PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, February 11-13, 2013 SGP-TR-198

A NEW DOUBLE FRACTAL MODEL FOR TEST ANALYSIS WITH TRANSIENT INTERPOROSITY TRANSFERENCE FOR PETROLEUM AND GEOTHERMAL SYSTEMS

Alex R. Valdes-Perez1, Hector Pulido2, Heber Cinco-Ley3, Leif Larsen1,4

1. University of Stavanger, Stavanger 4036, Norway. 2. Petróleos Mexicanos, Av. Marina Nacional 329, Col. Petróleos Mexicanos, México D.F. 3. Universidad Nacional Autónoma de México, 4. Kappa Petroleum Exploration & Production, Norway.

e-mail: [email protected]

behavior (late times). A characteristic of the transient ABSTRACT pressure behavior of these systems is that in a semilog plot two parallel straight lines will be shown. A new double porosity model for Naturally Fractured The first straight line correspond to the first radial Reservoirs (NFRs) assuming fractal fracture network flow due to fractures expansion and the second behavior and its solution is presented. Primary straight line is attributed to a radial flow when all porosity is idealized as Euclidian matrix blocks (slabs porous systems are acting as a single one. Besides, a or spheres) and Secondary porosity is defined by any third straight line non-parallel to the other two is post-depositional geological phenomenon such as observed, which describes the interaction between fractures and vugs. porous media (intermediate times). Shape of such In order to provide a framework, the generalized straight line depends on the flow regime conditions, radial flow model solution for well test analysis for i.e., transient or pseudosteady state interporous petroleum and geothermal systems in Laplace and transference. This topic has been under debate by Real space was developed. Development of an some authors and it is part of the discussion on the appropriate wellbore storage model for fractal present work. reservoirs is also shown. Nature of flow in multi-porous systems obeys to For this model, the dimensionless fractal fracture the fact that flow in each porous medium behaves area parameter was developed. In addition, differently in terms of gradient pressure from the interporosity skin factor between matrix blocks and other media. Such behavior is known as transient fractal fracture network is introduced. Relationship of interporous flow; this flow regime was studied convergence between interporosity skin under previously by de Swaan (1976), Najurieta (1980), transient transference regime and pseudosteady state Cinco-Ley et al. (1982), Serra et al. (1982) and transference regime is discussed. An analytical Streltsova (1982). general solution was obtained in Laplace space; On the other hand, practice has shown that besides, analytical solutions in real space that pressure gradients in all porous media apparently describe the behavior of NFRs at different stages and behave in the same way. Such flow transference is different cases of flow are also presented. Early, known as pseudosteady-state flow. Barenblatt et al. intermediate and late-time approximations are used to (1960) and Warren et al. (1963) developed this obtain reservoir and fractal fracture network theory. parameters. A synthetic example is presented to However, Cinco-Ley et al. (1985) showed that the illustrate the application of this model. apparent pseudosteady-state transference behavior can be attributed to a presence of interporous skin INTRODUCTION between matrix and fracture network. Such Behavior of NFRs interporous skin is produced by a film created by NFRs are multi-porous systems, caused by chemical mineralization or interaction between fluids in the or tectonic events or both. Porous media identified in face of the matrix blocks. Mineralization has been NFRs are matrix, fractures, faults and vugs, from observed in outcrops, where precipitation and other micro to mega scales. chemical phenomena create a skin between different Transient pressure behavior in non-fractal double porous media. Interporous skin is defined as: porosity systems has three flowing periods: fracture k x S  1 d . (1) network expansion (early times), interaction between intD kd h1 porous media (intermediate times) and single system Where: 3d d 1 k1  permeability of medium 1, V  b e r e r . (2) b de h1  characteristic length of medium 1, Where: k  permeability of damaged zone,  Area of a unit sphere of in d dimensions; it is d de e xd  thickness of damaged zone. defined as: 2 de / 2   , (3) de d   e   2  de  Euclidean dimension, b  extent of the flow region, r  radial distance from the centre of the source (measured along the flow paths). x gamma function of x . r  width between the surfaces. Since the term r in eq. 2 represents the width between the surfaces, it can be deduced that the exposed to flow area is given by: Figure 1: Scheme of the interporous skin between A  b3de rde1 . (4) matrix blocks and fracture. exp. flow de

