Water Coning for v h Well

SPE SPE 22931

Water Coning Calculations for Vertical and Horizontal Wells Weiping Yang and A.A. Wattenbarger, * Texas A&M U. E 'SPE Member Copyright t 991. Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Dallas. TX, October 6-9, 1991.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the authDr(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex, 730989 SPEDAL.

ABSTRACT

Most authors have concentrated on correlations for critical rate and breakthrough time in vertical and horizontal wells. WOR (water-oil ratio) has also been addressed in vertical wells. However, WOR perfonnance in horizontal wells has not received much attention. The purpose of this work is to develop a method suitable for either hand calculation or simulation to predict (1) critical rate, (2) breakthrough time, and (3) WOR after breakthrough in both vertical and horizontal wells.

An extensive sensitivity analysis of water coning was per-fonned using numerical simulation. From this analysis, an empirical coning correlation was developed based on the basic flow equations and regression analysis. The fonnat of the correlation is similar to Addington's gas-coning correlation. It predicts critical rate, break-through time and WOR after breakthrough.

WOR perfonnance at variable rate production conditions has also been evaluated in this work. It was found that WOR has hyster-esis, (i.e., WOR not only depends on the current production rate, but also the previous production history). However, given sufficient time after rate changes, hysteresis disappears. At such conditions, the correlations can also give a good estimation of WOR for variable rate cases.

This correlation provides a hand calculation method of coning prediction for both vertical and horizontal wells. It can also be used as a coning function for 3-D coarse grid reservoir simulation. The correlation was tested and found to be reliable and accurate in predict-ing WOR, as well as critical rate and breakthrough time when water-oil mobility ratio is smaller than 5 or viscous forces are not dominating.

References and illustrations at end of paper.

459

INTRODUCTION

Many wells produce from oil zones underlain by "bottom water". When the well is produced, water moves up toward the wellbore in a cone shape. At certain conditions, water breaks through into the well and concurrent oil and water production begins. This phenomena is referred to as water coning.

Many authors have addressed the coning problem in tenns of critical rate (the maximum production rate without producing water), water breakthrough time, and water-oil ratio (WOR) after water breakthrough. Many methods have been developed for predicting these behaviors.

Critical rate is probably the topic which has been discussed the most. Since the first paper from Muskat and Wyckoff in 1935, a number of correlations was developed for predicting critical rate. In general, these correlations can be divided into two categories.

The first category detennines critical rate analytically based on the equilibrium conditions of viscous forces and gravity forces. It started by developing an oil potential function and then solved for the critical rate by letting viscous forces equal the gravity forces. However, the methods of calculating oil potential are various. For example, Muskat and Wyckoff solved a Laplace equation for single phase flow, while Chaney et a1.4 and Chierici et a1.5 used potentiomet-ric models. Wheatly's15 method also falls into this category, but, he took into account the influence of cone shape on the oil potential, which others had not done before. Chaperonl6 and Gigerl? extended this method to horizontal wells.

The second category is empirical correlations. Schols8 developed a correlation from his lab experiment, while Hayland et al. 19 developed their correlation from computer simulation runs.

2 WATER CONING CALCULATIONS FOR VERTICAL AND HORIZONTAL WELLS SPE 22931

Addingtonl2 also discussed critical rate calculation. However, his concept of critical rate was different from others. Addington was solving a closed outer boundary problem that never reaches steady-state conditions, while others were dealing with open outer boundary problems at steady-state conditions. Furthermore, Addington's critical rate is decreasing with time or cumulative oil production, while others had a constant critical rate.

Methods are also available for predicting water breakthrough time. Sobocinski and Cornelius6, based on their experimental and computer simulation results, developed a dimensionless plot which traces the rise of cone apex from its build-up to breakthrough. Cone breakthrough time and critical rate can be determined from the plot. Bournazel and Jeanson7 evaluated this plot and developed a simple analytical expression to fit the plot. Papatzacos et a1. 20 investigated the cone breakthrough time in horizontal wells. Both single-cone and simultaneous two-cone cases were considered. The solutions were derived by a moving boundary method with constant pressure or gravity equilibrium assumed on the moving boundary. Their solution only applies for infmite acting reservoirs.

WOR after breakthrough in vertical wells was also addressed by some authors. Bournazel and J eanson 7 presented a method assuming that water is separated from oil, the oil-water interface rises and stays at some point of perforation interval. By calculating the length of the perforation interval in the water, WOR can be predicted.

Byrne and Morse9, Mungan10, Blades and Strightll investigat-ed the effects of various reservoir and well parameters on WOR performance using numerical simulation. However, they had not come up with a general predictive method.

Addington12 developed a set of gas-coning correlations for 3-D coarse grid simulation. The correlation can be used to predict critical coning rate and gas-oil ratio (GOR) after coning has been achieved. Even though the correlations were developed for specific data in Prudoe Bay field, the technique can be of use in water-coning evaluation.

