pressure-transient responses of horizontal and...

Pressure-Transient Responses of Horizontaland Curved Wells in Anticlines

and DomesN. Al-Mohannadi, SPE, E. Ozkan, SPE, and H. Kazemi, SPE, Colorado School of Mines

Summary

This paper presents a discussion of the pressure-transient re-sponses of horizontal wells in anticlinal structures and curved andundulating wells in slab reservoirs. It confirms that, in the absenceof a gas cap, conventional horizontal-well models may be used toapproximate the flow characteristics of the systems in which thetrajectory of the well does not conform to the curvature of theproducing structure. If a gas cap is present, however, the uncon-formity of the well trajectory and producing layer manifests itself,especially on derivative characteristics when the gas saturationincreases around the well. In general, the most significant devia-tions from the conventional horizontal-well behavior are observedduring the buildup periods following long drawdowns. In thesecases, the pressure-transient analysis is complicated and requiresdetailed numerical modeling of the well trajectory and reservoirgeometry in the vertical plane.

Introduction

Conventional horizontal-well pressure-transient models assumethat the top and bottom boundaries of the reservoir are horizontalplanes; that is, the producing stratum is a slab, and the well isstraight and parallel to the slab boundaries. Wells, however, maybe drilled horizontally in anticlines and domes, or they may becurved or undulating in a horizontal slab reservoir.

In the literature, several reservoir shapes have been consideredin the context of horizontal wells: infinite slab (Clonts and Ramey1986; Ozkan et al. 1989; Goode and Thambynayagam 1987; Rosaand Carvalho 1989; Kuchuk et al. 1990, 1991; Ozkan and Ragha-van 1990a), cylinder (Ozkan and Raghavan 1991a, 1991b), rect-angular parallelepiped (Ozkan and Raghavan 1991a, 1991b; Da-viau et al. 1988; Odeh and Babu 1990), and vertical no-flowboundary at an arbitrary orientation (Azar-Nejad et al. 1996a). Thecommon feature of these reservoir models is the assumption thatthe top and bottom boundaries are horizontal planes. Despite thefact that the conditions at the top and bottom boundaries stronglyinfluence the pressure-transient characteristics of horizontal wells(Clonts and Ramey 1986; Ozkan et al. 1989; Goode and Tham-bynayagam 1987; Ozkan 2001), the effect of the curvature of theseboundaries, as in the case of anticlines and domes, has not beendiscussed in the literature.

Similar to the curvature of the top and bottom boundaries, thecurvature or undulations of horizontally oriented wells (referred toas horizontal wells in this paper) have not attracted much attentionin the pressure-transient-analysis literature. Two studies have ad-dressed this issue specifically. Azar-Nejad et al. (1996b) consid-ered a curved well that was a quarter of a circle (from vertical tohorizontal) in a slab reservoir. They showed that especially inanisotropic reservoirs, the pressure-transient response of thecurved well could not be approximated by that of a straight hori-zontal well of equal drilled length. This study did not address theissue of effective well length and the effect of the aspect ratio (the

ratio of the distance from the well to the closest boundary andthickness of the formation).

Goktas and Ertekin (2003) discussed another common problemfor horizontal wells—undulations. Their study indicated that whenthe vertical window of undulations becomes comparable to theformation thickness, undulations might influence the characteris-tics of pressure-transient responses. For practical windows of un-dulations that commonly result from standard drilling practices,however, the pressure-transient responses could be closely ap-proximated by that of a straight horizontal well. This conclusionwas different from that of Azar-Nejad et al. (1996b). It also mustbe noted that Goktas and Ertekin (2003) used the straight distancebetween the tips of the undulating well in the comparisons withstraight horizontal wells, as opposed to the total drilled length usedby Azar-Nejad et al. (1996b).

Another issue, which has not been sufficiently discussed in theliterature, is the analysis of horizontal-well pressure-transient re-sponses in the presence of a gas cap. Two of the relevant studiesin this area are by Fleming et al. (1994, 1996). On the basis of theobservations from buildup tests in a fractured, vuggy, carbonatereservoir that contained a large gas cap, they noted that the exis-tence of a gas cap would cause oscillations on the derivatives ofthe pressure-buildup responses of horizontal wells. They also sup-ported their arguments through numerical studies.

This paper presents a discussion of the pressure-transient re-sponses of horizontal wells in anticlines and curved or undulatingwells in slab reservoirs. We document the conditions under whichthe conventional horizontal-well models can be used in the analy-sis of these responses. The pressure-transient responses of straightand curved wells in slab reservoirs (Figs. 1a through 1f) areinvestigated first. Our intention here is to answer the question ofwhether the curvature (or undulations) of the well should be ofconcern in the analysis of horizontal-well, pressure-transient re-sponses. For completeness, we consider the influence of frictionalpressure drop in the wellbore (finite wellbore conductivity). Wedocument the results for the cases with and without gas cap sepa-rately. For all cases considered in this study, we consider the effectof the free-gas movement emanating from the gas cap, knowingthat the pressure-transient test time is much smaller than the timeconstant for the free gas to come out of or go into the solution.Hereafter, we refer to a system without gas cap as single-phase-flow system.

