Research on the nonlinear spherical percolation model with quadratic pressure gradient and its percolation characteristics

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1 Vol., No., (010) htt://dx.doi.org/10.436/ns Natural Science Research on the nonlinear sherical ercolation model with quadratic ressure gradient and its ercolation characteristics Ren-Shi Nie 1, Yong ing 1 Petroleum Engineering eartment, Southwest Petroleum Universit, Chengdu, China; nierenshi000@163.com Luliang Oilfield Production Coman, Petrochina Xinjiang Oilfield Coman, Kelamai, China Received 18 November 009; revised 11 ecember 009; acceted 30 ecember 009. ABSTRACT For bottom water reservoir and the reservoir with a thick oil formation, there exists artial enetration comletion well and when the well roducts the oil flow in the orous media takes on sherical ercolation. The nonlinear sherical flow equation with the quadratic gradient term is deduced in detail based on the mass conservation rincile, and then it is found that the linear ercolation is the aroximation and simlification of nonlinear ercolation. The nonlinear sherical ercolation hsical and mathematical model under different external boundaries is established, considering the effect of wellbore storage. B variable substitution, the flow equation is linearized, then the Lalace sace analtic solution under different external boundaries is obtained and the real sace solution is also gotten b use of the numerical inversion, so the ressure and the ressure derivative bi-logarithmic nonlinear sherical ercolation te curves are drawn u at last. The characteristics of the nonlinear sherical ercolation are analzed, and it is found that the new nonlinear ercolation te curves are evidentl different from linear ercolation te curves in shae and characteristics, the ressure curve and ressure derivative curve of nonlinear ercolation deviate from those of linear ercolation. The theoretical offset of the ressure and the ressure derivative between the linear and the nonlinear solution are analzed, and it is also found that the influence of the quadratic ressure gradient is ver distinct, eseciall for the low ermeabilit and heav oil reservoirs. The influence of the non-linear term uon the sreading of ressure is ver distinct on the rocess of ercolation, and the nonlinear ercolation law stands for the actual oil ercolation law in reservoir, therefore the research on nonlinear ercolation theor should be strengthened and reinforced. Kewords: Nonlinear Sherical Percolation; Quadratic Pressure Gradient; Percolation Characteristics; Reservoir; Partial Penetration Comletion Well; Mathematic Model 1. INTROUCTION So far, the research on nonlinear ercolation has increasingl aroused widesread concern and attention. The nonlinear ercolation is the modern develoment of a new direction [1]. Retaining the nonlinear term was roosed b Odeh A S []. He thought ignoring the quadratic gradient term would cause larger error in hdraulic fracturing, big ressure dro flow, ST and large ressure dro ulse testing. Bai M Q [3] considered that ignoring the quadratic gradient term in flow equation is equivalent to ignoring convection flow term in diffusion-convection equation. Wang Y [4] established the nonlinear flow model in oroelastic media. Chakrabart C [5] derived the mathematical model with nonlinear diffusion equation and made the quantitative analsis of the quadratic term. Braeuning S [6] established the nonlinear radial flow model of the variable-rate well-test. Tong engke [7,8] solved the well test models of heterogeneous and dual orosit reservoir. Concerning the sherical flow, the linear sherical flow model was studied b William E. Brigham, Charles A. Kohlhaas and Mark A. Proett, et al. [9-11]. In their models, no nonlinear sherical flow model is found, so this aer resents the nonlinear sherical flow model and researches its ercolation characteristics for artial enetration comletion well in the formation. Coright 010 SciRes.

