horizontal fractures in single in multilayer reservoir

CSUG/SPE 147004 Horizontal Fractures in Single and Multilayer Reservoirs Leif Larsen, SPE, Kappa Engineering, University in Stavanger Copyright 2011, Society of Petroleum Engineers This paper was prepared for presentation at the Canadian Unconventional Resources Conference held in Calgary, Alberta, Canada, 15–17 November 2011 This paper was selected for presentation by a CSUG/SPE program committee following review of information contained in an abstract submitted by the author(s) Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s) The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied The abstract must contain conspicuous acknowledgment of SPE copyright Abstract Type curves for circular uniform-flux fractures in unbounded single-layer reservoirs were presented in the literature several decades ago Unfortunately, these results have restricted validity since the simple uniform-flux inner-boundary condition will cease to be physically meaningful when significant boundary effects from the top and bottom of the formation start to affect the data This is especially important for large fractures in formations of limited thickness For horizontal fractures it is therefore important to include finite conductivity as part of the fracture model Solutions addressing these issues are presented in this paper along with examples highlighting effects of different modeling options The solutions include both compressible and incompressible flow inside the fractures The solutions have also been extended to multi-layer reservoirs through a decoupling approach, with both unbounded and bounded circular models used in examples Solutions for layered reservoirs are important because layering is common with obvious effects on productivity and because layering can create unusual pressure-transient characteristics during buildups if there are shale streaks or other low-permeability flow barriers between layers acting as partially sealing boundaries Horizontal fractures are becoming more important due to an increasing number of shallow-depth disposal wells, for instance as part of carbon capture and storage developments Shallow gas developments with fracturing needed to obtain sufficient productivity are also adding to the importance of these fracture models Both for producers and disposal wells, layering will be an issue and has to be addressed Introduction Pressure-transient solutions for circular uniform-flux fractures in unbounded single-layer reservoirs were presented in the literature by Gringarten and Ramey (1974) The results, which were based on Gringarten’s dissertation (1972), were also reproduced as one of the type curves presented by Earlougher (1975) for cases with a centered fracture Figure shows an example of such pressure-transient data from a centered uniform-flux fracture with radius 10 times the thickness of a low permeability isotropic formation An important observation here is that the solution exhibits two log-cycles of apparent simple depletion (pseudosteady state) behavior between early linear flow and late radial flow This behavior was also discussed by Gringarten and Ramey (1974) Although the mathematical solution used to generate the data in Figure is correct, given the inner-boundary condition of uniform flux, it is clear that after nearby formation boundaries (top and bottom) start affecting the flow pattern, the uniform flux condition cannot be maintained for large fractures Only infinite-conductivity and finite-conductivity fractures are therefore realistic physical models in a strict sense Even so, since the uniform-flux and infinite-conductivity solutions will have identical early data, and both will approach late radial flow, the uniform-flux solutions will be adequate for many highconductivity cases However, with radial flow within fractures, it follows that fracture conductivity will tend to be more important for horizontal fractures than for vertical fractures Solutions for finite-conductivity fractures should therefore be used unless high-conductivity is likely Effects of compressible flow within horizontal fractures should also be considered if there is a possibility that this might be important For early data the latter case can be treated as a transient double-porosity model of the “slab” type, but this approach is only valid when flow from the formation to the fracture can be treated as purely one dimensional and perpendicular Outer boundaries, such as circular no-flow or constant pressure boundaries, or single and other boundaries that can be generated through simple image-well techniques based on line-source solutions, can be added in both the single and multilayer