In the second section of our results, we address the issuesregarding horizontal wells in anticlines (Figs. 2a and 2b). Here,we consider only straight horizontal wells. Again, we discuss thesingle-phase-flow systems first. Then, we delineate the effect of agas cap. Finally, we discuss the pressure-buildup responses.

Our discussions below are preceded by a brief introduction ofthe numerical simulator developed for this study. The details of thesimulator can be found in the Appendices and in Al-Mohannadi (2003).

Numerical Simulation ofHorizontal-Well ResponsesIn this paper, we have used two finite-difference formulations oftransient flow for a horizontal well (see Appendix A). The firstformulation was for single-phase flow, and the second formulationwas for two-phase flow of oil and gas (the relative permeabilitiesused in this work are presented in Appendix B). In the latter, wehave neglected gas solubility in oil for simplicity. (Solution gas

Copyright © 2007 Society of Petroleum Engineers

This paper (SPE 84378) was first presented at the 2003 SPE Annual Technical Conferenceand Exhibition, Denver, 5–8 October, and revised for publication. Original manuscriptreceived for review 10 May 2004. Revised manuscript received 26 June 2006. Paper peerapproved 5 November 2006.

66 February 2007 SPE Reservoir Evaluation & Engineering

can be included in the model, but when significant volumes ofsolution gas become free around the wellbore, free gas, rather thangas cap, dominates the pressure-transient behavior. In this study,we restrict our attention to the influence of the gas cap becausepressure-transient test times are normally much smaller than thetime constant for considerable free gas to be released from or gointo the solution.)

The two-phase formulation uses the classic implicit-pressure,explicit-saturation (IMPES) technique, which degenerates to thesingle-phase formulation when only one mobile phase exists. Thetimestepping is based on the logarithm of time to mimic earlytransient-flow behavior. Similarly, the spatial distribution of grid-blocks surrounding the wellbore has a logarithmic distribution tomimic radial flow. While these numerical formulations are wellknown, their proper use (by means of timestepping and grid design)is crucial in simulating early-time pressure-transient responseswith accuracy that is comparable to that of analytical solutions.

Three features of our numerical model are to be noted:• For calculating wellbore pressure, pwell, we treat the wellbore

as an additional node within the wellblock. In analogy to thetraditional radial-flow simulation, we calculate a wellblock radius,ro, from Eq. A-5 in Appendix A, at which the radial pressure isequal to the average pressure, po, in the wellblock. (We computero on the basis of the material balance, as is done in radial flow.The derivation of our wellblock radius is given in Appendix C.)This wellblock radius, ro, yields better results than the one sug-gested by Peaceman (1983) in simulating pressure transients. Wehave found much improvement by using the block radius in the

conventional well index and modifying the four transmissibilitiesof the adjacent nodes by replacing the distance from node to nodewith node-to-ro location (Al-Kobaisi et al. 2006).

• Both steady- and unsteady-state flow can be accommodatedin the wellbore, although our current results based on Eqs. A-24through A-36 assume steady-state flow.

• The wellbore model formulated by Eqs. A-24 through A-36accounts for the wellbore hydraulics (frictional pressure drop inthe horizontal section) and the wellbore-storage effects. The well-block and wellbore are linked by the steady-state well index givenby Eq. A-3, which uses the wellblock radius discussed above.

ResultsWe will present our results in three sections. The first section ofthe results concerns straight and curved wells in slab reservoirs(Figs. 1a through 1f). We also will use these results to verify ournumerical simulator with the analytical models presented in Ozkanet al. (1989), Ozkan and Raghavan (1990a, 1991a), and Ozkanet al. (1995, 1999). In the second section, we will consider straightwells in anticlines (Figs. 2a and 2b). The third section will bedevoted to the discussion of pressure-buildup responses.

The data used to simulate various cases discussed here aresummarized in Tables 1 through 4. As shown in Table 3, tohighlight the effect of frictional pressure drop (finite-conductivityeffect), we selected a set of reservoir properties and flow rates thatyield small reservoir drawdown and relatively high wellbore pres-sure drop. The grid systems used in the simulations are shown inFigs. 3a through 3c.

Fig. 1—Schematics of the wells in slab reservoirs simulated in this study.

67February 2007 SPE Reservoir Evaluation & Engineering

It must be noted that our interest in this paper is to investigatethe more common cases rather than the extremes. To discuss theeffect of well curvature in slab reservoirs, we choose a 10-ft ver-tical window of curvature in an 81-ft-thick reservoir (Figs. 1b, 1c,1e, and 1f). In addition, unless specified otherwise, all the undu-lating and curved wells were modeled with amplitude/wavelengthratio of less than or equal to 0.01. Therefore, our results should bemostly applicable to the cases in which the curvature or undula-tions of the well are a result of drilling practices, not that ofgeosteering. Also, in comparing the effects of a gas cap in slab andanticlinal reservoirs, we adjust the height of the gas cap to maintainthe gas material balance.

