Journal of Oil, Gas and Petrochemical Sciences (JOGPS)

Open Access Journal

Frequency: Bi-Monthly

ISSN 2630-8541

Volume : 1 | Issue : 3

Technical Paper

A new analytical model of ultimate water cut for light oil reservoirs with bottom-water

Samir Prasun,1 Sayantan Ghosh2

1Louisiana State University, Baton Rouge, LA, USA
2University of Oklahoma, Norman, OK, USA

Received: September 11, 2018 | Published: September 17, 2018

Correspondence: Samir Prasun, Louisiana State University, Baton Rouge, LA, USA, Email prasoonsamir@gmail.com

Citation: Prasun S, Ghosh S. A new analytical model of ultimate water cut for light oil reservoirs with bottom-water. J Oil Gas Petrochem Sci. (2018);1(3): 74–81. DOI: 10.30881/jogps.00015

Abstract

Ultimate water cut (WCult) defines well’s maximum water production for uncontained oil pay with bottom-water. The WCult is important to determine if the reservoir development is economical. Since presently-used WCult formula derives from simplifying assumption ignoring the effect of non-radial inflow, the formula needs to be redefined. A new analytical formula of WCult is developed by considering the inflow of oil and water into separate completions at the top of oil-zone and aquifer respectively. Then the formula is verified using the design of 46 simulated experiments representing wide variety of reservoir-bottomwater systems. It was found out that the for light-oil reservoirs, the presently-used theoretical formula may significantly diverge from the proposed formula which closely matches the simulated data and is more physics driven. Hence the proposed formula should be preferred. However, for the viscous oil reservoirs, the presently used formula conforms to the proposed formula, which is also proved mathematically.

Keywords: Ultimate water-cut, light oil, bottom-water reservoir, water coning, partial penetration

Introduction

Ultimate water-cut is a maximum stabilized water cut in an oil-pay affected by water coning. The scenario is physically modeled by setting a balanced-oil-rate (BOR) boundary of the well’s drainage area by replacing the produced oil at the the drainage boundary. After the water break-through time, there is an initial rapid increase of water-cut representing the water cone development stage, followed by the stabilization period until the WC value becomes constant, WCult.

Kuo and Desbrisay1 introduced the concept and formula of ultimate water-cut2:

WCult= M h w M h w + h o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadEfaca WGdbGaamyDaiaadYgacaWG0bGaeyypa0JcdaWcaaqaaKqzGeGaamyt aiaadIgakmaaBaaaleaajugWaiaadEhaaSqabaaakeaajugibiaad2 eacaWGObGcdaWgaaWcbaqcLbmacaWG3baaleqaaKqzGeGaey4kaSIa amiAaOWaaSbaaSqaaiaad+gaaeqaaaaaaaa@4A2E@ (1)

Shirman and Wojtanowicz3 showed that WCult in DWS wells is always lower than that in conventional wells. Their experimental results revealed that it is possible to completely reduce WCult to zero at high drainage rates. Other authors3–5 showed the dependence of ultimate water-cut on production rate. For production rates slightly higher than critical rates (maximum possible production rate without water breakthrough), water-cut would stabilize at value lower than that in Eq. (1). After conducting laboratory experiments, Shirman and Wojtanowicz3 found out that the water-cut stabilization value may not predict the Kuo and Desbrisay1 model at low production rate. They modified Eq. (1) by including the effect of production-rate as,

WCult=(1 q cr Q ) M h w M h w + h o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadEfaca WGdbGaamyDaiaadYgacaWG0bGaeyypa0JaaiikaiaaigdacqGHsisl juaGdaWcaaqaaiaadghadaWgaaqaaKqzadGaam4yaiaadkhaaKqbag qaaaqaaiaadgfaaaGaaiykamaalaaakeaajugibiaad2eacaWGObqc fa4aaSbaaSqaaKqzadGaam4DaaWcbeaaaOqaaKqzGeGaamytaiaadI gajuaGdaWgaaWcbaqcLbmacaWG3baaleqaaKqzGeGaey4kaSIaamiA aSWaaSbaaeaacaWGVbaabeaaaaaaaa@5453@ (2)

