clarkson_(2013)

International Journal of Coal Geology 109110 (2013) 101146

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International Journal of Coal Geology

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Review article

Production data analysis of unconventional gas wells: Review oftheory and best practices

C.R. Clarkson Department of Geoscience, University of Calgary, 2500 University Drive NW Calgary, Alberta, Canada T2N 1N4

Tel.: +1 403 220 6445.E-mail address: [email protected].

0166-5162/$ see front matter 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.coal.2013.01.002

a b s t r a c t
a r t i c l e i n f o
Article history:Received 28 June 2012Received in revised form 3 January 2013Accepted 3 January 2013Available online 11 January 2013

Keywords:Production data analysisRate transient analysisCoalbed methaneShale gasReviewBest practices

Unconventional gas reservoirs, including coalbed methane (CBM), tight gas (TG) and shale gas (SG), havebecome a significant source of hydrocarbon supply in North America, and interest in these resource playshas been generated globally. Despite a growing exploitation history, there is still much to be learned aboutfluid storage and transport properties of these reservoirs.A key task of petroleum engineers and geoscientists is to use historical production (reservoir fluid productionrate histories, and cumulative production) for the purposes of 1) reservoir and well stimulation characteriza-tion and 2) production forecasting for reserve estimation and development planning. Both of these subtasksfall within the domain of quantitative production data analysis (PDA). PDA can be performed analytically,where physical models are applied to historical production and flowing pressure data to first extract informa-tion about the reservoir (i.e. hydrocarbon-in-place, permeability-thickness product) and stimulation (i.e. skinor hydraulic fracture properties) and then generate a forecast using a model that has been calibrated to thedynamic data (i.e. rates and pressures). Analytical production data analysis methods, often referred to asrate-transient analysis (RTA), utilize concepts analogous to pressure-transient analysis (PTA) for their imple-mentation, and hence have a firm grounding in the physics of fluid storage and flow. Empirical methods, suchas decline curve analysis, rely on empirical curve fits to historical production data, and projections to the fu-ture. These methods do not rigorously account for dynamic changes in well operating conditions (i.e. flowingpressures), or reservoir or fluid property changes.Quantitative PDA is now routinely applied for conventional reservoirs, where the physics of fluid storage andflow are relatively well-understood. RTA has evolved extensively over the past four decades, and empiricalmethods are now applied with constraints and rules of thumb developed by researchers with some confi-dence. For unconventional reservoirs, these techniques continue to evolve according to our improved under-standing of the physics of fluid storage and flow.In this article, the latest techniques for quantitative PDA including type-curve analysis, straight-line(flow-regime) analysis, analytical and numerical simulation and empirical methods are briefly reviewed, spe-cifically addressing their adaptation for CBM and SG reservoirs. Simulated and field examples are provided todemonstrate application. It is hoped that this article will serve as practical guide to production analysis forunconventional reservoirs as well as reveal the latest advances in these techniques.

2013 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022. Concept of rate transient analysis: flow-regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033. Production analysis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.1. Straight-line (or flow-regime) analysis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.1.1. Additional considerations for CBM and shale reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.1.2. Simulated examples of CBM and shale straight-line analysisCases 1 and 2dry shale . . . . . . . . . . . . . . . . . . . 1133.1.3. Simulated examples of CBM and shale straight-line analysisCases 3 and 4wet coal . . . . . . . . . . . . . . . . . . . 116

3.2. Type-curve methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.2.1. Fetkovich type-curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.2.2. Blasingame type-curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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102 C.R. Clarkson / International Journal of Coal Geology 109110 (2013) 101146

3.2.3. Wattenbarger type-curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.2.4. Multi-fractured horizontal well type-curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.2.5. Other useful type-curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.2.6. Alternative type-curve matching approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.2.7. Additional considerations for CBM and shale reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.3. Analytical and numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273.4. Empirical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.4.1. Application of Arps decline-curve methodology for CBM wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.4.2. Application of Arps decline-curve methodology for tight gas and shale gas wells . . . . . . . . . . . . . . . . . . . . . . 1323.4.3. New empirical approaches for tight gas and shale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.5. Hybrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354. Field examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.1. Field example 1: 2-phase CBM well (vertical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.2. Field example 2: 1-phase shale gas well (MFHW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5. Discussion: future development of production data analysis techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.1. Analytical (type-curves, straight-line, analytical simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.2. Empirical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.3. Hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Nomenclature . 142

1. Introduction

Advances in reservoir evaluation and drilling/completion technol-ogy have enabled commercial production of hydrocarbons from un-conventional reservoirs over the past few decades, including naturalgas from coal (coalbed methane or CBM), low-permeability (tight)gas (TG) and shale gas (SG) reservoirs. In North America, these re-source types now contribute significantly to hydrocarbon supply;with recent suppression of gas prices, the focus of producers hasbeen on liquids-rich (i.e. oil, gas condensate) unconventional plays.The success of unconventional plays in North America has triggeredglobal activity in the area of unconventional reservoir explorationand exploitation.

Economics of unconventional reservoirs is tied closely to well-performance (Haskett and Brown, 2005). Operators have therefore fo-cused extensively on optimizing well performance and hydrocarbonrecovery in the past few decades. Advances in wellbore architecturedesign (i.e. horizontal wells), and stimulation technology (i.e. hydrau-lic fracturing) have enabled commercial production from reservoirspreviously thought of as only source rocks (i.e. coals and organic-rich shales). Additional improvements in the areas of core analysis,well-log analysis, well-test analysis, remote sensing, reservoir surveil-lance and production technology have allowed operators to betterlocate wellbores in the sub-surface to exploit these reservoirs, moreeffectively design stimulation programs, and more efficiently operatethe wells to achieve maximum performance.

An important reservoir engineering technique that is used for res-ervoir characterization and evaluation, and for production forecast-ing, is quantitative production data analysis (PDA). Rate-transientanalysis (RTA) is a relatively new form of PDA, where fluid productiondata, along with flowing pressure information, is used to extract res-ervoir (i.e. hydrocarbons-in-place and permeability-thickness prod-uct) and stimulation information (i.e. skin and hydraulic fractureproperties). The theory behind RTA is exactly analogous to pressuretransient analysis (PTA), and hence the techniques used for RTA arevery similar to PTA. These techniques include production type-curveanalysis, straight-line (flow-regime) analysis and analytical andnumerical simulation. RTA is essentially an inversion problemwhere reservoir attributes are obtained from the reservoir signal,i.e. rate and/or flowing pressure data. Once this inversion is solvedfor key attributes, these attributes along with other reservoir/fluid/

well/operating condition data can be used in a model that accuratelycaptures the physics of the reservoir/well system to generate a fore-cast of future hydrocarbon production. The RTA approach, combinedwith analytical and numerical simulation for forecasting, requiresmodels for fluid storage and transport. These techniques are referredto as advanced production data analysis methods and require bothproduction and flowing pressure data, along with fluid property andreservoir (volumetric) data to arrive at a quantitative result. Signifi-cant development of the RTAmethod has been made for conventionalreservoirs in the past four decades.

Empirical methods, such as decline curve analysis, can also beused for forecasting. For conventional reservoirs, the Arps (1945)hyperbolic/exponential decline forecasting method remains themost popular, but because it is an empirical technique, its use canlead to significant errors, even for conventional reservoirs. As willbe discussed in detail below, the assumptions behind the Arps de-cline curve method (i.e. constant operating conditions, static reser-voir behavior) are very often violated, causing the method to yielderroneous results. Ideally, advanced PDA methods should be usedin parallel with empirical methods to constrain the decline-curveforecast. Workflows to ensure this consistency have been providedin the literature for conventional reservoirs (i.e. Fetkovich et al.,1996; Mattar and Anderson, 2003) and were provided in an accom-panying article by the author (Clarkson, 2013-this volume).

Advanced and empirical production data analysis methods forunconventional reservoirs continue to evolve. The primary complica-tions associated with the adaptation of these methods for unconven-tional include complex reservoir behavior, as well as flow behaviorassociated with complex wellbore/hydraulic fracture geometry. Ana-lytical solutions to flow equations, on which the advanced PDA tech-niques are based, typically make many simplifying assumptionsabout the reservoir/fluids/and hydraulic fracture, including (amongothers) slightly-compressible fluid behavior, homogeneous and staticreservoir properties.