Fractal Models Porosity of a Unitary Fracture Fractal models for characterizing NFRs are the best Conceptually it represents the volumetric fraction way to represent randomness in the fracture network occupied by a single fracture regarding the total rock distribution within the reservoir. Figure 2 shows an volume. It is given by: idealization of a well intercepting a fractal fracture unitary fracture volume Vuf network within a NFR. uf   . (5) total bulk volume Vb Chang et al. (1990) proposed a model to characterize reservoirs of single and double porosity Fractal Fracture Network Porosity under pseudosteady-state transference conditions It represents the volumetric fraction of all fractures in with fractal geometry. On the other hand, Olewareju the rock. It is defined as: (1996) developed a double porosity model for fractal fracture network volume Vfb reservoirs, assuming free interporous transeference fb   . (6) (transient flow without interporous restriction), and a total bulk volume Vb fixed shape of Euclidean matrix blocks. Assuming fractures with the same characteristics Work developed by Acuña et al. (1995) and all over the bulk, fractal fracture network volume, Camacho-Velazquez et al. (2008) are also relevant can be expressed as: for this topic. V  n rV r , (7) fb f uf where: nf rnumber of fractures into fractured bulk,

Vuf  unitary fracture volume. Moreover, the number of fractures into fractured bulk can be expressed using a power-law model:

Dfb 1 n f r  ar . (8) Where: a  site density parameter, Dfb  fractal dimension of the fracture network. Therefore, fracture network volume is expressed as: D 1 V  ar fb V r . (9) Figure 2: Idealization of a fractal fracture network. fb uf

Combining eq. 2 and 7, porosity of the fracture Irregular Shape Volume of Rock network is given by: Barker (1988) defined that an irregular shape volume D d ar fb e V of rock can be represented by the volume of two uf  fb  . (10) equipotential surfaces, such region is defined as:  b3de de

2

Darcy’s Law in Fractal form k fb  fb  ; (17) Chang et al. (1990) proposed the following equation  fbctfb to express Darcy’s law in a fractal form: Where: kr p q   . (11) ctfb fractal fracture network total compressibility.  r Prior equation uses permeability as a function of Dimensionless variables radius; it can be expressed as: The following dimensionless variables in field units are used in the present study. aV k r p q  uf fb fb , (12) Dimensionless radius: fb r r rD  , (18) Where: rw p fb  pressure in fractal fracture network. Where: r wellbore radius. k fb  permeability of the fractal fracture network. w   viscosity of flowing fluid. Dimensionless time: 0.00026367k   grouping parameter, defined as: fb tD  t , (19) c  r 2   Dfb  1 , (13) t t w Where: Where:   conductivity index; related to the spectral ct t  fbctfb mactma, (20) exponent of the fractal fracture network. ma porosity of matrix blocks, c  matrix blocks total compressibility. PROPOSED MODEL tma For an oil-filled system, dimensionless pressure in Model development the fracture network: Based on eqs. 2, 10 and 12 the diffusivity equation for   D 0.03281 aV k p r,t a double porosity fractal reservoir with transient p r ,t  fb uf fb fb , (21) fbD D D  1 interporosity transfer is derived: 887.22 qBo orw  fb 1   p fb r,t Where: r    Dfb 1   q flow rate, r r  r  , Bo  formation volume factor of oil, t pi initial pressure. kmaAfb p   1 p fb r,t  ma p  d  Dimensionless pressure in the matrix: k  d um a sur  t fb 0 fb  0.03281  Dfb   aVufk fbpmar,t (14) p r ,t   . (22) maD D D 887.22 1 Where, qBoorw fb For gas reservoirs, dimensionless pressure in the pma pressure in matrix blocks. fracture network: k permeability of the Euclidean matrix blocks, ma  0.03281  Dfb   aVuf k fbmp fb  Afb  fractal fracture network area per unit of bulk p r ,t  , (23) fbD D D 8937.5 1 volume, it is defined for slab matrix blocks as: qg g ZTrw  fb 2 Where, pseudo-pressure fuction is given by: Afb  , (15) 2 2 hma  hf mpfb  pi  pfbr,t; (24)

Where: qg  gas flow rate, hma  strata height, Z  real gas deviation factor, hf  fracture width. T temperature.

