Well test, rate transient analysis and reservoir simulation for characterizing multifractured unconventional oil and gas reservoirs
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Abstract
Unconventional reservoirs such as shale gas and shale oil have become an increasingly important source of energy in the USA with potential reservoirs identified worldwide. Due to the insufficient permeability of the shale reservoirs, they require efficient stimulation using multistage hydraulic fractures to produce gas in commercial quantities. A critical challenge in the reservoirs is performance evaluation of the fracturing and characterization of the stimulated reservoir volume (SRV) for permeability and hydraulic fracture size. Conventional well test analysis in multistage fractured shale reservoirs may not provide reliable results due to the extensive wellbore storage effect, fracture complexities, and heterogeneity of the lowpermeability reservoir. To overcome such issues, advanced well test analysis techniques integrated with rate transient analysis can be used to reduce uncertainties associated with estimation of the reservoir and hydraulic fractures’ dynamic parameters. This paper proposes a practical methodology and workflow for characterizing the SRV parameters in multifractured wells in unconventional oil and gas reservoirs using well test and rate transient data analysis based on diffusivity equation solution for linear and elliptical flow regimes integrated with numerical reservoir simulation. A reservoir simulation model is built and run for a typical fractured shale reservoir to verify the reliability of the proposed simplified approach. Furthermore, multifractured unconventional reservoir field examples of well test analysis, reservoir simulation and history matching are presented to show how the stimulated reservoir volume can be characterized to perform a more reliable production forecast in shale oil and shale gas reservoirs.
Keywords
Multifractured wells Shale gas reservoirs Rate transient analysis Well testingNomenclatures
 L
Horizontal well length
 n
Number of fractures (stages)
 X_{f}
Fracture halflength size
 Y_{e}
Distance between a hydraulic fracture and the SRV boundaries in each fracturing stage (Y _{ e } = L/2/n)
 A
Drainage area of the SRV (2X _{f} × 2Y _{ e })
 A_{c}
Crosssectional area to linear flow in the SRV (2 × 2X _{f} × h)
 K_{SRV}
Permeability of stimulated rock volume
 h
Formation thickness
 μ
Fluid viscosity
 B_{o} and B_{g}
Formation volume factor for oil and gas, respectively
 φ_{m}
Porosity of matrix
 φ_{f}
Porosity of fracture
 C_{t}
Total compressibility
 δ
Shape factor
 a
Matrix block size
 L
Simulation grid block size
 b (h_{f})
Fracture aperture
 t
Time
 T
Temperature
 RTA
Rate transient analysis
 PBA
Pressure buildup analysis
 t_{BDF}
Elapsed time to reach boundarydominated flow effect
 m_{LF}
Pressure derivative value for infinite acting linear flow regime
 m_{Ell}
Pressure derivative value for infinite acting elliptical flow regime
 m_{RF}
Pressure derivative value for infinite acting radial flow regime
 GR
Gamma ray
 CGR
Corrected GR
 S_{w}
Water saturation
Introduction
Unconventional reservoirs such as shale gas and shale oil have become an increasingly important source of energy worldwide. Due to the insufficient permeability of the shale reservoirs, they require efficient stimulation using hydraulic fractures to produce hydrocarbon in commercial quantities (Bagherian et al. 2010).
The SRV is a network of hydraulic fractures and untreated matrix which are randomly distributed in the dualporosity and dualpermeability system. The main SRV parameters are SRV permeability, hydraulic fracture spacing (shale matrix block size) and SRV porosity (volume of the open fractures inside the stimulated reservoir volume). Characterizing the SRV parameters is a critical challenge as it indicates the effectiveness of the stimulation of the lowpermeability shale, the efficiency of the drained volume due to well spacing, and the well future production performance.