2 R. S. Nie et al. / Natural Science (010) EUCTION OF THE NONLINEAR SPHERICAL PERCOLATION EQUATION When single-hase fluid flow through orous medium, it would conform to the mass conservation rincile, so b this rincile the flow equation of continuit can be exressed b ( vx) ( v) ( vz) ( ) (1) x z t where: v is flow velocit, cm/s; is oil densit, g/cm 3 ; t is flow time, s; φ is rock orosit, fraction; x,, z reresent the Cartesian coordinates. If ignoring the imact of gravit and caillar forces, and the inertial resistance is not considered, it would conform to the arc s law, so the equation of motion is as follows k v () where: k is rock ermeabilit, μm ; μ is fluid viscosit, mpa s; is formation ressure, MPa. The fluid flow through orous medium is a rocess of ercolation, and is also a state of constantl changing rocess, in which the arameters related to ercolation are constantl changing with ressure and temerature. Usuall the change of temerature in reservoir is inareciable, so the flow is taken as isothermal flow. The rock and fluid are elastic and slightl comressible, the state equation of fluid and the state equation of rock are exressed as follows resectivel ( 0 ) 0 ec (3) 0 ec f ( 0) (4) where: C is oil comressibilit, MPa -1 ; C f is rock comressibilit, MPa -1 ; the subscrit 0 reresents reference value, usuall use the value in standard conditions. Substitute Eq.() into Eq.(1) kx k ( ) ( ) x x kz ( ) ( ) z z t kx kx ( ) x x x kx kx x x x x (5) (6) Changing the form of Eq.(3) 1 1 ln ln 0 0 (7) C C 1 x C x 1 t C t Substitute Eq.(8) into Eq.(6) kx kx ( ) x x x k x kxc ( ) x x x (8) (9) (10) B the same method, the following two equations can be deduced k k ( ) k kc ( ) kz kz ( ) z z z k z kz C ( ) z z z Changing the form of Eq.(4) f f (11) (1) 1 1 ln ln0 0 (13) C C 1 t C t f (14) Substitute Eq.(14) and Eq.(9) into Eq.(5), the right of Eq.(5) can be changed ( ) C t t t Cf Ct t t (15) Ct C Cf (16) where: C t is total comressibilit of rock and oil, MPa -1. Substitute Eqs.(10)-(1) and Eq.(15) into Eq.(5), we have kx ( kx k ) ( k z x z x x kx kx ) C[ k x ( ) z z x Coright 010 SciRes.

3 100 R. S. Nie et al. / Natural Science (010) ( ) z( k k ) ] C t z t (17) 3.SPHERICAL PERCOLATION MOELS AN ITS SOLUTION x If the ermeabilit is isotroic and constant, k / r 0, k / r 0, k / r 0, the Eq.(17) becomes C ( ) [( ) x z x Ct ( ) ( ) ] z k t (18) Eq.(18) is the governing differential equation in Cartesian coordinates, the equation in radial sherical coordinates becomes 1 Ct ( r ) C ( ) r r r r k t (19) where, r reresents the radial sherical coordinates. Eq.(19) is the nonlinear flow governing artial differential equation with quadratic ressure gradient term. We call the second ower of the ressure gradient as quadratic ressure gradient. The function ex(x) b use of Maclaurin series exansion is written b n ex( x) 1 x x / x / n! (0) If we use Maclaurin series exansion for Eqs.(3) and (4) and neglect the second order and the above higher order item, the Eqs.(3) and (4) can be rewritten b Eqs. (1) and () resectivel [1 C ( )] (1) 0 0 [1 C ( )] () 0 f 0 The aearance of quadratic ressure gradient term is siml because that we didn t make an simlification for the state Eqs. (3) and (4) in the deduction of the flow governing artial differential equation. If we use Eqs. (1) and (), instead of Eqs.(3) and (4), in the deduction of the flow governing artial differential equation, the quadratic ressure gradient term will not come u, and the deduced flow equation is the conventional linear flow equation, which is shown in almost an ercolation mechanics books and aers, so the deduction of the linear flow equation is certainl omitted here. Owing to the existence of quadratic ressure gradient, the flow equation takes on nonlinear roerties. Therefore it can be safel concluded that the conventional linear flow equation is the aroximation and simlification of nonlinear flow equation with quadratic ressure gradient term, and that the nonlinear ercolation law stands for the actual flow law of oil in reservoir Phsical Model For bottom water reservoir, the osition of drilling and comletion of oil well is usuall in the to of the oil formation, the flow diagram shown in Figure 1. For some reservoirs, the oil formation is ver thick, the osition of drilling and comletion of well is usuall in the middle of the formation, the flow diagram shown in Figure. For the two actual situations, the oil flow in the orous media is in the form of sherical ercolation. Phsical model assumtions are as follows: 1) A single well with artial enetration comletion in the formation like Figure 1 or Figure roducts at constant rate, the external boundar ma be infinite or closed or constant ressure; ) The rock and the single-hase fluid are slightl comressible, a constant comressibilit; 3) Isothermal and arc flow, the ermeabilit and orosit of isotroic roerties; h e sw r e r w Well r sw b Formation Figure 1. Sherical flow diagram for well comletion osition in the to of the formation. h w r sw Well b Formation Figure. Sherical flow diagram for well comletion osition in the middle of the formation. Coright 010 SciRes.