For cube matrix blocks is defined as: Afb Dimensionless pressure in the matrix: 6h2  0.03281  ma Dfb   aVufk fbmpma Afb  , (16) , (25) 3 pmaDrD,tD  1 hma hf  8937.5 qggZTrw fb

 fb fractal fracture network hydraulic diffusivity where: 2 2 coefficient, it is defined: mpma pi  pmar,t. (26) For geothermal reservoirs (steam), dimensionless pressure in the fracture network:

3 2 2  4maDn  tD  D 0.03281 aV Mk m p   fb uf fb  fb 4maD H pfbDrD,tD  , (27) F ,H ,t   e D . (38) 81361 1 maD D D  WsZTrw fb HD n1 Where: Constant Rate General Solution of an Infinite 2 2 Fractal Reservoir Model assuming Transient mpfb  pi  pfbr,t, (28) Interporosity Transference W  mass flow rate, In order to have a well test analysis model the M  Molecular weight. following conditions have been set. Dimensionless pressure in the matrix: Initial condition for fractal fracture network:  0.03281  Dfb   aVuf Mkfbmpma p r ,0  0, (39) p r ,t  , (29) fbD D maD D D  1 81361 Ws ZTrw  fb Inner Boundary: And:  p fbD1,tD  2 2 rD  1, (40) mpma pi  pmar,t. (30) rD For this work the oil-filled case was considered. Outer boundary: Similar expressions should be obtained for the other lim pDrD,tD   0 . (41) cases. The substitution of variables from eq. 18 to eq. rD 22 into Eq. 14 yields: Therefore, solution of eq. 31 in Laplace space is 2 given by:  p fbDrD ,tD   p fbDrD ,tD    1  2  2 2  2 sf s 2  rD rD rD r K r D 1    2 D  tD 2    pmaDrD ,  pD rD,s . (42)  rD AfD1 FmaD, HD ,tD  d      3/2 2 sf s 0 s f sK   Dfb    2  2   Pressure at wellbore is given by:  p fbDrD ,tD   rD   2  t K1  sf s D   2  (31) 2 pwDs  , (43) Where, dimensionless storativity ratio,  is defined: 3/ 2  2  s f sK D  sf s  c fb   2    fb tfb ; (32) 2 ct t where interporous transference function is defined Dimenssionless matrix hidraulic diffusivity: as: k c  AfD1Fm aD, HD ,s   ma t t ; (33) f s   , (44) maD HDSma fbD mactmak fb 1 Fm aD,HD ,ss Dimensionless block size, for slabs: m aD r 2 Where: w ; (34) HD  2 Sma fbD  Interporous skin between matrix and hma fractures. and for spheres: Moreover, for slab matrix blocks: r 2 H  w ; (35)   1 H s  D 2 F ,H ,s maD tanh D  , (45) dma maD D   HDs 2 maD Where:   And, for spheres as matrix blocks: dma  sphere diameter.    1 H s    F  , H ,s  maD coth D   2 maD . (46) Dimensionless fractal fracture area, AfD :  maD D    HDs   2 maD  HDs   AfbhmaVbrw It can be verified that, for radial flow and non- AfD  . (36) Vma fractal object, i.e., Dfb  2, and   0, eqs. 42 and Fluid transference function, assuming slabs: 43 converge to the model proposed by Cinco-Ley et 2 2 maD2n1  tD  al. (1985); in addition to the previous conditions, if 4   maD HD , (37) no interporous skin exists, the model converges to the FmaD,HD,tD   e HD n1 one proposed by Cinco-Ley et al. (1982). Figure 3 Or, if spheres as matrix blocks, fluid transference shows the convergence of this model to the one function is: developed by Cinco-Ley et al. (1982) assuming slabs.