Kuo and DesBrisay13 investigated the sensitivity of water coning performance to various reservoir parameters using numerical simulation. A correlation of predicting water cut performance was developed from the sensitivity analysis.

Kabir14 studied water coning into gas wells using simulation. The sensitivity of reservoir and fluid properties on water-gas ratio was discussed.

This paper presents a water-coning correlation to predict critical rate, water breakthrough time and WOR after breakthrough for both vertical and horizontal wells. The correlation was developed following the same procedure as Addington's. It can be used either as a hand calculation method or a coning function for 3-D coarse grid reservoir simulation.

APPROACH TO THE PROBLEM

Addington12, when studying gas coning into an oil well, observed that a straight line results when gas-oil ratio (GOR) is plotted

460

against the average oil column height above perforations after gas breakthrough on a semi-log scale.

Based on this observation, Addington performed an extensive parameter sensitivity analysis, from which slope and intercept of the straight line was correlated with various reservoir and fluid properties affecting coning performance. From this correlation, not only the GOR can be predicted, but also the critical rate can be calculated.

We followed the same procedure as Addington did and developed a water-coning correlation. As the first step of developing the coning correlation, a one well model was simulated at a constant total production rate. The one well model was run on a two-dimensional simulator. For vertical wells, a r-z radial model was used and for a horizontal well, a 2-D x-z model was used. The well was simulated with a wide range of properties.

Fig. 1 shows a sketch of a reservoir with a bottom aquifer and a well perforated above the aquifer. As production begins, water cones up toward the wellbore. If assuming that water is displacing oil in a piston-like manner, then an imaginary current water-oil contact can be defmed. Fig. 1 shows this contact by a dashed line. The oil column height between the current contact and the bottom of the perforation is defmed as the average oil column height below perfora-tion, denoted by hbp. It can be calculated by writing a material balance equation. The calculation is discussed in the Appendix.

As production proceeds, hbp decreases. At some point of time, water breaks into the wellbore, the average oil column height below perforation at this time is termed average oil column height below perforation at breakthrough, denoted by hwb. After water breaks into the well, WOR increases as t"p decreases.

After simulating a one well model at different properties for both vertical and horizontal wells, we found that Addington's correla-tion form, with a slight modification, applies to water coning. That is, the plot of WOR plus a constant, c, as a function of t"p is a straight line after water breakthrough on a semi-log scale, as shown by Fig. 2. The straight line relationship can be described mathematically as:

WOR = 0 h >h bp wb ...... (1) Log(WOR +c) = m(hbp -hwb) +Log(c) hbp S; hWb

c is a constant, depending on whether it is a vertical or a horizontal well. Therefore, if the breakthrough height hwb' slope of the straight line m and constant c can be determined, then, the whole process of coning can be predicted.

As we have mentioned in the Appendix, for a tank reservoir, ~p is linearly related to the cumulative oil production Np' the WOR + c vs. t"p plot can be easily converted to a WOR + c vs. Np plot.

The method of determining hwb' m and c was developed from a stepwise procedure. First, a number of simulation runs was made to investigate the coning performance at different reservoir and fluid properties both for vertical and horizontal wells. Then, for each simulation run, WOR + c was plotted against t"p on a semi-log scale, from which m and hwb were determined. Once the hwb and m data

SPE 22931 WEIPING YANG AND R.A. WATIENBARGER 3

was obtained for all the simulation runs, regression analysis was then used to defme the relationship between m, hWb and various reservoir and fluid properties.

We followed this procedure and developed a coning correlation for both vertical and horizontal wells, respectively, the results will be discussed in the following sections.

VERTICAL WELLS

The water-coning performance at different reservoir and fluid properties was investigated using a 2-D r-z numerical simulator. Fig. 3 sketches the reservoir geometry, grid size and boundary conditions. Following assumptions were made during the simulation:

1. No flow across the outer boundary. 2. Formation is underlain by a recharged bottom aquifer. 3. Only one perforation interval. 4. Reservoir is homogeneous but anisotropic. 5. Only water and oil are present at reservoir conditions. 6. Capillary pressure can be ignored.

Parameter Sensitivity Analysis

The parameter sensitivity analysis was made to provide data for developing a predictive correlation of calculating breakthrough height hwb and slope m.

To begin the parameter sensitivity analysis, a base case was set up first and all the simulation runs were conducted by varying base case data. Eleven parameters were varied to establish the 48 simulation cases. The relative permeability data is tabulated in Table 1. The input data for base case and all other runs are summarized in Table 2.

From these simulation runs, it was found that the constant, c, for vertical wells is 0.02. Therefore, the WOR changes can be described by the following equation:

WOR =0 h > h bp wb (2) Log(WOR+0.02) =m(hbp-hwb) + Log(0.02) hbp S hwb .....