For shale gas and coal reservoirs, fluid storage and flow propertiesthat complicate analysis include (Clarkson et al., 2012a):

1. Desorption of gas from organic matter and clays (CBM andorganic-rich shale)

2. Ultra-lowmatrix permeability, which causes transient flow periodsto be extensive (SG)

103C.R. Clarkson / International Journal of Coal Geology 109110 (2013) 101146

3. Dual porosity or dual permeability behavior caused by natural frac-tures or induced hydraulic fractures (or both) (CBM and SG)

4. Additional reservoir heterogeneities, such as multi-layers (ex.contracting permeability and gas contents in coal seams) andlateral heterogeneity (ex. coal seam pinchouts)

5. Stress-dependent porosity and permeability caused by a highly-compressible fracture pore volume (CBM and SG)

6. Desorption-dependent fracture porosity and permeability due tomatrix shrinkage effects (CBM)

7. Multi-phase flow of gas and water (2-phase CBM and SG), or gasand condensate (SG)

8. Non-Darcy flow, including slip-flow and diffusion (CBM and SG)

Aswewill see, correction for some of these affects (1., 5., 6., 7., 8.) re-quires alteration of primary variables used for RTA, primarily pressureand time, to pseudovariables that incorporate changes in fluid proper-ties, desorption and reservoir properties. If this is done carefully, datamay be converted using pseudovariables allowing the use of RTAmethods designed to analyze single-phase production of slightly-compressiblefluids,which is the assumptionmade in existing analyticalsolutions to the flow equations. Item 2 suggests that in many shale gaswells, boundary-dominated flowwill not be reached in manywells in areasonable time frame, meaning that estimates of hydrocarbon-in-place ultimately contacted by the well will not be accurate. Reservoirheterogeneities can be accommodated with the choice of solutionsand methods that incorporate these effects; we note however, thatthe solutions often greatly oversimplify the true geologic characteristicsof the reservoir.

Not only do reservoir characteristics of SG and CBM impact RTA,but also the hydraulic fracture network that is created during stimu-lation. Short and long-term production characteristics of unconven-tional gas reservoirs can be impacted by the fracture geometrycreated through the stimulation process. Because multi-fracturedhorizontal wells are now primarily used for exploitation of SG reser-voirs, quantitative hydraulic fracture characterization has provendifficultcreated fracture geometries often do not conform to theconventional bi-wing planar fracture geometry assumed in conven-tional reservoirs (Fig. 1). More complex geometries can be created,often by design, to maximize reservoir contacted in ultra-low perme-ability reservoirs. When complex geometries are created, resulting ina stimulated reservoir volume (SRV) (Mayerhofer et al., 2010), hy-draulic fracture and reservoir characterization are inseparable, asthe induced hydraulic fracture network often serves to define the

Fracture width

Simple Fracture

Complex FractureWith Fissure Opening

C

CN

Fig. 1. Spectrum of fracture geometries expected for conventional and unconventioModified from Warpinski et al. (2008).

reservoir. This is a significant challenge in RTA of SG wells: the sepa-ration of reservoir from hydraulic fracture effects.

In CBM, the stimulation treatment can also significantly affect pro-duction characteristics. The two most common methods of stimulatingvertical wells are through the use of hydraulic fracture stimulation andcavity completions (Palmer, 2010). Horizontal well completions arealso used to exploit lower permeability coals, with or without stimula-tion of the wellbore.

Production data analysis involves the analysis of production ratesignatures. These signatures are a consequence of the fluid flow dy-namics in the reservoir, which are often referred to as flow-regimes.As we will see, the unique reservoir properties, along with comple-tion and stimulation style of coal and shale have a profound impacton the type and sequence of flow-regimes that are seen, and hencethe methods used for analysis. We begin with an introduction to thetypes of flow-regimes encountered for CBM and SG, followed by a dis-cussion of the different types of production data analysis methodsthat are currently being used to analyze them. Lastly, we will demon-strate application of the PDA techniques to CBM well and shale gaswells.

2. Concept of rate transient analysis: flow-regimes

As mentioned in the Introduction, advanced analytical methodsused for quantitative production analysis are classified as rate-transient analysis methods, which are analogous to pressure-transientanalysis methods used to analyze well-test data. RTA essentially in-volves the analysis of a particular form of well-test, referred to as adrawdown test, where the well is produced against known wellboreconstraints over a long period of time. In conventional well-test analy-sis, a short drawdown period typically precedes a lengthier shut-in/pressure buildup periodone or both of these test periods may be ana-lyzed for reservoir and/or hydraulic fracture properties for stimulatedwells. These tests are typically conducted for a period of days, or per-hapsweeks, under highly controlled conditions, with data collected fre-quently (often on the order of seconds) and accurately (with testseparators for flow periods, and quartz gauges for pressure measure-ments). For RTA, the drawdown period is essentially the producinglife of the well, possibly interrupted by shut-in periods, with less fre-quently acquired datarates and flowing pressures (if measured) areoften only recorded on a daily basis (at best), but often less frequently.The data quality for analysis is therefore rarely as good as for PTA, but itdoes represent a longer period of time.

Fracture Half-Length

omplex Fracture

omplex Fractureetwork

nal reservoirs. Complex fracture networks have nearly equal width and length.


Both RTA and PTA start with the identification of flow-regimes,which are characteristic flow patterns or geometries in the reservoir,which may be analyzed for reservoir and/or hydraulic fracture prop-erties for stimulated wells. These flow patterns in the short-termare affected by flow to the wellbore or hydraulic fracture networkfor stimulated wells, and in the long term by heterogeneities in thereservoir and flow boundaries. One of the simplest sequences thatcan be observed is for a non- or slightly-stimulated vertical well com-pleted in a homogeneous, isotropic reservoir bounded on all sides(Fig. 2, left side). In the example of Fig. 2, the flow rate at the well isassumed to be constant (drawdown test), resulting in an early tran-sient radial flow period as the pressure drops in the well-bore (seeFig. 2a cross-section view, green dashed lines), and a pressure tran-sient propagates out radially away from the well (see Fig. 2b planview). This early transient radial period can be interpreted for perme-ability and skin, if it can be observed in the RTA signature. Slightlystimulated, or cavity completed CBMwells may exhibit this signature,for example. Once the pressure transient contacts all boundaries, theboundary-dominated flow period is entered, and the reservoir de-pletes like a tank. During boundary-dominated flow, the pressuredrops uniformly throughout the reservoir and can be interpreted toderive reservoir volume, and most importantly, hydrocarbons-in-place. Observation of this flow-regime is critical to obtain quantitativeestimates of reserves.

The most common way to identify flow-regimes for pressure-transient analysis is to use a pressure-derivative, or rate-normalizedpressure-derivative, versus time on a loglog plot (Fig. 2, right side).The example in Fig. 2 is for a numerically-simulated vertical wellcompleted in a dry (gas-production only), homogeneous, isotropiccoal reservoir with instantaneous desorption producing against aconstant flowing pressure. The derivative was calculated using themethod of Bourdet (Bourdet et al., 1983, 1989). The time function inthis example has been modified to allow the constant flowing

2000

2200

2400

2600

2800

3000

3200

3400

3600

Transient Flowderive k, skin

Boundary Dominated Flowderive volume

a)

b)

Fig. 2. Left side: flow-regimes associated with a slightly-stimulated vertical well subject to coa) cross-section and b) plan views. Lines correspond to isopotential lines; arrows are strearadially away from the well in plan view (b) and the well-bore pressure drops (cross-secinterpreted for reservoir permeability and near wellbore skin. During boundary-dominatedand material-balance-like calculations can be performed to interpret the wellbore pressure d(non-stimulated) vertical dry coal well.Left side: from Clarkson (2011), modified from Fekete RTA short course notes. Right side: m

pressure scenario to be converted to the equivalent constant rate con-dition, and to account for desorption and gas property changes withpressure (discussed below). The transient radial flow period is identi-fied as a zero slope, and the boundary-dominated flow period, as aunit slope. For noisy data, as is often seen with production data,other techniques for flow-regime identification must be considered(discussed below). Once the flow-regimes have been identified, thevarious production analysis methods can be applied.

For hydraulically-fractured wells, the sequence of flow-regimesmay be more complex (Fig. 3, left side) (Cinco-Ley and Samaniego-V,1981). The example in Fig. 3 is for a vertical well with an infinite-conductivity planar hydraulic fracture, completed in a homogeneoustight gas reservoir with closed boundaries. The sequence of flow-regimes includes: early (formation) linear flow to the hydraulic frac-ture; elliptical flow as the pressure transient moves past the ends ofthe fracture; pseudoradial flow (analogous to transient radial flow inthe previous example); and boundary-dominated flow. If the fracturesare of finite conductivity (discussed below), at early time, there maybe an additional pressure drop/linear flow along the fractures occur-ring simultaneously with formation linear flowthis flow period isreferred to as bilinear flow. At very early times in the finite conductiv-ity fracture case, there may be a fracture linear flow period, butthis flow period is generally too short to be observed with productiondata.

As with the slightly-stimulated well case described above, deriva-tive techniques can be used to identify flow-regimes (Fig. 3, rightside). The example in Fig. 3 is a numerically-simulated vertical wellwith a single planar, infinite conductivity fracture, completed in a ho-mogeneous, isotropic tight gas reservoir producing against a constantflowing pressure. The time function in this example has been modi-fied to convert the constant flowing pressure scenario to the equiva-lent constant rate condition, and to account for gas property changeswith pressure (discussed below). The formation linear flow period is

1.0E+3

1.0E+4

1.0E+5

1.0E+6

1.0E+0 1.0E+1 1.0E+2 1.0E+3 1.0E+4

d(

m(p

)/q

)/d

lnt*

ca

t*ca, days

Radial Derivative Plot

Boundary-Dominated flow(unit-slope)

Radial Flow (zero-slope)

nstant flow-rate constraint. Pressure changes at the well and the reservoir are shown inmlines. During transient flow (green dashed lines), the pressure transient propagatestion view). During this flow period, the flowing pressure signature over time can beflow (red solid lines), the pressure drops at the same rate everywhere in the reservoir,rop for hydrocarbon-in-place. Right side: identification of flow-regimes for a simulated

odified from Clarkson (2009).