4 R. S. Nie et al. / Natural Science (010) ) Considering wellbore storage effects (in the beginning of oening well, the fluid stored in the wellbore starts to flow, the oil in the formation does not flow); 5) At time t=0, ressure is uniforml distributed in the reservoir, equal to the initial ressure i ; 6) Ignoring the imact of gravit and caillar forces. 3.. Mathematic Model The governing differential equation in radial sherical coordinate sstem 1 Ct ( r ) C ( ) r r r r 3.6k t (3) where: r is the radial sherical distance from well, m; the unit of well roduction time (t) becomes h, so the coefficient 3.6 aears in the Eq.(3). Initial conditions (4) t 0 i where, i is initial formation ressure, MPa. Inner boundar condition k 3 ( r ) rr qb sw r d C w s (5) dt where: the r ws is called the seudo well radius of radial sherical flow, and r ws =b/(ln(b/r w )) [1], b is the formation enetration thickness of well comletion, m; r w is the real well radius m; q is oil rate at wellhead, m 3 /d; B is oil volume factor, dimensionless; C s is wellbore storage coefficient, m 3 /MPa; w is wellbore ressure, MPa. External boundar condition lim (infinite) (6) r r r e i i (constant ressure) (7) r rre 0 (closed) where, r e is external boundar radius, m Solution to Mathematic Model The dimensionless definitions are as follows: imensionless ressure 3 krsw i / ( qb) ; (8) imensionless radius based on seudo sherical flow radius r r/ rsw ; imensionless wellbore storage coefficient C C / ( C r ) ; 3 s t sw imensionless time t 3.6 kt/ ( Ctrsw ) ; imensionless quadratic ressure gradient coefficient qbc / ( krsw ) The dimensionless model is as follows: The governing differential equation in radial sherical coordinate sstem ( ) r r r r t Initial conditions t 0 0 (9) (30) Inner boundar condition dw C ( r ) r 1 1 dt r (31) External boundar condition lim ( r, t ) 0 (infinite) (3) Take r r re 0 (constant ressure) (33) r rre 0 (closed) (34) 1 ln x (35) where, x is substitution variable between variables. Making the uer variable substitutions for Eqs. (9-34), the model can be converted to The governing differential equation Initial conditions x x x r r r t ( ) (36) x t 0 1 (37) Inner boundar condition x x ( C x) r 1 0 t r (38) External boundar condition lim xr (, t ) 1 (infinite) (39) Take r r re x 1 (constant ressure) (40) x r r re 0 (closed) (41) Coright 010 SciRes.

5 10 R. S. Nie et al. / Natural Science (010) x / r 1 (4) where, is substitution variable between variables. Making the uer variable substitutions for Eqs. (36-41), the model can be converted to The governing differential equation Initial conditions Inner boundar condition (43) r t t 0 0 (44) [ C ( 1) ] r 1 t r External boundar condition r r (45) lim 0 (infinite) (46) 0 (constant ressure) (47) re 1 ( ) r 0 (closed) re r r Introduce the Lalace transform based on t, that is zt 0 (48) L [ ( t )] ( z) ( t )e dt (49) where, z is Lalace sace variable. So, making the Lalace transform of Eqs.(43-48), the model becomes: The governing differential equation in Lalace sace d z 0 (50) dr Inner boundar condition in Lalace sace d [( C z 1) ] (51) z r 1 dr External boundar condition in Lalace sace lim 0 (infinite) (5) r So the general solution of Eq.(50) becomes Substitute Eq.(57) into Eq.(51), have e zr B (57) B zc ( z 1 z)e z The general solution of Eq.(50) can be got b zc ( z 1 z)e z e zr (58) (59) At the wellbore bottom, r=r w, r =1, = w, = w, x=x w, = w, therefore, the solution of the sherical ercolation model with infinite external boundar in Lalace sace can be got b w r 1 zc ( z 1 z) (60) The real sace solution w and the derivative (d w /dt ) can be easil obtained b use of Stehfest numerical inversion [13] for Eq.(60). Substitute the values of inversion into variable substitution relationshis, Eq.(35) and Eq.(4), so the real sace solution w and the derivative (d w /dt ) can be certainl gained. Accordingl, the ressure and the ressure derivative bi-logarithmic te curves of nonlinear sherical ercolation can be drawn u (see Figure 3). For constant ressure boundar: At the wellbore bottom r =1, = w, the Eq.(55) becomes z z e Ae B 0 (61) Substitute Eq.(61) into Eq.(51) and Eq.(53), have resectivel w z z ze A ze B ( Cz 1) w (6) z 0 (constant ressure) (53) r re 1 ( ) r 0 (closed) re r r The general solution of Eq.(50) can be exressed b (54) zr zr Ae Be (55) For infinite boundar: Substitute Eq.(55) into Eq.(5), have A 0 (56) Figure 3. Te curves of nonlinear sherical ercolation affected b β under infinite external boundar. Coright 010 SciRes.