The WOR from each simulation run was least square fitted by the above equation, from which the height hWb and slope m was determined. The last two columns in Table 2 list the m and hwb for each run.

Generalized Correlations

Parameter sensitivity analysis shows that height hWb and slope m are functions of the various reservoir and fluid properties. These functions were defmed using the regression analysis.

As Table 2 shows, hwb increases with production rate qt and oil viscosity, etc. However, the increase of hWb is limited by a natural constraint:

hWb S h - hp - hap. . . . . . . . . . . . . . . . . . .. (3)

With this in mind, we came up with the following results:

461

1 + 39.0633 X 10-4

. ... (4)

(5)

m = 0.015 [1 +485.7757 [_1_] 0.5 [~] 0.5 1 (1-45)(1-)..)] rne qn 1+Mo.03 h1.7

(6)

The parameters were grouped together based on the basic flow equations and the grouping was confirmed by regression analysis. Eq. 4 guarantees that hwb can never go beyond h - ~ - hap.

HORIZONTAL WELLS

The same procedure of developing correlations for vertical wells was followed here for horizontal wells. First, the WOR behavior at different reservoir and fluid properties was investigated by numerical simulation, then the breakthrough height hwb and slope m were determined, fmally, the regression analysis was used to correlate hwb and m with various reservoir and fluid properties.

A 2-D x-z model was used in the simulation. Fig. 4 sketches the reservoir geometry, grid and boundary conditions. In addition to the assumptions made for vertical wells, it was further assumed that the horizontal well is long and fully penetrated so that a 2-D x-z geometry can be used.

Parameter Sensitivity Analysis

The sensitivity of various reservoir and fluid properties on the coning behavior in horizontal wells was investigated extensively by varying the base case data. Eleven parameters were varied and evaluated by 47 simulation runs. The input data for base case and all these runs are summarized in Table 3. The relative permeability curve is the same as in vertical weU


From the sensitivity analysis, it was found that the best way of presenting WOR data is to plot WOR + 0.25 as a function of average oil column height below perforation hbp' The resulted plot is a straight line on a semi-log scale, which can be described mathematically by the following equation:

The WOR results from each simulation run was curve fitted by the above equation, from which the breakthrough height hWb and slope m were determined. The last two columns of Table 3 list the hwb and m for each run.

Generalized Correlations

Parameter sensitivity analysis shows that the breakthrough height hWb and slope m are functions of reservoir and fluid properties. The height hwb increases with production rate, oil viscosity, etc. However, the same argument for hwb in vertical wells still applies here, that is, the increase in hwb is limited by a natural constraint:

hWb S h - hap' ....................... (8)

thus: h-hap

hWb ~ 1

With this in mind, we came up with the following results:

(9)

[h-hap ]2=1 + 4.7921X1O-4X~.32[_1 ]0.65 [..:...] [_1_] (10)

hWb xD qD 1 +MO.4

[ 0.18 [ ] 04 [ ] 0.5 ](11)

m=0.004 1+2.7496 Xah

X~ q~ (I+Mo.25)(I-A)0.3

(12)

Again, the parameters were grouped together based on the basic flow equations and the grouping was confirmed by regression analysis.

462

HOW TO CALCULATE CRITICAL RATE

The correlation for hwb can be used as a critical rate correla-tion. Assuming that a well is produced at a rate of qt> then, right at the height hwb' water breaks into the well. To see this process other way around, assuming that the height is at hwb, then, if the production rate is above 'It, the well produces water; if rate is below 'It, the well does not produce water. Therefore, the rate solved from Eq. 4 or 10 is actually the critical rate at height hwb'

To demonstrate that this is the case, we made five simulation runs, the input data for these runs are the same as in base case except production rate. Fig. 5 shows the five production schedules and the corresponding WOR performance. Schedule A, Band C have a constant production rate of 1000, 2500 and 4500 RBID, respectively. Schedule D and E have a variable rate which starts at 2500 RB/D, then, when hbp drops to 65.12, production rate is increased to 4500 RB/D in schedule E; decreased to 1000 RB/D in schedule D. The figure depicts that at the height of 65.12, when rate is higher than 2500 RB/D, the well is coning water; when rate is below 2500 RB/D, the well is not coning water. Therefore, the critical rate at the height of 65.12 is 2500 RB/D. Of course, at different height, the critical coning rate is different, which can be solved from Eq. 4:

(13)

k k' h2~'Y h co qeD' . . . . . . . . . . . . . . . . . . .. (14) JLo

for vertical wells.

Similarly, for horizontal wells, critical rate can be solved from Eq. 10 as:

= 4.7921X1O-4x~.32 [_1_] 0.65 1 h~p xD I+M.4 (h-h )2_h2

ap bp

(15)

M k:oLh~'Y q .................. (16) eD

JLo

These equations show that critical rate is decreasing with height ~P' thus, critical rate is decreasing with time or cumulative oil production.