1.

2.

3.

4.

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04

d(

m(p

)/q

)/d

lnt c

a

tca, days


Boundary-Dominatedflow (unit-slope)

Pseudo -Radialflow (zero-slope)

Linear flow(1/2-slope)

Elliptical flow

Fig. 3. Left side: sequence of flow-regimes for a hydraulically-fractured vertical tight-gas well with infinite-conductivity fracture. Lines correspond to isopotential lines; arrows arestreamlines. Flow regime 1. (inner black arrows) corresponds to linear flow; flow-regime 2. (elliptical green dashed lines) corresponds to elliptical flow; flow-regime 3. (circulargreen dashed lines) corresponds to pseudoradial flow; and flow-regime 4. (circular red line) corresponds to boundary-dominated flow. Right side: identification of flow-regimes fora simulated hydraulically-fractured vertical tight gas well.Right side: modified from Clarkson and Beierle (2011).


identified as a 1/2 slope, the radial flow period as a zero slope, and theboundary-dominated flow period, as a unit slope. A transitional ellip-tical flow period corresponds to the non-linear portion of the deriv-ative between linear and radial flow.

The sequence of flow-regimes for hydraulically-fractured wellsdescribed above may be visualized by observing gridblock pressurechanges over time in a numerical simulator (Fig. 4). Flow-regimes13 (transient flow regimes) in Fig. 3 are identified by pressure gradi-ent geometries in Fig. 4.

For multi-fractured horizontal wells (MFHW), the sequence offlow-regimes may be even more complex (Fig. 5, left side) (Chen andRaghavan, 1997; Raghavan et al., 1997). The example in Fig. 5 is for amulti-fractured horizontal well with infinite-conductivity planar hy-draulic fractures, that are widely spaced, completed in an infinite-acting (no outer boundaries) homogeneous tight gas reservoir. Theearly flow-regimes are similar to the vertical, hydraulically-fracturedwell described above, where flow is restricted to between fractures. Be-cause the latest trend inMFHWs is close hydraulic fracture spacing and/or multiple perforation clusters per fracturing stage, the transitionalflow-regimes including early elliptical and radial flow may not appear.Note that it is possible to see 2 linear flow periodsan early linearflow period where flow is perpendicular to the hydraulic fractures,and a late or compound linear flow, where flow is perpendicular tothe effective wellbore length (Chen and Raghavan, 1997; Raghavan etal., 1997; van Kruysdijk and Dullaert, 1989). At some point after earlylinear flow, the fractures will interfere, resulting in a flow period thatmay appear like boundary-dominated flow, followed by late or com-pound linear flow.

Derivative techniques can be used to identify flow-regimes (Fig. 5,right side), but with a great deal of potential variation in the sequenceof flow-regimes seen, as well as their length, depending on both res-ervoir properties and completion design. As mentioned, some transi-tional flow-regimes may be absent, depending on hydraulic fracturespacing and fracture length. Further, early linear flow may be tooshort to observe in higher permeability reservoirs and late/compoundlinear flow and the second formation radial flow period may beabsent, depending on formation permeability and well-spacing. Anearly, sub-linear (appearing as bilinear flow, or skin effects) may

occur as a result of flow-convergence to the horizontal well, cleanupeffects after stimulation, fracture face skin and finite-conductivityfractures (Nobakht and Mattar, 2012). Finally, the position ofthe late linear flow period relative to early linear flow is affected byfracture spacing (Nobakht et al., 2012a). As with hydraulically-fractured vertical wells, simulation gridblock pressure gradients area useful way to visualize the sequence of flow-regimes over time(not shown).

The MFHW flow-regime sequence for shale gas wells has recentlybeen discussed by several authors (i.e. Cheng, 2011; Song andEhlig-Economides, 2011). A conceptual diagram showing this se-quence is given in Fig. 6, and the corresponding derivative signatureis given in Fig. 7.

The primary difference between the sequence of flow-regimes ob-served in shales and for conventional reservoirs is the absence oftransitional (elliptical and radial) after the first linear flow period(Fig. 6). Fracture storage is usually too short to observe with typicalproduction data, or masked by cleanup effects. Song et al. (2011)coined the term pseudo pseudosteady-state for the post-fractureinterferencethey noted that during this timeframe, each fracturewill produce from its own drainage volume, but that the slope onthe derivative is not quite unity because flow is unbounded beyondthe fracture tips. Note that the reservoir volume defined by the hy-draulic fracture stimulation is often referred to as a stimulated reser-voir volume, or SRV, which many authors believe to be the practicalextent of drainage in shale reservoirs (i.e. Ozkan et al., 2011).

Bello (2009) noted that multi-fractured horizontal wells in shalecan exhibit transient dual porosity characteristics. Moghadam et al.(2010) similarly documented dual porosity characteristics, and illus-trated several sequence of flow-regimes that may occur in such sys-tems, of which two possibilities are shown in Fig. 8. Note thatbilinear flow is possible with this model (scenario b of Fig. 8) due tosimultaneous fracture-linear and matrix-linear flow.

As noted above, a combination of reservoir properties andwellbore/fracture geometry has a significant impact on the sequenceof flow-regimes encountered in unconventional reservoirs. For hori-zontal wells, there are many combinations that may be encountered,several of which are shown in Fig. 9. These 8 scenarios form the basis

b)

c)

Approximate extent of elliptical flow pattern

Approximate extent of radial flow pattern

Hydraulic Fracture

a)

Fig. 4. Simulation case illustrating the transient flow-regimes associated with a verticalwell with an infinite conductivity hydraulic fracture. Pressures range from 3300 psi(red) to 1000 psi (blue). a) Linear flow; b) elliptical flow; c) pseudo-radial flow.Modified from Clarkson and Beierle (2011).


of several models that have been used to describe production perfor-mance in unconventional reservoirs. Scenarios 1 and 2 illustrate anopenhole-completed single horizontal lateral completed in a single-porosity reservoir, and dual-porosity reservoir, respectively. Thiscompletion style continues to be used in low-permeability CBMplays, such as the Mannville coals of Western Canada (Gentzis andBolen, 2008; Gentzis et al., 2009) and the fringe-Fairway Fruitlandcoals of the San Juan Basin (Clarkson et al., 2009), but multi-lateralwells are becoming more common. Scenario 2 has been used tomodel multi-fractured horizontal wells where a complex fracture ge-ometry is created, and the SRV exhibits dual porosity characteristics.Scenarios 3 and 4 are cases where the SRV occupies a region immedi-ately surrounding the horizontal well, but in Scenario 4, the SRV andbackground (naturally-fractured) reservoir have differing dual poros-ity characteristics. Scenarios 5 and 6 correspond to cases where alow-complexity, primary hydraulic fracture has been created. Scenar-ios 7 and 8 also contain primary hydraulic fractures, but these are

embedded within an SRV that has also been created as a result ofstimulation; in these cases the conductivity of the primary hydraulicfracture is greater than the conductivity of induced/natural fractureswithin the SRV. Scenario 7 is the basis of the popular trilinear flow an-alytical model (Brown et al., 2011; Ozkan et al., 2011) discussedbelow.

Understanding the flow regime sequence caused by hydraulicfracture geometry and reservoir properties is critically importantwhen interpreting rate-transient/decline characteristics of unconven-tional gas wells. Once the flow-regimes are confidently identified,there are several production analysis methods that may be appliedto obtain hydraulic fracture/reservoir information from them. Thesetechniques will be discussed next.

3. Production analysis methods

Production analysis can be used to derive the following informa-tion about the reservoir and the wellbore or fracture geometry:

1. Estimated ultimate recovery (EUR) and original gas-in-place (OGIP)from boundary-dominated flow data

2. Fracture or matrix permeability, hydraulic fracture half-length andfracture conductivity or contacted matrix surface area (for com-plex fracturing cases), effective wellbore lengthfrom transientflow data.

Advanced production data analysis (or rate-transient analysis)methods use both production rates and flowing pressures in the anal-ysis to account for variable operating conditions of thewells. Empiricalmethods use only production data. As discussed by Clarkson andBeierle (2011) there are several production analysis methods thathave been commonly applied for unconventional gas reservoirsincluding:

1. Straight-line (or flow-regime) analysis methods2. Type-curve methods3. Analytical and numerical simulation4. Empirical methods5. Hybrid (analytical and empirical) methods.

In the following, we detail these methods and provide simulatedexamples for CBM, TG and SG. The focus will be on straight-line,type-curve (advanced production analysis methods) and empiricalmethods. Analytical simulation will be briefly discussed, as willhybrid methods. Numerical simulation is beyond the scope of thecurrent discussion.

3.1. Straight-line (or flow-regime) analysis methods

These techniques for production data are analogous to those usedin pressure transient analysis (Lee et al., 2003). Once the flow-regimes have been identified using derivative analysis or other loglog diagnostic techniques (Figs. 2, 3, 5, and 7) the data correspondingto specific flow-regimes is analyzed on specialty plots designed tolinearize the dataset for that flow geometry. Depending on theflow-regime identified, hydraulic-fracture properties or reservoirproperties may be obtained.