models if the fracture does not extend too close to the boundary or boundaries The restriction concerning nearby boundaries is caused by a lack of symmetry in these models 2 CSUG/SPE 147004 Figure Log-log diagnostic plot of data from a large horizontal uniform-flux fracture in a low-permeability formation Mathematical Models for Horizontal Fractures with Uniform Flux For circular horizontal fractures with uniform flux we only need the fracture radius rf and location z z f along with the permeability ratio k z / kr Gringarten (1972) used a Greens function-based approach to develop real-space solutions directly for this model, while the main solutions presented in this paper are based on integrals of Laplace-transformed point-source solutions and superposition schemes to generate formation boundaries, with the Stehfest (1970) algorithm used to generate real-space data This is quite straightforward for horizontal uniform-flux fractures with pressures computed on the axis of the fracture, i.e., at r for any z location Solutions for fractures with uniform flux can in turn be used to generate solutions for horizontal fractures with infinite or finite conductivity by an approach similar to that used for vertical fractures The wellbore pressure for horizontal fractures with infinite conductivity can also be approximated by using an equivalent pressure point on a uniform-flux fracture, similar to the approach used for vertical fractures Details of the development of Laplace transformed solutions for uniform-flux horizontal fractures (radial symmetry always assumed) based on integrals of point-source solutions for unbounded 3D models are presented in Appendix A, with superposition schemes used to generate upper and lower boundaries with no flow or constant pressure It is also shown how integrals of point-source solutions in formations of finite thickness can be used to the same end In dimensionless form the solutions for points on the axis of the fracture, i.e., at rD for any zD , can be expressed in either of the two “closed forms” pD (rD 0, z D , s) pD (rD 0, z D , s ) hD s sr e fD j zD z fD jhD s e z D z fD jhD s r fD ( z D z fD jhD )2 s e e r fD ( zD z fD jhD )2 s (1) and rfD sK1 (rfD s ) s 2 cos( z D n ) cos( z fD n n ) rfD n s K1 (rfD n ) (2) n n 2 / hD and n s n k z / kr The definition of n is standard, while n and the choice of cosine or sine in Eq depend on the type of formation boundaries Eq can also be modified to change the type of formation boundaries by specific choices of signs of the terms This, which is a standard approach, is described in detail in Appendix A Note also that the k z / kr ratio has been used explicitly in Eq 2, while it is only used indirectly in Eq through an initial coordinate scaling Also note that the definitions of dimensionless pressure and time used for the solutions above are based on radial flow and formation properties with horizontal permeability ( kr ) as reference if both formation boundaries are sealing (top and bottom), where n For cases with rD , i.e., for observation points away from the axis of the fracture, which we need to generate solutions for fractures with finite conductivity, the integral expressions cannot be evaluated in closed form We therefore have to resort to numerical integration, which for the current study solely have been based on point-source solutions in unbounded 3D models, with superposition used afterwards to include boundaries Details to this end are described in Appendix A CSUG/SPE 147004 Examples and Analyses of Data from Horizontal Fractures with Uniform Flux From data sets that exhibit the three flow regimes early (vertical) formation linear, pseudo-steady state and late radial, it is possible to determine the three key unknowns: kr , k z and rf , provided the formation thickness and other basic formation and fluid parameters are known The point is that the pseudo-steady state (pss) data should correspond to the identity p 0.2339qB t a ct rf2 h 0.074466qB t a (5) ct rf2 h in oil field units for some constant a, where rf2 h represents the affected rock volume Either the derivate from the log-log plot at a chosen time or the slope from a Cartesian plot can therefore be used to determine the fracture radius rf from the slope m 0.074466qB (6) ct rf2 h Since early linear-flow data should correspond to the identity p 4.064qB t rf2 k z ct 2.5873qB t (7) rf2 k z ct in oil fields units if there is no fracture skin, the slope of a square-root of time plot can be used to determine k z if rf is known Eq is obtained by recognizing that the product hx f appearing in the equation for vertical fractures represents the area of one side of half the fracture Analysis of late radial