Slab Reservoirs. We first consider straight and curved horizontalwells in slab reservoirs.

Straight Horizontal Wells—Model Verification. To start ourdiscussions, we first present the conventional horizontal-wellmodel: single-phase flow to a straight horizontal well in a slabreservoir (Fig. 1a). Figs. 4 and 5 show the comparisons of thepressure and derivative responses from numerical and analyticalmodels for the infinite-conductivity (Table 2) and finite-conductivity (Table 3) cases, respectively. [The analytical models

for the infinite- and finite-conductivity horizontal-well cases werepresented in Ozkan and Raghavan (1991b) and Ozkan et al. (1995,1999), respectively.] The agreement between the analytical andnumerical results shown in Figs. 4 and 5 verifies our numerical-model, grid-system, and timestep selection. (The late-time differ-ence between the analytical and numerical results in Fig. 5 resultsfrom the fact that the analytical model used in the comparison isfor an infinite slab reservoir.)

Effect of Well Curvature—Single-Phase Flow. Figs. 6through 9 are intended for the discussion of the effect of wellcurvature in slab reservoirs. In Figs. 6 and 7, the well is concavedown (Fig. 1b) without and with finite-conductivity effects, re-spectively. Figs. 8 and 9 are the counterparts of Figs. 6 and 7 fora well that is concave up (Fig. 1c). As in Goktas and Ertekin(2003), we assume that the effective length for a curved well maybe closely approximated by the straight distance between the tipsof the well. The good match between the results for the straight andcurved wells shown in Figs. 6 through 9 indicates that the effect ofundulations is negligible and the straight distance between the tipsof the well can be used as the effective horizontal-well length in aslab reservoir.

Because the amplitude/wavelength ratio for the examples inFigs. 6 through 9 is small (and, thus, the difference between thedrilled length and straight distance between the tips is insignifi-cant), we have tested other cases to verify the conclusion that theeffect of undulations is negligible and the straight distance be-

Fig. 2—Schematics of the wells in anticlines and slab reservoirs used in comparisons.


tween the tips can be used as the effective horizontal-well length.As an example, Fig. 10 shows a case of 2½-cycle sinusoidal un-dulations along a drilled length of 1,451.8 ft in a slab reservoir of81-ft thickness. The undulations have the maximum amplitude of10 ft and a wavelength of 350 ft (the maximum amplitude/wavelength ratio is 0.03). The simulation grid used for this case isshown in Fig. 11. In this case, the straight horizontal distancebetween the tips of the well is 1,000 ft, which is only 69% of thedrilled length of 1,451.8 ft. As shown in Fig. 12, however, thepressure-transient responses of the undulating well match those ofa 1,000-ft straight horizontal well closely. Other results not pre-sented here also verify the conclusion that the undulations do notsignificantly influence the pressure-transient responses and the ef-fective horizontal-well length is the straight distance between thetips of the well.

The above conclusion agrees with the observations of Goktasand Ertekin (2003), but is different from the conclusion of Azar-Nejad et al. (1996b) (note that Azar-Nejad et al. used the totaldrilled length as the effective length in their work). For clarity, itshould be emphasized that the conclusion noted above is valid forrelatively small amplitudes of undulations that are more commonin practice. As noted by Goktas and Ertekin (2003), however, theeffect of undulations may become noticeable when the amplitude

of the undulations becomes comparable to the formation thickness.(Even in this case, the noticeable difference is only at early andintermediate times.)

On the basis of the results above, we conclude that, providedthat the straight distance between the tips of the well is used as theeffective well length, the conventional horizontal-well models(straight well in a slab reservoir) may be used to analyze thepressure-transient responses of curved (and undulating) wells. Thisconclusion is also valid for the cases of considerable wellborepressure drop as long as a straight-horizontal-well model that takesinto account the wellbore friction effects is used. However, itexcludes wells that come too close to the top and bottom bound-aries because of high curvature or wide undulations. In addition,care must be taken in extending the above conclusion to the casesof nonuniform skin distribution and selectively open segmentsbecause the distribution of high- and low-influx regions alongthe well may affect the pressure distributions in and around thewell severely.

Effect of the Gas Cap. The discussion above has been limitedto single-phase flow. Fig. 13 shows the effect of a gas cap on thepressure-transient responses of straight horizontal wells in slabreservoirs (Fig. 1d). The figure compares the drawdown responsesof straight horizontal wells with and without a gas cap (Figs. 1aand 1d). As also observed in Fleming et al. (1994, 1996), gas

Fig. 3—Grid systems used to simulate the four different well/reservoir configurations modeled in this study.


encroachment toward the well affects the pressure and derivativeresponses significantly.

