roceedings World Geothermal Congress 00 Bali, Indonesia, 5-9 Aril 00 Analysis of ressure Transient Resonse for an Injector under Hydraulic Stimulation at the Salak Geothermal Field, Indonesia Jorge A. Acuna Chevron Geothermal and ower, Sentral Senayan II, 6 th Floor, Jl. Asia Afrika, Jakarta 070, Indonesia jacuna@chevron.com Keywords: Fall off, fractional flow, fractal flow, stimulation, Salak, Awibengkok. ABSTRACT Four ressure fall-off ressure transient tests were conducted in a well being stimulated by high ressure injection. The tests were analyzed with a technique based on fractional flow in a comosite reservoir. The results show the rogress of the stimulation rocess as the measured reservoir roerties change from test to test. Novel techniques for analysis of ressure transient tests are demonstrated. The reason for the very large wellbore storage effect usually found in injection fall-off tests is discussed and considered in the analysis.. INTROUCTION The sequence of ressure fall-off tests (FO s shown in this aer have been reviously analyzed by Yoshioka et al. (009. Their analysis, however only covers the radial flow art of the tests. In this aer an alternative interretation is made that includes the entire test. It relies on fractional flow theory. Fractional flow behavior, and the closely related fractal flow behavior and its relevance to naturally fractured systems has been discussed by Acuna et al. (995. In linear, radial or sherical flow the ressure derivative has the characteristic sloe of 0.5, 0 and -0.5 resectively. Fractional flow allows for the derivative to have any sloe in between. A densely and uniformly fractured media in which a vertical well is comleted should show tyical radial flow. When many of the fractures are missing or sealed, the network has un-fractured or unconnected areas of many sizes. In this case fully develoed radial flow may not be observed but rather fractional flow. Chang and Yortsos (990 resented solutions for an infinite fractured fractal medium with and without matrix articiation. Barker (988 resented a solution very similar in the context of fractional dimension flow. The solutions without matrix articiation have been exlored numerically by Acuna (993 and alike (998 and alied to actual cases by Acuna et al. (995 and Leveinen (000. Solutions without matrix articiation are suitable for fractured systems with very low matrix ermeability such as geothermal reservoirs, aquifers in crystalline rock and ossibly tight gas as well as shale gas reservoirs.. BACKGROUN An injector well located in a geothermal field in Indonesia named Salak, also known as Awibengkok, was subjected to high ressure injection with the objective of increasing its injectivity from October 007 to Setember 008. Four ressure fall-off (FO tests were erformed to determine the effectiveness of the stimulation. The FO s are articularly useful because they were done while injecting in a colder low ermeability area in the marginal area of the reservoir. Therefore they do not have the usual drawbacks associated to ressure transient tests in geothermal systems, namely the resence of steam or two-hases in the reservoir as well as thermal transients that take lace when injecting in very hot regions. The diagnostic ressure derivative lots of these tests show a change in the sloe of the derivative at a given time. The interretation of this behavior is that there is a change in the roerties of the medium at a corresonding distance that results in different flow dimensionality. This behavior can be reroduced by a radial comosite model with inner and outer areas with different roerties. This case was qualitatively exlored by Acuna et al. (995 and actual ressure tests showing this behavior have been analyzed by Leveinen (000. The objective of this aer is to resent a technique for analysis of ressure transient tests in fractional dimension comosite models. It comlements and extends the work of Leveinen (000 by including exressions for drawdown based on the transition time as well as the corresonding exressions for buildu tests. 