data, if present, is standard and will yield kr and the pseudo-radial skin value Although straight-line analyses will yield the key results, matching the data with a complete model should always be tried, if available, to check the consistency of both the analyses and the data To check results from Fig 1, the data were generated with k z kr md , h 30 ft , rf 300 ft , q 100 STB/D , B RB/STB , cp , 0.1 and ct 10 psi With reduced vertical permeability the ratio between the fracture radius and the effective thickness h h kr / k z will be reduced, and hence also the likelihood of observing the pss period This is illustrated in Fig with cases based on data from Fig with different k z / kr ratios Since rf / h 10 if k z / kr , note that k z / kr 0.01 corresponds to the effective ratio rf / h This is essentially the lower end of the ratios that might indicate the presence of pss data for centered fractures For off-center fractures the ratio must be higher for the pss period to appear For shorter fractures it will not be present, and for very short fractures we might instead observe a period with spherical flow between early linear and late radial periods kz/kr = 0.0001 0.001 0.01 0.1 Figure Sensitivity of horizontal uniform-flux fractures to anisotropy 4 CSUG/SPE 147004 The importance of the ratio rf / h on the pressure response is also illustrated in Fig by changing only rf in the basic isotropic model with thickness 30 ft For the middle case with rf / h we see again that the onset of the pss period is just indicated, while the case with rf / h 0.1 has a clear segment of spherical flow data between the early linear and late radial flow periods Note that the shift in the data between the Figs and is caused by delayed onset of boundary effects (top and bottom) when the k z / kr ratio is reduced in Fig 2, while the onset of radial flow shows up earlier in Fig for shorter fractures, but only after the onset of boundary effects (top and bottom) rf = ft rf = 30 ft rf = 300 ft Figure – Sensitivity of horizontal uniform-flux fractures to fracture radius (isotropic data) Another situation of interest is how the pressure response is affected by off-center fractures This is illustrated in Fig with the fracture moved to just ft from the nearest formation boundary, corresponding to a radius of investigation at slightly less than 0.004 hrs in any direction The data illustrate the key points – that very early data exhibit linear flow from both sides, that data between 0.1 and hrs are dominated by linear flow from just one side (other side trapped), with the onset of pss behavior afterwards and the rest similar to the behavior of the centered reference fracture (markers) Markers: Centered fracture Curves: Fracture ft from bottom Linear flow from one side Linear flow from both sides Figure – Effects of moving the fracture close to a formation boundry By using the solution for fractures with uniform flux we can also approximate the wellbore pressure for fractures with infinite conductivity by computing the pressure at the equaivalent pressure point rD 0.628rfD in dimensionless coodinates Choosing the fraction 0.628 for horizontal fractures differs from the fraction 0.74 normally used for vertical fractures Fig illustrates the difference between wellbore pressures obtained from fractures with uniform flux (markers) and infinite CSUG/SPE 147004 onductivity (curves) by this approach To justify the choice 0.628rfD for the equivalent pressure point to approximate the wellbore behavior for horizontal fractures with infinite conductivity we need to compare with solutions for fractures with fracture conductivity included in the model Markers: Uniform flux Curves: Infinite conductivity Figure – Infinite conductivity handled through the use of an equivalent pressure point Horizontal Fractures with Finite Conductivity There are two main differences between the treatment of horizontal and vertical fractures with finite conductivity, but both are based on subdivision and determination of the rate to each element with the constraint that the relevant flow equation is honored inside the fracture The differences involve radial vs linear flow inside the fracture and the use of uniform-flux circular flat rings vs uniform-flux vertical strips Even so, the computational set-ups are quite similar Similar to vertical fractures the fracture conductivity is defined by the product Fc k f w , where k f denotes the fracture permeability and w its width, or thickness With kr denoting horizontal permeability in the formation, we shall also define the dimensionless fracture conductivity by the expression FcD Fc k r rf kf w k r rf , (8) which is similar to that used for vertical fractures If flow