One of the common methods for characterizing the dynamic reservoir parameters is well testing, in which the pressure transient data are recorded and analyzed using the plot of transient pressure and its derivative versus time on Log–Log scale. The test data are normally analyzed using a diagnostic plot to identify the radial flow regime and calculate the average permeability of the reservoir accordingly. In a conventional reservoir, wellbore storage effect may last few hours, and then in few days the reservoir will exhibit its average properties upon reaching infinite acting radial flow regime. However, for multifractured unconventional reservoirs, things are radically different because of the large drop in mobility, fracture complexities, and heterogeneity of the lowpermeability formation that cause an extensive wellbore storage effect and slow propagating pressure pulse into the formation. As a result, we may have to wait months or years to detect the complicated flow regimes in the drainage area around the wellbore and fractures, and maybe centuries to eventually detect the equivalent infinite acting radial flow regime (KAPPA Engineering 2015). Therefore, using the conventional well test analysis to characterize the SRV parameters in multifractured shale reservoirs may fail to provide reliable results, and advanced techniques may be required.
Reservoir flow regimes in hydraulically fractured wells
A pressure transient test breaks into several flow regimes, each seeing deeper in the reservoir than the last. Depending on the well completion type and the reservoir geological and geometric attributes, different flow regime might be revealed on pressure transient diagnostic plots. In conventional well test analysis, diagnosis of the radial flow regime is critical in quantitative well test interpretation, since a reliable value for reservoir permeability can be estimated when the latetime radial flow regime is established in the reservoir (Badazhkov 2008; Bourdarot 1998).
After the reservoir flow regimes in the SRV, the SRV boundarydominated flow effect is then followed by linear flow regime inside the untreated reservoir zone towards the drainage area of the multistage fractured well (Fig. 4c), then elliptical flow regime, and finally at late time when pressure disturbance propagates deep enough into the reservoir, a pseudo radial flow regime is established (Fig. 4d), with slope of zero on pressure derivative. On the pressure derivative curve based on solution of radial flow diffusivity equation, the slope of +1 shows wellbore storage effect, the slopes of −0.5, +0.5, +0.25 and +0.36 indicate spherical, linear, bilinear and elliptical flow regimes, respectively, and the slope of zero indicates radial flow regime. In typical testing time duration in multifractured shale reservoirs, only the earlytime flow regimes might be the ones that can be detected on the diagnostic plots, but not the latetime flow regimes.
In multifractured shale reservoirs, field observations indicate that the earlytime linear flow regime and the earlytime elliptical flow regime are established in the SRV, but radial flow regime cannot be established in the SRV. The reservoir flow regimes in SRV are followed by the SRV boundarydominated effect. Well test analysis methods such as rate transient analysis (RTA) and pressure buildup analysis (PBA) in the multifractured wells require analyzing the reservoir flow regimes that are detected on well test analysis diagnostic plots, to characterize the SRV dynamic parameters.
Rate transient analysis (RTA) for multifractured shale reservoirs
 A plot of the normalized pressure (d[P _{wf}]/d[Q]) versus time (d[t]) on the Log–Log scales as shown in Fig. 5a can identify the data points related the linear flow regime based on the line slope of ½ on the diagnostic plot. When the slope increases from ½ to higher values, this indicates the time at which the boundarydominated flow effect starts (t _{BDF}). For gas wells, normalized pseudo pressure should be plotted versus time as shown in Fig. 5c.

