Geosciences and Engineering, Vol. 1, No. 1 (01), pp. 65 74. MEASUREMENT OF POROSITY AND GAS PERMEABILITY OF TIGHT ROCKS BY THE PULSE DECAY METHOD ANDRÁS GILICZ TIBOR BÓDI EON Földgáz Storage, H-1051Budapest, Széchenyi István sqr. 7 8, andras.gilicz@eon-foldgaz.com Miskolc University, Applied Geo Science Research Institute, H-3515 Miskolc-Egyetemváros, P.O.Box. bodit@akki.hu Abstract Due to rising gas demand there is an increasing focus towards unconventional tight and shale gas reservoirs, which contain by several order of magnitude more gas, than conventional reservoirs although under more difficult circumstances. For production their key petrophysical parameters (porosity and permeability) have to be determined. The main difficulty is, that their permeability is low, typically in the nano and 0.1 md range, therefore conventional steady state measurement techniques cannot be applied. For this type of measurements the so called pressure pulse decay method is used. In this technique a pressure wave propagates through the core. Its decline time is inversely proportional to permeability, whereas the final equilibrium pressure is characteristic for the porosity of the core. In the Applied Geo Science Research Institute of Miskolc University a unique apparatus was developed a few years ago, which can carry out the pulse decay technique on full diameter radial cores. Complete measurement technique and interpretation software was developed for this purpose. In laboratories worldwide however the conventional linear plug measurement is still used widespread, so it was decided to extend the existing interpretation software with the algorithms of the linear plug measurements to obtain a flexible tool, which can interpret both types of measurements with the same easiness. The extension took place by solving the governing partial differential equation of the measurement with linear geometry, and implement the solution into the existing software. The software developed simulates the measurement itself, so no simplifying assumptions are needed for evaluation. Practical examples show the applicability of the new tool. 1. Methodology The scheme of the measurement is shown in Figure 1. As can be seen two vessels (V 1, V ) are connected to the inlet and outlet face of the core having a pore volume of V p. The whole system is filled up initially with gas or fluid all the valves being open. After equilibrium has been reached, Valve 1 is closed, and the pressure of the inlet vessel V 1 is increased slightly (c.5%). By opening Valve 1 the pressure difference between vessels V 1 and V declines through the core. Decline is measured in time with a data acquisition system. Besides other input values this recording is used to evaluate the measurement for permeability and possibly porosity.
66 András Gilicz Tibor Bódi Figure 1 In the figure above a linear plug is visible. The corresponding laboratory equipment is shown in Figure. If the core plug is a full diameter whole core, the measurement principle is still the same, but the core holder is different. This is shown in Figure [1], the corresponding lab apparatus is depicted in Figure 3. This core holder is slightly more complex than the conventional linear one but has several advantages [1]: The radial whole core is larger than the conventional linear plug, so it represents the reservoir more correctly. Radial flow takes place in the core, just like the flow in the vicinity of wells. Both porosity and permeability can be obtained from the same measurement simultaneously. The measurement is quick and easily reproducible, reproducibility is excellent. Core preparation works are simpler. The measurement can be carried out under reservoir conditions. Core load can be regulated in both axial and radial directions, so reservoir stress conditions can be simulated. Description of the radial measurement method was described in an earlier paper [1]. It has to be noted however that this unique radial measurement technique is not widely used in laboratories worldwide; rather the conventional linear plug technique is used. So the idea came to develop a flexible tool, which combines the two measurement techniques, i.e. both radial and linear. The physical development of such a device is however a complicated and costly task, which can be accomplished on longer term only, but the development of the software, was already possible, this paper describes this intermediate situation. So the problem to be solved was to extend the existing radial interpretation software with the conventional linear one.
Measurement of Porosity and Gas Permeability of Tight Rocks 67 Figure Figure 3
68 András Gilicz Tibor Bódi Figure 4. Evaluation, interpretation methods. The radial pulse decay technique has been discussed in other papers already [1, ], this is not the scope of this work. In this paper the linear pulse decay technique is discussed only. To evaluate a linear pulse decay measurement, Jones has proposed a technique [4]. According to his method the logarithm of the normalized pressure difference change is linear in time: p ln = b + m1 t p 0 (1) Permeability can be calculated from the m 1 slope. kw ( ) m1µ wl cw + cv1 = 1 1 A + V 1 V () So the principle of the Jones method is to depict measurement points according to Equation, and calculate permeability from the slope of the fitted straight line. As an example, Figure 5. depicts such an evaluation, which was carried out with water on a core sample of a tight, unconventional reservoir. Blue dots depict measurement points, whereas the blue line is fitted to the points.
