SPE-173330-MS

Application of Linear Flow Volume to Rate Transient Analysis

Jorge A. Acua, Chevron U.S.A. Inc.

Copyright 2015, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Hydraulic Fracturing Technology Conference held in The Woodlands, Texas, USA, 35 February 2015.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

This paper explores the application of two techniques to rate and pressure transient analysis of fracturedwells in tight reservoirs. The first one is the Linear Flow Volume (LFV), defined as a region with sizedetermined by a characteristic distance proportional to but smaller than the radius of investigation.Similarly to it, this distance and the LFV increase with time without dependence on flow rate or pressure.The LFV can be calculated analytically for an infinite linear system. It can also be calculated in terms ofcumulative production, pressure change and compressibility. When calculated this way, it gives resultsthat are useful to characterize the well and reservoir before and after linear flow. It gives fracture storagevolume for early time and reservoir volume when there is boundary dominated flow. It can also be usedto help assess whether there is a closed or open boundary condition. The second technique is anormalization method for tests with variable pressure and flow. This is an alternative to the conventionalapproach of using p/qB with equivalent time (Q/q) as normalizing variables. The alternative technique issimple and provides normalized variables with much less noise and a better defined straight line in linearflow plots with clear indication of the end of linear flow when present. More importantly, it retainsdependence on actual production time as opposed to equivalent time. This eliminates distortions causedby the use of equivalent time. The combined use of LFV and the alternative normalization techniqueprovides an improved methodology for analysis of fractured wells in tight reservoirs. Validation withsimulated examples is presented and application to actual cases is discussed.

IntroductionConventional pressure transient tests may be impractical in fractured wells in very tight reservoirs becauseof the very long shut-in time required to observe a meaningful response. Rate Transient Analysis uses flowrate and pressure measurements taken while the well produces and offers a practical alternative to evaluatethese reservoirs. In order to calculate reservoir parameters, the conventional approach relies on theconversion of production and pressure records into an equivalent pressure transient test at constant flow.This is commonly done by creating a normalized pressure obtained by dividing pressure by flow rate(p/qB). It is necessary, however to use an equivalent time also named mass balance time, instead of actualproduction time. This equivalent time is defined as the ratio of cumulative production to actual flow (Q/q)(Blasingame et al, 1991). This technique has been used for many years. It has however some problems

that are well known. One of them is that noise contained in real measurements of flow rate and pressureis amplified when dividing one by the other in the calculation of normalized pressure.

Although the noise problem can be mitigated by the use of integrals, the conventional way to calculatenormalized pressure has the additional problem that the resulting data does not follow the time sequenceof the original data. Low flow rates, for example, associate large equivalent time to points correspondingto earlier times while at the same time give large values of p/qB. This causes distortion of the resultingnormalized pressure that is particularly evident at late times. Even for sets of data that do not contain databeyond linear flow, this distortion may cause an apparent transition from linear flow at late equivalenttimes as well as the appearance of boundary dominated flow. Thus, common methodologies for volumeestimation that rely on the existence of boundary dominated flow may be incorrectly applied to wells thatare still in transient linear flow.

Quantifying reserves inside a stimulated reservoir volume or determining the type of boundarycondition beyond linear flow depend on late time data and it is in this region where conventional methodsintroduces the largest uncertainty. Using reservoir simulation may not solve the problem becauseinformation used to constraint the model may be in error if determined by conventional methods.

The solution presented here starts from a derivation of the linear flow solution using fractal diffusivityprinciples. This solution is applied to the definition and calculation of the reservoir volume dominated bylinear flow (LFV) as it changes with time. This volume makes possible to derive a pressure normalizationtechnique that reduces noise and retains dependence to real time. The volumetric calculation of the LFVprovides information before and after linear flow.

Theoretical DevelopmentWe start by assuming that we have a vertically fractured well with a fully penetrating, infinite conductivityfracture of length equal to 2Xf. The reservoir is homogenous with slightly compressible flow. This linearflow problem will be solved using a radial embedding space. We start by distorting the shape of the linearelemental volumes into radially concentric ones (Figure 1). We have to honor the pore space in eachelemental volume and also the product of permeability and area perpendicular to flow (kA). This mapping

Figure 1Mapping a linear medium into a radial one by honoring pore volume and product of permeability by area perpendicular to flow.