The critical rate calculated in this manner is different from the rate calculated from the classic steady-state methods. The reasons are that classic methods are associated with the open outer boundaries under steady-state conditions. And the critical rate is the rate below which there is no water production at any time. This method is for a closed boundary problem, which never reaches steady-state conditions. Critical rate is the rate below which there is no water production at a particular time.


HOW TO CALCULATE BREAKTHROUGH TIME

Fo.r a tank reservo.ir, the average o.il co.lumn height belo.w perfo.ra-tio.n hbp is linearly related to the cumulative o.il pro.ductio.n Np. Then, the cumulative o.il pro.ductio.n at breakthro.ugh can be calculated fro.m the breakthro.ugh height hwb:

(Np)bt hWb = h - - hap - hp ' . . . . . . . .. (17) Atp(1-swc -sor) so.lve fo.r (Np)bt' we have:

the breakthrough time can be predicted as:

tbt = (Np)bt ..................... . qt

(19)

This pro.cedure applies to. bo.th vertical and ho.rizo.ntal wells.

The breakthro.ugh time calculated fro.m this co.rrelatio.n is co.mpared with o.ther metho.ds and simulatio.n results. Fig. 6 sho.ws the co.mpari-so.n fo.r a vertical well, in which co.rrelatio.n breakthro.ugh time is co.mpared with So.bo.cinski's metho.d and simulatio.n results, (the breakthrough time fro.m simulatio.n was taken as the time when water cut equals 0.01). Fo.r this case, the co.rrelatio.n gives a very go.o.d appro.ximatio.n to. the simulatio.n results. But, So.bo.cinski's metho.d is o.bvio.usly to.o. high. The reaso.n eQuId be that So.bo.cinski's co.rrelatio.n is o.nly fo.r o.pen bo.undary problems.

Fig. 7 sho.ws the co.mpariso.n fo.r a ho.rizo.ntal well, where co.rrela-tio.n is co.mpared with Papatzaco.s's metho.d and simulatio.n. Again,o.ur co.rrelatio.n result matches the simulatio.n result. Ho.wever, Papatzaco.s's breakthro.ugh time is to.o. high, the reaso.n co.uld be that his metho.d o.nly applies to' infinite acting reservo.irs.

HOW TO CALCULATE WOR AFTER WATER BREAKTHROUGH

To. fmd the WOR at height hbp fo.r a given pro.ductio.n rate, first, calculate the breakthro.ugh height hwb fro.m Eq. 4 o.r 10 and slo.pe m fro.m Eq. 5 o.r 11, then use the fo.llo.wing equatio.n to' fmd WOR fo.r a vertical well:

WOR = 0 hbp > hWb (20) Lo.g(WOR+0.02) = m (hbp-hwb) + Lo.g(0.02) hbp:S;;hwb

and use the fo.llo.wing fo.r a ho.rizo.ntal well:

A sample calculatio.n fo.r a co.nstant rate case was made fo.r a vertical and ho.rizo.ntal well respectively. The results were co.mpared with the simulatio.n results. The co.mpariso.ns are sho.wn in Figs. 8 and 9. The figures sho.w that co.rrelatio.n gives a go.o.d match to' the simulatio.n results.

463

DISCUSSION OF RESULTS

The co.rrelatio.n can also. be used to' predict WOR fo.r variable rate cases. The predictio.n is based o.n the assumptio.n that WOR has no. hysteresis, i.e., WOR is o.nlya functio.n o.f current height hbp and current pro.ductio.n rate, previo.us pro.ductio.n histo.ry has no. influence o.n the current WOR. Under such an assumptio.n, the co.rrelatio.ns are valid fo.r variable rate case, o.nly hwb and slo.pe m have to. be recalcu-lated each time when rate changes.

A sample calculatio.n fo.r a vertical well is sho.wn in Fig. 10, where so.lid line represents the WOR calculated fro.m co.rrelatio.n while circle represents simulatio.n WOR. The pro.ductio.n rate starts at 2500 RBID, decreased to. 1000 RBID at height o.f 42 ft, then increased to. 4500 RBID at height o.f 21.6 ft.

A similar sample calculatio.n was made fo.r a ho.rizo.ntal well. Fig. 11 sho.ws the co.mpariso.n o.f co.rrelatio.n with simulatio.n results. Again, so.lid line represents the WOR calculated fro.m co.rrelatio.n while circle represents simulatio.n WOR. The pro.ductio.n rate starts at 2500 RBID, decreased to. 1000 RBID at height o.f 30.5 ft, then increased to. 4500 RBID at height o.f 12.3 ft.