3. THEORETICAL EVELOMENT The solution of Chang and Yortsos (990 for a well in an infinite fractal medium can be written as (+ ( + r r ( r, t =, ( + ( + t Where, t and r are dimensionless ressure, time and radius resectively. x,y is the incomlete Gamma function and x is the Gamma function. In Acuna et al. (995 there is an error in this equation, the value - is shown instead of - in the first argument of the incomlete Gamma function, this error was also found by alike (998. The correct exression shown above was resented in Acuna (993. Fractional flow can be caused by a orous medium of variable flow area as shown by oe (99. In this case the connectivity is erfect and diffusion occurs normally. It can also be caused by fracture systems that cover the available sace with gas of many sizes as shown by Acuna et al. (995. In this case diffusion is delayed as the network is not erfectly connected, the arameter is greater than zero and we have fractal flow. Equation for fractal flow converges to the solution for fractional flow resented by Barker (988 when is zero. The arameter combines geometric and fluid transort roerties of the fracture network. The flow dimension equals. For the case of radial flow (, ~0 the incomlete Gamma function -,x equals the (
exonential integral function Ei(-x, the Gamma function aroaches one and the familiar solution for a radial systems is obtained from Equation. For long enough time, or short times at the wellbore, the following aroximation derived from using the first two terms of the ower series exansion of the incomlete Gamma function can be used. (+ ( ( + t r ( r, t = ( ( ( ( + 3. rawdown analysis When dealing with a comosite medium the time of change in derivative sloe is called the transition time τ. For times less than then transition time τ the system behaves as an infinite medium with the characteristics of the inner medium. Therefore drawdown for t < τ is given by ( + t ( t = + S ( ( + Where is dimensionless ressure for the inner region or region and t is dimensionless time. For times larger than the transition time (t τ, the resence of the inner region can be accounted for as a seudo-skin. The exression for the drawdown equation for t τ in terms of the contact radius ρ is as follows ( t (+ = ( (+ ( ρ + ( + S ( ( + (+ ( t ρ ( ( + Where subscrits and on the lower right side of the arenthesis mean that roerties inside corresond to those regions. i is the ratio between actual ressure and dimensionless ressure for region i. Equation 4 can also be exressed in terms of the transition time τ as follows ( t ( + = ( ( + + ( t τ τ + S ( ( + By insection of the two exressions it follows that the distance to the interface between the two regions ρ can be calculated by equating the seudo-skin. The exression valid for the two regions is + ( + + ρ = τ (+ ( Equation 6 can be interreted as a radius of investigation. 3. Buildu analysis Buildu equations can be obtained from the drawdown equations and again there are two exressions, one for times (3 (4 (5 (6 before the time of change in derivative sloe τ and another for afterward. For t τ the exression is the following ( t ( + + ( ( + = ( ( t τ τ t (7 As before subscrits and on the lower right side of the arenthesis mean that roerties inside corresond to those regions. For times t > τ the analysis shows that region can be analyzed as if it were an infinite medium with characteristics equal to region. The equation is ( + ( t t ( t = (8 The distance to the interface is obtained from Equation 6. For the case of buildus the technique to solve the roblem starts by solving the outer region first. The fractional flow time is defined as ( t t T = and the derivative f is calculated. A grah of ressure and ressure dt f derivative versus T f will show a horizontal ressure derivative lot when the correct value of is selected. The sloe m of the ressure versus T f grah is given by. ( ( + m (9 = T and T reresent the ratio of dimensional to dimensionless ressure and time resectively defined as qµ r + µ r w = and w Φch T =. CF and ( CF π ( kh 0 CFT kh 0 CF T are conversion factors for ressure and time; CF is for consistent units and.70-3 for oil and gas customary units. CF T equals for consistent units and.6370-4 for oil and gas customary units with time in hours. The flow rate q is volumetric flow rate at reservoir conditions (qb o in oil and gas customary units. The subscrit 0 next to kh and φch mean that these values are only valid at r =. t In radial flow as T f converges to ln(, x t aroaches and the lot becomes a common Horner lot with Equation 9 used to estimate kh directly. Contrary to the case of radial flow, however the sloe of the fractional flow grah contains also information about φch and kh may not be calculated directly. 4. EXAMLE OF ANALYSIS The diagnostic lot of the first FO tests is shown in Figure. There is a dimensionality transition that occurs at a time close to 0 hours. The early art of the test seems to behave as tyical radial flow. The second art of the test is fractional flow with a derivative sloe close to 0.35. The comosite fractional