Figure 2Constant flow run (q 10 BPD) made for a MFHW (left) and constant pressure run (pwf 500 psi) for the same reservoir (right). Wellproperties are included in the Appendix.

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also requires no pressure gradient parallel to theelemental volumes and, therefore, no flow parallelto elemental volumes. Both conditions are met inthe linear and radial media shown in Figure 1.

By equating pore volumes between the two me-dia in Figure 1 we derive a function for porosity inthe radial medium. By equating kA we define also afunction for the permeability in the radial medium.Both have the same form

(1)

where is replaced by porosity or permeability k. These functions of porosity and permeability havethe form suggested by Chang and Yortsos (1990) and also discussed by Beier (1994) and Cossio et al.(2012), (r) or

Dd and k(r). In our case D1 is the dimension of the medium and d2 is theembedding dimension that corresponds to the dimension in which the problem is being solved. Theparameter relates to the internal connectivity of the medium and equals zero for perfectly connectedsystems (Acua et al., 1995). We set it to zero because we are dealing with a homogeneous porousmedium. The apparent radial dependence of properties is the mathematical consequence of solving a flowproblem using an embedding space of different dimension. Inserting these two functions into the radialdiffusivity equation we obtain the fractal diffusivity equation (Chang and Yortsos, 1990) as shown in theAppendix. The solution for a reservoir of width 2Xf and infinite length is

(2)

where (x, y) is the upper, or complementary, incomplete gamma function, (x) is the gamma functionand dimensionless variables are given in the Appendix. This solution can also be written as

(3)

where . After some time, the exponential and the ratio of gamma functions converge to oneand we have a long term expression for pressure that, as expected, changes linearly with distance asfollows.

(4)

At any time, we can define a region with linear flow by extrapolating the long term pressure profile(Eq. 4) to zero. Thus, the distance to the linear edge is given by

(5)

This distance is smaller than most definitions for radial or linear distances of investigation (Kuchuk,2009). We will call it linear edge distance to avoid any confusion with radius of investigation. Adimensional expression for this distance is given in the Appendix. The linear flow volume (LFV) is givenby , where rD is just the linear edge distance divided by Xf. A dimensional expression forLFV derived from for the linear edge distance is also given in the Appendix.

Assuming no wellbore storage, or fracture storage as will be denoted here, and no skin and usingdefinitions shown in the Appendix we find that the LFV for a distance rD can be expressed as

Figure 3Linear flow volume (LFV) for the two runs in Figure 2including log-time derivatives to indicate slope changes. The known SRVvolume and the calculated telf are shown by the dot.

SPE-173330-MS 3

(6)

where p pi pwf. The first expression corre-sponds to constant flow rate qB and the second oneto constant pressure p. For a well at constant flowrate qB, using equations 5 and 6 and the equation forpressure in the Appendix, the LFV can be written as

(7)

where QB is cumulative production and c iscompressibility. For a well flowing at constantdownhole pressure, the cumulative production is QB 2qBt. Using equations 5 and 6 and the equationfor flow in the Appendix we get

(8)

The two expressions provide a similar volume as long as the system behaves as an infinite linearsystem. For variable flow and pressure cases we have found that Eq. 7 should be applied for constant andincreasing flow and Eq. 8 for decreasing flow. Since decreasing flow is the most common case we willcontinue the discussion with Eq. 8. The equation shown in the Appendix for LFV corresponds to aninfinite linear reservoir with no wellbore storage or skin. Eq. 8 diverges from this solution when the endof linear flow is reached. Outside of the linear flow range, Eq. 8 gives the correct fracture storage volume

Figure 4Comparison of p/qB and derivative calculated using Eq. 10 and variable flow data with data from a constant flow simulation (left). Thesame plot using conventional p/qB and equivalent time (Q/q) (right) for comparison.