For the particular case shown in Fig. 13, the effect of the gascone starts while the well is in the early-time radial-flow periodand destroys the conventional flow characteristics of the interme-diate- and late-time flow (Ozkan 2001). The start of the deviationfrom the conventional horizontal-well behavior (single-phase flowtoward a straight horizontal well in a slab reservoir) is dictated bythe physical parameters governing gas coning toward a horizontalwell (Chaperon 1986; Giger 1989; Ozkan and Raghavan 1990b;Umnuayponwiwat and Ozkan 2000).

It is clear from the results shown in Fig. 13 that the failure torecognize the effects of gas coning would perplex the analystsduring the interpretation of horizontal-well tests. In addition, theloss of the characteristics of the intermediate- and late-time flowregimes makes it impossible to estimate the directional permeabili-ties by use of the conventional models (Ozkan 2001). Therefore, tobe able to analyze horizontal-well tests under the influence of gascap, detailed numerical models, such as the one developed in thisstudy, are crucial.

The next issue to be addressed is the effect of well curvature inthe presence of a gas cap. Figs. 14 and 15 compare the responsesof concave-down and concave-up wells (Figs. 1e and 1f), respec-tively, with that of a straight horizontal well (Fig. 1d) in the pres-ence of a gas cap. As mentioned before, we used the straightdistance between the tips of the curved wells as the effectivehorizontal-well length in the simulation of the straight-well case.The results in Figs. 14 and 15 indicate that the curvature of the welldoes not influence the pressure and derivative responses. As dis-cussed in Fig. 13, however, the effect of the gas cap should betaken into account in the interpretation of the pressure and deriva-tive responses.

Anticlines. In this section, we concentrate on the effect of thecurvature of the reservoir. On the basis of our findings in theprevious section, we consider only infinite-conductivity, straighthorizontal wells in an anticline under single-phase flow and gas-cap conditions.

Single-Phase Flow. We first examine the effect of the curva-ture of the reservoir under single-phase-flow conditions. Fig. 16compares the pressure-transient responses of (straight) horizontalwells in anticlines (Fig. 2a) and slab reservoirs (Fig. 2c). There isno discernable influence of the curvature of the pay zone on thepressure-transient responses. Therefore, the conventional horizon-tal-well models (which consider a slab reservoir) can be used toanalyze the pressure-transient responses of horizontal wells in an-ticlines and domes under single-phase-flow conditions. Below,however, this conclusion is shown to be invalid if a gas cap existsin the system.

Gas Cap. On the basis of the results for slab reservoirs notedearlier, we expect to have some influence of the gas cap on thepressure-transient responses of horizontal wells in anticlines.Fig. 17 shows the pressure and derivative responses of (straight)horizontal wells in anticlines with and without a gas cap (Figs. 2band 2a, respectively). As expected, there is significant difference inthe pressure and derivative characteristics when the gas cone ad-vances toward the well.

An interesting observation is also made from Fig. 18, where thepressure and derivative responses of horizontal wells in anticlines(Fig. 2b) and slab reservoirs (Fig. 2d) are compared in the presenceof a gas cap. After the gas breakthrough, some difference is ob-served, especially between the derivative responses of horizontalwells in slab reservoirs and anticlines. This indicates that in thepresence of a gas cap, the approximation of an anticline by anequivalent slab reservoir may not be valid for the analysis ofhorizontal-well tests.

Fig. 5—Verification of the numerical results for single-phase-flow, finite-conductivity horizontal-well case.

Fig. 6—Comparison of the pressure-transient responses forstraight and curved (concave-down) wells: slab reservoir,single-phase flow, and infinite-conductivity well.

Fig. 7—Comparison of the pressure-transient responses forstraight and curved (concave down) wells: slab reservoir,single-phase flow, and finite-conductivity well.

Fig. 4—Verification of the numerical results for single-phase-flow, infinite-conductivity horizontal-well case.


Pressure Buildup. So far, we have discussed only the drawdownresponses. For the cases in which the well or reservoir curvaturehas no effect on the pressure-transient characteristics of horizontalwells, the buildup responses should also follow the conventionalmodels. Therefore, we consider the buildup responses for twocases only: horizontal wells in slab reservoirs and anticlines in thepresence of a gas cap (Figs. 2d and 2b, respectively).

Fig. 19 shows the buildup responses of horizontal wells in slabreservoirs and anticlines following a long producing period of16,400 hours at a constant rate of 2,000 sft3 (the data for this caseare given in Tables 1 and 4). The producing time is long enoughfor the gas to break through by the time the well is shut in. For bothcases shown in Fig. 19, derivative responses especially displayoscillations after the initial radial-flow period.