flow model is a very aealing model in our case because this well has been stimulated with water injection at a temerature 00F less than the rock. Injection ressure is above the fracture closure ressure. The injectivity has increased with time and it is exected that by the time this test was erformed a region close to the wellbore has been fractured by thermal shock as well as ressure. The analysis technique resented here is alicable even to the first art of the test as radial flow is just a secial case of fractional flow. 0000 000 tdt (si 00 tdt ( t m dtf ( t τ τ t = m ( 800 700 600 500 400 300 00 00 0 y =.89x + 0.00 y = 0.00x +.89 0 0 0 30 40 Tf dtf 0 0.0 0. 0 00 000 dt (hr Figure. iagnostic lot of first fall-off test showing a dimensional transition characterized by the change in sloe in the derivative curve. The first ste in analyzing the buildu test is to evaluate the late art that is unaffected by the resence of the inner region. Rewriting Equation 7 in dimensional form and using Equation 9 we have ( t = mt f ( t = m t (0 Calculating Tf as well as and iterating on until the dt f derivative is flat we get Figure. The ressure difference is calculated with resect to a final ressure such that the difference becomes zero when Tf equals 0. From the sloe and using Equation 9 we get = 39. 03. T As resented in Chang and Yortsos (990 fluid flow roerties change with distance from the well as (+ ( ( k ( r = k 0r and Φ( r = Φ 0r. The value of cannot be determined without additional information, however it is not exected to be large. An assumtion of ~ 0 will be made in this aer. Le Borgne et al. (004 shows how multi-well tests can be used to obtain a value for. In a fractured medium it is exected that the change in ermeability and orosity with distance from the well is due to a change in the number and connectivity of fractures transorting fluid. To kee the analysis arallel to conventional ressure transient derivations it is assumed that the embedding geometry is cylindrical. For the analysis of the early art of the test we reformulate Equation 7 in dimensional form as follows 3 Figure. Fractional flow lot for the later art of the FO shown in Figure for a selection of = 0.653. Only data after the transition time τ = 8 hrs are shown. Best fitting lines are shown. All values in the left hand side are known after solving for the late time art of the test. Thus we calculate the left hand side and lot versus T f as defined in the right hand side. τ is the transition time equals 8 hours. Figure 3 shows the resulting fractional flow lot for the early art of the test. In this well where injection has taken lace for weeks and near wellbore fracturing is resent it is safe to assume that there is no skin factor. Any near wellbore imrovement will be catured by the inner region characterization. dtf 400 350 300 50 00 50 00 50 0 y = 6.30x - 0.9 y = 0.00x + 6.7 dtf 0 4 6 8 Tf Figure 3. Fractional flow lot for the early art of the FO shown in Figure showing the fit for a selection of = 0.95. Only data before the transition time τ = 8 hrs are shown. With this assumtion the drawdown Equation 5 and the buildu Equation 7 are subtracted and used to estimate for region. In dimensional form this equation valid for t τ is ( ( t t = ( t m ( + t + m (