within the fracture is treated as incompressible, then the fracture radius, location and dimensionless conductivity fully define the model However, if flow within the frature is treated as compressible, then we also need to include the fracture diffusivity kf f (9) ctf f as part of the model, either directly, or through the dimensionless fracture diffusivity kf f fD f ct ctf kr kf ct (10) ctf kr f Note that with fracture conductivity Fc and rf given we cannot vary fD arbitrarily in a physical model, but we can let fD approach infinity in a mathematical model corresponding to incompressible flow within the fracture as a limiting solution Fig illustrates the effect of fracture conductivity on the pressure response for the case with incompressible flow assumed in the fracture Two key observations can be made – that the effective length is reduced when the conductivity is reduced, and that early data exhibit a semi-log response The latter corresponds to half the slope we would observe in early data with compressible flow inside the fracture The reason is explained in Appendix C Fig illustrates how the data are affected by CSUG/SPE 147004 changes in the diffusivity ratio with compressible flow inside the fracture Note that the response approaches that of incompressible flow when fD is increased, and that we for low values of fD start observing a discrepancy during the pss 100 in Fig 7, is caused by increased fracture storativity in addition to that of the formation, and hence an artifact of the approach for unrealistic diffusivity ratios period This, which is barely indicated for the case fD Fc = md·ft 50 500 5000 5E4 5E5 Figure – Effect of conductivity on fractures with incompressible flow Fc = 500 md·ft 10000 1000 Diffusivity ratio 100 Figure – Effect of the diffusivity ratio with compressible flow and finite conductivity In Appendix C it is shown how a double-porosity slab model can be used to model early data frm fractures with compressible flow These results serve to explain the bahavior of early data from fractures with incompressible flow, and in addition can be used to verify the validity of the full model This is illustrated in Fig 8, with data based on Fig with a fracture of length 300 ft and fD values 100 and 1000 CSUG/SPE 147004 Markers: FC with compressible flow Curves: “Double-porosity” model Diffusivity ratio 100 1000 Figure – Comparison of data from full solutions and from infinite-acting double-porosity type early data Horizontal Fractures in Layered Reservoirs Only cases with a single fracture in a layered reservoir have been considered in this paper, with mathematical solution presented in Appendix E in terms of Laplace transforms The approach is based on decoupling as described by Larsen (1999), with standard dimensionless model description used in terms of ’s (flow capacity fractions), ’s (storativity fractions) and ’s (cross-flow parameters) The approach can be used for any fracture type, and with outer boundaries of standard types if the boundaries not come too close to the fracture (there is a symmetry problem for small distances) Fig has been used to verify that the mathematical approach is valid, with three identical, isotropic layers of thickness 30 ft (each) with permeability md used in the layered model with a 300 ft fracture in the middle of the bottom layer What we see is that the layered model with default lambdas based on the layer properties reproduces almost exactly the single-layer solution with thickness 90 ft and the fracture 15 ft from the bottom With a very low lambda value (1E-8) the layered model follows that of just the bottom layer for almost 1000 hours before we start seeing the effect of crossflow With lambda 1E-6 the layered solution starts deviating from that of just the bottom layer after less than 10 hours Markers: Single layer, 30 and 90 ft Curves: Three layers, varied lambdas Lambdas = 1E-8 Lambdas = 1E-6 Default isotropic Figure – Comparing single-layer and multi-layer solutions Fig 10 illustrates a re-charge scenario during an extended buildup following 300 hours production The case is based on two fracture types, one with uniform flux and one with finite conductivity ( Fc 50000 md.ft), both of length 300 ft in the middle of a 3-layer 90 ft model The crossflow parameter between the middle layer and the others is very low, at 1E-9 The outer radius is 500 ft and closed The key point to make is that both fractures exhibit pss behavior during late buldup data, similar to what can appear in buildup data from wells in segmented reservoirs with partially sealing boundaries 8 CSUG/SPE 147004 Circular 3-layer model Finite conductivity Uniform flux Figure 10 – Re-charge effect in buildup data in