The oscillations in the derivative responses shown in Fig. 19have also been noted in Fleming et al. (1994, 1996) and attributedto two-phase-flow effects around the well owing to gas encroach-ment from a large gas cap. The tests reported in Fleming et al.(1994, 1996) displayed more cyclic oscillations in the derivativeresponses than those observed in Fig. 19. The difference in theintensity of the oscillations should be attributed to the difference inthe reservoir heterogeneity. In this study, we modeled the reservoiras a homogeneous system, whereas the well tests discussed inFleming et al. (1994, 1996) were from a naturally fractured reser-voir. In general, as the degree of heterogeneity increases, the os-cillations in the derivative responses should be expected to increase.

It is important to note that when the oscillations begin, thederivatives of the pressure-buildup responses for the slab reservoirand anticline do not follow each other. This behavior is a result ofthe differences seen during the drawdown period, as displayed inFig. 18. Therefore, similar to our comment for the correspondingdrawdown case, we emphasize the use of a numerical model that

takes into account the correct geometry of the reservoir and theexistence of the gas cap.

ConclusionsIn this paper, we have presented new results pertaining to thepressure-transient responses of horizontal wells in anticlines andcurved wells in slab reservoirs, with and without a gas cap. Theseresults should help clarify questions on the choice of the appro-priate models or the adequacy of the assumptions used in thepressure-transient analysis of such systems. The following specificconclusions have been drawn on the basis of the results of this study:1. Unless the curvature or undulations of the well are extremely

large, their effect is negligible. In these cases, the straight horizon-tal distance between the tips of the well may be used as effectivewell length in conventional pressure-transient-analysis methods.

2. In cases in which high frictional pressure losses are expected,the use of a finite-conductivity horizontal-well model should besufficient to analyze the pressure-transient responses of curvedand undulating wells accurately.

3. The existence of a gas cap causes significant deviation from theconventional, single-phase-flow horizontal-well models. Inthese cases, numerical models should be used to interpret thepressure-transient responses.

4. Conventional horizontal-well models for slab reservoirs can beused to analyze horizontal-well responses in anticlines anddomes provided that flow is single-phase.

5. In the presence of a gas cap, an anticline cannot be approxi-mated by a slab reservoir for the analysis of pressure-transientresponses of horizontal wells. For these cases, both the existence

Fig. 8—Comparison of the pressure-transient responses forstraight and curved (concave up) wells: slab reservoir, single-phase flow, and infinite-conductivity well.

Fig. 9—Comparison of the pressure-transient responses forstraight and curved (concave up) wells: slab reservoir, single-phase flow, and finite-conductivity well.

Fig. 10—2½-cycle sinusoidal, undulated well in an 81-ft-thickslab reservoir.

Fig. 11—Grid structure for 2½-cycle sinusoidal, undulated wellin an 81-ft-thick slab reservoir.


of the gas cap and the curvature of the reservoir should beincorporated rigorously into a numerical model.

6. Buildup derivative responses of horizontal wells in the presenceof a gas cap display oscillations after the early-time radial flowif the producing time is sufficiently long for gas breakthrough.In these cases, anticlines should not be approximated by slabreservoirs, and detailed numerical models should be used forpressure-buildup analysis of horizontal wells.

NomenclatureB � formation volume factor, rft3/sft3

b � reciprocal formation volume factor, sft3/rft3

c � compressibility, psi−1

Cw � wellbore-storage coefficient, STB/psiD � depth from datum, ft

dpipe � pipe diameter, ftf � friction factorh � formation thickness, ft

HF � frictional head loss, ftk � permeability, md

kr � relative permeability, fractionL � horizontal-well length, ft

ng � Corey exponent for gasno � Corey exponent for oil

NRe � Reynolds numberp � pressure, psia

pc � capillary pressure, psiapo � wellblock pressure, psia

q � flow rate, sft3/Dro � wellblock radius, ftrw � wellbore radius, ft

s � skin factorS � saturationt � time, hours

T � temperature, °F� � transmissibility, md-ft/cp

VR � rock volume for a grid, ft3

Vwell � velocity in the well, ft/secWI � well index, sft3/D/psixe � formation size in the x-direction, ftye � formation size in the y-direction, ft� � porosity, fraction� � density gradient, psi/ft� � viscosity, cp

�D � relative roughness of the well surface� � density, lbm/ft3

�t � timestep, hours�x � grid size in the x-direction, ft�y � grid size in the y-direction, ft�z � grid size in the z-direction, ft

Subscripts

b � baseg � gasi � x-directionj � y-directionk � z-directionn � old time

Fig. 12—Pressure-transient responses for straight and 2½-cycle-sinusoidal, undulated wells in an 81-ft-thick slab reser-voir; single-phase flow, infinite conductivity.

Fig. 13—Effect of gas cap on pressure-transient responses:slab reservoir and straight well.

Fig. 14—Effect of the curvature of the well (concave down) onpressure-transient responses with an overlying gas cap: slabreservoir.

Fig. 15—Effect of the curvature of the well (concave up) onpressure-transient responses with an overlying gas cap: slabreservoir.


n+1 � current timeo � oilr � residualt � time

T � totalwell � wellbore

wf � wellbore flowingx � x-directiony � y-directionz � z –direction

� � formation

AcknowledgmentsParts of this work have been completed to fulfill the PhD degreerequirements of Nasser Al-Mohannadi at the Colorado School ofMines. The State of Qatar has provided the financial support forthe studies of Nasser Al-Mohannadi.