is ( wf - si for buildu and ( si - wf for fall-off tests. The second term to the right is negligible when t >> t. We get = 03.95 and with Equation 9 we get T = 0.049. To solve the radius of contact between the two regions we use Equation 6 with the known value of T for region to obtain ρ = 35.. Since the wellbore radius equals 0.5 ft we have a radius of contact of 8 ft. The values of T for region can be obtained by using Equation 6 again with the value of ρ just found. In this case we obtain T = 0.036 and from Equation 9 we get =.30. Once and T are calculated values of kh and φch are obtained and shown in Table. 5. RESULTS AN ISCUSSION Figure 4 shows the ressure and ressure derivative match for the 4 FO s analyzed. Very early time deviations can be due to using Equation as an aroximation to Equation. It is also aarent that there is no wellbore storage eriod. This could be due to the earliest data being recorded 5 minutes after shut-in. The derivative shae of test 3 and 4 suggest that more that regions may be needed to imrove the match. The technique shown here can be easily extended to more than regions. Table resents the arameters calculated for the four FO s. The tests show an increasing ermeability in the two regions from test to test. Additionally the asymtotic ressure (inf kees decreasing which may be showing a better connection to the main geothermal reservoir that is at a much lower ressure that this marginal region. Test T inf kh 0 Фch 0 kh 0 Фch 0 Interface days si m-ft ft/si m-ft ft/si ft FO 74 966 0.95 7979.58 0.65 6745 6.0 7.9 FO 84 84 0.95 8746.65 0.6 97997 0.8 7.4 FO 3 876 0.90 5077.07 0.6 35895 3.8 5.4 FO 4 07 753 0.85 30670 0.99 0.64 0585 5. 5. Table. T is the effective time injecting at 4 bm (64 l/s. inf is the ressure at which the test converges at very long times. Φch includes fracture dilation effects. The value of the fractional flow arameter in both regions is remarkably consistent. The inner region (Region is not only increasing in ermeability but also in size. The distance to the interface with the outer region (Region increases from test to test. Figure 5 (to shows the ermeability thickness roduct kh versus radial distance. esite large values of kh shown in Table for the maximum value for Region kh is never larger than 0000 m-ft. Assuming a thickness of 3000 ft which should be tyical of a geothermal reservoir this reresents a ermeability of less than 4 m at the interface between the two regions. This shows how the restriction to flow occurs in the outer region as should be exected in a ermeability stimulation rocess. The φch values obtained are remarkably high. Figure 5 (bottom shows the variation with distance from the well. Using the same tyical reservoir thickness and a tyical orosity the resulting comressibility is a few orders of magnitude too high comared to that of water at 350 F. Injection at ressure above fracture closure ressure as occurred in these tests, results in fracture dilation. The associated comressibility of a dilated system is that of the entire system as the fracture volume changes with ressure. As discussed by Van den Hoek (00, when analyzed conventionally this results in a wellbore storage coefficient that is orders of magnitude too high. In reality the effect of wellbore storage itself is negligible in comarison to the dilation effect of the fracture system. This effect is known as fracture comliance. Fracture comliance can still be interreted as a roduct of fracture volume and total comressibility including fracture dilation effects, therefore the notation will be ket as φch. Fracture comliance is treated in the equations as a large wellbore storage coefficient. By using using t instead of t it is observed C that values of φch obtained in these tests should be interreted as C where C is the fracture comliance in π r w ft 3 /si. As the fracture volume changes with radius it is intuitively exected that the volume of reservoir subject to dilation also changes with radius, but the exlanation of the variation of fracture comliance with radius is beyond the scoe of this aer. Fracture comliance decreases in later tests signaling that the relative volume change of the ore sace with ressure is less. This should be consistent with more fractures remaining oen and the observed increased ermeability. The values of φch, actually C r π w, in Figure 5 (bottom result in a maximum fracture comliance value of 4. ft 3 /si (0.95 bbl/si comarable to values reorted in the literature. The wellbore storage is less than % of the fracture comliance value. The variation of ermeability-thickness with radius we have adoted in this discussion is just a convenient assumtion. In reality what we interret as a change in kh could be due to a change in flow area in a medium with constant ermeability as shown by oe (99, or as