Figure 5Pressure, log time derivative and pressure derivative for a vertically fractured well with fracture storage and skin at constant flow (1STBPD) in a finite system (left). Constant pressure run (pwf 500 psi) for the same reservoir with maximum flow during fracture storage periodlimited to 50000 bpd.

Figure 6LFV for the constant pressure run (Eq. 8) gives the correctfracture storage volume (3330 bbl) and reservoir volume (1.23E6 bbl) atlate times. The constant flow curve (Eq. 7) overestimates the fracturestorage volume by 4/. The dot shows the known system volume and thetiming at which it should be reached (see Appendix).

4 SPE-173330-MS

at early times and approaches the correct reservoir volume in case of boundary dominated flow at latetimes.

Fracture storage effects disappear with time from LFV but skin effects last much longer and acorrection needs to be made. The reason is that not all pressure depletion occurs in the reservoir asassumed in Eq. 8. To correct for skin, a plot of p/qB versus is made that should be linear

. Using the values of m and b from the linear fit we correct the LFV as follows

(9)

This correction is strictly valid only for the transient period. The values of m and b can also be usedto calculate reservoir properties including skin factor (see Appendix). If the end of linear flow is visibleand an assumption for distance to linear edge can be made then the permeability k can be calculated. Eq.

Figure 7Plot of p/qB versus where slope m and intercept b are obtained (left). The LFV curves (right) shown uncorrected (Eq. 8) and corrected(Eq. 9). Correct fracture storage volume is seen in the uncorrected curve. LFV for infinite linear system is also shown as well as the known reservoirvolume at calculated telf (dot).

Figure 8Comparison of ratio p/qB and derivative from a constant flow run derived with Eq. 10 (left). The same plot using the conventional methodof normalizing with p/qB and using equivalent time (Q/q) (right).

Figure 9Comparison of p/q calculated with Eq. 10 versus production time (left) and p/qB calculated conventionally versus equivalent time (right)for a typical tight oil well. All scales (linear), best fit and telf (dot) are the same for both graphs.

SPE-173330-MS 5

9 is applied to the entire data set and this causes thecorrection to modify the fracture storage volumethat should be correct to begin with. The compari-son of corrected and uncorrected LFV plots shownbelow illustrates this.

Eq. 7 and 8 can be used to derive a synthetic setof p/qB for constant flow from the variable pressureand flow data. The equation is

(10)

where t is time in units required to convert theratio Q/t into the proper units for flow rate. Eq. 10can be used as an alternative to the actual ratio of pressure to flow rate to normalize pressure. It resultsin much less noise because it uses cumulative production instead of flow, and more importantly, it workswith actual production time without requiring equivalent time (Q/q).

ValidationWe will validate this formulation with two simulated cases. The first case is a multi-fractured horizontalwell (MFHW) with no fracture storage or skin in an infinite reservoir so conditions are always transient.The second case is a vertically fractured well in a finite system with fracture storage and skin. Propertiesfor the two cases are presented in the Appendix.

MFHW in an infinite systemHere we show that for an infinite acting reservoir the formulation presented above has equal or greaterprecision than the conventional p/qB methodology. For this case two runs were made: a constant flow runand a constant pressure run. Each run was made with two commercial reservoir simulators that providedvirtually the same result. Figure 2 (left) shows the resulting pressure and pressure derivative plot for theconstant flow run. The shape is typical for MFHW giving an initial linear flow that transition into asecondary pseudo linear flow that occurs outside of the stimulated reservoir volume (SRV). This is thevolume that contains all the artificial fractures and finally radial flow as the system is infinite.

Figure 2 (right) shows the result for the constant pressure run. This run is presented using cumulativeproduction QB and the log derivative of cumulative production that is simply qBt. This format uses

the same units of volume for both curves and makes possible to see the similarities with the constant flowrun.