Measurement of Porosity and Gas Permeability of Tight Rocks 69 Evaluation of "Pulse Decay" permeability measurement with Jones method porosity of tested sample φ = 6.7 % 1.0 p/ p 0 p ln = 0.19607 0.000331 t p0 k w ( c + c ) m1µ wl w = 1 1 A + V1 V1 V1 n = 3.05 10 4 md 0.1 0 00 400 600 800 1000 Time, s Figure 5 There is an other evaluation method also, where the mathematical model simulating the whole measurement is fitted to measurement points. Such a solution was published by Haskett et al [3]. Their solution however is full with dimensionless variables, conversion factors, etc. which makes it difficult to understand and apply, so it was decided to develop our own solution, which was called RIAES LPD. The governing equation of flow within the linear core is the partial differential equation of hydraulic diffusivity: where P P = C x t (3) P = p p ini (4) Initial conditions are: p(x; t = -0) = p ini (5) Boundary conditions are for x = 0, and t > 0 p u (t = 0) = p1 (6) p d (t = 0) = p (7) p u (t) = p(x = 0; t). (8) dp p C 1 1 = (9) dt x x = 0
70 András Gilicz Tibor Bódi where i.e. for x = L, t > 0 C1 µ cvu ka = (10) p d (t) = p(x = L; t) (11) further dp p C = (1) dt x x = L C µ cvd =, (13) ka C = ϕµc. (14) k In general p = p ini, but not necessarily. With this notation the solution can be given quite generally. Equations 8 and 11 express pressure continuity between core ends and attached vessels, whereas equations 9 and 1 express mass balance, i.e. the rate of pressure change is proportional to in- and outlet fluxes of the core. Equation 3 was solved via Laplace transform. The general solution is as follows: where n M t 1 M C α (15) n= 1 αn 1 P( x, t ) = P + e C F F M1 = CP cos( α n x) + C1P1 cos αn (L x) (16) C 1 C α M n = P sin( α nx) + P1 sin α n (L x) C (17) L (C1 + C ) C1C L F1 = cos( α nl) αn Cαn C (C1 + C )L F = + sin( α 3 nl) α n nc α and α n s are the roots of the following transcendent equation: (C1 + C )Cα tg( α n nl) = C1C αn C α n s can be calculated numerically. (18) (19) (0)
Measurement of Porosity and Gas Permeability of Tight Rocks 71 Pressures of the vessels are measured at the in- and outlet end of the core (i.e. at x = 0, and x = L) so the mathematical solution has to be taken at these locations. Accordingly: M 1 x 0 C P C 1 P 1 cos( n L) = = + α (1) M1 CP x L cos( nl) C1P = 1 = α + () C1Cα M n P x 0 1 sin( nl) = C = α (3) C1Cα M n P x L sin( nl) = C = α (4) The above formulas were implemented into the software the graphical interface of the software is shown below in Figure 6. Figure 6 An example application with the new software is shown in Figure 7.