A plot of the normalized pressure (d[P _{wf}]/d[Q]) versus (d[t ^{½}]) as shown in Fig. 5b can provide slope of the linear flow regime straight line (m _{LF}). For gas wells, normalized pseudo pressure should be plotted versus the time function as shown in Fig. 5d.
In the above equations, P is pressure (psia), t is time (days), Q _{o} is oil flow rate (STBD), Q _{g} is gas flow rate (MSCFD), B is formation volume factor, h is reservoir thickness (ft), µ is viscosity (cp), φ _{SRV} is the effective average porosity of the porous media in SRV that contributes to the flow (fraction), C _{t} is total compressibility, Y _{ e } is distance to the boundary (optimistically, Y _{ e } is half distance between each two adjacent hydraulic fractures), L is horizontal well length (ft), K _{SRV} is permeability of the stimulated reservoir volume, T is reservoir temperature (R), and n is number of the hydraulic fracturing stages.
The RTA method requires recording of surface production rates and bottomhole pressure data. However, if only the wellhead pressure data are available instead of bottomhole pressure data, then WHP data should be converted to BHP using multiphase flow well model correlations, and then be input into the RTA models. RTA can practically be used in multifractured shale reservoirs to characterize the SRV parameters.
Pressure buildup analysis (PBA) for multifractured shale reservoirs
Integrating the above two Eqs. 9 and 10 can provide the two unknowns X _{f} and K _{SRV} for an oil well in shale reservoirs (assuming that the effective average porosity of SRV to be known).
In the above equations, P is pressure (psia), t is time (h), q is oil flow rate (STBD), B is formation volume factor, h is reservoir thickness (ft), µ is viscosity (cp), φ _{SRV} is the effective average porosity of SRV (fraction) that may be dominated by matrix porosity effect since the fractures may have very small porosity, C _{t} is total compressibility, n is number of hydraulic fractures, and K _{SRV} is SRV average permeability that may be mainly controlled by permeability of the fractures. Based on the analytical derivations, the value of m _{LF} can be a good indicator of well deliverability. The production data for the first few months can be displayed on RTA plots to evaluate well deliverability of producing wells in a field.
For multifractured wells in unconventional lowpermeability gas reservoirs, only the linear flow regime may be observed on the diagnostic plots (testing time is not long enough to have radial flow regime detected). To get some estimates of the SRV parameters, one can predict a theoretical latetime radial flow regime inside the SRV using advanced methods such as the second derivative of transient pressure, to integrate the theoretical radial flow equation with the linear flow equation, and solve the two equations for the two unknowns with some uncertainties (Bahrami and Siavoshi 2013).
It should be noted that the above simplified equations do not take into account the changing gas viscosity and gas compressibility with the use of pseudo time, and they are based on singlephase flow. To get more accurate well test analysis results, gas slippage effect with pseudo time and pseudo pressure and also multiphase parameters can be considered in the analytical approach of well test data analysis.
Uncertainties of SRV characterization using analytical methods
The main advantage of RTA method is that in RTA, using just WHP and Q data which can always be recorded and available during production, the SRV can be characterized. However, the RTA equations may have some uncertainties as well. The disadvantage of RTA method is that the earlytime data specially the linear flow time period may be affected by well cleanup period, when oil and/or gas are produced with water and fracturing fluids. This causes the earlytime linear flow data points on RTA plot to be scattered, and therefore m _{LF} cannot be identified very accurately. In other words, m _{LF} on RTA plot may not be the actual representative of the fracture performance. On RTA plots, the data points may also be scattered because of the changes in well conditions during production (well head, choke size, liquid loading in wellbore, etc.). These affect RTA plots and calculation of slope of the lines. Also if the effective number of producing fractures (n) is not known, it makes uncertainties to be more. Running a production log is required to identify the effective number of hydraulic fractures, and improve reliability of RTA results.
The main advantage of pressure buildup test data analysis is that the flow rate is constant during the test (Q = 0); therefore, the data points are significantly less scattered compared to the data on RTA plots, and m _{LF} can be identified more accurately using PBA plots. The accurate estimation of m _{LF} is very important, as it is an indicator of the well’s future production performance. Another advantage of PBA is that K _{SRV} is calculated independent from Y _{ e } (that maybe unknown), and therefore the calculation of X _{f} will be more reliable. The uncertainties associated with the PBA method are when only a linear flow regime is observed on the diagnostic plots (one equation with two unknowns). It should be noted that the units of m _{LF} in RTA and PBA equations are different. Use m _{LF,RTA} = 4.9/Q × m _{LF,PBA} to convert m _{LF} from RTA units to m _{LF} from PBA units.
RTA equations are based on assuming Y _{ e } = L/n/2 (i.e., the reservoir volume around the horizontal well between each two fracturing stages, being fully stimulated). If the actual Y _{ e } is smaller, then RTA underestimates the X _{f} value. Also the RTA method is based on assuming the whole SRV being productive shale with 100 % net volume, whereas in SRV there is productive net shale volume, as well as nonproductive shale volume, and in reality, NTG volume may not be 100 %.
Also SRV permeability is not uniform and it is higher near the wellbore (larger fractures), and it is lower away from wellbore (smaller fractures). Using the analytical methods to estimate SRV parameters, it assumes a homogeneous singleporosity system and a fully stimulated pore volume with 100 % productive shale (optimistic assumption), which may result in underestimation of hydraulic fracture size.
Therefore, in designing the well spacing based on the calculated X _{f}, the actual well spacing should be considered larger than the one determined from RTA.