The LFV for both runs is shown in Figure 3. Equations 7 and 8 were used respectively for the constantflow and constant pressure runs. The log time derivatives of volume are also presented in Figure 3

to magnify changes in slope. The two curves are very close to each other. Also shown as a dot is theknown size of the stimulated reservoir volume (SRV 4hnfXfr 2hLwXf) as well as the time to endof linear flow (telf) given by both calculated with r Lw/nf/2 100 ft (See Appendix).The shape of the LFV is the same than that of the cumulative production for constant downhole pressureas they both differ by just a constant.

For reservoirs that extend beyond the SRV, the LFV shows a smaller rate of increase in volume aftertelf. Thus, a small decrease in slope in the LFV plot suggests the existence of an open boundary. The actualvalue of the slope after telf may depend of the geometry of the SRV.

Figure 4 (left) shows a synthetic p/qB and derivative calculated using data from the constant pressure(variable flow) run and Eq. 10 and its comparison with p/qB from the constant flow run. Also shown in

Figure 10The corrected, uncorrected and calculated LVF for the caseof Figure 9. Scales are linear and time axis is the same as that of Figure9. Reservoir size and telf are shown by a dot.

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Figure 4 (right) are the same curves using the conventional method of p/qB and equivalent time (Q/q). Forthis purely transient example both techniques give a good approximation to the actual solution for entiredata range. The conventional curves are off during linear flow by a factor of in pressure. Anderson and

Mattar (2003) recommend shifting equivalent time by which also makes the two lines coincide. The

alternative methodology, however, preserves dependence on actual production time.

Vertically fractured well in a finite system with fracture storage and skinHere we illustrate the effect of wellbore or fracture storage and skin as well as boundary dominated flowto demonstrate the usefulness of the LFV. This is a fully penetrating vertically fractured well with infinitefracture conductivity. The reservoir is a square box with length equal to fracture length and a smallerwidth. The geometry and reservoir properties are specified in the Appendix. We also made two runs forthis case one at constant flow and one at constant pressure. The constant flow run is shown in Figure 5(left). We show pressure, pressure derivative as well as the primary derivative . The log derivative

shows fracture storage corresponding to a volume of 3330 bbl. We then see the half-slope of linear flowbetween 24 and 600 hours approximately and finally the unitary slope of boundary dominated flow. Thevolume of the reservoir is 1.23E6 bbl.

The constant pressure run is shown in Figure 5 (right). Cumulative production QB and the log timederivative are shown together with the primary derivative that equals flow rate qB. There is a large

discontinuity in flow rate when the system transitions from fracture storage to reservoir flow. Whensimulating systems with fracture storage the maximum flow rate must be specified during this period andit was set at 50000 BPD in this case. During this period pressure is not constant.

Figure 6 shows the LFV for both runs using Eq. 8 for the constant pressure run and Eq. 7 for theconstant flow run. The constant pressure curve correctly gives the fracture storage volume and approachesthe correct reservoir volume as flow rate becomes smaller at late times. The constant flow curveoverestimates both volumes, the fracture storage volume by a factor of 4/. Both curves agree during thetransient linear period. The slope of the LFV curve after telf decreases significantly. No increase of LFVwith time relates to the existence of a closed boundary. Thus, the slope change after telf can be used to helpidentify the boundary conditions at the edge of the linear flow region.

Figure 7 (left) shows the plot of p/qB versus . As mentioned before the slope and intercept can beused to correct the LFV using Eq. 9. Figure 7 (right) shows the effect of the correction. Also shown is theLFV for an infinite linear system calculated using the equation in the Appendix. The known size of thereservoir is also shown as a dot at telf. The corrected expression for the LFV coincides with the onecalculated for an infinite system once the fracture storage effect dies off after approximately 10 hours. The

Figure 11Comparison of p/q calculated with Eq. 10 and production time (left) and p/qB calculated conventionally with equivalent time (right) fora typical shale gas well. Scales (linear), best fit and telf are the same for both graphs.

SPE-173330-MS 7

uncorrected curve can be used to see the fracturestorage volume and help determine the reservoirvolume for the case of boundary dominated flow.