4 between media. Such behavior has been attributed to pseudosteady state transference conditions.

Figure 3: Pressure and pressure derivate function behavior for some values of D , and  fb Figure 5:Impact of interporous skin in pressure and without interporous skin and its pressure derivative function, when convergence to the model proposed by . Cinco-Ley et al., (1982). It is important to point out that in general, the relationship Dfb   2 in double porosity fractal models will provide a response like the typical shown by double porosity reservoirs assuming radial flow, i.e., a semilog straight line during fractal fracture network expansion (short times) parallel to the semilog straight line during single system behavior (late times). Therefore, even when the typical double porosity pressure behavior -which assumes radial flow - be observed, a fractal behavior might be taking place. Figure 4 shows the pressure behavior on a semilog plot, for , and values that satisfies this Figure 6:Impact of interporous skin in pressure and condition, assuming slabs as matrix blocks. pressure derivative function, for Dfb   2. For practical application, phenomena around wellbore such as skin at wellbore and wellbore storage are incorporated in Laplace space as follows: Swell pwD s p s,C ,S  s . (47) wD  D well  2 1 SwellsCD  CDs pwD s

Figure 7 and Figure 8 show the effect of wellbore storage and skin around wellbore in the examples shown in Figure 5 and Figure 6, Figure 4: Semilog plot of the pressure behavior for respectively. some values of that satisfy the condition of .

If there is no interporous skin, i.e., Sma fbD 0 , this model acquires a general shape of transient interporous transfer. A particular case of this model in terms of the matrix-fracture interaction parameter,  , was presented by Olewareju (1996). Figure 5 and Figure 6 show the impact of interporous skin in the response of pressure and pressure derivative function, when Dfb  2 and Dfb   2, respectively; assuming slabs as matrix blocks. It can be observed that, even for low interporous skin Figure 6: Behavior of pressure and pressure values ( Sma fbD  0.1), the pressure derivative derivative function, when , function shows a valley shape during interaction considering phenomena around wellbore. 5 a) For Transient flow with low interporous skin, interporous transference function is approximated as: 1   S H s  f s  A 1 maD 1 ma fbD D  , (52) fD H s   D  maD  Then, approximated solution for Dfb  2 is:    2 1 1  maD  pwDtD   ln tD  ln   2 ln AfD1      2 4 2  HD 

1 1 H    S D t 2  0.4329 Figure 7: Behavior of pressure and pressure ma fbD  D  2  maD  derivative function, when Dfb  2 , (53) considering phenomena around wellbore. For Dfb  2:

v1 Solutions at early, intermediate and late times   1    maD  In order to provide models to compare transient    AfD1    2   HD  pressure data during fracture expansion, interaction pwDtD     3 v    22v1 between matrix and fractal fracture network and v  . (54) single system behavior, particular solutions for each  2  one of these periods were developed. In addition, such solutions are useful to calibrate the numerical  HD  3v    1v Sma fbD1 v   v  inversion of eq. 43.   2    t 2  maD t 2   D  2  v  D  Solution at early times     At early times, only fracture network expansion is   2   acting; during this period interporous transference In order to have a straight line shape, previous function is approximated as: expression can be written as: v1 f s ,   1    maD   AfD1 v    H  2   2   D  Hence, solution at wellbore is given by: tD pwDtD     3v    22v1 v  . (55)  Dfb    2   1 ,    2   22 t   D  , (48) pwD tD     HD  3v   D   Sma fbD1v    fb    2    2    t  maD    2  D  2v       where:   2   a, x Incomplete Gamma Function. If D  2 pressure derivative function will b) On the other hand, for severe damage between fb matrix blocks and fractal fracture network, i.e., yield a slope equals to zero. Then eq. 48 is pseudosteady-state transference equivalence, approximated: interporous transference function is given by: lnt  2ln  2ln0.57721 p t   D . (49) AfDmaD wD D   2 f s  , (56) Sma fbDHDs Otherwise: Then, if Dfb  2 : 12v v1 1   2  1v , (50)   S H   pwDtD   tD 2ln   2 ma fbD D   0.57721   2v 1  v 1 v      AfDm aD   p t     , (57) Where: wD D   2 D Otherwise: v  fb . (51) 1     2 12v 1v   2       2 Sma fbDHD p t      . (58) Solution at intermediate times wD D    v  AfDm aD  At intermediate times, beginning of matrix flow contribution takes place. This phenomenon occurs under transient-state conditions.