a closed 3-layer model with fracture in middle layer and low lambdas Conclusions Analytical solutions can be derived and used to model and analyze data from horizontal fractures of the basic types such as uniform flux, infinite conductivity, and finite conductivity with both incompressible and compressible flow Outer boundaries can be generated by standard image-well techniques based on line-source wells or Bessel function solutions, provided the boundaries not come too close to the fracture The solutions can be extended to layered models Boundaries can be added in layered models by the same techniques used for single layers Nomenclature A = tri-diagonal matrix, Eq E-1 aij = elemenst of A B = formation volume factor, RB/STB ct = total compressibility, psi-1 H = orthogonal matrix, Eq E-3 h = thickness, ft K = diagonal matrix flow capacity fractions, Eq E-1 k = permeability, md p = pressure, psia q = flow rate, STB/D r = radius, ft S = radius, ft s = Laplace variable, dimensionless t = time, hours w = fracture width (thickness), ft z = vertical coordinate, ft = diffusivity, md·psi/cp = viscosity, cp = porosity = s / Df = Special Laplace variable, Eq C-4 j = flow capacity fraction, Layer j j = storativity fraction, Layer j j = crossflow parameters Subscripts D = dimensionless d = damage i = initial f = fracture CSUG/SPE 147004 r = radial w = wellbore z = vertical (direction) Acknowledgment The author would like to thank Kappa Engineering for support to publish this paper References Earlougher, R.C., Jr 1977 Advances in Well Test Analysis SPE Monograph Series, Vol Gringarten, A.C 1971 Unsteady-State Pressure Distributions Created by a Well with a Single Horizontal Fracture, Partial Penetration, or Restricted Flow Entry PhD dissertation , Stanford U Gringarten, A.C and Ramey, H.J., Jr 1974 Unsteady-State Pressure Distributions Created by a Well with a Single Horizontal Fracture Soc Pet Eng J: 413–426; Trans., AIME, 257 Hartsock, J.H and Warren, J.E 1961 The Effect of Horizontal Hydraulic Fracturig on Well Performance J Pet Technology: 1050–1056; Trans., AIME, 222 Larsen, L 1999 Determination of Pressure-Transient and Productivity Data for Deviated Wells in Layered Reservoirs SPEREE: 95–103 Ozkan, E and Raghavan, R 1991 New Solutions for Well-Test-Analysis Problems: Part 1–Analytical Considerations SPEFE: 429–438 Stehfest, H 1970 Algorithm 368: Numerical Inversion of Laplace Transforms Communications of the ACM 13 (1): 47–49 Appendix A–Horizontal Fractures with Uniform Flux Two somewhat different approaches will be presented, starting with the Laplace transformed solution for a point source in an unbounded 3D model, which in cylindrical coordinates can be expressed in the form hD pD (rD , z D , s ) 2 s rD with the source at z z fD ) ( zD rD ( z D z fD )2 s e z f and h a chosen reference thickness For an unbounded 3D model we can determine the solution for a uniform-flux disc of radius rf in the plane z pDuf (rD hD srfD 0, z D , s ) r fD e z f by integration In particular, for any point on the z axis we just get rD ( z D z fD )2 s rD z fD ) ( zD rD drD For observation points with radial coordinate ro pDuf (roD , z D , s) if ro (A-1) hD s srfD e zD zwD s e hD s sr e fD z D z fD s r fD ( zD z fD )2 s e (A-2) it is more challenging For such cases we can use the solution hD srfD ( r fD roD )2 ( z D zwD )2 s r fD roD r fD roD e rD ( zD zwD )2 s rD ( zD zwD )2 (rD )rD drD (A-3) rf , where (rD ) sin 1 2roD rD 2 oD fD 4r r D (r oD r 2 fD r ) sin rfD rD rD 2roD roD 2rD rfD 2roD rD (A-4) represents the angle of the arc in the upper half of the fracture area If ro rf , then the first expression on the right-hand side ( rD ) If ro rf , then we can then use Eq A-3 without the first will be and it is also possible to simplify the expression for expression on the right-hand side if we replace the lower bound of integration by roD rfD , along with ( rD ) from Eq A-4 The integrals above must be computed numerically This can be done by using the subsitution rD rD e D r rD ( z D zwD )2 s ( zD zwD ) u2 (rD )rD drD e u1 u ( f (u ))du (A-5) 10 CSUG/SPE 147004 with u rD zwD ) s and f (u ) ( zD u / s ( zD zwD )2 The solutions above apply to single circular fractures in a 3D infinite model, with image “wells” used to generate formation boundaries at z (bottom) and z h (top) This can be done with standard “image fractures” patterns at z jh zw for all integer values of j The boundary type is controlled by the signature of neighboring fractures, with + for producers and – for injectors, i.e., with multipliers for producers and for injectors No-Flow at Both Boundaries: This is achieved if all multipliers are , i.e., if all fractures are treated as producers No-Flow at z = and Constant-Pressure at z = h : This is achieved with multipliers for both images at z jh for each j Constant-Pressure at z = and No-Flow at z = h : This is achieved with multipliers even, and at z w and at z w if j is odd for each j Constant-Pressure at Both Boundaries: This is achieved with multipliers at if j is even & at z w and zw and at z w at z at if j is odd z w if j is jh for each j It is also possible to follow the general approach of Ozkan and Raghavan (1991) and start with a Laplace-transformed point-source solution for a formation of finite thickness in the form K (rD s ) cos( zD s n pD (rD , zD , s) n ) cos( zwD n ) K (rD s n ) (A-6) for unbounded models, where n s n with k z / kr and the parameters n along with the choice between the sine and cosine functions in Eq A-6 are used generate no-flow or constant pressure boundaries at the top and bottom of the formation Note also that the first term in Eq A-6 is only included if both boundaries are sealing To generate a solution for a fracture with uniform flux from Eq A-6 we can use the identity b 1 abK1 (ab) (A-7) a2 xK (ax)dx to derive the solution pwD ( s) 2 rfD r fD pD (rD , zwD , s )rD drD rfD sK1 (rfD s ) s cos ( zwD n n ) rfD n s K1 (rfD n ) (A-8) n for the transformed wellbore pressure, again with the first term of the final expression only included if both the formation boundaries are sealing Note that the first term on the “right-hand side” of Eq A-8 represents the solution for a fully penetrating full cylindrical “well” with production at all points of the cylinder (on the surface and inside) This is not a meaningful physical model, but it is a useful mathematical solution as just demonstrated It is also used below for layered reservoirs A similar real-space model also played a role in the approach used by Gringarten (1972) Appendix B–Horizontal Fractures with Finite Conductivity and Compressible Flow Only radial-symmetric horizontal fractures will be considered with isotropic internal properties and hence flow governed by the equation pf r r r r pf z pf f t (B-1) for the fracture medium with thickness w in the z direction If we for the time being assume that the fracture is placed in the plane defined by z , then we get the boundary condition CSUG/SPE 147004 w/ kf qB w 11 r w/ pf dz .(B-2) r r rw at the wellbore In a larger perspective it is clear that pressure variations across the thickness of the fracture can be ignored Pressures within the fracture can then be replaced by the average pressure w/ w p f (r , t ) p f (r , z, t )dz .(B-3) w/ across the fracture If the integral from Eq B-3 is applied to both sides of Eq B-1, then we get the new equation pf r r r r pf w z pf z z w/ w/ z pf f t (B-4) The same integral applied to Eq B-2 yields the new boundary condition kf qB w r pf r (B-5) r rw With flow from the formation to the fracture we must have continuity in pressure and flux at the fracture surface, and hence p f (r, z, t ) p(r, z, t ) and (k f / )( p f / z ) (k z / )( p / z ) at z w / for all r rf if there is no fracture face skin Since flow near the fracture surface will be linear except near the fracture tips, the gradients for small values of z can be considered to be independent of z It follows that we can set k f ( p f / z ) z w / k z ( p / z ) z for the positive side of the fracture in terms of the z axis and similarly for the negative side We can therefore rewrite Eq B-4 in the form pf r r r r since ( p / z ) z set p f 2k z kf w p z ( p / z)z f z pf t (B-6) because of symmetry The next step is to ignore variations in p f with respect to z and just p f in the key equations above If we in addition move to dimensionless variables based on the wellbore radius and formation properties with the horizontal permeability kr as reference and apply the the Laplace transform with respect to t D is, then we get the transformed flow equation dpDf d rD rD drD drD rfD FcD pD zD s pDf (B-7) Df zD for the fracture along with a standard flow equation for the formation (also in cylindrical coordinates), together with the boundary condition rD dpDf drD rD hD (B-8) srfD FcD at the wellbore, where FcD k f w / kr rf , rfD rf / rw , hD h / rw , and fD f / We shall also use the modified continuity condition pDf (rD , s) pD (rD , 0, s) 2S f pD zD .