). We perform the same exercises for Field B (see Figs ) and for Field C (see Figs ) and observe similar trends as we observe in the case of Field A the results for the power-law exponential and stretched exponential models are almost always the same. The results for the power-law exponential and the logistic growth models are also very similar, except for a few cases. In general, the Duong model always predicts the highest EUR values. The degree of discrepancy in the results obtained using the Duong model is higher in the case of Field B and Field C than Field A. This is most likely due to the flow regimes exhibited by the production data in Field A where "power-law" flow regimes (linear/bi-linear flow, etc.) appear to dominate the time-rate performance. Based on these results, we can conclude that the Duong model could be more applicable to plays such as Field A where long term power-law flow regimes exist however, additional constraints might be required to prevent over-estimation of EUR values when the Duong model is being applied. Summary and Conclusions Summary: This work presents a workflow that can be used to analyze and forecast time-rate data of wells in low/ultra-low permeability reservoirs. The key component of the workflow is the application of the "Db," "!-derivative," and "q/g p " diagnostic plots to guide the analysis and obtain model parameters for a given time-rate relation. Once model parameters are obtained, the production profile is extrapolated to yield the "estimated ultimate recovery" (EUR) at a specified time limit or abandonment rate. In this work we have presented diagnostic and time-rate model-based analyses of long-term production data from 17 wells taken from three different shale plays. We have also provided a tight-gas example which has a somewhat long duration and high-quality production history to further demonstrate the methodology. Our overall assessment, based on the wells used in this study, is that production data (i.e., time-rate data) of reasonable quality can effectively drive the proposed diagnosticbased analysis. We are confident that a practicing reservoir engineer can effectively and efficiently utilize the diagnostic workflow process to ensure the most comprehensive and representative analyses of a given set of production data. We also provide the usual caveat that there must be sufficient production history available to perform a competent analysis; analysis of short-time data (no matter how pristine the quality of the data) will lead to overly optimistic estimates of reserves and predictions of future performance.

9 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 9 Conclusions: The following conclusions are established from observations made in this work:! The "Db" diagnostic plot should be the primary diagnostic used to establish the well-reservoir character.! The "q/g p " diagnostic plot is an excellent data check, and should be incorporated into diagnostic analyses however; the expectation of a completely linear trend [i.e., Duong (2011)] is optimistic.! The "!-derivative" diagnostic plot is useful for establishing the existence of "power-law" flow regimes. References Arps, J.J Analysis of Decline Curves. Trans. AIME 160: Clark, A.J., Lake, L.W., and Patzek, T.W Production Forecasting with Logistic Growth Models. Paper SPE presented at the SPE Annual Technical Conference and Exhibition, Denver, CO, 30 October-02 November. Duong, A.N Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs SPE Reservoir Evaluation and Engineering 14 (3): Ilk, D., Perego, A.D., Rushing, J.A., and Blasingame, T.A Exponential vs. Hyperbolic Decline in Tight Gas Sands Understanding the Origin and Implications for Reserve Estimates Using Arps' Decline Curves. Paper SPE presented at the SPE Annual Technical Conference and Exhibition, Denver, CO, September. Ilk, D., Rushing, J.A., and Blasingame, T.A Decline Curve Analysis for HP/HT Gas Wells: Theory and Applications. Paper SPE presented at the SPE Annual Technical Conference and Exhibition, New Orleans, LA, October. Johnson, R.H. and Bollens, A.L The Loss Ratio Method of Extrapolating Oil Well Decline Curves. Trans. AIME 77: 771. Jones, P.J Estimating Oil Reserves from Production-Decline Rates. Oil and Gas Journal 40 (35): Kisslinger, C The Stretched Exponential Function as an Alternative Model for Aftershock Decay Rate. Journal of Geophysical Research 98 (2): Kohlrausch, R Theorie des elektrischen Rückstandes in der Leidner Flasche. Poggendorff 91: Lee, W.J. and Sidle, R.E Gas Reserves Estimation in Resource Plays. Paper SPE presented at the 2010 SPE Unconventional Reservoirs Conference, Pittsburgh, PA, USA, February.doi: / MS. Rushing, J.A., Perego, A.D., Sullivan, R.B., and Blasingame, T.A Estimating Reserves in Tight Gas Sands at HP/HT Reservoir Conditions: Use and Misuse of an Arps Decline Curve Methodology. Paper SPE presented at the SPE Annual Technical Conference and Exhibition, Anaheim, CA, November. Valkó, P.P Assigning Value to Stimulation in the Barnett Shale: A Simultaneous Analysis of 7000 Plus Production Histories and Well Completion Records. Paper SPE presented at the SPE Hydraulic Fracturing Technology Conference, College Station, TX, January.