6 Solution at late times At late times, the flow is dominated by matrix under pseudosteady-state flow, it yields a single system behavior; for this period of time, interporous transference function is given by: f s 1, (59) Hence solution of eq. 31 is given by:  D 1   fb 1 ,    2   22t  p t     D  . (60) wD D  D   fb    2     2 If D  2 , eq. 60 can be approximated: Figure 9: Convergence from the short, intermediate fb and long times solutions to the general ln t  2ln  20.57721 solution, when Dfb   2 , with no p t   D . (61) wD D   2 interporous skin.

Otherwise: Transient interporous transference with low interporous skin 1  212v p t   t1v . (62) Figure 10 and Figure 11 show the pressure behavior wD D  2 v1  v1v D of a double porosity fractal reservoir, in semilog scale Adjust of analytical solutions in real space shown for and log-log scale for , in previous section, to the general solution respectively. For this case, interporous skin is low. numerically inverted for different scenarios are As it was shown in previous section, the presence of shown from Figure 8 to Figure 13. a valley shape in the pressure derivative function may

Transient interporous transference without propitiate an interpretation that flow transference interporous skin occurs under pseudosteady state conditions. In both Figure 8 and Figure 9 show the pressure behavior of cases, it can be observed that straight line, which a double porosity fractal reservoir, in semilog scale corresponds to the intermediate times, has a non-flat slope, which indicates that flow transference is for and log-log scale for , Dfb    2 occurring under transient interporous transference, respectively. For this case, the interporous even when interporous skin is present. transference is assumed to be free, i.e, no interporous skin. Both figures show two parallel straight lines at early and late times. Interaction between porous media is represented by a non-flat straight line between the two parallel straight lines.

Figure 10: Convergence from the short, intermediate and long times solutions to the general solution, when , with low Figure 8: Convergence from the short, intermediate interporous skin. and long times solutions to the general solution, when Dfb  2 , with no Transient interporous transference with severe interporous skin interporous skin. Figure 12 and Figure 13 show the convergence of this model to the pseudosteady-state interporous transference (severe interporous skin) for and Dfb    2 , respectively. A flat

7 straight line when matrix blocks and fractal fracture network are interacting is the characteristic behavior EXAMPLE OF APPLICATION of the media interaction under apparent pseudosteady-state conditions. A test was carried out in Well A. The behavior of pressure and pressure derivative function are shown in Figure 14. Reservoir and well data are given in Table 1.

Table 1: Reservoir and well data for the example. Parameter Quantity q [bpd] 2,000

Bo [[email protected]/ [email protected]] 1.6 o [cp] 6 rw [ft] 0.5  fb [fraction] 0.01

Figure 11: Convergence from the short, intermediate and long times solutions to the general solution, when Dfb  2, with low interporous skin.

Figure 14: Pressure and pressure derivative function behavior for synthetic example.

Pressure derivative function in Figure 14 does not show the fractal fracture expansion, i.e., behavior at early times. Then, analyzing late time response (see Figure 15) and comparing such behavior with eq. 62, it can be conclude that, v  0.8362 , hence, following Figure 12: Convergence from the short, intermediate relation between fractal parameters was deduced: and long times solutions to the general Dfb  0.8362 1.6724. Then, the methodology solution, when Dfb  2 , with severe described by Flamenco et al. (2003) was applied and interporous skin (pseudosteady-state). resulting parameters are:   0,  1.588 , min max c v1 aV k v  0.000127 and  t t uf fb min c v1 aV kv  0.451.  t t uf fbmax

Figure 13: Convergence from the short, intermediate and long times solutions to the general solution, when D  2, with severe Figure 15: Log-log plot of the late time pressure fb behavior for synthetic example. interporous skin (pseudosteady-state).