The figures sho.w that every time when rate is changed, co.rrelatio.n predicts a mo.re abrupt jump o.f WOR. Ho.wever, as time go.es o.n after rate changes, co.rrelatio.n WOR gradually appro.aches simulatio.n WOR. This trend is o.bserved in bo.th figures. The deviatio.n o.f co.rrelatio.n fro.m simulatio.n WOR is the result o.f hysteresis assumptio.n. Right after rate changes, previo.us pro.ductio.n rate is still playing its ro.le, the WOR deviatio.n is mo.st severe, WOR has hysteresis. But, given sufficient time after rate changes, the influence fro.m previo.us pro.ductio.n histo.ry is diminishing, and co.rrelatio.n WOR is appro.aching simulatio.n WOR, which implies that WOR hysteresis disappears.

WOR hysteresis can also. be seen fro.m Fig. 5. After pro.duc-tio.n rate is decreased to. 1000 RBID in schedule D, WOR do.es no.t fo.llo.w schedule A curve, indicating that pro.ductio.n histo.ry befo.re rate change do.es have so.me influence o.n the WOR after rate change, i.e., WOR has hysteresis. Ho.wever, WOR difference between two. schedules is really small, hysteresis is no.t severe here. The same trend can also. be o.bserved by co.mparing schedule C and schedule E. Since rate o.nly changes o.nce in schedule D and E, hysteresis is no.t very important, co.nsequently, co.rrelatio.n can give a go.o.d appro.xima-tio.n fo.r such cases.

CONCLUSIONS

This paper presents a water co.ning co.rrelatio.n to. predict critical rate, breakthro.ugh time and WOR after breakthro.ugh fo.r bo.th vertical and ho.rizo.ntal wells. The co.rrelatio.n was develo.ped based o.n the basic flo.w equatio.ns and regressio.n analysis using the data fro.m numerical simulatio.ns. The fo.rmat o.f the co.rrelatio.n is similar to. Addingto.n's gas co.ning co.rrelatio.n and it can be used in a similar way, i.e., either as a hand calculatio.n metho.d o.r a co.ning functio.n fo.r a 3-D co.arse grid simulatio.n. Fro.m o.ur experience, the co.rrelatio.n can give meaningful approximatio.n when water-oil mo.bility ratio. is smaller than 5 o.r visco.us fo.rces are no.t do.minating. The accuracy


may become less for values outside this range. With this in mind and recalling other assumptions made, we draw the following conclusions:

1. As water cone moves up, critical rate gradually decreases. Eqs. 14 and Eq. 16 predict this critical rate for vertical and horizontal wells, respectively.

2. For a tank reservoir, the ilwb correlation, Eqs. 4 and 10 can be used to calculate water breakthrough time for vertical and horizontal wells, respectively. The calculation procedure is described by Eq. 19.

3. For constant rate cases, WOR after breakthrough can be predicted from Eq. 20 or Eq. 21 by calculating ilwb and m from Eqs. 4 and 5 or Eqs. 10 and 11.

4. This study found that WOR has hysteresis. That is, previous rates or rate changes do have some effects on the current WOR. But, given sufficient time, these effects disappear.

5. If rate does not change very frequently, that is, there is enough time for hysteresis to disappear, the method can be used to predict WOR for variable rate cases. The prediction is only approximate since it is based on the non-hysteresis assumption. The approximation is more accurate at times long after the rate changes occur.

NOMENCLATURE

A cross sectional area, ft2 Bo oil formation volume factor, stb/rb h initial oil formation thickness, ft hap oil column height above perforations, ft hbp average oil column height below perforation, ft ho current oil zone thickness, ft

~ perforation length, ft ht total formation thickness, ft hw current water zone thickness, ft hWb breakthrough height, ft kh horizontal permeability, md ley vertical permeability, md ko oil effective permeability, md kro' oil relative permeability at Swe

~ , water relative permeability at 1-Sor L horizontal well length, ft LOG LOG of base 10 m slope M water oil mobility ratio Np cumulative oil production, stb p pressure, psi PI parameter groups P2 parameter groups

~ critical coning rate, stb/D qD dimensionless production rate qeD dimen~ionless critical coning rate qt total fluid production rate, RB/D rw wellbore radius, ft rDe dimensionless drainage radius

re

Swe Sor t

~t tD tDBT WC WOR X. xD Jl.o JI.w 'Yo 'Yw q, Il:y o A

drainage radius, ft connate water saturation residual oil saturation time, days breakthrough time, days dimensionless time dimensionless breakthrough time water cut water-oil ratio drainage width; ft dimensionless drainage width oil viscosity, cp water viscosity, cp oil gravity, psi/ft water gravity, psi/ft porosity, fraction water-oil gravity difference, psi/ft fraction of perforated interval fraction of oil column height above perforation

REFERENCES

1. Muskat, M. and Wyckoff, R.D.: "An Approximate Theory of Water Coning in Oil Production," Trans. AlME (1935), 114, 144-161.