(B-9) zD 12 CSUG/SPE 147004 for all rD rfD with a fracture (surface) skin Sf added If the fracture is divided into a sufficient number of circular (flat) rings, then each ring can be assumed to have uniform flux in terms of flow from the formation With n rings we get n unknown rates q j (t ) for j 1, , n that must be determined by solving a linear system of equations With the subdivision rw must have (q j / 2) B (r j kz p z j r ) for each j 1, r0 r1 rn rf it follows from Darcy’s law that we , n If we change these expressions to dimensionless form, apply the z Laplace transform to each and subsititute the resulting indentites pD zD zD hD qDj ( s) with q Dj (r rD , j ) Dj qj / q denoting dimensionless rate from Ring j into B-7, then we get the transformed equation dpDf d rD rD drD drD 2hD qDj ( s) 2 rfD FcD (rDj rD , j ) s pDf (B-10) Df for Ring j of the fracture A general solution of Eq B-10 is known and given by pDf (rD , s) Aj K (rD for Ring j, where ) B j I (rD s/ Df 2hD qDj (s) (B-11) 2 rfD FcD (rDj rD, j ) ) and A j and B j are parameters that must be determined from boundary and continuity conditions, and the last term on the right-hand side is a direct particular solution It follws the Laplace transform of the dimensionless wellbore pressure will be given by pwD ( s) A1 K0 ( ) B1 I ( ) 2hD qD1 (s) , (B-12) rfD FcD (rD1 1) but to use this identity we need to know A1 and B1 and the rate transform qD1 To determine these quantities we need to determine the parameters A j and B j and rate transforms q Dj for all j 1, , n , i.e., a total of 3n unknowns To this end we need 3n independent equations in these unknowns Conditions that can be used to set up such equations are continuity in pressure and flux within the fracture at each rDj for j 1, , n , together with the boundary condition at the wellbore To simplify the mathematical model we shall also assume that the fracture has a no-flow outer boundary at rD rfD , and hence not allow flow directly into the edge of the fracture With any fracture of non-trivial length, this restriction has no impact on the final solution For the fracture we can set up 2n equations, starting with the boundary condition A1 K1 ( ) B1 I1 ( ) hD / ( s rfD FcD ) at the wellbore, the no-flow condition An K1 (rfD ) Bn I1 (rfD ) at the outer edge of the fracture along with identities obtained from pressure and flux continuity at the internal fracture interfaces for j 1, , n The rest of the equations can be derived from the condition of pressure continuity given by Eq B-9 by restricting it to the midpoint of each fracture element, i.e., by using the transformed dimensionless identities ˆ pDfj (rDj , s) ˆ where rDj (rD , j ˆ Aj K (rDj ˆ ) B j I (rDj rDj ) / for j 1, ) 2hD qDj (s) , (B-13) 2 rfD FcD (rDj rD, j ) ,n Note that the skin value introduced in Eq B-9 can be defined in terms of the identity Sf wd k z 2rw k zd (B-14) CSUG/SPE 147004 13 with k zd denoting vertical permeability in the damaged zone and wd the thickness of the zone on each side of the fracture In terms of the uniform flux rings with areas 2 (rDj rD, j ) used in the subdivision above, it follows hat Eq B-14 can be replaced by pDf (rD , s ) S f hD pD (rD , 0, s ) in the range rD , j Dj rD , j ) (r qDj ( s ) (B-15) ˆ ˆ rDj In particular, note that this identity applies to the midpoints rDj , with p D ( rDj , z D rD 0, s ) ˆ obtained by adding the response from all j rings at rDj To this end we need a general Laplace transformed solution for flow to horizontal uniform-flux rings This is a non-standard solution which is obtained by using differences of two circular fractures Appendix C–Infinite-Acting Fracture Radial and Formation Linear Flow For early data with infinite-acting flow inside the fracture (edge not yet reached) and infinite-acting vertical flow in the formation (top and bottom not yet reached) we can assume that we, at least initially, have 1D linear flow in the formation With infinite-acting linear vertical flow in the formation we get the Laplace transformed solution pD (rD , zD , s) a(rD , s)e zD s / (C-1) outside the fracture (in the upper half) for some unknown function a(rD , s ) , with pD / z D zD a(rD , s ) s / It follows from Eq B-9 that we must have pDf (rD , s) pD (rD , 0, s) Sf pD zD Sf s / a(rD , s ) , (C-2) zD and hence that the Laplace transformed flow equation (Eq B-7) for the fracture can be expressed in the form d dpDf rD drD drD s rfD FcD (2 / ) S f s / s pDf (C-3) Df For the infinite acting period we then get the general solution pDf (rD , s) s rfD FcD (2 / ) S f s / AK0 (rD ) with s (C-4) Df From a standard boundary condition at the wellbore we can determine A and derive the solution pwD ( s) Since rfD FcD pDf (rD hD K0 ( 1, s) srfD FcD ) K1 ( ) (C-5) (k f w) / (kr rw ) , note that the dependence on the fracture radius is only apparent Approximate solutions of the dimensionless wellbore pressure can now be deduced from the identity above with Schapery’s approximation g (t ) sg ( s ) s 1/ ( e t ) To this end we shall use the small-argument approximations of the Bessel functions and assume that pwD ( s) hD K ( srfD FcD ) K1 ( ) hD ln srfD FcD (C-6) 14 