A synthetic set of p/qB for constant flow isobtained from variable flow data using Eq. 10. Thecases we have analyzed show that the alternativemethodology is an excellent approximation duringtransient flow but it is off by a factor of 4/ duringwellbore storage. We can apply Eq. 10 but changingthe constant from 4/ to 1 during fracture storageand boundary dominated flow. This gives a goodapproximation during boundary dominated flow.Figure 8 (left) shows the resulting comparison be-tween the synthetic p/qB data versus the actual plot from the constant flow run. The breaks in thederivative show the points where the constant was changed.

Figure 8 (right) shows the p/qB plot obtained using conventional methods. The result is comparable tothe alternative methodology except during the transition from fracture storage to linear flow and duringthe boundary dominated flow. The conventional methodology is a very good approximation duringwellbore storage and has the correct slope during boundary dominated flow, but it is off by the factor

during transient flow. Another feature of the conventional methodology is that boundary dominated

flow period is greatly distorted making it look much longer than the 4.25E4 hours (4.85 years) that theflow and pressure record last. Using production time eliminates this distortion and makes possible to treatp/qB as a pressure transient test with flow equal to 1 and use conventional pressure transient analysistechniques.

Discussion of application to actual casesThe methodology presented has been applied to tight oil wells as well as shale gas wells. We haveobserved that the apparent equivalence of the conventional method with the alternative one observed inthe two simulated examples previously shown, breaks down when dealing with real data. The twoexamples that follow are intended to illustrate how the alternative methodology improves the identifica-tion of linear flow and its transition to a different flow regime.

Figure 9 (left) shows a linear flow plot constructed with the alternative normalization technique(Eq. 10) for a typical tight oil well. The expected linear behavior and its end are evident. Best linear fitand telf are shown. Figure 9 (right), constructed conventionally, makes it difficult to identify the expectedlinear behavior or the end of linear flow. The same best linear fit is shown for reference.

The slope m of the plot in Figure 9 (left) can be used to calculate with the same equation usedin pressure transient analysis (see Appendix). The intercept b, negative in this case, can be used tocalculate skin factor (see Appendix). The telf allows calculation of permeability provided that anassumption with respect to distance to the linear edge can be made (see Appendix).

Figure 10 shows the LFV corrected with m and b from Figure 9 (left) using Eq. 9. It also shows theuncorrected LFV and the calculated LFV for an infinite linear system (see Appendix). The fracture storagevolume is not well defined but the reservoir pore volume can be seen at telf. Despite the noise in the data,the change in slope after telf suggests a closed boundary. Our experience indicates that data from tight oilis usually noisier that data from shale gas wells as illustrated in the next example.

Figure 11 (left) shows the linear flow graph for a typical shale gas well. The alternative methodologyshows a linear flow period as well as a departure that signals the end of linear flow (telf). The slope aftertelf is slightly different indicating the existence of an open boundary after the SRV. Figure 11 (right),

Figure 12Uncorrected, corrected and infinite linear LVF curves forthe case of Figure 11. Scales are linear and the time axis is the same asthat of Figure 11. SRV volume and telf are shown by the dot.

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constructed with the conventional p/qB and equivalent time does not show a unique slope during linearflow and the end of linear flow is not as clear. The equivalent time range is greatly increased and onlypart of it is shown. Both graphs have the same scales and the best fit of Figure 11 (left) is shown in Figure11 (right) just for reference.

The LFV is shown in Figure 12 in both corrected and uncorrected forms. Also shown is the calculatedLFV for an infinite linear system (see Appendix). The early flat part in the uncorrected curve should helpcalculate fracture storage. The corrected and infinite linear LFV curves coincide for the duration oftransient linear flow. The reservoir volume at telf corresponds to the SRV and flow after telf is still transientas there seems to be an open boundary, or continuation of the permeable medium, at the edge of the SRV.

ConclusionsA linear flow solution was derived using fractal diffusivity principles. Although this approach has beenavailable since the early 90s, it had little relevance because it requires the assumption that porosity andpermeability change with distance from the well. This behavior although valid for fractal objects (Acunaand Yortsos, 1991), is physically unreasonable for porous media. We have demonstrated that for the caseof linear flow reservoirs, these variations are not related to physical characteristics of the reservoir but arethe mathematical consequence of solving a linear flow problem within an embedding radial space. Thesame might be true for other flow dimensions. This understanding may help develop the full potential ofthe fractal diffusivity formalism.