7 András Gilicz Tibor Bódi Figure 7 3. Conclusions The pulse decay technique was used to determine the permeability of a core originating from a Hungarian unconventional gas reservoir. Measurements were done with water. Interpretation was done with both the Jones and our method, and results were compared. The comparison is shown in Figure 8 and Table 1. 4.0E-03 3.5E-03 Water permeability ( RIAES LPD), md 3.0E-03.5E-03.0E-03 1.5E-03 1.0E-03 5.0E-04 Regression coefficient R = 0.9998 0.0E+00 0.0E+00 5.0E-04 1.0E-03 1.5E-03.0E-03.5E-03 3.0E-03 3.5E-03 4.0E-03 Water permeability (Jones), md Figure 8
Measurement of Porosity and Gas Permeability of Tight Rocks 73 Number Porosity (%) Permeability (Jones) (md) Table 1 Permeability (AFKI) (md) 1 6.700 3.05E-04 3.606E-04 5.900 4.36E-04 5.48E-04 3 4.100 4.84E-04 4.95E-04 4 7.400 4.041E-04 4.719E-04 5 6.900.65E-04 3.171E-04 6 6.700 3.999E-04 4.68E-04 7 6.100 3.95E-04 3.970E-04 8 6.600 1.385E-04 1.517E-04 9 6.500.994E-04.957E-04 10 3.800 8.979E-05 1.037E-04 11 6.900 3.579E-04 3.908E-04 1 6.700 1.780E-04.009E-04 13 6.600.437E-04.739E-04 14 3.400.787E-04 3.485E-04 15 6.400 3.1E-04 3.873E-04 16 6.00 3.788E-04 4.496E-04 17 7.000 3.70E-04 4.415E-04 18 6.900 3.63E-04 3.995E-04 19 6.400.63E-04 3.080E-04 0 6.400.960E-04 3.498E-04 1 7.500 8.59E-04 9.915E-04 7.000 3.707E-04 4.546E-04 3 8.100 3.138E-03 3.733E-03 4 6.900.571E-04 3.303E-04 5 6.00 1.764E-04.03E-04 6 6.00.941E-04 3.553E-04 7 6.400.39E-04.969E-04 8 6.600 1.916E-04.359E-04 The following conclusions can be drawn: There is a strong correlation between the two measurement methods. The Jones method slightly underestimates the values obtained by the numerical method. This is most probably because the Jones method has some limiting assumptions. In case of using water, porosity measurement was not possible unlike in the case of gas as a measurement medium. The two methods give consistent results in cases, when the straight line fit of the Jones method is easy and straightforward. If the match of the Jones line is good, than the match of the numerical model is good, and vice versa.
74 András Gilicz Tibor Bódi In case of a linear fit there is some degree of freedom to fit the straight line, which however is not possible with the numerical method. The ideal medium for measurement is gas, not water, or any other fluid. Gas does not contaminate the core, there is a large compressibility contrast between the gas and the steel core holder, and the real reservoir fluid is also gas. Nomenclature p pressure difference p 0 pressure difference at t = 0 t time b intersection m 1 slope µ viscosity l length A cross section c w isothermal compressibility of water c 1 isothermal compressibility of the measurement system V 1 volume of the inlet pressure V volume of the outlet vessel c compressibility C see eq. 14. C 1 see eq. 10. C see eq. 13. F 1, see eq. 15. k permeability M numerator M 1, members in eq. 15. P p p ini p pressure p u inlet pressure p d outlet pressure p 1 initial inlet pressure p initial outlet pressure x distance V u inlet vessel volume V p pore volume V d outlet vessel volume Greek letters φ porosity µ viscosity α n positive roots of eqn. 0 Acknowledgement: This works was carried out ad part of the TÁMOP-4..1.B-10//KONV-010-0001project in t5he framework of the New Hungarian Development Plan. The realization of this project is supported by the European Union, co-financed by the European Social Fund. REFERENCES [1] Bódi, Tibor Gilicz, András: Tömött kőzetek porozitásának és gázpermeabilitásának egyidejű meghatározása a dinamikus nyomásváltozás mérésével. XXVII. Nemzetközi Olaj- és Gézipari Konferencia Hungary, Siófok 008. szeptember 16 19. CD A 06 1 40. [] Gilicz, András: Application of the Pulse Decay Technique. SPE 688, presented at the 66 th Annual Technical Conference and Exhibition of the SPE held in US Texas, Oct. 6 9, 1991. [3] Haskett Narahara: A Method for Simultaneous Determination of Permeability and Porosity in Low Permeability Cores. SPE Formation Evaluation, September 1988, 651 658. [4] Jones, S. C.: A Technique for Faster Pulse Decay Permeability Measurement in Tight Rock. SPE Formation Evaluation, 1997 March. [5] Serag El Din, et al.: Whole Core Versus Plugs: Integratin Log and Core Data to Decrease Uncertainty in Petrophysical Interpretation and STOIP Calculations. SPE 137679