a variation of the ermeability area roduct as described by Barker (988 and Leveinen (000. In naturally fractured reservoirs such as Awibengkok we exect that the ermeability of the individual fractures can change dramatically from fracture to fracture. Over large areas, however the combined effect of fractures roduces a bulk ermeability that can be effectively measured with multi and single well tests. The concet of fracture ermeability as a bulk roerty distributed over large areas is consistent with the aroach taken here. Work with numerical models of fractured networks (Acuna, 993 show that there is direct correlation between ressure derivative sloe changes and fracture network roerty known as the radial mass even at very small distances. Radial mass is just the volume of fractures as it changes with distance from the well. The radial variation of roerties at short distances from the well may be unique to a well osition but as the ressure ulse exands away from the well and engages more fractures. Other wells should have similar late time resonses as they engage larger reservoir area with time. 4
0000 000 tdt (si 00 tdt 00000 0000 kh (m-ft 000 FO FO FO3 FO4 0 0000 000 tdt (si 00 0.0 0. 0 00 000 dt (hr tdt 00 0 Φch (ft/ si 0. 0 00 000 0000 r FO FO FO3 FO4 0 0000 000 tdt (si 00 0 0000 000 tdt (si 00 0 0.0 0. 0 00 000 dt (hr 0.0 0. 0 00 000 dt (hr tdt 0.0 0. 0 00 000 dt (hr tdt Figure 4. ressure and ressure derivative match for the 4 FO tests erformed during the injection stimulation. Test is at the to and test 4 at the bottom. 0.0 0 00 000 0000 r Figure 5. ermeability-thickness (to and φch (bottom versus radial distance from the well for all the four FO tests. 6. CONCLUSIONS A method for analysis of drawdown and buildus for fractional flow comosite reservoirs with low matrix ermeability has been resented. For buildus the method relies on the use of diagnostic grahs that is a generalization of the Horner lot for fractional dimensions in each region searately. A relationshi was derived to calculate the interface radius when the transition time is known from the derivative lot. The method seems robust and sensitive and could rovide a new way to interret ressure transient tests in wells where the ressure derivative changes with time. The method was alied to a sequence of four FO s for a well under injection stimulation. The results show how the stimulation leads to rogressively large ermeability as well as extension in size of an inner region of imroved conductivity. The ermeability values obtained are consistent with a medium of modest conductivity and their change shows the effectiveness of the stimulation. Very large values of storativity were obtained which are consequence of dilation in the fracture system. The storativity of the total system increases due to the resonse of the fracture volume to ressure. ACKNOWLEGEMENTS The ermission of Chevron Geothermal and ower to ublish this aer is gratefully acknowledged. 5
REFERENCES Acuna J.A., Ershaghi I and Yortsos Y.C. ractical Alication of Fractal ressure-transient Analysis in Naturally Fractured Reservoirs. SE aer 4705. SE Formation Evaluation. 995. Acuna J.A. Numerical Construction and Fluid Flow Simulation in Networks of Fractures Using Fractal Geometry. h issertation. University of Southern California. 993. Barker J.A. A generalized radial flow model for hydraulic tests in fractured rocks. Water Resources Research 4(0, 796-804. 988. Chang J. and Yortsos Y.C. ressure Transient Analysis of Fractal reservoirs. SE Formation Evaluation 89:3, 990. oe T.W. Fractional imension Analysis of Constant- ressure Well Tests. SE aer 70. 99. Le Borgne, T., O. Bour, J. R. de reuzy,. avy, and F. Touchard. Equivalent mean flow models for fractured aquifers: Insights from a uming tests scaling interretation, Water Resour. Res., 40, 004. Leveinen J. Comosite model with fractional flow dimensions for well test analysis in fractured reservoirs. Journal of Hydrology 34, 6-4. 000. alike H. uming Tests in Fractal Media. MSc Thesis. University College London. 998. Van den Hoek. ressure Transient Analysis in Fractured roduced Water Injection Wells. SE aer 77946. 00. Yoshioka K., asikki R. and Riedel K. Hydraulic Stimulation Techniques Alied to Injection Wells at the Salak Geothermal Field, Indonesia. SE aer 84. March 009. 6