10 10 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Nomenclature a Dng = Model coefficient for the Duong time-rate model, D -1 a LGM = Model coefficient for the Logistic Growth time-rate model, D -1 b = Arps' decline exponent (hyperbolic time-rate relation), dimensionless! = "Beta" function (relates rate and derivative functions), dimensionless D = Reciprocal of the loss ratio, D -1 D lim = Terminal decline constant for the exponential time-rate relation, D -1 D i = Initial decline constant for the exponential and hyperbolic time-rate relations, D -1 Dˆ i = Decline coefficient for the Power-Law Exponential time-rate model, D -1 D! = Terminal decline coefficient for the Power- Law Exponential time-rate model, D -1 EUR = Estimated ultimate recovery, BSCF G p = Cumulative gas production, MSCF or BSCF G p,max = Maximum gas production (at a specified time limi, MSCF or BSCF K = Carrying Capacity for the Logistic Growth time-rate model, MSCF or BSCF m Dng = Time exponent for the Duong time-rate model, dimensionless n = Time exponent for the Power-Law and Stretched Exponential time-rate models, dimensionless n LGM = Time exponent for the Logistic Growth time-rate model, dimensionless q = Production rate, MSCF/D q i = Initial rate for the exponential and hyperbolic time-rate models, MSCF/D qˆ i = Initial rate coefficient for the Power-Law and Stretched Exponential time-rate models, MSCF/D q 1 = Initial rate coefficient for the Duong time-rate model, MSCF/D t = Production time, days " = Time coefficient for the Stretched Exponential time-rate model, D -1! = Micro (i.e uni

11 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 11 Fig. 1 Production rate and time plot for all wells (Field A) Fig. 2 Diagnostic plot! Gas rate/gas cumulative production plot for all wells (Field A).

12 12 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 3 Diagnostic plot! Computed D- and b- parameters and production time for all wells (Field A). Fig. 4 Diagnostic plot!!-derivative and production time for all wells (Field A).

13 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 13 Fig. 5 Production rate and time plot for all wells (Field B) Fig. 6 Diagnostic plot! Gas rate/gas cumulative production plot for all wells (Field B).

14 14 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 7 Diagnostic plot! Computed D- and b- parameters and production time for all wells (Field B). Fig. 8 Diagnostic plot!!-derivative and production time for all wells (Field B).

15 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 15 Fig. 9 Production rate and time plot for all wells (Field C) Fig. 10 Diagnostic plot! Gas rate/gas cumulative production plot for all wells (Field C).

16 16 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 11 Diagnostic plot! Computed D- and b- parameters and production time for all wells (Field C). Fig. 12 Diagnostic plot!!-derivative and production time for all wells (Field C).

17 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 17 Fig. 13 Time-rate analysis for East TX tight gas well! All models (rate and production time). Fig. 14 Time-rate analysis for East TX tight gas well! All models (gas rate/gas cumulative production and production time).

18 18 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 15 Time-rate analysis for East TX tight gas well! All models (computed D- and b-parameters and production time). Fig. 16 Time-rate analysis for East TX tight gas well! All models (!-derivative and production time).

19 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 19 Fig. 17 Time-rate analysis for Well A.1! All models (rate and production time). Fig. 18 Time-rate analysis for Well A.1! All models (gas rate/gas cumulative production and production time).