8 Similarly, intermediate times can be analyzed CSwellD Dimensionless storage around damaged plotting tv 2pt vs t , and comparing it with the zone at wellbore straight line given by eq. 55, after doing the ctfb fractal fracture network total correspondent transformations to field units. Figure compressibility. 16 shows the vs plot for this example. ctma Matrix blocks total compressibility. D fb Fractal dimension of the fracture network.

de Euclidean dimension. dma Sphere diameter. FmaD, HD,tD Fluid transference function. f s Interporous transference function.

HD Dimensionless block size. h1 Characteristic length of medium 1. h f Fracture width. h Strata height. Figure 16: Specialized plot for intermediate times of ma the pressure behavior for synthetic k1 Permeability of medium 1. example. kd Permeability of interporous damaged zone. k Permeability of the fractal fracture CONCLUSIONS fb network. 1. A fractal flow model that describes transient k Permeability of the Euclidean matrix interporous behavior in double porosity ma systems was developed. With this model is blocks. possible to consider spheres or slabs as M Molecular weight. matrix blocks. n f r Number of fractures into fractured bulk. 2. Main advantage of using fractal models is p Pressure in fractal fracture network. that these are the best way to represent fb randomness in the fracture network pi Initial pressure. distribution within the reservoir. pma pressure in matrix blocks. 3. Analytical solutions in real space to describe q Flow rate. pressure behavior during early, intermediate and late times were developed. Solutions qg Gas flow rate. during intermediate times are used to r radial distance from the centre of the characterize parameters useful for reservoir source (measured along the flow paths). engineering studies; such as matrix block, r Dimensionless radius. fractal fracture network area per unit of bulk D volume and interporous skin. rw Wellbore radius. 4. Advantages of using transient interporous rwe Effective wellbore radius. transference models with interporous skin S Interporous skin. were discussed. intD Sma fbD Interporous skin between matrix and NOMENCLATURE fractures. S Skin around wellbore. A Exposed to flow area. well exp. flow s Laplace space parameter. Afb fractal fracture network area per unit of T Temperature. bulk volume. t Time. AfD Dimensionless fractal fracture area. tD Dimensionless time. a Site density parameter. xd thickness of damaged zone. Bo Formation volume factor of oil. Vb Bulk volume. b Extent of the flow region. Vfb Fractal fracture network volume. C Wellbore storage. Vuf Unitary fracture volume. CD Dimensionless wellbore storage. v Grouping parameter. C Storage around damaged zone at wellbore. Swell W Mass flow rate.