From RTA diagnostic plot, estimate t _{BDF}.

From PBA Log–Log diagnostic plot, estimate m _{LF} and m _{Ell}.

To estimate X _{f} and K _{SRV} (2 unknowns, 2 equations) for oil wells, use the values of m _{LF} and m _{Ell}, and integrate Eqs. 9 and 10. For gas wells, integrate Eqs. 11 and 12.

To estimate X _{f}, K _{SRV} and Y _{ e } (3 unknowns, 3 equations) for oil wells, use the values of t _{BDF}, m _{LF} and m _{Ell}, and integrate Eqs. 3, 9 and 10. For gas wells, integrate Eqs. 7, 11 and 12.
Reservoir simulation model input data for RTA and PBA evaluation in the shale gas reservoir
r _{w}  0.25  ft  Gas S.G.  0.7  _  T  300  F 
h  150  ft  μg  0.031  cp  Pi  7500  psia 
Porosity  0.06  fraction  Ct  7.68E − 05  1/psi  q  300  MSCFD 
Comparison of RTA and PBA results (assuming singleporosity SRV)
It should be noted that SRV is a dualporosity and dualpermeability system, but the RTA and PBA equations are based on singleporosity system. Using the matrix porosity in RTA calculations as the effective SRV porosity, it assumes a singleporosity system and a fully stimulated pore volume, in which the volume occupied by hydraulic fractures is equal to porosity of matrix (this is an optimistic assumption, since fracture porosity is significantly less than matrix porosity). The assumption may result in underestimation of hydraulic fracture size, and therefore RTA results may be optimistic. In this paper, reservoir simulation and history matching approach in a dualporosity model are presented to characterize SRV more accurately.
Characterizing SRV using dualporosity simulation model
The RTA and PBA equations are based on singleporosity models, whereas the SRV in multifractured shale reservoirs is actually a dualporosity system, and the stimulated rock volume is a random network of untreated shale matrix blocks and fracture planes. Characterization of a dualporosity SRV in multifractured shale reservoirs generally includes estimating the dynamic parameters such as average SRV permeability and hydraulic fracture size.
Well test analysis of multifractured shale reservoirs using dualporosity concept is challenging and may not be practical due to the extensive wellbore storage effect in a long fractured wellbore, heterogeneity of the reservoir, and complexities in the dualporosity stimulated rock volume (Restrepo and Tiab 2009), and therefore conventional well test analysis for dualporosity behavior using interporosity flow coefficient (λ) and fracture storativity ratio (ω) may not be meaningful.
The main input parameters that are required to model fluid flow in a fractured system are fracture permeability, fracture compressibility, fracture porosity, shape factor, and the productive shale net volume in SRV (NTG_{shale}). Shape factor (δ) can be defined for Kazemi model, as a function of matrix block size (a) that can be simplified to δ = 4/a ^{2} (Saeedi 2012).
For characterizing fracture porosity (φ _{f}), fracture opening (b) and matrix block size (or fracture spacing: a), processing of image log data if it is an openhole well can be used (Dashti et al. 2009; Luthi 1990). However, in multifractured casedhole wells, logging applications for this purpose are very limited and running fracture characterization image logs may not be practical in fractured casedhole wells. The alternative method is to use history matching approach and tuning the values to estimate the fracture parameters φ _{f} and δ.
In the above equations, K _{f,actual} is the actual intrinsic permeability of fractures, K _{f,DP} is the equivalent fracture permeability for dualporosity reservoir simulation model (permeability for the fracture grids), K _{well test} is average well test permeability (from RTA or PBA), b is average fracture aperture, a is average matrix block size, and L is average grid size in the SRV section of the simulation model.
The downhole stresses applied from different directions to the formation rock can control aperture (and, therefore, permeability) of the hydraulically created fractures. The fractures that are perpendicular to the minimum stress direction may have significantly better permeability than the fractures perpendicular to the maximum stress direction. The permeability estimated from RTA or PBA is equivalent to SRV K _{average}, the average permeability in the SRV model (K _{average} = K _{ x } ^{1/3} × K _{ y } ^{1/3} × K _{ z } ^{1/3} ).
To characterize SRV, reservoir simulation and history matching can be done using dualporosity dualpermeability model. The reservoir model should be built considering that the given dualporosity SRV volume is surrounded by the untreated singleporosity rock. Then the cumulative volume of injected water, oil production, water production, gas production, and flowing bottomhole pressure should all be matched during history matched using the matching parameters such as SRV size and SRV permeability. The history match is achieved when the correct values for matrix and fractures parameters in the SRV are used in the reservoir model.
Field example: Eagle Ford Shale
Typical core data have shown matrix permeability of 1E−6 md. For well test analysis, some wells with pressure and rate data available are selected to guessestimate reservoir characteristics, and then reservoir simulation and history matching are used to characterize the reservoir as dualporosity system, and therefore perform more accurate production forecast from the multifractured shale reservoir.
PBA for a shale gas multifractured horizontal well
Using the diagnostic plot shown in Fig. 11, the linear and elliptical flow regimes are both detected, and therefore the proposed well test analysis method can be used based on linear flow (LF) derivative and elliptical flow (EF) derivative. By integrating the diffusivity equation solution for the two reservoir flow regimes, the two unknowns K and X _{f} are calculated (Eqs. 11, 12): SRV permeability of 0.0005 mD and hydraulic fracture halflength of 48 ft.
RTA for shale gas and shale oil multifractured horizontal wells
The rate and pressure transient data are studied for two multifractured wells, one in a shale oil reservoir (well O1) and one in a shale gas reservoir (well G2), to show applications of RTA in characterizing SRV in multifractured wells.
Characterizing multifractured shale oil reservoir using dualporosity simulation model
Input data into the reservoir simulation model of the multifractured SRV
Matrix porosity  8  % 
Initial S _{w}  20  % 
Matrix compressibility  3E−6  1/psi 
Matrix permeability K _{m} in X, Y and Z directions  1e−6  md 
Reservoir simulation and history match results for the multifractured SRV
Fracture porosity  0.5  % 
Fracture compressibility  1E−5  1/psi 
Transmissibility multipliers for the grids that face fracturing pressure (in X, Y and Z directions, respectively)  45,000, 5000, 45,000  – 
Dualporo fracture permeability (K _{f}, DP) in X direction  5e−5  md 
Permeability multipliers for the fractured grids (in X, Y and Z directions)  1 0.02 50  – 
Conclusions