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Fig. 16—Comparison of pressure-transient responses ofstraight wells in slab reservoirs and anticlines, single-phaseflow.

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Fig. 19—Effect of reservoir curvature and gas cap on pressure-buildup responses of straight wells.


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Ozkan, E., Sarica, C., and Haci, M. 1999. Influence of Pressure DropAlong the Wellbore on Horizontal-Well Productivity. SPEJ 4 (3): 288–301. SPE-57687-PA. DOI: 10.2118/57687-PA.

Ozkan, E., Sarica, C., Haciislamoglu, M., and Raghavan, R. 1995. Effect ofConductivity on Horizontal Well Pressure Behaviour. SPE AdvancedTechnology Series 3 (1): 85–94. SPE-24683-PA. DOI: 10.2118/24683-PA.

Peaceman, D.W. 1983. Interpretation of Well-Block Pressures in Numeri-cal Reservoir Simulation With Nonsquare Grid Blocks and AnisotropicPermeability. SPEJ 23 (3): 531–543. SPE-10528-PA. DOI: 10.2118/10528-PA.

Rosa, A.J. and Carvalho, R.S. 1989. A Mathematical Model for PressureEvaluation in an Infinite-Conductivity Horizontal Well. SPEFE 4 (4):559–566. SPE-15967-PA. DOI: 10.2118/15967-PA.

Umnuayponwiwat, S. and Ozkan, E. 2000. Water and Gas Coning TowardFinite-Conductivity Horizontal Wells; Cone Build-up and Break-through. Paper SPE 60308 presented at the SPE Rocky Mountain Re-gional/Low-Permeability Reservoirs Symposium and Exhibition, Den-ver, 12–15 March. DOI: 10.2118/60308-MS.

Appendix A3D Finite-Difference Formulation of Oil and Gas Flow.

Oil Flow.

�xTox��x po − �o�xD� + �yToy��y po − �o�y D�

+ �zToz��zpo − �o�zD�

+ qo =VR

�t�t��boSo�, . . . . . . . . . . . . . (A-1)

where

qo = qoi, j,k

n+1 = −WIoi, j,k

n �poi, j,k

n+1 − pwelli, j,k

n+1 � , . . . . . . . . . . . . . . . . . . (A-2)

WI oi, j,k

n = 2��0.006328�� kkro

�oBo�

i, j,k

n �xi, j,k

�lnro

rw+ s� , . . . . . . . (A-3)

k = �ky kz, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4)

ro = ��z�y�3.142 e−0.5, . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-5)

(see Appendix C for the derivation of Eq. A-5)

VR

�t�t�� boSo� = C11�tSo + C12�t p, . . . . . . . . . . . . . . . . . . . . (A-6)

VRi, j,k= �x

i, j,k�y

i, j,k�z

i, j,k, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-7)

C11 =VRi, j, k

�t�24�boi, j,k

n+1 �i, j,kn+1�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-8)

and C12 =VRi, j, k

�t�24× ��So�i, j,k

n �cobo�i, j,k

n+1

2 + bon+1So

n��c��i, j,k

n+1

2� . . (A-9)

Gas Flow.

�xTgx��xpg − �g�xD� + �yTgy

��ypg − �g�yD�

+ �zTgz��zpg − �g�zD�

+ qg =VR

�t�t�� bgSg�, . . . . . . . . . . . . (A-10)

where

qgi, j,k

n+1 = −WIgi, j,k

n �pgi, j,k

n+1 − pwelli, j,k

n+1 � , . . . . . . . . . . . . . . . . . . . . . (A-11)

WIgi, j,k

n = 2��0.006328�� kkrg

�gBg�

i, j,k

n �xi, j,k

�lnro

rw+ s� , . . . . . . (A-12)

VR

�t�t��bgSg� = C21�tSo + C22�t p, . . . . . . . . . . . . . . . . . . . (A-13)

C21 = −VRi, j, k

�t�24�bgi, j,k

n+1 �i, j,kn+1�, . . . . . . . . . . . . . . . . . . . . . . . . . . (A-14)

and C22 =VRi, j, k

�t�24 ��Sg�i, j,kn �cgbg�i, j,k

n+1

2 + bgn+1Sg

n��c��i, j,k

n+1

2� . . . (A-15)

Eqs. A-1 and A-10 were combined to eliminate the saturationsin the classic IMPES formulation. The outcome, the pressure equa-tion, is presented in the following form and solved by a Gaussroutine while accounting for the boundary conditions.