9 Z Real gas deviation factor. Cinco-Ley, H., Samaniego V., F. (1982), “Pressure Transient Analysis for Naturally Fractured d Area of a unit sphere of in de dimensions. e Reservoirs,” paper SPE 11026 presented at the  Grouping parameter, SPE 57th Annual Fall Technical Conference and x Gamma function of x . Exhibition, New Orleans, LA, Sept. 26-29. a, x Incomplete Gamma Function. Cinco-Ley, H., Samaniego V., F., Kucuk, F., (1985), mp Pseudo-pressure fuction. “The Pressure Transient Behavior for Naturally Fractured Reservoirs with Multiple Block Size,” r Width between surfaces. th  Viscosity of flowing fluid. paper SPE 14168 presented at the SPE 60 Annual Fall Technical Conference and o Oil viscosity. Exhibition, Las Vegas, NV, Sept. 22-25. Porosity of matrix blocks. ma Cinco-Ley, H., Economides, M.J., Miller, F.G. uf Porosity of a unitary fracture. (1979), “A parallelepiped Model to Analyze the  Fractal fracture network porosity. Pressure Behavior of Geothermal Steam fb Penetrating Vertical Fractures,” paper SPE 8231 th ct t System total compressibility. presented at the SPE 54 Annual Technical  Fractal fracture network hydraulic Conference and Exhibition, Las Vegas, Sept. 23- fb 26. diffusivity coefficient.  Dimenssionless matrix hidraulic De Swaan, O.A., (1976), “Analytic Solutions for maD Determining Naturally Fractured Reservoir diffusivity. Properties by Well Testing,” SPEJ Trans. AIME  Conductivity index; related to the spectral 261, 117-122, June. exponent of the fractal fracture network.  dimensionless storativity ratio. Doe, T.W. (1991), “Fractional Dimension Analysis of Constant-Pressure Well Tests,” paper SPE 22702 presented at the SPE 66th Annual REFERENCES Technical Conference and Exhibition, Dallas, TX, October 6-9. Acuña, J.A., Ershaghi, I. and Yortsos, Y.C., (1995), “Practical Application of Fractal Pressure- Flamenco-López, F., Camacho-Velázquez, R. (2003), Transient Analysis in Naturally Fractured “Determination of Fractal Parameters of Fracture Reservoirs,” SPEFE 10 (3): 173-179; Trans Networks Using Pressure Transient Data,” paper AIME, 299. SPE-24705-PA. SPE 82607, SPE Reservoir Evaluation & Engineering, February. Barenblatt, G.I., Zheltov, Iu, P. and Kochina, I.N., (1960), “Basic Concepts in the Theory of Gringarten, A.C., Bourdet, D.P., Landel., P.A., Seepage of Homogeneous Liquids in Fissured Kniazeff, V.J. (1979), “A Comparison Between Rocks (strata),” PMM Vol. 24, No. 5, 852-864, Different Skin and Wellbore Storage Type- in Russian. Curves for Early-Time Transient Analysis,” paper SPE 8205 presented at the SPE 54th Barenblatt, G.I., Zheltov, Yu, P., (1960), Annual Technical Conference and Exhibition, “Fundamental equations of Filtration of Las Vegas, Sept. 23-26. Homogeneous Liquids in Fissured Rocks,” Soviet Physics Doklady Vol. 5, 522-525. Gringarten, A.C., (1987), “How To Recognize “Double Porosity” Systems from Well Tests,” Barker, J.A. (1988), “A Generalized Radial Flow paper SPE 16437, JPT, June. Model for Hydraulic Tests in Fractured Rock,” Water Resources Research, Vol. 24, No. 10, Moench, A.F. (1984), “Double-Porosity Models for a 1796-1804. Fissured Groundwater Reservoir with Fracture Skin,” Water Resources Research, Vol. 20, No. Camacho-Velazquez, R., Fuentes-Cruz, G., and 7, 831-846. Vasquez-Cruz, M., (2008), “Decline-Curve Analysis of Fractured Reservoirs with Fractal Najurieta, H.L., (1980), “A Theory for Pressure Geometry.” SPE Res Eval & Eng11 (3): 606- Transient Analysis in Naturally Fractured 619. SPE-104009-PA. Reservoirs,” JPT 1241-1250, July. Chang, J., Yortsos, Y.C. (1990), “Pressure-Transient Olarewaju, J. (1996), “Modeling Fractured Reservoirs with Stochastic Fractals,” paper SPE Analysis of Fractal Reservoirs,” SPE Formation th Evaluation, 289, 631. 36207 presented at the 7 Abu Dhabi International Petroleum Exhibition and Conference, Abu Dhabi October 13-16.