CSUG/SPE 147004 For sufficiently early data dominated by fracture flow we can set s/ Df and proceed with the ideas above and derive a simple infinite-acting solution for radial flow in a porous medium of thickness w with permeability k f and diffusivity This result is obvious and does not need to be elaborated For later data we shall only consider the case with S f rfD FcD 2kr rw k z / kr s s kf w k z kr rw Df and assume that we can set s (C-7) kf w with negligible error We can then derive the semilog solution pwD (tD ) spwD ( s) s 1/( e tD ) hD ln s 2rfD FcD ln kz krw 2k f w s 1/( e tD ) kh ln tD 2k f w ln kz krw 2k f w (C-8) with slope equal to one half of the semilog slope of early data dominated by compressible flow in the fracture alone Appendix D–Horizontal Fractures with Finite Conductivity and Incompressible Flow With incompressible flow inside the fracture we get the Laplace transformed flow equation dpDf d rD rD drD drD 2hD qDj ( s) 2 rfD FcD (rDj rD , j ) (D-1) for Ring j of the fracture with general solution hD 2rfD FcD (rDj pDf (rD , s) rD , j ) qDj (s)rD Aj ln rD B j (D-2) obtained by direct integration It follows that we must have pwD ( s) pDf (rD 1, s) hD qD1 (s) B1 (D-3) 2rfD FcD (rD1 1) Now, in order to use the identity above we need to know B1 and the transform of the rate into the first ring In order to determine these unknowns we need to determine all the 3n unknowns form boundary and continuity conditions at internal fracture interfaces and midpoint pressures at the midpoints of the fracture rings This is similar to the approach in Apendix B, only simpler In particular, note that we must have hD srfD FcD rD dpDf drD rD hD qD1 ( s) A1 (D-4) rfD FcD (rD1 1) at the wellbore together with the condition dpDfn drD rD r fD hD qDn ( s )rfD 2 rfD FcD (rDn rD , n ) An rfD (D-5) at the outer edge of the fracture Appendix E–Horizontal Fractures in Layered Reservoirs Basic developments for layered reservoirs can follow the approach used by Larsen (1999), except for a switch to radial coordinates With n layers we get a system of n flow equations that can be expressed in the matrix form CSUG/SPE 147004 15 p1D d d K (rD rD drD drD p1D A ) (E-1) pnD pnD where K is a diagonal matrix with entries along the diagonal and entries a j a j, j 1, j j along the diagonal, and A is a tri-diagonal matrix with entries j s j j on either side of the diagonal Following the approach by Larsen (1999) we j can decouple the the system of equations above and instead consider the system U1 d d (rD rD drD drD U1 ) E Un , (E-2) Un with E being a diagonal matrix of of eigenvalues j , and the solutions from Eq E-2 being related to those from Eq E-1 through the identity p1D U1 ( K) H pnD , (E-3) Un where H is an orthogonal matrix that must be determined simultaneously with the eigenvalues Since Eq E-3 is decoupled we get n modified Bessel equations, each with a general solution in the form U b (rD , s) j U j (rD , s) where j j j K0 (rD j ) I (rD j j ) , (E-4) for each j if all layers are unbounded, boundary at rD reD , the coefficients j K0 (reD j j ) / I (reD j K1 (reD j j ) / I1 (reD j ) for each j if all layers have a no-flow ) for each j if all layers have a constant-pressure boundary at rD reD , and are unknowns that must be determined from inner-boundary conditions of the model For the special case of an undamaged line-source well (no skin values), we first need to solve the set of equations 1 R , (E-5) n where and rij spwD n i i / pwD for each i, hij for j 1, n i and rij hij K0 ( j ) I ( j j ) for j 1, , n if Layer i is perforated, and i , n otherwise From these solutions we get p wD from the identity n h jm j j i m (E-6) m The only modification needed for a horizontal fracture is to replace K0 ( j ) by the corresponding solution for a horizontal fracture in the perforated layer based on proper layer properties If a single linear boundary is wanted in each layer, then we must replace j I ( j ) by an image line-source well at twice the distance for each j This approach can also be generalized to other outer boundaries based on image-well techniques  ... solutions and from infinite-acting double-porosity type early data Horizontal Fractures in Layered Reservoirs Only cases with a single fracture in a layered reservoir have been considered in this paper,... pressure for fractures with infinite conductivity by computing the pressure at the equaivalent pressure point rD 0.628rfD in dimensionless coodinates Choosing the fraction 0.628 for horizontal fractures. .. flux Curves: Infinite conductivity Figure – Infinite conductivity handled through the use of an equivalent pressure point Horizontal Fractures with Finite Conductivity There are two main differences