Ai, j,kpi, j+1,kn+1 + Bi, j,kpi, j−1,k

n+1 + Di, j,kpi−1, j,kn+1 + Ei, j,kpi, j,k

n+1 + Fi, j,kpi+1, j,kn+1

+ Hi, j,kpi, j+1,kn+1 + Li, j,kpi, j,k+1

n+1 = Ri, j,k, . . . . . . . . . . . (A-16)

where, for example,

Fi, j,k = i, j,kn+1T

oxi +1

2, j,k

n + Tgxi +

1

2, j,k

n, . . . . . . . . . . . . . . . . . . . . . . (A-17)

and

Toxi�

1

2, j,k

n = 0.006328� kxkro

�oBo�

i�1

2, j,k

n �yi, j,k�zi, j,k

�xi�1

2, j,k

, . . . . . . . . (A-18)

Ei, j,k = −�Ai, j,k + Bi, j,k + Di, j,k

+ Fi, j,k + Hi, j,k + Li, j,k

+ i, j,kn+1C12 + C22 + i, j,k

n+1WIoi, j,k

n � , . . . . . . . . . . . (A-19)

i, j,kn+1 = −

C21

C11=

bgi, j,k

n+1

boi, j,k

n+1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-20)

and poi, j,k

n+1 = pgi, j,k

n+1 − pcogi, j,k

n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-21)

Fluid Saturations. The oil and gas saturations are calculatedexplicitly as follows:


Sgi, j,k

n+1 =�t�24

VRi, j, kbgi, j,k

n+1 �i, j,kn+1

×

Tgxi +

1

2, j,k

n ��pgi+1, j,k

n+1 − pgi, j,k

n+1 �

− �gi +

1

2, j,k

n�Di+1, j,k − Di, j,k��

− Tgxi −

1

2, j,k

n ��pgi, j,k

n+1 − pgi−1, j,k

n+1 �

− �gi −

1

2, j,k

n�Di, j,k − Di−1, j,k��

+ Tgyi, j +

1

2,k

n ��pgi, j+1,k

n+1 − pgi, j,k

n+1 �

− �gi, j +

1

2,k

n�Di, j+1,k − Di, j,k��

− Tgyi, j −

1

2,k

n ��pgi, j,k

n+1 − pgi, j−1,k

n+1 �

− �gi, j −

1

2,k

n�Di, j,k − Di, j−1,k��

+ Tgzi, j,k +

1

2

n ��pgi, j,k+1

n+1 − pgi, j,k

n+1 �

− �gi, j,k +

1

2

n�Di, j,k+1 − Di, j,k��

− Tgzi, j,k −

1

2

n ��pgi, j,k

n+1 − pgi, j,k−1

n+1 �

− �gi, j,k −

1

2

n�Di, j,k − Di, j,k−1��

+ qgi, j,k

n

+ 24VRi, j, k

�t�bgi, j,k

n Sgi, j,k

n �i, j,kn �

, . . . (A-22)

Son+1 = 1 − Sg

n+1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-23)

Wellbore Flow.

pwelli, j,k

n+1 − pwelli+1, j,k

n+1 = �i +

1

2, j,k

n��Di, j,k − Di+1, j,k� − HFi +

1

2, j,k�,

. . . . . . . . . . . . . . . . . . . . . . . (A-24)

HFi +1

2, j,k = f

�xi +1

2, j,k V

wellTi +1

2, j,k

2

2 dpipe g, . . . . . . . . . . . . . . . . . . . . (A-25)

f =64

NRefor NRe 2,000, . . . . . . . . . . . . . . . . . . . . . . . . . . (A-26)

and

1

�f= 1.74 − 2 log�2�D

d+

18.7

NRe�f� for NRe � 2,000,

. . . . . . . . . . . . . . . . . . . . . . . (A-27)

VwellTi +

1

2, j,k

2 = VoTi +

1

2, j,k

2 + VgTi +

1

2, j,k

2, . . . . . . . . . . . . . . . . . . . . . (A-28)

VoTi +1

2, j,k

= qoTi+1, j,k

n

�

4d2 �86,400, . . . . . . . . . . . . . . . . . . . . . (A-29)

VgTi +1

2, j,k

= qgTi+1, j,k

n

�

4d2 �86,400, . . . . . . . . . . . . . . . . . . . . . (A-30)

NRei +1

2, j,k

n=

0.06157 �i +

1

2, j,k

n

5.615 rw× ��qoT

n Bon

�o�

i+1, j,k

+�qgTn Bg

n

�g�

i+1, j,k�,

. . . . . . . . . . . . . . . . . . . . . . . (A-31)

�i+1, j,kn = �

i +1

2, j,k

n�144, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-32)

�i +

1

2, j,k

n=

�qoTn Bo

n�i+1, j,k�oi+1, j,k

n + �qgTn Bg

n�i+1, j,k�gi+1, j,k

n

�qoTn Bo

n�i+1, j,k + �qgTn Bg

n�i+1, j,k

, . . . . . . (A-33)

