A New Method for Calculating Oil-Water Relative Permeabilities with Consideration of Capillary Pressure

Transcription

1 A Ne Method for Calculating Oil-Water Relative Permeabilities ith Consideration of Capillary Pressure K. Li, P. Shen, & T. Qing Research Institute of Petroleum Exploration and Development (RIPED), P.O.B. 910, Beijing, China Abstract To-phase relative permeabilities from unsteady-state core flooding data are usually computed using JBN method by neglecting capillary pressure force. A ne analytical method has been developed to calculate oil-ater relative permeabilities ith capillary pressure included. Introduction The unsteady-state (USS) displacement in reservoir cores is very often used to measure oil-ater to-phase relative permeabilities. A lot of methods 1,,3 have been developed in order to calculate the relative permeabilities from the experimental core flooding data. The most often used method for calculating relative permeabilities is the so-called JBN method 1. Hoever, JBN method is derived assuming that capillary pressure is neglected. The usual ay to satisfy this assumption in an immiscible oil-ater-rock system is to perform the experiments at a very high flooding rate. Certain problems appear hen the experiments are run at very high flooding rates. First, the relative permeabilities measured at high injection rates do not represent the real value of reservoir relative permeabilities. The flo rates in reservoirs are generally believed very lo. The effect of flo rates on the relative permeabilities is not very clear. Secondly, for some lo permeable cores and unconsolidated sandstone, high injection rates may not be applied. The needed orking pressure to apply high flo rates in core samples ith lo permeability may be over the maximum pressure of the instrument. The serious formation damage may occur in unconsolidated sandstone if the high injection rate is applied in order to overcome the effect of capillary pressure on relative permeabilities. The numerical simulation method, Automatic History Matching 4, can be used to calculate the relative permeabilities ith the consideration of capillary pressure. This type of method is complicated and much time consuming. Civan and Donaldson 5,6 proposed a semianalytical method to calculate relative permeabilities ith capillary pressure included. In their method, it is still necessary to use numerical iteration and optimal method. As revieed briefly above, it is necessary to develop an analytical method 7 to calculate relative permeabilities ith capillary pressure included. Therefore, the relative permeabilities can be computed from the USS displacement performed at a lo flooding rate. The main purpose of this paper is to derive the equations to calculate the relative permeabilities of oil and ater hen the capillary pressure is included. Basic Theory Given that oil and ater are incompressible, the volumetric continuity equations are expressed as: 46 Mechanics and Practice, V. 16, No.,1994

2 vi + φ Si = 0 (1) t here i represents oil or ater phase, v i and S i are the floing velocity and the saturation of phase i, respectively. x and t represent the distance from the inlet end of the core and the injection time, respectively. φ is the porosity of the core. The floing velocity v i can be calculated as follos based on the extended Darcys La: v i = kkri µ i pi ( ρ g sinα) i () here k and k ri represent the absolute permeability of the core and the relative permeability of phase i. µ i and p i are the viscosity and the pressure of phase i. ρ i is the density, g the gravity constant, and α the angle beteen the central axis of a cylindrical core and the horizontal direction. Capillary pressure is defined as: P c = po p (3) Capillary pressure P c is a function of ater saturation S. The floing velocity of oil phase is obtained by substituting Equation 3 in Equation : kk v ro p ( P c o = + ρ sinα) µ o g (4) o x The floing velocity of phase i, v i (, at the production end face of the core is calculated in the folloing: 1 dn pi ( A dt v i ( t = (5) ) here A is cross section area of the core, N pi ( is the production of phase i at the injection time t. The fraction flo of phase i is defined as: vi ( f ( ) = (6) i t here f i ( is the fraction flo; v i ( and are the floing velocity of phase i and the total floing velocity at time t, respectively. 47 Mechanics and Practice, V. 16, No.,1994

3 The ater fraction at the outlet end face, f (, can be obtained by substituting Equation and 4 in Equation 6: µ o k k P + ( ρ c ρ v t o) g sinα + k ro ( ) f( = (7) µ µ + o k r k ro Equation 1 can be expressed in the folloing for the ater phase using Equation 6: f ( ) v t + φ S = 0 (8) t The so-called Buckley-Leverett equation can be obtained from Equation 8: dx dt = df (9) S φ ds ( ) The ater fraction term in Equation 9, hich is expressed by Equation 7, is much more complicated in the case ith the capillary pressure included than that ithout the capillary pressure. Calculation of Relative Permeabilities ith Capillary Pressure Included It is necessary, at first, to get the relationship beteen the differential pressure p applied at the to ends of the core sample and relative permeability in order to calculate the relative permeabilities. Substituting Equation in Equation 6: pi fi µ i kkri = + ρ sinα (10) i g The differential pressure p beteen the to ends of the core can be calculated using the folloing equation: p i dx 0 p = L x (11) here L is the length of the core sample. The value of the differential pressure p beteen the to ends of the core at a given time is unique. For a ater-et rock-oil-ater system ith initial ater saturation, it is L p reasonable to calculate p referred to ater phase. That is, p = dx. This ill be 0 discussed later in more detail. 48 Mechanics and Practice, V. 16, No.,1994