Both Eqs. (1) and (2) assume the radial flow in the oil-zone and aquifer having a BOR boundary depicted in Figure 1, and thereby ignores any nonradial distorted inflows (in oil-zone and aquifer) to a partially penetrating well. Prasun and Wojtanowciz6,7 attempted to include the effect of partial-penetration in the closed-boundary reservoirs. However, they found that the new modified WCult formula reduces back to the original formula (Eq. (1)); thus disapproving any effect of partial-penetration on ultimate water-cut in these reservoirs. Apparently, they verified the effect of partial penetration by comparing the formula with the results from the wide variety of NFRs. However, they failed to understand that the generalized consideration of all attributes of reservoirs while verification, may conceal the partial-penetration effects for certain types of reservoirs. So, this study derives a new model of ultimate water-cut for the BOR systems considering the non-radial inflow to a partial-penetrating well, and then verifies it with particular types of reservoirs classified as light oil and viscous oil reservoirs. A good match for the particular reservoir, would justify the relevance of the partial penetration effects for this reservoir.

<strong>Figure 1 </strong>  Oil and water horizontal flow in their respective zones.

Figure 1 Oil and water horizontal flow in their respective zones.

Modified analytical formula of ultimate water-cut

In derivation of a new ultimate water-cut model for a partially penetrating well in BOR system, we consider the following assumptions:

  1. There is a piston-like displacement of oil by coned water flowing into the well. So, the rising water cone development covers larger area of oil completion before final stabilization. Eventually, the ratio of well completion producing oil and water becomes equal to the ratio of oil and water zone thickness, when ultimate water-cut is reached.3
  2. In a piston-like displacement, there is almost no mixing between the flow regions of oil and water. Assumption 1 follows that the partially penetrating oil completion region (producing only oil) is at the top of oil-zone, whereas, for simplicity, we assume the partially penetrating water completion region (producing only water) is displaced from the oil-zone to the top of aquifer as shown in Figure 2. This assumption ignores the additional skin due to the water inflow from aquifer to the completion in oil-zone.

Darcy-law flow-rate equations of oil ( q o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghakm aaBaaaleaacaWGVbaakeqaaaaa@39AC@ ) ( q o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghakm aaBaaaleaacaWGVbaakeqaaaaa@39AC@ ) and water ( q w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaabaGaam4Daaqabaaaaa@3897@ ) ( q w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaBaaabaGaam4Daaqabaaaaa@3897@ ) well-inflow (into their respective completions) during ultimate water-cut stage, at surface conditions, can be given by (Appendix A),

q o = 2π k h k ro h o ( p e p w ) μ o B o (ln r e r w +so) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaaaaaa aapeGaamyCaKqbaoaaBaaabaqcLbmacaWGVbaajuaGbeaajugibiab g2da9OWaaSaaaeaajugibiaaikdacqaHapaCcaWGRbGcdaWgaaWcba qcLbmacaWGObaaleqaaKqzGeGaam4AaSWaaSbaaeaajugWaiaadkha caWGVbaaleqaaKqzGeGaamiAaSWaaSbaaeaajugWaiaad+gaaSqaba qcLbsacaGGOaGaamiCaOWaaSbaaSqaaiaadwgaaeqaaKqzGeGaeyOe I0IaamiCaOWaaSbaaSqaaKqzadGaam4DaaWcbeaajugibiaacMcaaO qaaKqzGeGaeqiVd02cdaWgaaqaaKqzadGaam4BaaWcbeaajugibiaa dkeammaaBaaabaGaam4BaaqabaqcLbsacaGGOaGaciiBaiaac6gakm aaliaabaqcLbsacaWGYbGcdaWgaaWcbaGaamyzaaqabaaakeaajugi biaadkhakmaaBaaameaacaWG3baaleqaaaaajugibiabgUcaRiaado haoiaad+gajugibiaacMcaaaaaaa@6A7B@ (3)