To illustrate the application of straight-line analysis for a tight gasreservoir (CBM and shale gas will be discussed below), we have pro-vided a summary of the expected flow-regimes, diagnostic signatureof the flow-regime on a derivative, the plotting variables for theflow regime, and the extracted properties for a vertical, hydraulically-fractured well (Table 1). Note that the flow-regimes in Table 1 arethe same as those identified in Fig. 3, except for bilinear flow(simultaneous linear flow within the fracture and perpendicular tothe fracture, which may be observed in finite conductivity fracturesdiscussed below). The specialty plots for each flow-regime (columns 3and 4, Table 1) are derived from constant rate solutions to the flow

1.

2.

3.

4.

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04

d(

m(p

)/q

)/d

lnt

t, days


B-D flow

Early Linear flow

Late Radialflow

FractureInterference

Early Radialflow

Late Linear flow

Fig. 5. Left side: sequence of flow-regimes for a multi-fractured horizontal well with planar infinite conductivity fractures with wide fracture spacing, completed in a tight gasreservoir. Lines correspond to isopotential lines; arrows are streamlines. Flow regime 1. (inner black arrows) corresponds to linear flow; flow-regime 2. (elliptical green dashedlines) corresponds to elliptical flow; flow-regime 3. (elliptical red solid lines) corresponds to fracture interference; and flow-regime 4. (outer black arrows) corresponds to late(compound) linear. Right side: identification of flow-regimes for a simulated multi-fractured horizontal well completed in a tight gas reservoir.Right side: modified from Clarkson and Beierle (2011).


equations. Most of these solutions were derived for drawdownwell-tests, as discussed in Lee et al. (2003), except the flowing materialbalance (boundary-dominated flow) equation which is a modificationof the equation given by Agarwal et al. (1999). The ellipticalflow-regime solution was given in Cheng et al. (2009). In Table 1, t isreal time (hours), ta is pseudotime (hours), m(p) is pseudopressure(psi2/cp) and mFL and bFL are the slope and intercept, respectively, ofthe corresponding specialty plotall other parameters are definedin the Nomenclature section. As will be discussed below, rate-normalizedm(p) function and bilinear, linear and radial superpositiontime functions are used to account for variable rates and flowing

Pseudolinear FlowFracture StoragePseudo

Pseudosteady State Flow

Derivative Slope = 1

Derivative Slope = 1/2 Derivative

Slope = -1

Pressure Depletion in SRVTransient Flow

to Each Fracture

Inter-fractu

re pressu

re Interferen

ce

Linear flow normal to transverse fractures

Pseudo pseudosteady state flow

~ 1

Fig. 6. Sequence of flow-regimes that are possible for MFHW completed in shale reservoirs. Nperiod.Modified from Song and Ehlig-Economides (2011).

pressures, and pseudovariables (pressure and time) are used toaccount for gas property changeswith pressure. These data transforma-tions allow solutions derived for constant rate production ofslightly-compressible fluids (the basis for most well test solutions) tobe used.

From the bilinear flow-regime (row 1, Table 1), hydraulic fractureconductivity (width of fracture, wf, times fracture permeability, kf)can be estimated from the slope of the bilinear specialty plot (column3 of Table 1), if permeability is known, along with standard volumet-ric inputs for the reservoir (thickness, porosity, fluid properties).From the formation linear flow (row 2, Table 1), hydraulic fracture

Compound Linear Flow

Pseudoradial Flow

Pseudosteady State

(Interwell Interference)


Derivative Slope = 1/2


Steady State (External Pressure Support)

P = Constant

Transient Flow to SRV

Compound linear flow Pseudoradial

flow

Boundary Behavior

ote the absence of transitional (elliptical or radial) flow after the first pseudolinear flow

1 1010-4

1

10

100

RN

P o

r R

NP

slope,

Compound linear flow

Nearly unit slope,Pseudo pseudosteady state

slope,Pseudolinear flow

0 slope,Pseudo radial flow

Boundary behavior:Unit slope,

Pseudosteady state

10-1

10-3

10-2

10-1

102 103 104 105 106 107 108

te, hours

Fig. 7. Rate-normalized pressure (RNP) and rate-normalized pressure derivative (RNP) signature for MFHW in shale. Note te is material balance time in the terminology of Song andEhlig-Economides (2011).Modified from Song and Ehlig-Economides (2011).


half-length (xf) can be estimated from the slope of the linear specialtyplot (column 3 of Table 1), if permeability is known, along with stan-dard volumetric inputs.

The elliptical flow regime (row 3, Table 1) analysis procedurewas de-veloped recently by Cheng et al. (2009), and is iterative; separate iterative

Bou

ndar

y-D

omin

ated

Horizontal Well Horizonta

Fracture Linear Flow (FL) Fracture Boundary-(FBD)

Bou

ndar

y-D

omin

ateda)

Horizontal Well

Fracture Linear Flow (FL) FL + ML => Bilin

Horizonta

b)

Fig. 8. 2 possible scenarios (a) and (b) for flow-regime sequences in a transient dual pointerporosity flow coefficient and storativity ratio, respectively, and yeD is a dimensionlessModified from Moghadam et al. (2010).

procedures were developed for wells with infinite-conductivity andfinite-conductivity fractures. In the finite-conductivity case, A and B aremodified, and bilinear flow regime analysis is also required. The uniqueaspect of elliptical flow is that it does not appear as a straight-line on asemi-log (radial) derivative, and that permeability and xf can be obtained

Horizontal Welll Well

Dominated Flow

Bou

ndar

y-D

omin

ated

Bou

ndar

y-D

omin

ated

Bou

ndar

y-D

omin

ated

Bou

ndar

y-D

omin

ated

Matrix Linear + FBD Flow

Horizontal Well

ear (BL) Flow

Bou

ndar

y-D

omin

ated

Bou

ndar

y-D

omin

ated

Bou

ndar

y-D

omin

ated

ML + FBD Flow

l Well

rosity system. For a), >3/(yeD)2 and for b), b3 /(yeD)2, where and are thereservoir length.

Scenario 1 Scenario 2



Scenario 7

Single Porosity Reservoir Dual Porosity Reservoir

Stimulated Reservoir Volume

Discrete Hydraulic Fractures Ex. Nobakht and Clarkson type-curves (Nobakht et al. (2011)

Trilinear flow model (Brown et al. 2011)

Scenario 8

Horizontal WellEx. Horizontal well type-curves of

Fig. 9. Possible combinations of reservoir/hydraulic fracture encountered for shale/tight gas reservoirs.Modified from Clarkson and Pedersen (2010).

Table 1Summary of flow-regime analysis for hydraulically-fractured gas wells. Note that the unit of time is in hours in this table.Modified from Clarkson and Beierle (2011).

Flow regime Loglog diagnostic Specialty plot Extracted properties

Bilinear Radial derivative: 1/4 slopeBilinear derivative: zero slope

m pi m pwf q vs. (ta)

1/4 or ta;BLS Xnj1

qjqj1

qnta;nta;j1 1=4 wf kf

1=2 k 1=4 443:2TmBLh gicti 1=4

Linear Radial derivative: 1/2 slopeLinear derivative: zero slope

m pi m pwf q vs. (ta)

1/2 or ta;LS Xnj1

qjqj1

qnta;nta;j1 1=2 ffiffiffikp xf 40:93T

mLh gicti 1=2

Elliptical Radial derivative: variable slopem pi m pwf

q vs. ln(A+B) k 1422TmEhxf exp bE

Pseudoradial Radial derivative: zero slope

m pi m pwf q vs. log(ta) or ta;RS

Xnj1

qjqj1

qnlog ta;nta;j1

k 1637TmRhs 1:1513 bRmR log kgicti r2w

3:23

Boundary-dominated Radial derivative: unit slope qm pi m pwf vs. Gi

m pi m pR m pi m pwf Gi from x-intercept



from the analysis. Prior to the development of the straight-line techniqueintroduced by Cheng et al., permeability could only be uniquely deter-mined from radial flow, which may take a long time to develop fortight reservoirs.

From the radial flow-regime (row 4, Table 1), permeability can beestimated from the slope of the radial specialty plot (column 3 ofTable 1), if standard volumetric inputs for the reservoir are known,and skin from the y-intercept.

From the boundary-dominated flow period, OGIP can be obtainedfrom the x-intercept of the specialty plot. Derivation of OGIP fromflowing (or dynamic) material balance is analogous to obtainingOGIP from conventional (static) material balance, except flowingpressures are used instead of static (shut-in) pressures. The flowingmaterial balance (FMB) procedure is also iterative, as discussed inMattar and Anderson (2003); kh may also be extracted from they-intercept if skin is known.