−WIoi, j,k

n �poi, j,k

n+1 − pwelli, j,k

n+1 � − WIoi+1, j,k

n �poi+1, j,k


n+1 � − · · ·

+5.615Cw

�t�24�pwelli, j,k

n+1 − pwelli, j,k

n � = qoTn+1,

. . . . . . . . . . . . . . . . . . . . . . . (A-34)

−WIgi, j,k

n �pgi, j,k

n+1 − pwelli, j,k

n+1 � − WIgi+1, j,k

n �pgi+1, j,k


n+1 � − · · ·

+5.615Cw

�t�24�pwelli, j,k

n+1 − pwelli, j,k

n � = qgTn+1,

. . . . . . . . . . . . . . . . . . . . . . . (A-35)

and qTn+1 = qgT

n+1 + qoTn+1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-36)

Appendix BRelative Permeability Curves. The relative permeability curvesused in this study have been generated by use of the Brooks andCorey equation (Brooks and Corey 1964) as shown below:

kro�so� = k*ro�So − Sor

1 − Sor�no

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1)

and

krg�sg� = k*rg�Sg − Sgr

1 − Sor�ng

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-2)

where Sgc�0.0, Sor�0.2, kro* �1.0, k*rg�0.8, no�2.5, andng�2.0.

Appendix CDerivation of Wellblock Radius, ro. The simulated pressure for agridblock corresponds to the average pressure, p, in the gridblock.If we denote po as the simulated pressure for a radial wellblock ofarea �(re

2−rw2 ), then the steady-state-flow equation is given by

q =2�kh

�

po − pwf

lnro

rw−

1

2

=2�kh

�

p − pwf

lnro

rw−

1

2

. . . . . . . . . . . . . . . . . (C-1)

The assumption of steady-state flow is justified because of thesmall volume of the wellblock leading to negligible fluid storage.If we replace the radial wellblock by a rectangular block of area�z×�y, then we can write the following approximation:

q =2�kh

�

po − pwf

ln��z�y��

rw−

1

2

. . . . . . . . . . . . . . . . . . . . . . . . . (C-2)

The accuracy of this approximation increases when the aspect ratioof the rectangle approaches unity.

If ro is the radial distance from the well at which the pressureis equal to po (that is, ro is the radius at which the average pressureexists in the gridblock), then we can write

q =2�kh

�

po − pwf

lnro

rw

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (C-3)

Because the rates predicted by Eqs. C-2 and C-3 must be the same,equating Eqs. C-2 and C-3, we obtain

ro = ��z�y�� e−0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (C-4)

To apply this formulation to an anisotropic gridblock, �z and �yshould be replaced by �z√k/kz and �y√k/ky, respectively, andk�√kykz should be used to satisfy the material balance in thegridblock.


SI Metric Conversion Factorsbbl × 1.589 873 E−01 � m3

cp × 1.0* E−03 � Pa·sft × 3.048* E−01 � m

ft3 × 2.831 685 E−02 � m3

°F (°F−32)/1.8 � °Cpsi × 6.894 757 E+00 � kPa

*Conversion factor is exact.

Nasser Al-Mohannadi is a faculty member at Qatar U. and aresearcher at Qatar Shell Research and Technology Center.He holds a BS degree from Sultan Qaboos U. and MS and PhDdegrees from the Colorado School of Mines, all in petroleumengineering. Erdal Ozkan is a professor and codirector of theMarathon Center of Excellence for Reservoir Studies in the Pe-troleum Engineering Dept., Colorado School of Mines. Previ-ously, he was on the faculty of Istanbul Technical U. Ozkanholds BS and MS degrees from Istanbul Technical U. and a PhDdegree from the U. of Tulsa, all in petroleum engineering. He

has served on the editorial committees of several SPE journalsand has been a Review Chairperson and Executive Editor forSPEREE. Ozkan is also a member of the SPE Reservoir Descrip-tion and Dynamics Advisory Committee. Hossein Kazemi is theChesebro’ Chair Professor and codirector of the MarathonCenter of Excellence for Reservoir Studies in the Petroleum En-gineering Dept., Colorado School of Mines. He was the man-ager of Reservoir Technology and Associate Director of Mara-thon Oil Co.’s Petroleum Technology Center from 1969 to 2000.Kazemi holds BS and PhD degrees in petroleum engineeringfrom the U. of Texas at Austin. He is a member of the Natl.Academy of Engineering and an Honorary Member of theAmerican Inst. of Mining, Metallurgical, and Petroleum Engi-neers. Kazemi is also a Distinguished Member of SPE. He hasserved on numerous SPE committees and has authored or co-authored some 60 technical papers in the SPE literature. Ka-zemi has been both a Distinguished Author and DistinguishedSpeaker for SPE and has given many technical presentationsworldwide. His SPE awards include Honorary and DistinguishedMembership, the 1987 John Franklin Carll Award, the 1995 De-Golyer Medal, the 1991 SPE Distinguished Service Award, the1980 Denver Section Henry Mattson Technical Service Award,and the 2006 IOR Pioneer Award.


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