6 R. S. Nie et al. / Natural Science (010) zre zre e A e B 0 (63) For closed boundar: Substitute Eq.(61) into Eq.(54), have e e ( e 1)e zr zr zr A ( zre 1)e B 0 (64) Combining Eqs.(61-64), the coefficients A and B and the function at wellbore w in Lalace sace can be easil obtained b use of some linear algebra method (such as Gauss-Jordan reduction, etc), then nonlinear sherical ercolation te curves can also be drawn u (see Figure 4 and Figure 5) b use of the same method. 4. CHARACTERISTICS OF THE NONLINEAR PERCOLATION Figure 4. Te curves of nonlinear sherical ercolation affected b β under constant ressure external boundar Parameter Sensitivit Analsis to Te Curves Figure 3 shows the te curves of nonlinear sherical ercolation affected b β under infinite external boundar. Can be seen from the figure, the curves var with the value of the dimensionless quadratic ressure gradient coefficient β (from u to down, β=0, 0., 0.4), when β=0 it is just the curve of linear ercolation model. It can be easil seen that the curves have the trait of unit sloe in the wellbore storage stage, which shows that there is no influence of quadratic ressure gradient in this flow stage, and that the location of the ressure and the ressure derivative curves is lower than that of the conventional linear model curve in the stage of infinite-acting radial sherical flow. The bigger the β is, the greater the offset is. Figure 4 and Figure 5 show the te curves of nonlinear sherical ercolation affected b β under constant ressure external boundar and closed external boundar resectivel. Can be seen from the figures, the trait of unit sloe in the wellbore storage stage still exist and there still exists a offset due to the effect of β, but in the late flow stage of boundar resonse the ressure derivative curves is going down until focusing on a oint for constant ressure boundar and the ressure derivative curves is going u until focusing on a line together with the ressure curves for closed boundar, which is comletel different from Figure 1. Figure 6 shows the te curves of nonlinear sherical ercolation affected b C under infinite external boundar. Can be seen from the figure, the curves var with the value of the dimensionless wellbore storage coefficient C, and the bigger the C is, the lower theressure derivative curve is. Figure 7 shows the te curves of nonlinear sherical ercolation affected b r e under different external boundaries. Can be seen from the figure, the curves var with the value of the dimen- Figure 5. Te curves of nonlinear sherical ercolation affected b β under closed external boundar. Figure 6. Te curves of nonlinear sherical ercolation affected b C. sionless radial sherical radius r e, and the bigger the r e is, the later the time of going u or going down is. According to the definition of β and the robable values of β (Table 1), it is clearl demonstrated that β is roortional to oil viscosit μ, and inversel roortional to formation ermeabilit k. So there is usuall a bigger β for the low ermeabilit, heav oil reservoirs, and the influence of the quadratic ressure gradient nonlinear term is ver distinct, the quadratic ressure gradient should not be neglected. For the fixed grou of arameters (q, B, C ), the seed of ressure wave roagation Coright 010 SciRes.