Because most of the analytical solutions derived for fluid flow to asinglewell, which are used in straight-line analysis, assume: 1) constantrate or constant pressure inner (well) boundary-conditions, 2) singlephase flow of a slightly-compressible or non-compressible fluid,3) Darcy (laminar) flow, 4) no chemical reactions between fluid andsolid (i.e. adsorption); and 5) a porousmediumwith negligible porosityor permeability changes, efforts to linearize the data for each flowregime must focus on addressing the assumptions of the analyticalsolutions. To make the solutions applicable to gas, pseudopressureand pseudotime are used to account for gas property variations withpressure:

m pi m pwf

pi

pwf

pgz

dp 1

ta gct

it

0

dt

gct: 2

Note that the traditional definition of pseudotime (ta) for produc-tion analysis involves the evaluation of gas properties (viscosity andcompressibility) at pore-volume average (reservoir) pressure; asdemonstrated by Nobakht and Clarkson (2012a,b), this may lead toinaccuracy in reservoir/hydraulic fracture property determinationfor ultra-low permeability reservoirs, such as shales; for these cases,Nobakht and Clarkson (2012a,b) recommend use of a correctedpseudotime, where the gas properties are evaluated at the averagepressure in the region of influence during transient flow. Correctionsfor non-Darcy flow, adsorption and non-static permeability arediscussed in a later section.

To account for variable production rates and flowing pressures at thewell due to operational changes, superposition time should be used forthe x-axis variable and rate-normalized pseudopressure for the y-axisvariable for the specialty plots in Table 1. Use of superposition time,the form of which is flow-regime specific, allows variable-rate data tobe analyzed using constant-rate solutions to the flow-equations. Thedynamic flowing material balance method (Agarwal et al., 1999;Mattar and Anderson, 2005) was derived using a superposition timefunction that is applicable to boundary-dominated flow, which is com-monly referred to as material balance time (material balancepseudotime for gas):

tca gct

i

qgt

0

qgdtgct

: 3

Note that for slightly-compressible fluids, assuming pore volumecompressibility is constant, the viscosity-total compressibility prod-uct is approximately constant. Further, the integral of q over time issimply the cumulative production. The combination of these allows

a simpler version of material balance time to be used: tc=Q /q.Some authors have used this simpler form of material balance timeas a first approximation for gas.

An example application of flow-regime (straight-line) analysis tothe simulated hydraulically-fractured vertical well (tight gas, matrixpermeability=0.01 mD) case given in Fig. 3 is provided in Fig. 10.The four flow-regimes identified in Fig. 3 are analyzed on their corre-sponding specialty plots using the equations given in Table 1; becausethe hydraulic fracture was assumed to be infinite conductivity, anearly bilinear flow period is not seen or analyzed. Because pseudo-radial (Fig. 10c) and boundary-dominated flow periods (Fig. 10d)are observed in this example, they would first be analyzed for perme-ability and OGIP, respectively. The permeability estimate obtainedfrom radial flow analysis would then be used to calculate fracturehalf-length from linear flow analysis (Fig. 10a), which yields an esti-mate of xf

ffiffiffik

p. The elliptical flow period (Fig. 10b) can also be analyzed

to yield an estimate of permeability and fracture half-length to confirmthe estimates from radial and linear flow, respectively. In the absence ofradial or elliptical flow, as is often the case in shale reservoirs, perme-ability would have to be estimated independently from core data,pre-fracture stimulation well-testing, or from fracture interferencetime associated with multi-fractured horizontal wells, as discussedbelow. As noted by Clarkson and Beierle (2011), the parameters derivedfrom straight-line analysis of this example are in reasonable agreementwith simulation input values, although fracture half-length from linearflowanalysis is somewhat over-estimated due to the use of convention-al pseudotime (Eq. (2)), instead of the corrected pseudotime, suggestedby Nobakht and Clarkson (2012a,b). As will be seen, the correctedpseudotime results in more accurate estimates of xf, particularly forultra-low permeability reservoirs.

Multi-fractured horizontal wells can be examined in an analogousfashion to what is presented above for a vertical well with a single hy-draulic fracture, but we note that it is possible to have multiple linearflow periods (ex. early formation linear flow and/or late compoundlinear flow). If the fractures are spaced far enough apart, transitional(early elliptical and early radial) flow-regimes may be observed,and if the wells are spaced far enough apart, late elliptical and late ra-dial flow may appear. The equations provided in Table 1 can still beapplied, but if multiple linear flow periods occur, for example, theearly linear flow period can be used to extract a total fracturehalf-length (nxf, where n is the number of fractures) for the well (ifpermeability is known), and the late linear flow period may be usedto estimate effective well-length (if permeability is known), or per-meability, if effective well-length is assumed. Similarly, for radialflow, early radial flow will yield an estimate of nkh, and late radialflow will yield an estimate of the true formation kh.

An example is provided below from Clarkson and Beierle (2011);the example is for a simulatedMFHWcompleted in a tight gas reservoir(Fig. 11). The derivative signature for this simulated example is shownin Fig. 5two radial and two linear flow periods are obtained. Analysisof the late-radial flow period yields a permeability value within reason-able agreement (within 10%)with that input into the numerical simula-tor (0.01 mD); the total fracture half-length is also within 10%. As withthe previous simulated example, some error in the derived parametersis due to the use of conventional pseudotimethe estimates withcorrected pseudotime are within 5%.

The simulated examples provided above are for a relativelystraight-forward tight gas (single porosity, permeability=0.01 mD)case, with no issues related to desorption, multi-phase flow, non-Darcy flow, or non-static permeability. We now discuss modificationsof the straight-line analysis method for CBM and shale wells withcomplex reservoir behavior.

3.1.1. Additional considerations for CBM and shale reservoirsThe most common way to incorporate complex reservoir behavior

into straight-line analysis for CBM and shale reservoirs is through

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

[m(p

i)-m

(pw

f)]/

qg

Superposition Time

Linear Flow Plot

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

3.5E+06

4.0E+06

4.5E+06

5.0E+06

[m(p

i)-m

(pw

f)]/

qg

ln(A+B)

Elliptical Flow Plot

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

3.5E+06

4.0E+06

4.5E+06

5.0E+06

6420

[m(p

i)-m

(pw

f)]/

qg

Superposition Time

Radial Flow Plot

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

7.0E-04

8.0E-04

9.0E-04

1.0E-03

No

rmal

ized

Rat

e, s

cf/D

/psi

2/cp

Normalized Cumulative Production, MMscf

Flowing Material Balance

Use to calculate xf,when k is known [from plot b) or c)].

Use to calculate k and skin. Skin can be used to estimate infinite-conductivity-equivalentxf. k can be used with plot a) to obtain xf.

Use to calculate OGIP and drainage area (A).Pressures obtained from FMB can be used in conventional pseudotime calculations.

Use to calculate k and xfand compare with results of plot c) and a), respectively.

0 10 20 30 40 50 5.0 6.0 7.0 8.0 9.0

0 1000 2000 3000 4000 5000 6000

c) d)

b)a)

Fig. 10. Use of specialty plots (linear, a, elliptical, b, radial, c, and boundary-dominated, d) for analyzing flow-regimes associated with simulated hydraulically-fractured vertical wellin a tight gas reservoir (see Fig. 4). Ideally, if either or both elliptical and radial flows are observed, then permeability is used in linear flow analysis to derive an independent es-timate of xf. Analysis tips are also provided (orange boxes). Note that transitional flow-regimes (elliptical and radial) are often not observed in unconventional gas reservoir such asshale.Modified from Clarkson and Beierle (2011).


alteration of the pseudovariables, pseudotime and pseudopressure, inTable 1. For example, for a dry gas reservoir, alterations of pseudotimefor a) adsorption, b) non-static permeability, and c) non-Darcy flow,are given with Eqs. (4)(6), respectively:

ta gct

it

0

dtgc

t

4

ta;NSP gct

i

kit

0

k p dtgct

5

ta;NDF gct

i

kt

0

ka p dtgct

6

where:

ct total compressibility including desorption compressibilityk p absolute permeability at the pore volume average pressure

of the reservoirki absolute permeability at initial pressureka p apparent permeability at the pore volume average pressure

of the reservoirk liquid-equivalent permeability.

To correct pseudotime for adsorption, the total compressibility termis altered (Bumb and McKee, 1988; Clarkson et al., 2007; Gerami et al.,

2007). To account for non-static permeability, a relationship betweenpermeability and pore pressure is first established, and is used to calcu-late permeability at the pore volume average pressure of the reservoir.In coal reservoirs, permeability may both decrease due to increase instress during depletion, and increase due to matrix-shrinkage effectsduring desorptionanalytical models, for which a comprehensive sum-mary was recently provided by Pan and Connell (2012), may be used tomodel permeability changes in these cases. In shale reservoirs, it is gen-erally assumed that permeability decreases with an increase in stress(decrease in pore pressure)Thompson et al. (2010) modeled perme-ability changes with pore pressure. Clarkson et al. (2012b), Nobakhtet al. (2012b) and Ozkan et al. (2010) recently demonstrated thatnon-Darcy flow effects, such as slip flow and diffusion, can be incorpo-rated into rate-transient analysis by including an apparent permeabilitychange with pressure into the analysis.