2. Buckley, S.E. and Leverett, M.C.: "Mechanisms of Fluid Dis-placement in Sands," Trans. AlME (1942) 146, 107-116.

3. Meyer, H.I., and Garder, A.O.: "Mechanics of Two Immiscible Fluids in Porous Media," Journal of Applied Physics, November 1954, Vol. 25, No. 11, p. 1400.

4. Chaney, P.E., Noble, M.D., Henson, W.L., and Rice, T.D.: "How to Perforate Your Well to Prevent Water and Gas Coning," Oil &: Gas Journal, May 7, 1956, p. 108.

5. Chierici, G.L., Ciucci, G.M., and Pizzi, G.: "A Systematic Study of Gas and Water Coning By Potentionmetric Models," JPT, August 1964, pp.923-29.

6. Sobocinski, D.P., and Cornelius, A.I.: "A Correlation for Predicting Water Coning time," JPT, May 1965, pp.594-600.

7. Bournazel, C. and Jeanson, B.: "Fast Water Coning Evaluation," Paper APE 3628 presented at the SPE 46th Annual Fall Meeting, New Orleans, October 3-6, 1971.

8. Schols, R.S.: "An Empirical Formula for the. Critical Oil Produc-tion Rate," Erdoel Erdgas, Z., January 1972, Vol. 88, No.1, pp. 6-11.

9. Byrne, W.B. and Morse, R.A., "The Effects of Various Reservoir and Well Parameters on Water Coning Performance," paper SPE 4287 presented at the SPE 3rd Numerical Simulation of Reservoir Simulation of Reservoir Performance Symposium, Houston, January 10-12, 1973.

464

SPE 22931 WEIPING YANG AND R.A. WAITENBARGER 7

10. Mungan, N.: "A Theoretical and Experimental Coning Study," Soc. Pet. Eng. J. (Iune 1975) 221-236.

11. Blades, D.N. and Stright, D.H., Ir., "Predicting High Volume Lift Performance in Wells Coning Water," J. Can. Pet. Tech. (October-December 1975) 62-70.

12. Addington, D.V.: "An Approach to Gas-Coning Correlations for a Large Grid Cell Reservoir Simulator," JPT (November 1981) 2267-74.

13. Kuo, M.C.T., and DesBrisay, C.L.: "A Simplified Method for Water Coning Predictions," Paper SPE 12067, SPE 58th Annual Fall Meeting, San Francisco, October 5-8, 1983.

14. Kabir, C.S.: "Predicting Gas Well Performance Coning Water in Bottom-Water-Drive Reservoirs," SPE Paper 12068, presented at the 58th Annual Fall Meeting, San Francisco, October 5-8, 1983.

15. Wheatly, M.I., "An Approximate Theory of Oil Water Coning," SPE Paper 14210, SPE 60th Annual Fall Meeting, Las Vegas, NV, September 22-25, 1985.

16. Chaperon, I.: "Theoretical Study of Coning Toward Horizontal and Vertical Wells in Anisotropic Formations: Sub critical and Critical Rates," SPE Paper 15377, SPE 61st Annual Fall Meeting, New Orleans, LA, October 5-8, 1986.

17. Giger, F.M.: "Analytical 2-D Models of Water Cresting Before Breakthrough for Horizontal Wells," SPE Paper 15378, SPE 61st Annual Fall Meeting, New Orleans, LA, October 5-8, 1986.

18. Piper, L.D., Gonzalez, L.M.: "Calculation of the Critical Oil Production Rate and Optimum Completion Interval," SPE paper 16206, presented at the SPE Production Operations Symposium held in Oklahoma, March, 87-10, 1987.

19. Heyland, L.A., Papatzacos, P., Skjaeveland, S.M.: "Critical Rate for Water Coning: Correlation and Analytical Solution," SPE Reservoir Engineering, November 1989.

20. Papatzacos, P., Herring, T.R., Martinsen, R., Skjaeveland, S.M.: "Cone Breakthrough Time for Horizontal Wells," Paper SPE 19822, SPE 64th Annual Fall Meeting, San Antonio, TX, October 8-11, 1989.

21. Yang, W.: "Water Coning Calculations for Vertical and Horizontal Wells," MS thesis, Texas A&M University, August 1990.

APPENDIX

For a tank reservoir, there is no flow across the outer boundary. The height hbp is uniquely related to the cumulative oil production. The relationship can be derived from a material balance equation. As shown by Fig. 1, three regions have to be included when writing a material balance equation, the aquifer, water invaded region and the oil column between top of the reservoir and current water oil contact. In the aquifer, it is assumed that oil saturation is zero, the region between initial water oil contact and the current water-oil contact is defined as the water invaded region, in which oil saturation equals the residual oil saturation. In the region above the current water-oil contact, it was assumed that oil saturation is still at its initial level 1 - !we.