7 104 R. S. Nie et al. / Natural Science (010) Figure 7. Te curves of nonlinear sherical ercolation affected b r e. Table 1. The robable values of β. k/( 10-3 μm ) μ/(mpa s) β becomes slower when k/μ decreases with the increasing of β. Comared with the conventional linear model, given a fixed roduction time, ressure decline slows down and the seed of decline is inversel roortional to β, which is comletel accordant with the theoretical curves as Figures 3-5. In conclusion, for a concrete reservoir, due to the effect of quadratic ressure gradient, comared with conventional linear model, the time of stable roduction of the nonlinear model is rolonged on condition that the same decline of reservoir ressure. 4.. The Influence Analsis of Nonlinear Term Table and Table 3 exhibit the results of the ressure offset and ressure derivative offset analses vs. Figure 3. As shown the data in these tables, the offset increase with the increasing of time at a constant β, and ressure derivative relative offset is greater than the ressure relative offset at a fixed time. From the data in the table, it is found that the imact of the quadratic gradient is extremel intense when time is articularl long and the quadratic coefficient β is articularl large, so the quadratic ressure gradient should be retained in flow equation. After all, the nonlinear ercolation law is the actual flow law of oil in orous medium, so the research on the nonlinear ercolation model and its ercolation law with quadratic ressure gradient should be strengthened and reinforced. 5. CONCLUSIONS In this aer it is demonstrated that the quadratic ressure gradient has some distinct influence on the wellbore1) The linear ercolation is the aroximation and simlification of nonlinear ercolation with quadratic ressure gradient term. Table. The offset analsis of nonlinear term(β=0.). t /C linear P w nonlinear offset relative offset /% linear P w t /C nonlinear offset relative offset /% Table 3. The offset analsis of nonlinear term(β=0.4). t /C linear P w nonlinear offset relative offset /% linear P w t /C nonlinear offset relative offset /% Coright 010 SciRes.

8 R. S. Nie et al. / Natural Science (010) ) The new-stle te curves of nonlinear sherical ercolation with quadratic ressure gradient effect in shae and characteristics are obviousl different from the te curves of linear model, the location of the ressure and the ressure derivative curves is lower than that of the conventional linear model curve. 3) The te curves are affected b the quadratic gradient coefficient β, the offset of ressure and ressure derivative is directl roortional to β and time. 4) For a concrete reservoir, due to the effect of quadratic ressure gradient, comared with conventional linear model, the time of stable roduction of the nonlinear model is rolonged on condition that the same decline of reservoir ressure. 5) The imact of the quadratic ressure gradient under certain conditions is extremel intense, eseciall for the low ermeabilit and heav oil reservoirs, and the quadratic ressure gradient term should not be neglected and should be retained in flow equation. 6) The nonlinear ercolation law is the actual flow law of oil in orous medium, so the research on the nonlinear flow model and its alication with quadratic ressure gradient should be strengthened and reinforced. REFERENCES [1] Yan, B.S. and Ge, J.L. (003) New advances of modern reservoir and fluid flow in orous media. Journal of Southwest Petroleum Institute, 5(1), 9-3, (in Chinese). [] Odeh, A.S and Babu,.K. (1998) Comrising of solutions for the nonlinear and linearized diffusion equations. SPE Reservoir Engineering, 3(4), [3] Bai, M.Q. and Roegiers, J.C. (1994) A nonlinear dual orosit model. Al Math Moclellin, 18(9), [4] Wang, Y. and usseault, M.B. (1991) The effect of quadratic gradient terms on the borehole solution in oroelastic media. Water Resource Research, 7(1), [5] Chakrabart, C., Farouq, A.S.M. and Tortike, W.S. (1993) Analtical solutions for radial ressure distribution including the effects of the quadratic gradient term. Water Resource Research, 9(4), [6] Braeuning, S., Jelmert, T.A. and Sven, A.V. (1998) The effect of the quadratic gradient term on variable-rate well-tests, Journal of Petroleum Science and Engineering, 1(), 03-. [7] Tong,.K. (003) The fluid mechanics of nonlinear flow in orous media. Beijing: Petroleum Industr Press in Chinese, (in Chinese). [8] Tong,.K., Zhang, Q.H. and Wang, R.H. (005) Exact solution and its behavior characteristic of the nonlinear dual-orosit model. Alied Mathematic and Mechanics, 6(10), , (in Chinese). [9] William, E.B., James, M. and Peden, K.F.N. (1980) The analsis of sherical flow with wellbore storage. SPE 994. [10] Charles, A.K. and William, A.A. (198) Alication of linear sherical flow analsis techniques to field roblems-case studies. SPE [11] Mark, A. and Proett, W.C.C. (1998) New exact sherical flow solution with storage and skin for earl-time interretation with alications to wireline formation and earl-evaluation drillstem testing. SPE [1] Joseh, J.A. and Koederitz, L.F. (1985) Unsread-state sherical flow with storage and skin. SPEJ, 5(6), [13] Stehfest H. (1970) Numerical inversion of Lalace transform algorithm 368, Communication of the ACM, 13(1), Coright 010 SciRes.