Pseudopressure may be corrected for a) non-static permeabilityand b) non-Darcy flow, using Eqs. (7) and (8), respectively:

m pi NSPm pwf

NSP 2

kipi

pwf

k p gz

pdp 7

m pi NDFm pwf

NDF 2

kpi

pwf

ka p gz

pdp: 8

0.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

1.0E+05

[m(p

i)-m

(pw

f)]/

qg

Superposition Time

Early Linear Flow Plot

0.0E+00

1.0E+06

2.0E+06

3.0E+06

4.0E+06

0 5 10 15 20

3 4 5 6 7

[m(p

i)-m

(pw

f)]/

qg

Superposition Time

Late Radial Flow Plot

Use to calculate total xf,when k is known [from plot c)]

Use to calculate kand skin. k can be used with plot a) to obtain total xf.

c)

b)

a)

Fig. 11. (a) Simulation grid for numerically simulated MFHW example and analysis ofearly linear (b) and late radial (b) flow-regimes using specialty plots.Modified from Clarkson and Beierle (2011).


Procedurally, Eqs. (1)(8) can be used in the flow-regime analysisequations of Table 1, and analysis can be performed analogously tothe tight gas examples provided above.

As noted by Nobakht and Clarkson (2012a,b), the estimation ofpermeability and fluid properties in pseudotime at pore-volume aver-age pressure for the reservoir can lead to errors in hydraulic fracture/reservoir properties derived from rate-transient analysis for ultra-lowpermeability reservoirs. They proposed that average pressure inthe region of influence first be evaluated, and then the pressure-dependent properties (fluid properties, adsorption, and permeability)evaluated at that pressureevaluation procedures for constant pres-sure (Nobakht and Clarkson, 2012b) and constant rate (Nobakhtand Clarkson, 2012a) well operating constraints were provided.

Corrections for multi-phase flow are more complex. In 2-phaseCBM and shale reservoirs, effective permeability to gas and waterchanges as a function of fluid saturation, which requires relativepermeability for each phase to be known. Further, alteration ofpseudopressure requires that a relationship between relative perme-ability and pressure be knownstrictly-speaking, relative permeabil-ity is a function of saturation, not pressure. Mavor and Saulsberry(1996) reviewed the approaches that have been used to performtwo-phase CBM pressure-transient analysis, including: 1) an adapta-tion of Perrine's (1956) approach which utilizes a multi-phase totalmobility function; 2) Kamal and Six's approach (1993) which utilizesa multi-phase flow potential with an assumed relationship betweensaturation and pressure and 3) Mavor's approach, which also utilizesa multi-phase flow potential, with a different (from Kamal and Six)assumption for the relationship between saturation and pressure.The multi-phase flow potential, as defined by Mavor and Saulsberry(1996) is:

M p p

pb

krggBg

krwwBw

!dp 9

where pb is a base pressure (psia). Mavor and Saulsberry assumed a lin-ear relationship between flowing pressure and saturation to allowM(p)to be calculated. To extend this approach to production analysis, thepseudotime functionwould also have to be adjusted, inwhich case a re-lationship between pore-volumepressure and saturationwould have tobe known.Mohaghegh and Ertekin (1991), who developed type-curvesfor two-phase CBM reservoirs, evaluated saturation-dependent vari-ables at the initial gas saturation, which considerably simplifies theanalysis. Clarkson et al. (2008) utilized a form of Eq. (9) for productionanalysis (flowing material balance), but assumed that gas mobilitydominated the two-phase pseudopressure calculationthe equationused in that work is given below:

m p MP 2p

pb

krggz

pdp 10

where MP refers to multi-phase in the modified pseudopressure.Clarkson et al. (2008) did not utilize a modified pseudotime in theirwork.

To linearize the partial-differential equation governing flow of gasthrough a gas-condensate reservoir, Sureshjani and Gerami (2011)utilized a two-phase pseudopressure and pseudotime. Those authorsalso developed a modified material balance pseudotime and flowingmaterial balance plot to analyze boundary-dominated flow data.Use of modified pseudotime and pseudopressure to perform rate-transient analysis on wells exhibitingmulti-phase flow characteristicsrepresents the most advanced approach; similar modifications to thepseudovariables can be performed for multi-phase oil and CBM wells.

While the approach developed by Sureshjani and Gerami (2011) forgas-condensate reservoirs is themost rigorous for rate-transient analy-sis of multi-phase cases, it is not terribly practical as it is applied to2-phase CBM or shale wells. Use of modified pseudovariables requires1) that relative permeability be known and 2) relative permeability berelated to pressure, to facilitate pseudopressure calculations. In recent

xe

ye

xf

Fig. 12. Vertical hydraulically-fractured well centered in a rectangular reservoir.

Table 2Input parameters for simulated 1-phase (shale gas) vertical well case.

Input parameter Value

Thickness (ft) 100Bulk density (g/cm3) 2.47Porosity (%) 10Gas gravity 0.69Initial absolute permeability (mD) 0.1Initial reservoir pressure (psia) 3500Initial water saturation (%) 0Reservoir temperature (F) 200Langmuir Volume (scf/ton, in-situ) 89Langmuir Pressure (psia) 535.6Drainage Area (ac) 57Fracture half-length (ft) 250Wellbore Diameter (in.) 8.4Skin factor 0Flowing Bottomhole Pressure (psia) 250


work by Clarkson et al. (2012c), an empirical approach was suggestedfor 2-phase CBM reservoirs. They first represented the y-axis variable(of the specialty plot) for rate-transient analysis of gas data for CBMwells producing both gas and water as:

y krg Sg

m pi m pwf h i

qg: 11

Note that the inverse of this variable is used for flowing materialbalance analysis. Moving krg outside of the pseudopressure calcula-tion (see Eq. (11)) removes the need to relate relative permeabilityto pressure, but additionally assumes that saturation gradients aresmall because a pore-volume average saturation is used in krg calcula-tions. Therefore, Eq. (11) is strictly applicable to high permeabilitycases, with the approximation expected to worsen as permeabilitydecreases. It is further assumed that the production data is dominatedby gas. The need to define relative permeability for gas as a functionof saturation remains.

The x-axis variable (of the specialty plot) includes a modifiedpseudotime defined as follows:

ta;MF gc

t

i

krg

i

t

0

krg Sg

dt

gct

12

where MF refers to multi-phase. As above, the starred variables arecorrected for desorption. For variable rate data, Eq. (12) would beincluded in the superposition time calculation (see Table 1).

In order to improve consistency in the analysis of 2-phase CBMwells, which requires knowledge of relative permeability, Clarksonet al. (2012c) suggested that pressure- and saturation-dependent vari-ables in straight-line (e.g. Eqs. (11) and (12)) and type-curve analysis(see below) be derived from outputs of analytical or numerical simula-tion. The rate-transient and simulation results are therefore linked, pro-viding improved consistency, but not necessarily uniqueness.

In the following section, we demonstrate the application of thestraight-line modifications for simulated CBM and shale gas examples.

3.1.2. Simulated examples of CBM and shale straight-lineanalysisCases 1 and 2dry shale

The first example is designed to illustrate the application of cor-rections to straight-line analysis so that CBM and shale wells maybe analyzed. In particular, the corrections for 1) adsorption and 2)non-static permeability are illustrated. The simulated example is fora vertical well producing through a single planar hydraulic fractureof infinite conductivity from a dry (single-phase) shale gas reservoirwith a permeability of 0.1 mD (Fig. 12). Other model inputs aregiven in Table 2. In this example, the permeability is assumed to be1) static (Case 1), or 2) exponentially decreasing (Case 2). For Case2, the permeability change was modeled using the approach ofOzkan et al. (2010) and Raghavan and Chin (2004).

k kie ppi 13

Where is the permeability modulus, assumed=4104 psi1

for this example.Simulated production rates for Case 1 (static permeability) and Case

2 (changing permeability) are given in Fig. 13. Linear flow is followeddirectly by boundary-dominatedflow in both cases.We see that the pri-mary differences between the two cases are 1) the initial rates and2) the character of the boundary-dominated (B-D) flow data.

To illustrate the impact of adsorption on the analysis, Case 1 is an-alyzed by 1) ignoring adsorption and 2) applying the correctionsdiscussed in Section 3.1.1. The flow-regimes are first identified withderivatives (Fig. 14a, b) to determine which straight-line plots to

apply to the data. In Case 1 and Case 2, only linear and boundary-dominated flow appear. Another tool for flow-regime identificationis the transient productivity index (Araya and Ozkan, 2002;Medeiros et al., 2008, 2010). Historical workers have used an averagereservoir pressure in the transient PI calculation, but recentlyWilliams-Kovacs et al. (2012) derived a transient PI that utilizes theaverage pressure in the region of influence for transient linear flow,which is useful for ultra-low permeability reservoirsthis new PI isgiven in Fig. 14c for Case 1.

The primary impact of ignoring adsorption is the OGIP estimatefrom flowing material balance (Fig. 15a). The flowing material bal-ance analysis requires a material balance equation (MBE) to allowpore-volume average pressure to be calculatedas discussed byWilliams-Kovacs et al. (2012), there are several MBEs available foranalysis of shale gas wells. For the No Corrections Made FMB plotin Fig. 15a, a conventional gas (p/z-cum) MBE was used that doesnot include corrections for adsorption; the Corrections Made FMBplot uses King's (1993) MBE equation for dry CBM/shale reservoirs,and includes corrections for adsorption in a modified z-factor calcula-tion (z*). Further sensitivities to MBE selection were discussed byWilliams-Kovacs et al. (2012). We see that when the corrections aremade for adsorption (using King MBE equation) in the FMB plot, thecorrect (simulator input) OGIP and drainage area are obtained(~6.7 bscf and 57 ac, respectively). When the corrections for

10

100

1000

10000

100000

1 10 100 1000 10000 100000

Gas

Rat

e, M

scf/

D

Time, days

Gas Production Rates

Static k Case (Case 1) Changing k Case (Case 2)

Linear flow B-D flow

Fig. 13. Gas production rates for Case 1 and Case 2.