With these assumptions, the oil material balance equation can be written as:

htso = (hCh) 0.0 + (h-li)(I-swe) + Iisor ...... (A-I)

multiplying both sides by the cross-sectional area A and the porosity, we have:

h~~so = (h -li)A~(1-swc) + Iiso~~ . . . . . . . . . . (A-2)

the left-hand side equals the oil left in the reservoir, it should equal the original oil in place minus the cumulative oil production Np;

substitute this equation into Eq. (A-2), we have:

Solve for ii, we have:

Ii = NpB ................... . (A-5) A~(1 swe sor)

And hbp = h -Ii -hap -hp . . . . . . . . . . . . . . . . . . .. (A-6)

TABLE 1. Relative permeability data

!w ~ ~o 0.1500 O.OOOOE+OO 0.9500 0.2000 4.0000E-03 0.7500 0.2500 1.0200E-02 0.5876 0.3000 1. 6600E-02 0.4462 0.3500 2.3200E-02 0.3325 0.4000 3.0500E-02 0.2450 0.4500 3.9200E-02 0.1770 0.5000 4.9700E-02 0.1200 0.5500 6.3000E-02 7. 2400E-02 0.6000 7.9800E-02 3.7400E-02 0.6500 0.1000 1.6300E-02 0.7000 0.1244 5.6400E-03 0.7500 0.1525 7.7000E-04 0.7750 0.1698 3.8000E-04 0.7880 0.1784 1.9000E-04 0.8000 0.1870 O.OOOOE+OO 1.000 0.1870 O.OOOOE+OO

465


TABLE 2. Simulation inQut data and results - vertical wells

case Ich lev r. h h"p hp iLo iLw fl:y f/J ~ m hWb 1 4000 200 1300 160 3.75 16.25 1.5 0.31 0.0996 0.207 2500 -0.0366 65.12 2 2000 -0.0271 92.84 3 3000 -0.0323 75.26 4 4000 -0.0366 65.12 5 6000 -0.0432 53.34

6 50 -0.0445 58.90 7 100 -0.0394 61.50 8 200 -0.0366 65.12 9 400 -0.0329 70.88 10 800 -0.0298 77.40

11 1000 -0.0379 63.24 12 1300 -0.0366 65.12 13 1600 -0.0351 68.34 14 1800 -0.0340 70.17

15 100 -0.0381 60.34 16 160 -0.0366 65.12 17 200 -0.0364 68.10 18 260 -0.0361 71.71

19 3.75 -0.0366 65.12 20 13.75 -0.0339 62.16 21 23.75 -0.0324 60.02 22 43.75 -0.0319 55.46

23 8.75 -0.0375 71.20 24 16.25 -0.0366 65.12 25 26.25 -0.0342 60.00 26 36.25 -0.0329 54.61

27 0.5 -0.0460 36.05 28 1.5 -0.0366 65.12 29 3.0 -0.0294 92.61 30 4.0 -0.0271 105.13

31 0.20 -0.0364 69.13 32 J.31 -0.0366 65.12 33 0.40 -0.0364 63.68 34 0.50 -0.0366 61.68 35 0.70 -0.0366 58.68

36 0.0779 -0.0338 71.76 37 0.0893 -0.0354 68.47 38 0.1102 -0.0377 62.63 49 0.1198 -0.0386 60.42

40 0.1 -0.0362 66.38 41 0.207 -0.0366 65.12 42 0.30 -0.0366 65.53 43 0.40 -0.0367 65.37

44 1000 -0.0481 43.40 45 1500 -0.0429 52.34 46 3500 -0.0329 74.58 47 4500 -0.0304 81.67

Note: a blank entry in the table indicates that the parameter has the same value as base case or case 1.

466


TABLE 3. Simulation in~ut data and results - horizontal wens

case kh ~ xa h hap L /Lo /Lw A-y tP ~ m hWb 1 4000 200 1151.5 160 20 2303 1.5 0.31 0.0996 0.207 2500 -0.0392 36.04 2 1000 -0.0229 66.23 3 2000 -0.0303 48.02 4 3000 -0.0353 40.40 5 6000 -0.0452 31.10

6 50 -0.0378 42.98 7 100 -0.0366 39.75 8 200 -0.0392 36.04 9 400 -0.0406 33.45 10 800 -0.0419 30.73

11 600 -0.0441 33.60 12 800 -0.0424 33.74 13 1300 -0.0382 37.23 14 1500 -0.0364 39.07

15 100 -0.0406 31.79 16 200 -0.0390 38.42 17 260 -0.0387 41.82 18 300 -0.0383 44.13

19 1 -0.0464 46.19 20 10 -0.0414 40.29 21 40 -0.0386 30.73 22 60 -0.0377 28.01

23 1200 -0.0290 49.87 24 1600 -0.0331 43.20 25 2600 -0.0419 33.58 26 3000 -0.0447 31.22