10 Olarewaju, J. (1997), “Pressure Transient Analysis of Workshop on Geothermal Reservoir Engineering Naturally Fractured Reservoirs,” paper SPE Stanford University, Stanford, California, 37801 presented at the SPE Middle East Oil January 30 – February 1. Show and Conference, Manama, Bahrein, March Warren, J.E. and Root, P.J., (1963), “The Behavior of 15-18. Naturally Fractured Reservoirs,” SPEJ Trans Pulido, H., Samaniego, F., Rivera, J., Díaz, F., AIME, Vol. 228, 245-255, September. Galicia-Muñoz, G., (2006), “Well Test Analysis for Naturally Fractured Reservoirs with Transient Interporosity Matrix, Microfractures APPENDIX A and Fractures Flow,” Proceedings Thirty-first Generalized Radial Flow Model for Well Test Workshop on Geothermal Reservoir Engineering analysis Stanford University, Stanford, California, January 22 – 24. Barker (1988) proposed a general radial flow model with the shape: Pulido, H., Samaniego, F., Galicia-Muñoz, G., 2  p r ,t  d 1 p r ,t  p r ,t  Rivera, J., Velez, C. (2007), “Petrophysical D D D  e D D D  D D D ; (A.1) 2 r r t Characterization of Carbonate Naturally rD D D D Fractured Reservoirs for use in dual porosity following boundary conditions are established: simulators,” Proceedings Thirty-Second Initial condition: Workshop on Geothermal Reservoir Engineering pDrD,0 0 , (A.2) Stanford University, Stanford, California, Inner boundary: January 22 – 24. p 1,t  D D  1, (A.3) Pulido, H., Galicia-Muñoz, G., Valdes-Perez. A.R., tD Díaz-García F. (2011), “Improve Reserves Outer boundary: Estimation using Interporosity Skin in Naturally lim pD rD ,tD  0. (A.4) Fractured Reservoirs ,” Proceedings Thirty-Sixth rD  Workshop on Geothermal Reservoir Engineering Solution in Laplace Space of eq. A.1 is given by: Stanford University, Stanford, California, 2de January 31 – February 2. r 2 K r s D 2de D Serra, K. V., Reynolds, A.C. and Raghavan, R., 2 . (A.5) pD rD,s 3 (1982), “New Pressure Transient Analysis s 2 K  s Methods for Naturally Fractured Reservoirs,” de paper SPE 10780 presented at the SPE 1982 2 California Regional Meeting, San Francisco, In Real Space: 2de California, March 24-26. rD pD rD ,tD    Stehfest, H. (1970), “Numerical Inversion of Laplace  de   4de  2   Transforms,” Communication of the ACM, 13,  2   2  No.1, 47, January.  . (A.6) Streltsova, T.D., (1982), “Well Pressure Behavior of  2  rD 2de 4   a Naturally Fractured Reservoirs,” paper SPE 42d 4t  e  r 2  2 e  D  10782 presented at the SPE 1982 California  4  D  d Regional Meeting, San Francisco, California,   4t  2 ´0  D   r  March 24-26.  D     4tD Valdes-Perez. A.R., Pulido, H., Cinco-Ley., H.,   Galicia-Muñoz, G. (2011), “A New Bilinear Dimensionless variables are the same established in Flow Model For Naturally Fractured Reservoirs the main text, with   0 and taking the convergence with Transient Interporosity Transfer,” of fractal fracture network dimension to the Proceedings Thirty-Sixth Workshop on Euclidean dimension, i.e. Dfb  de . Geothermal Reservoir Engineering Stanford University, Stanford, California, January 31 – APPENDIX B February 2. Wellbore Storage for Fractal Models Valdes-Perez. A.R., Pulido, H., Cinco-Ley., H., Galicia-Muñoz, G. (2012), “Discretization of the Total flow rate is given by: 3de Resistivity, Capillary Pressure and Relative dpwf t d b k fb  p fb  e  de  1  (B.1) Permeability for Naturally Fractured qBo  C  r  dt   r  Reservoirs,” Proceedings Thirty-Seventh rrw

11 Using dimensionless variables and fractal fracture network definitions:  p r ,t  p t   fbD D D  wD  D  rD   CD 1, (B.2) rD tD  rD 1 Where dimensionless Wellbore Storage for fractal reservoirs is defined as:

 fbC CD  . (B.3) Dfb aVuf ct t rw Wellbore Storage in Damaged Zone Wellbore Storage in the entire damaged zone in a fractal reservoir is given by:

Dfb Dfb aVuf ct t rwe aVuf ct t rw CSwell  C   , (B.4) fb fb Where effective wellbore radius is defined as: Swell rwe  rwe . (B.5) Substituting effective wellbore radius definition in B.4, the following expression results:

DfbSwell DfbSwell CSwellDe CD 1e 1, (B.7) Where:  fbCSwell CSwellD  . (B.8) Dfb aVuf ct t rw

12