1.0E+04

1.0E+05

1.0E+06

1.0E+07

d(

m(p

)/q

)/d

lnt


Linear flow

Boundary-dominated flow

1.0E+04

1.0E+05

1.0E+06

d(

m(p

)/q

)/d

t^0.

5

Linear Derivative Plot

Linear flow


1.0E-04

1.0E-03

1.0E-02

1.0E-01

PI

Time, days

Transient Productivity Index Plot

Linear flow


1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

Time, days1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

Time, days1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

a)

b)

c)

Fig. 14. Identification of flow-regimes for Case 1. On semi-log (radial) derivative, linearflow appears as 1/2 slope, on the linear derivative (b), linear flow appears as a zeroslope. On the modified transient-PI plot (c), linear flow appears as a 1/2 slope,while boundary-dominated flow is a zero slope. Although real time was used in this ex-ample, for noisy data, use of material balance pseudotime or linear superpositionpseudotime would yield a smoother derivative.


adsorption are not made (conventional MBE), the FMB underpredictsthe OGIP (~5.9 bscf) and overpredicts the drainage area (~64 ac).

For linear flow analysis, we use the linear flow plot (see Table 1),with and without corrections for adsorption (Fig. 15b). For theCorrections Made linear flow analysis, corrections for adsorptionare made by including adsorption in the pseudotime calculations(Eq. (4)), whereas for the No Corrections Made linear flow analysis,Eq. (2) was used. Further, the average pressure used in Eqs. (2) and(4) was obtained using the procedures outlined by Nobakht andClarkson (2012b) and Nobakht et al. (2012b), respectivelyi.e. theaverage pressure in the region of influence for constant flowing pres-sure production. From Fig. 15b, we see that the corrections foradsorption do notmake a significant difference in the analysis; the frac-ture half-length derived from the slope of the plot is 254 ft using thecorrected plot and 260 ft using the uncorrected plot, both of whichare in reasonable agreement with the simulator input value of 250 ft.We note that correctionwill dependon the adsorption isothermcharac-teristics as well as drawdownNobakht and Clarkson (2012c) andNobakht et al. (2012b) analyze additional cases. We note that the im-pact of adsorption is to act as a negative skin (Clarkson et al., 2007),which explains why the uncorrected case yields a longer half-length.

An alternative method for analyzing the linear flow data in thisconstant flowing pressure case is to use the square-root of time plot(Wattenbarger et al., 1998) with real time, but using a drawdown cor-rection that accounts for fluid property and desorption with pressure:

xfffiffiffik

p f cp

315:4Tffiffiffiffiffiffiffiffiffiffiffiffigct

q 1mCP

14

where mCP is the slope of the square-root of time plot for constantflowing-pressure conditions, obtained by plotting

m pi m pwf q vs.

time (days) and:

f cp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigc

t

i

gct

vuuut 15

where fcp is an analytical drawdown correction derived by Nobakht andClarkson (2012c) to correct for fluid property variation with pressureand for desorption. The drawdown correction for the constant flowingpressure conventional gas case was derived by Nobakht and Clarkson(2012b) (Eq. (16)); a drawdown correction for non-Darcy flow wasalso derived in Nobakht et al. (2012b). For the complex fracture

geometry case xf should be replaced by contacted matrix surface area(Eq. (17)), as will be discussed for one of the field cases below.

f cp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigct

i

gct

vuut 16

Acmffiffiffik

p f cp

1262Tffiffiffiffiffiffiffiffiffiffiffiffigct

q 1mCP

17

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

0

qg

/[m

(pi)

-m(p

wf)

]



Corrections Made No Corrections Made

0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

500

[m(p

i)-m

(pw

f)]/

qg

Superposition Time


No Corrections Made Corrections Made

1000 2000 3000 4000 5000 6000 7000 8000 150100

a) b)

Fig. 15. Flowing material balance (a) and linear flow analysis (b) of Case 1, with and without corrections for desorption. The vertical lines on the linear flow plot indicate the portionof the plot through which a straight line was fitted.


Note that Ibrahim and Wattenbarger (2006) provided an empiri-cal calculation for fcp, but this was found to be less accurate than theanalytical method provided by Nobakht and Clarkson (2012b), andno corrections for adsorption were provided.

The square-root of time plot for Case 1 is given in Fig. 16notethat the plot is not affected by the corrections as real-time is used.The xf

ffiffiffik

pcalculation is however affected by the choice of drawdown

correction (Eq. (15) for adsorption, Eq. (16) for no adsorption)useof Eq. (15) combined with Eq. (14) yields a fracture half-length of255 ft (given permeability) using the corrected plot and 263 ftusing the uncorrected plot, both of which are in reasonable agree-ment with the simulator input value of 250 ft, and with the linearflow plot analysis above.

Use of the square-root time plot also allows an independent (toFMB) estimate of OGIP. The OGIP calculation, correcting for adsorp-tion, is:

OGIP f cp200:8Sgi

ctBg

i

ffiffiffiffiffiffiffiffitehs

pmCP

: 18

For conventional gas reservoirs with no adsorbed gas, the follow-ing equation would be used (Ibrahim and Wattenbarger, 2006):

OGIP f cp200:8Sgi

ctBg

i

ffiffiffiffiffiffiffiffitehs

pmCP

19

0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

1.2E+06

1.4E+06

1.6E+06

1.8E+06

2.0E+06

0 5 10 15 20 25 30 35 40 45 50

[m(p

i)-m

(pw

f)]/

qg

Sqrt Time (Days)^0.5

Square-Root Time Plot

tehs

Fig. 16. Square-root of time plot for Case 1. A straight-line is fit to the linear portion of thisplotthe green dashed vertical line indicates the end of the linear flow period (tehs).

where fcp and fcp are defined in Eqs. (15) and (16) above. Eqs. (18) and(19) (combined with Eqs. (15) and (16), respectively), yield OGIP es-timates of ~6.8 bscf and ~5.6 bscf; Eq. (18) (which corrects for ad-sorption) is in very good agreement with the corrected FMBanalysis, but Eq. (19) yields a substantially smaller OGIP as expected.

Finally, with respect to Case 1, we note that the square-root oftime plot, in combination with the distance of investigation calcula-tion provided below, can be used to estimate permeability of the res-ervoir:

y 0:159ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiktehsgct

i

vuut 20

which, corrected for adsorbed gas is:

y 0:159ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ktehsgc

t

i

vuut : 21

For Case 1, Eq. (21), when solved for permeability, and knowingy=ye /2 (Fig. 12) yields a permeability estimate of 0.10 mD whileEq. (20) yields a permeability of 0.096 mDthe non-corrected (foradsorption) equation yields a smaller value of permeability becausect>ct. As will be illustrated with a field example later, this approachfor permeability estimation can be used for multi-fractured horizon-tal wells where the fracture spacing can be ascertained and fractureinterference is evident from the square-root of time plot (Ambroseet al., 2011).

Turning our attention to Case 2, we again apply the FMB, linearflow and square-root time analysis with and without correction forpermeability changes, as illustrated in Fig. 17. Unlike the previouscase, the slope of the square-root of time plot changes due to inclu-sion of permeability change in the pseudopressure calculation. Wesee that the impact of not including corrections for permeability inthe FMB plot is underestimation of OGIP (~6 bscf), whereas the cor-rections yield OGIP=6.7 bscf, which is close to the simulator input.Both the corrected linear flow plot (with corrected linear superposi-tion pseudotime and corrected pseudotime) and the correctedsquare-root of time plot (with corrected pseudopressure and applica-tion of fcp with pressure-dependent properties evaluated at averagepressure in region of influence) yield fracture half-lengths of 271 ftand 270 ft, respectively, which are larger, but in reasonable agree-ment with actual value of 250 ft. The uncorrected cases, however,yield much smaller half-lengths of 182 ft and 183 ft, respectively.

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

qg

/[m

(pi)

-m(p

wf)

]




0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

105100500

[m(p

i)-m

(pw

f)]/

qg

Superposition Time



0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

1.2E+06

1.4E+06

1.6E+06

1.8E+06

2.0E+06

0 1000 2000 3000 4000 5000 6000 7000 8000

0 5 10 15 20 25 30 35 40 45 50

[m(p

i)-m

(pw

f)]/

qg

Sqrt Time (Days)^0.5

Square-Root Time Plot


c)

b)

a)

Fig. 17. Flowing material balance (a), linear flow (b) and square-root plot (c) for Case2, with and without corrections for permeability change. The vertical lines on the linearflow plot indicate the portion of the plot through which a straight line was fitted.