20 20 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 19 Time-rate analysis for Well A.1! All models (computed D- and b-parameters and production time). Fig. 20 Time-rate analysis for Well A.1! All models (!-derivative and production time).

21 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 21 Fig. 21 Time-rate analysis for Well A.2! All models (rate and production time). Fig. 22 Time-rate analysis for Well A.2! All models (gas rate/gas cumulative production and production time).

22 22 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 23 Time-rate analysis for Well A.2! All models (computed D- and b-parameters and production time). Fig. 24 Time-rate analysis for Well A.2! All models (!-derivative and production time).

23 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 23 Fig. 25 Time-rate analysis for Well A.3! All models (rate and production time). Fig. 26 Time-rate analysis for Well A.3! All models (gas rate/gas cumulative production and production time).

24 24 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 27 Time-rate analysis for Well A.3! All models (computed D- and b-parameters and production time). Fig. 28 Time-rate analysis for Well A.3! All models (!-derivative and production time).

25 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 25 Fig. 29 Time-rate analysis for Well A.4! All models (rate and production time). Fig. 30 Time-rate analysis for Well A.4! All models (gas rate/gas cumulative production and production time).

26 26 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 31 Time-rate analysis for Well A.4! All models (computed D- and b-parameters and production time). Fig. 32 Time-rate analysis for Well A.4! All models (!-derivative and production time).

27 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 27 Fig. 33 Time-rate analysis for Well A.5! All models (rate and production time). Fig. 34 Time-rate analysis for Well A.5! All models (gas rate/gas cumulative production and production time).

28 28 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 35 Time-rate analysis for Well A.5! All models (computed D- and b-parameters and production time). Fig. 36 Time-rate analysis for Well A.5! All models (!-derivative and production time).

29 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 29 Fig. 37 Time-rate analysis for Well A.6! All models (rate and production time). Fig. 38 Time-rate analysis for Well A.6! All models (gas rate/gas cumulative production and production time).

30 30 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 39 Time-rate analysis for Well A.6! All models (computed D- and b-parameters and production time). Fig. 40 Time-rate analysis for Well A.6! All models (!-derivative and production time).

31 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 31 Fig. 41 Time-rate analysis for Well B.1! All models (rate and production time). Fig. 42 Time-rate analysis for Well B.1! All models (gas rate/gas cumulative production and production time).

32 32 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 43 Time-rate analysis for Well B.1! All models (computed D- and b-parameters and production time). Fig. 44 Time-rate analysis for Well B.1! All models (!-derivative and production time).

33 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 33 Fig. 45 Time-rate analysis for Well B.2! All models (rate and production time). Fig. 46 Time-rate analysis for Well B.2! All models (gas rate/gas cumulative production and production time).

34 34 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 47 Time-rate analysis for Well B.2! All models (computed D- and b-parameters and production time). Fig. 48 Time-rate analysis for Well B.2! All models (!-derivative and production time).

35 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 35 Fig. 49 Time-rate analysis for Well B.3! All models (rate and production time). Fig. 50 Time-rate analysis for Well B.3! All models (gas rate/gas cumulative production and production time).

36 36 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 51 Time-rate analysis for Well B.3! All models (computed D- and b-parameters and production time). Fig. 52 Time-rate analysis for Well B.3! All models (!-derivative and production time).

37 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 37 Fig. 53 Time-rate analysis for Well B.4! All models (rate and production time). Fig. 54 Time-rate analysis for Well B.4! All models (gas rate/gas cumulative production and production time).

38 38 V. Okouma, D. Symmons, N. Hosseinpour-Zonoozi, D. Ilk, and T.A. Blasingame SPE Fig. 55 Time-rate analysis for Well B.4! All models (computed D- and b-parameters and production time). Fig. 56 Time-rate analysis for Well B.4! All models (!-derivative and production time).

39 SPE Practical Considerations for Decline Curve Analysis in Unconventional Reservoirs 39 Fig. 57 Time-rate analysis for Well B.5! All models (rate and production time). Fig. 58 Time-rate analysis for Well B.5! All models (gas rate/gas cumulative production and production time).