27 0.5 -0.0489 17.46 28 3.0 -0.0300 54.31 29 4.0 -0.0272 63.61 30 5.0 -0.0253 71.29

31 0.20 -0.0417 37.30 32 0.31 -0.0392 36.04 33 0.40 -0.0381 35.04 34 0.50 -0.0376 33.98 35 0.70 -0.0364 32.51

36 0.0779 -0.0348 40.86 37 0.0893 -0.0370 38.27 38 0.1102 -0.0411 34.21 39 0.1198 -0.0428 32.73

40 0.1 -0.0392 36.15 41 0.30 -0.0391 35.83 42 0.40 -0.0407 35.48 43 0.45 -0.0409 35.37

44 1000 -0.0641 20.83 45 1500 -0.0503 27.34 46 3500 -0.0335 42.70 47 4500 -0.0298 48.37

Note: a blank entry in the table indicates that the parameter has the same value as base case or case 1.

467


WaR + c 10

hap

1 - swc hp ,

I 1 hbp h

I ht 0.1

h sor

1 initial wac

V V V V 'if Fig. I-A sketch of well configurations for calculating 0.01 L-_----L __ .L..-_--L __ .l-.-_--L. __ ..l...-_--'-_----.J

average oil column height below perforations 0 20 40 60 80 100 120 140 160 hbp (ft)

Fig. 2-WOR + c vs. hbp plot from a simulation run

~ ~~~ . /'/

/

...

HH~!Gl! VV InH al " I\..

I~ ~ ~ I ~ I ~

Fig. 3-Simulation grid for a vertical well Fig. 4-Simulation grid for a horizontal well

468

SPE 22931 WEIPING YANG AND R.A. WATTENBARGER 11

W .. ~O~R_+~0~.0~2~ ____________________________ ~ 10 ~

Schedule C

-+- Schedule B -*"" Schedule A

o Schedule D o Schedule E

P,oduction Schedule

A 1000 B 2500 C 4500 D 2500 e 2500

1000 2500 4500 1000 4500

0.01 L_...L.-_....l.-_---L_--'-_---ll-_-'--_...L----' o 20 40 60 80 100 120 140 160

hbp (ft)

Fig. 5-Critical rate analysis WOR at different production schedules

Breakthrough time (days) 10000~~~-~----~~~----------------

1000

kt. .. 1000 md 100 Ie., - 50md

'. .. 1151.5 ft

h .. 160 ft hop -Oft II. - 1.5 cp

Correlation IIw .. 0.31 .y .. 0.0996 psi/ft /j,. Papalzcous ~ .. 0.207 L .. 1500 ft 0 Simulation

4000 10L-~--~----~--~----~--~----~~

5000 6000 7000 8000 9000 10000

Production rate (BBLs/D)

Fig. 7-Horizontal well breakthrough time comparison between correlation, simulation and Papatzcous' method

Breakthrough Time (days) 3000r-------~----~~~----------------_.

-A- Correlation ~ Sobocinski

2500 -e- Simulation

2000 kh .. 4000 md Ie., = 200 md '.

= 1300 ft h = 100 ft

1500 hop =Oft hp = 20ft II. = 1.5 cp IIw = 0.31 .y = 0.0996 psi/ft

~ .. 0.207 1000

500

469

oL-_---L __ L-_-L __ L-__ -L __ ~ __ ~

3 4 5 6 7 B 9 10

Production Rate (x1000 BBLs/D)

Fig. 6-Vertical well breakthrough time comparison between correlation, simulation and Sobocinski's method

WOR + 0.02 100~=-~~--------------------------~ Correlation

o Simulation

10

1

0,1

kh = 2000md Ie., = 100 md '. .. 1300 ft h .. 160ft hop = 0 ft

~ = 20 ft II. = 1.5 cp IIw = 0.31 .y = 0.0996 psi/ft

~ = 0.207 'It .. 6000 RBID

0.0 1 L ______ ---L ________ ..L--_____ --'-______ ~

o 0.2 0.4 0.6 Recovery (% original oil in place)

Fig. 8-WOR comparison between correlation and simulation for a vertical well

0.8


WOR + 0.25 1YOR + 0.02 10r-------------------------------------i 10

0 Simulation 0 Simulation Correlation Correlation

kh = 1000 md kh = 4000 md k., '" 200 md k., = 200md r. = 1151.5 It r. = 1300 It h = 160 It 0 h '" 160 It hOI' '" 20 It hop '" 3.75 It Po '" 1.5 cp h. = 16.25 It Pw '" 0.31 p. '" 1.5 cp .. y = 0.0996 psi/lt Pw = 0.31

~ '" 0.207 "Y = 0.0996 psi/lt

l = 2303 It ~ = 0.207 q. = 2500 RB/D

{2S00 h",,>42.0 0.1 qt. 1000 h"p>21.6

4500 h",,

Documents

Water Coning for v h Well