3.1.3. Simulated examples of CBM and shale straight-lineanalysisCases 3 and 4wet coal

In Clarkson et al. (2012c), 2-phase CBM cases were analyzed usingthe straight-line methodology discussed above. Case 3 corresponds toan analytically-simulated example, where both gas production andwater production were simulated and analyzed using a 2-phase ver-sion of the flowing material balance equation (Clarkson et al., 2008).FMB alone was used for analysis because the simulated data are inboundary-dominated flow. As mentioned previously, the pressure-and saturation-dependent variables used in the straight-line analysis

are obtained from output generated from a model used to history-match the gas production and water production data. In this case(Fig. 18), an analytical simulator was used to match the gas produc-tion and water production rates of another (commercial) analyticalsimulator (Fig. 18a), and output from the match used to create theFMB plot (Fig. 18b). We see that the 2-phase FMB plot is a straight-line pointing towards an OGIP of ~540 MMscf, which is consistentwith model input.

Case 4 was also shown by Clarkson et al. (2012c), and illustrates theimpact of relative permeability on transient radial flow analysis(Fig. 19). The gas production and water production curves were gener-ated using a numerical simulator, and were matched using an analyticalsimulator (Fig. 19a).We note that the history-match is not quite as goodas in Fig. 18 because an analytical model was used to matchnumerically-generated data (as discussed in Clarkson et al., 2012c).The radial flow plot, which is corrected for relative permeabilitychanges, using outputs from the analytical simulator, is shown inFig. 19b. The derived absolute permeability (~112 mD) is somewhathigher than the numerical model input (100 mD), but in reasonableagreementthe derived skin is slightly negative (1.2), despite themodel input being zero-skin. The errors are related to the slightmismatch between the analytical model and the numerical model, andthe fact that saturation gradients are ignoredstill, a reasonable first ap-proximation to permeability, using a method that does not require therelationship between relative permeability and pressure to be known.

The impact of not correcting for relative permeability changes inCases 3 and 4 is shown in Fig. 20. If the relative permeability correc-tions are not included in FMB analysis, the FMB plot is non-linearuntil late in time (cumulative production), where relative permeabil-ity effects are not as strong (Fig. 20a). If the radial flow plot is notcorrected for relative permeability to gas (Fig. 20b), a straight-linefit to the data will yield permeability values that are closer to effectivepermeability, not absolute permeability.

3.2. Type-curve methods

Type-curvemethods involvematching of production data to dimen-sionless solutions toflowequations,which correspond to differentwell/fracture geometries, reservoir types and boundary conditions. Unlikestraight-linemethods, type-curves are usually designed to capturemul-tiple flow-regimes associated with a particular well geometry/fracturegeometry/reservoir type, and are cast in dimensionless, not dimension-al form. Note that the use of the term type-curve in this work refersstrictly to the use of dimensionless type-curves, analogous to type-curve matching performed for pressure-transient analysis (ex. see text-book by Lee et al., 2003). In petroleum literature, the term type-curvehas also been used to denote ratetime or ratecumulative productionplots for wells representing a certain field or portion of a fieldwe usethe PTA-analog in this work.

The development of type-curve solutions for production analysishas been a popular activity among researchers and practitioners inthe past several decades, and this review cannot hope to capture allof that activity. The focus is instead on type-curves that the authorhas found particularly useful for the analysis of unconventional gaswells. Researchers and their affiliated students, co-workers and col-leagues, that have been particularly fruitful in the development ofproduction type-curve techniques that have proven useful for uncon-ventional gas analysis include: Fetkovich (i.e. Fetkovich, 1980),Blasingame (i.e. Amini et al., 2007; Palacio and Blasingame, 1993;Pratikno et al., 2003), Wattenbarger (i.e. Abdulal et al., 2011;Wattenbarger et al., 1998), Agarwal (i.e. Agarwal et al., 1999), andOzkan (i.e. Araya and Ozkan, 2002). Some of these researchers and af-filiates have developed the analytical solutions to the flow equations,in addition to providing the type-curves and type-curve matchingprocedures. In this work, we show a few examples of type-curve

0

100

200

300

400

500

0

200

400

600

800

1000

0 200 400 600 800 1000 1200

Wat

er R

ate,

ST

B/D

Gas

Rat

e, M

scf/

D

Time, days

2-Phase History-Match

Modelqg Actualqg Actualqw Modelqw

0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

0 200 400 600

qg

/[m

(pi)

-m(p

wf)

]

Normalized Cumulative Production , MMscf

2-Phase Flowing Material Balance a) b)

Fig. 18. Analytical model history-match (a) and flowing material balance analysis (b) of simulated 2-phase (gas+water) CBM case.Modified from Clarkson et al. (2012c).


usage for unconventional gas reservoirs, most of them generatedfrom this list of researchers.

As with the straight-line methods described above, the analyticalsolutions to flow equations to support type-curve generation are usu-ally derived using liquid solutions, static reservoir properties, and sim-ple boundary-conditions (i.e. either constant flow rate or pressure).Pseudovariables (defined above) are therefore often employed to ac-count for fluid and reservoir property variation with pressure, andnormalized production and superposition time (see Table 1) to ac-count for variable flow-rates and pressures. Solution techniques forthe flow equations will not be described in this workthe reader is re-ferred to comprehensive discussions in texts on pressure-transientanalysis (ex. Lee et al., 2003) and, for production type-curves, the indi-vidual papers by researchers cited above.

A critical step in dimensionless type-curve generation is the defini-tion of dimensionless variables. Well-defined dimensionless variablesserve the purpose of condensing many solutions to the flow equationsinto afinite set and allow for uniquematching of the data. The definitionof the dimensionless variables themselves determines which reservoir,and well/fracture properties can be obtained from the type-curvematch. It should be noted that dimensionless ratetime, cumulativeproduction-time, and ratecumulative production type-curves havebeen generated in the literature. Auxiliary curves involving derivativesand integrals have also been developed. The focus here is on ratetime type-curves, but it is noted that the auxiliary curves serve to im-prove the uniqueness of type-curve matching.

10

100

1000

10

100

1000

0 1000 2000 3000 4000

Wat

er R

ate,

ST

B/D

Gas

Rat

e, M

scf/

D

Time, days

2-Phase History-Match

Modelqg Actualqg Actualqw Modelqw

a) b

Fig. 19. Analytical model history-match (a) and radial flow analysisModified from Clarkson et al. (2012c).

A common well-test style definition for dimensionless-rate andtime is given below, assuming a gas-dominated (single-phase) reser-voir (Clarkson et al., 2012c):

qD T

0:000703 kgh m pi m pwf h i qg 22

tD 0:00634kg

gct

iL2c

t 23

tDA 0:00634kg

gct

iAt 24

where:

Lc is some characteristic dimension of the system used in tDdefinitionfor non- or slightly-stimulated wells exhibitingradial flow, Lc=rwa (apparent wellbore radius) and forhydraulically-fractured wells Lc=xf (fracture half-length).

kg is the effective permeability to gas, kg=krg k, where krg isassumed constant for the single-phase gas case.

A is area used in tDA definition.

Note that the unit of time in Eqs. (23) and (24) is days.

0.0E+00

5.0E+02

1.0E+03

1.5E+03

2.0E+03

0 2 4 6

krg

*[m

(pi)

-m(p

wf)

]/q

g

Superposition Time

Radial Flow Plot)

(b) of numerically-simulated 2-phase (gas+water) CBM case.

0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

0

qg

/[m

(pi)

-m(p

wf)

]




0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

4321

krg

*[m

(pi)

-m(p

wf)

]/q

g o

r [m

(pi)

-m(p

wf)

]/q

g

Superposition Time

Radial Flow Plot


200 400 600

Fig. 20. Flowing material balance analysis of Case 3 and radial flow analysis of Case 4, with and without corrections for relative permeability to gas (krg) changes.


As will be discussed further below for type-curve application toCBM and shale, and was discussed previously for straight-linemethods, the time and pseudo-pressure functions may require alter-ation, depending on the combination of fluid/reservoir properties,and boundary conditions. For production analysis, many researchershave altered these dimensionless variable definitions to improve theuniqueness and usefulness of the type-curves. We demonstrate theapplication of several type-curves for vertical and horizontal wellsbelow.

3.2.1. Fetkovich type-curvesFetkovich (1980) generated the first generation of production

type-curves that continue to be useful today. Fetkovich combined ana-lytical solutions for constant flowing-pressure radial flow of liquidswith the empirical decline-curve results of Arps (1945) for boundary-dominated flow. He altered the dimensionless rate and time definitionsas follows:

qDd qD ln re=rwa 1=2 25

tDd 2

re=rwa 21

ln re=rwa 1=2 tD: 26

Note that Fetkovich originally used the liquid-version ofdimensionless-rate and time variables instead of Eqs. 2223. Theresulting type-curve set is given in Fig. 21. To extend the applicationof the type-curves to gas, Eqs. 2223 are used, which include gasproperties and pseudopressure.

To use the type-curves, dimensional ratetime data from a well areplotted on the same loglog scale as the dimensionless type-curves,and then adjusted to overlay the dimensionless type-curves. Detailsof the type-curve matching procedure are provided in Lee andWattenbarger (1996) and Poston and Poe (2008). Assuming that

clarkson_(2013)

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