4 From Equation 9, one can obtain: dx = L df (1) f here f and f are ater fraction gradient at x and L (outlet end of the core), respectively. f is calculated in the folloing: 1 ALφ f = = (13) Q Q( here Q( is the total liquid production and Q is the dimensionless total liquid production. Substituting Equations 10 (using ater phase) and 1 in Equation 11: f f [ µ ρ sin L 0 kk g α df r f p = (14) The above equation can be reduced using Equations 5 and 13. The reduced Equation 14 is then differentiated and the relative permeability of ater phase can be obtained: kr d[ 1 Q = f ( sin ) [ gl (15) k P+ ρ α d µ LQ The relative permeability of oil phase can be obtained using Equation 7: kro µ o f = o (16) µ f k [( ) sin dpc ρ S k ρo g α + r ds If the gravity and capillary pressure are neglected, Equations. 15 and 16 are reduced to the equations of the most often used JBN method. It can be seen from Equations 15 and 16 that the relative permeabilities of ater as a et phase are not affected by the capillary pressure. The relative permeabilities of oil as a non-et phase, hoever, are affected by the capillary pressure. A lot of experimental results 8 sho that the imbibition relative permeabilities of et phase are the same as the drainage relative permeabilities of et phase. This implies that there is no effect of the capillary pressure on the relative permeabilities of et phase but the effect of the capillary pressure on the relative permeabilities of non-et phase is significant. This is also confirmed by the fact that there is an evident hysteresis beteen imbibition and drainage relative permeabilities of non-et 49 Mechanics and Practice, V. 16, No.,1994

5 phase. Qadeer et al 4 shoed that there is almost no effect of the capillary pressure on the relative permeabilities of et phase, similarly, the effect of the capillary pressure on the relative permeabilities of non-et phase is significant using automatic history matching method. From hat described above, our ne equations for calculating relative permeabilities of both et and non-et phases are consistent ith both the experimental data and the numerical simulation results. Calculation of Water Saturation Gradient at the Core Outlet 7 It is necessary to derive the expression of the ater saturation gradient at the outlet of a core (S /) in order to calculate the relative permeabilities of oil phase based on Equation 16. The ater saturation gradient at the outlet of a core may be expressed as: S S /t = (17) dx / dt By substituting Equation 9 in Equation 17: S Aφ = S (18) f Given that S ( x, represents the average ater saturation in the range of [0, x, S ( x, t ) can be expressed in the folloing: S S ( x, = S ( x, Q (19) ( x, By differentiating Equation 19: S ( x, S ( x, = Q (0) For ater flooding experiment, the average ater saturation in the range of [0, x, S ( x,, is calculated using the folloing equation: S [1 f ( x, Q ( x, = S i + (1) Axφ here S i is the initial ater saturation in the core. The folloing equation can be obtained from Equation 1: 50 Mechanics and Practice, V. 16, No.,1994

6 S ( x, 1 ( x, f ( x, [ f = + Q () Axφ Substituting Equation in Equation 0: S ( x, Q f ( x, f ( x, [ = + Q (3) Axφ The ater saturation gradient at the outlet of a core can be obtained by substituting Equation 3 in Equation 18: S ( x, Q df = [ x= L AφL dq d f + Q dq (4) Equation 19 can be also suitable at the outlet of a core, that is, x=l. Therefore, the ater saturation S at the outlet of a core can be calculated using folloing equation: S S = S Q (5) here S is the total average ater saturation in the hole core. No all the equations needed for calculating ater saturation (Equation 5) and relative permeabilities of both oil and ater phases (Equations. 15 and 16) have been developed ith capillary pressure included in the derivations. Conclusions A ne analytical method has been developed to calculate relative permeabilities of both oil and ater phases ith capillary pressure included. With the ne method, the core flooding experiments may be performed at the real reservoir flo rate. References 1. Johnson, E. F., Bossler, D.P., and Naumann, V.O.: Calculation of Relative Permeability from Displacement Experiments, Trans., AIME, 1959, 16, Jones, S. C. and Roszelle, W.O.: Graphical Techniques for Determining Relative Permeability from Displacement Experiments, JPT, May Huan, G. and Shen, P.: A Method for Calculating Unsteady-State Oil-Water Relative Permeability, J. of Petroleum Exploration & Development, 198 (), Qadeer, S., Dehghani, K., Ogbe, D.O., and Ostermann, R.D.: Correcting Oil/Water Relative Displacement Experiments, SPE 1743, presented at the SPE California Regional Meeting held in Long Beach, California, March 3-5, Civan, F. and Donaldson, E.C.: Relative Permeability from Unsteady-State Displacements: An Analytical Interpretation, SPE 1600, Mechanics and Practice, V. 16, No.,1994

7 6. Civan, F. and Donaldson, E.C.: Relative Permeability from Unsteady-State Displacements ith Capillary Pressure Included, J. of SPE Formation Evaluation, June 1989, Li, K.: Tomographic Study of Multi-phase Fluid Flo in Porous Media and the Systematic Investigation on the Calculations and Measurements of Relative Permeability, Ph. D. Thesis, Graduate School of RIPED, Beijing, China, Hong, S.: Petrophysics, China Petroleum University Press, 1980, Mechanics and Practice, V. 16, No.,1994