We applied the linear flow solution to the calculation of size of the region dominated by linear flowand found that it can be defined by a linear edge distance that is smaller than radial or linear distances ofinvestigation. The linear edge distance, similarly to the radius of investigation, does not depend on flowor pressure, but only on production time and diffusivity. The LFV can be analytically calculated for aninfinite linear system using this distance. It can also be calculated using cumulative production, pressureand time. When calculated this way, the LFV provides not only a description of the variation in time ofvolume of the region under linear flow but also fracture storage volume, akin to wellbore storage, at earlytimes and approximates reservoir volume when boundary dominated flow if present. The LFV can also beuseful in determining whether there is a closed boundary or not after the end of linear flow time.

We have modeled different conditions of flow and pressure and found that the LFV is not exactly thesame for all cases. This is expected because this simple approach is not equivalent to the convolutionmethods required to deal properly with variable flow and variable pressure cases. Similarly to theconventional method to normalize pressure, the alternative method presented here is an approximation. Itoffers however added advantages of substantial noise reduction and use of production time as opposed toequivalent time. These advantages make it superior in our opinion to the conventional method. We havefound that the precision of the method is good for practical applications. When applied to actual tight oiland shale gas cases, the results of the methodology presented in this paper has been validated withnumerical models and it has been found to produce good agreement between actual data and modeledresults.

The combined use of the alternative pressure normalization and LFV adds clarity to the analysis of allflow regimes. This is particularly valuable for the period after linear flow where the removal of noise andtime distortion is crucial for reliable test interpretation and reserves estimation.

AcknowledgementsThe permission of Chevron to publish this paper is gratefully acknowledged.

Nomenclature

B formation volume factor, L3/L3, [RB/STB]

SPE-173330-MS 9

c compressibility, Lt2/m [psi-1]LFV linear flow volume, L3 [RB]p pressure, m/Lt2, [psi]q volumetric flow rate at surface conditions, L3/t, [STB/D]Q cumulative production at surface conditions, L3, [STB]p* ratio of dimensional to dimensionless pressure, m/Lt2, [psi]q* ratio of dimensional to dimensionless flow rate, L3/t, [DTB/D]t* ratio of dimensional to dimensionless time, t, [hr]r* reference distance, L, [ft]t time, t, [hr]teq equivalent time (Q/q), t, [hr]telf time of end of linear flow, t, [hr]r distance, L, [ft]S skin factor [dimensionless]SRV Stimulated reservoir volume, L3 [RB]h thickness, L, [ft]

Greek symbols

(x) Gamma function(x,y) Upper incomplete gamma function porosity [dimensionless] viscosity, m/Lt, [cP]

Subscripts

D dimensionlessi initialwf wellbore flowing

ReferencesAcua, J.A., 1993. Numerical Construction and Fluid Flow Simulation in Networks of Fractures Using

Fractal Geometry. Ph.D. Dissertation. University of Southern California. Los Angeles, California.Acuna, J.A. and Yortsos, Y.C., 1991. Numerical Construction and Flow Simulation in Networks of

Fractures Using Fractal Geometry. SPE paper 22703Acuna, J.A., Ershaghi I and Yortsos Y.C., 1995. Practical Application of Fractal Pressure-Transient

Analysis in Naturally Fractured Reservoirs. SPE paper 24705.Anderson, D.M. and Mattar, L., 2003. Material-Balance-Time During Linear and Radial Flow.

Canadian International Petroleum Conference, Calgary, PETSOC paper 2003-201Barker, J.A., 1988. A Generalized Radial Flow Model for Hydraulic Tests in Fractured Rocks. Water

Resources Research 24(10), 17961804.Beier, R.A. Pressure-Transient Model for a Vertically Fractured Well in a Fractal Reservoir. SPE

Formation Evaluation, June, 1994.Blasingame, T.A., McCray, T.L. and Lee, W.J., 1991. Decline Curve Analysis for Variable Pressure

Drop/Variable Flowrate Systems, SPE paper 21513.Chang, J. and Yortsos, Y.C., 1990. Pressure Transient Analysis of Fractal Reservoirs. SPE Formation

Evaluation 289:311.Cossio, M., Moridis, G.J., Blasingame, T.A., 2012 A Semi-Analytic Solution for Flow in Finite-

Conductivity Vertical Fractures Using Fractal Theory. SPE paper 153715.