Rate transient analysis (RTA) and pressure buildup analysis (PBA) can practically be used to have estimations of the stimulated reservoir volume (SRV) parameters in shale reservoirs.

In RTA method, the slope of the linear flow regime data (m _{LF}) and the time at which boundarydominated flow starts (t _{BDF}) are used for K _{SRV} and X _{f} estimation.

In PBA method, integration of diffusivity equations solutions for linear and elliptical flow regimes can provide estimation of SRV characteristics. The values of m _{Ell} and m _{LF} can be determined, respectively, from zero slope line on elliptical flow derivative and zero slope line on linear flow derivative.

The value of m _{LF} can be a good indicator of well deliverability. The m _{LF} from PBA may be more reliable than the value of m _{LF} from RTA, since during a pressure buildup test, production rate is constant (Q = 0) and, therefore, the data quality is better. However, in terms of data availability in producing wells, most wells have production data for RTA, but few wells may have the pressure buildup data required for PBA.

Integration of rate transient analysis and pressure buildup analysis methods can reduce uncertainties in SRV characterization.

Reservoir simulation and history matching using dualporosity SRV model is more reliable method for SRV characterization, compared with the analytical methods.
Notes
Acknowledgments
The authors would like to thank Aurora Oil and Gas management for supporting the technical research study, Chet Ozgen and Basar Basbug from Nitec LLC for the technical discussions and support on simulation of multifractured shale reservoirs, and Baharak Ghaffari Nik (Schlumberger) for providing useful technical information on this topic. We acknowledge KAPPAEngineering for use of KAPPAECRIN, Coat’s Engineering for use of SENSOR reservoir simulator, and also LYNX pre and postprocessing reservoir simulation tool in this research study.
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