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Kuchuk, F.J., 2009. Radius of Investigation for Reserves Estimation from Pressure Transient WellTests. SPE paper 120515.

Palike, H., 1998. Pumping Tests in Fractal Media. M.Sc. Thesis. University College London.

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Appendix

Solution for other dimensions and dimensional equations for linear flow

The diffusivity equation for a well in a porous medium (0) with dimension D2 embedded in a medium of radialdimension is

(A-1)

pD, tD and rD are dimensionless pressure, time and radius respectively. This expression is similar to that presented by Changand Yortsos (1990, 1993) but different from that by Cossio et al. (2012). The solution is

(A-2)

where , (x,y) is the upper incomplete gamma function and (x) is the gamma function. (In Acua et al. (1995)

there is a typo in this equation: the value 1 is shown instead of 1 in the first argument of the incomplete Gammafunction, as also mentioned by Palike (1998). The correct expression shown above is in Acua (1993). This solution is similarin form to the generalized radial flow model presented by Barker (1988). Eq. A-2 gives the solution for linear, radial andspherical flow for values of equal to 0.5, 1 and 1.5 respectively and using the identity presented below it can be expressedas

(A-3)

For radial flow, this equation converges to the known logarithmic expression as 1 and .

The solution for linear flow (0.5) is shown in the paper and the dimensionless variables are defined as ,, and where

(A-4)

where p (pi pwf) with pi being initial pressure and pwf bottomhole flowing pressure. CFp and CFt are 0.001127 and0.0002637 for customary units and both equal to one for consistent units. Other symbols are defined in the Nomenclature

In dimensional form the distance to the edge of linear flow is

(A-5)

The linear flow volume (LFV) in bbl for an infinite linear reservoir at a given time t in hours is given by

(A-6)

where m is the slope p/qB versus as defined below.In dimensional form, pressure at the wellbore (r0) for constant flow (qB) is given by

(A-7)

The slope m of p/qB versus plot is given by

(A-8)

The intercept b of the p/qB versus plot in terms of the skin factor S is given by

12 SPE-173330-MS

(A-9)

Flow qB for constant pressure (pi pwf) is given by

(A-10)

Properties of the MFHW in an infinite reservoir

Properties of the vertically fractured well in finite reservoir

The gamma function and the upper incomplete gamma functionThe gamma function is defined as

(A-11)

A useful property is () ( 1)( 1)The upper incomplete gamma function, also known as complementary incomplete gamma function, is defined as

(A-12)

0, -1, -2, . . .For negative values of the first argument, the following expression can be applied recursively to calculate the upper

incomplete gamma function

(A-13)

Some special values are:(0, x) Ei(x)(0.5, x) (1, x) ex

Author BiographyJorge A. Acua is a consultant reservoir engineer at Chevron Energy Technology Company. He is involved in well testing andreservoir characterization, geothermal reservoir simulation and wellbore dynamics. He joined Unocal, now Chevron, in 1996.He holds a BS degree in Civil Engineering from the Universidad de Costa Rica and MS and PhD degrees in PetroleumEngineering from the University of Southern California. His research interests are pressure transient analysis in fracturedreservoirs and geothermal reservoir simulation.

SPE-173330-MS 13

Application of Linear Flow Volume to Rate Transient AnalysisIntroductionTheoretical DevelopmentValidationMFHW in an infinite systemVertically fractured well in a finite system with fracture storage and skinDiscussion of application to actual casesConclusionsAcknowledgementsReferencesProperties of the MFHW in an infinite reservoir

Properties of the vertically fractured well in finite reservoirThe gamma function and the upper incomplete gamma function