q w = 2π k h k rw hw( p e p w ) μ w B w (ln r e r w + s w ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaaaaaa aapeGaamyCaKqbaoaaBaaabaGaam4DaaqabaqcLbsacqGH9aqpkmaa laaabaqcLbsacaaIYaGaeqiWdaNaam4AaOWaaSbaaSqaaKqzadGaam iAaaWcbeaajugibiaadUgalmaaBaaabaqcLbmacaWGYbGaam4DaaWc beaajugibiaadIgakiaadEhajugibiaacIcacaWGWbGcdaWgaaWcba GaamyzaaqabaqcLbsacqGHsislcaWGWbGcdaWgaaWcbaqcLbmacaWG 3baaleqaaKqzGeGaaiykaaGcbaqcLbsacqaH8oqBlmaaBaaabaGaam 4DaaqabaqcLbsacaWGcbaddaWgaaqaaiaadEhaaeqaaKqzGeGaaiik aiGacYgacaGGUbGcdaWccaqaaKqzGeGaamOCaOWaaSbaaSqaaiaadw gaaeqaaaGcbaqcLbsacaWGYbGcdaWgaaadbaGaam4DaaWcbeaaaaqc LbsacqGHRaWkcaWGZbGcdaWgaaWcbaGaam4DaaqabaqcLbsacaGGPa aaaaaa@6683@ (4)

where, r e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaadwgaaOqabaaaaa@390A@ is the radial size of reservoir, ft; S o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaad+gaaOqabaaaaa@38F5@ is the skin factor due to oil-inflow defined by Eq. (A-4); S w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaadEhaaOqabaaaaa@38FD@ is the skin factor due to water-inflow defined by Eq. (A-7); r w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaadEhaaOqabaaaaa@391C@ is the well radius, ft. Now, after incorporating the above formulas into the ultimate water-cut equation (as shown in Appendix A), a new model of ultimate water-cut is developed, given by,

<strong>Figure 2 </strong>  Equivalence of oil and water inflow schematic between combined and separate systems.

Figure 2 Equivalence of oil and water inflow schematic between combined and separate systems.

WCult=(1 q cr Q ) M h w ln r e r w + s w M h w ln r e r w + s w + h o (ln r e r w + s o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadEfaca WGdbGaamyDaiaadYgacaWG0bGaeyypa0JaaiikaiaaigdacqGHsisl juaGdaWcaaqaaiaadghadaWgaaqaaKqzadGaam4yaiaadkhaaKqbag qaaaqaaiaadgfaaaGaaiykamaalaaakeaadaWcaaqaaKqzGeGaamyt aiaadIgajuaGdaWgaaWcbaqcLbmacaWG3baaleqaaaGcbaGaciiBai aac6gadaWccaqaaiaadkhadaWgaaWcbaGaamyzaaqabaaakeaacaWG YbWaaSbaaSqaaiaadEhaaeqaaaaakiabgUcaRiaadohadaWgaaWcba Gaam4DaaqabaaaaaGcbaWaaSaaaeaajugibiaad2eacaWGObqcfa4a aSbaaSqaaKqzadGaam4DaaWcbeaaaOqaaiGacYgacaGGUbWaaSGaae aacaWGYbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaamOCamaaBaaaleaa caWG3baabeaaaaGccqGHRaWkcaWGZbWaaSbaaSqaaiaadEhaaeqaaa aajugibiabgUcaROWaaSaaaeaajugibiaadIgalmaaBaaabaGaam4B aaqabaaakeaacaGGOaGaciiBaiaac6gadaWccaqaaiaadkhadaWgaa WcbaGaamyzaaqabaaakeaacaWGYbWaaSbaaSqaaiaadEhaaeqaaaaa kiabgUcaRiaadohadaWgaaWcbaGaam4BaaqabaGccaGGPaaaaaaaaa a@723D@ (5)

Validation of the proposed models using experiments

For simulation experiments, a 2-D radial-cylindrical model is built with IMEX simulation model depicted in Figure 3 using the base case reservoir properties, PVT and simulation grid data presented in Appendix C. In the model, transition zone is neglected and the produced oil and water is injected back to the oil drainage boundary and aquifer respectively at the constant pressure boundary (representing BOR boundary). The production well is completed in 50% of the total oil-zone thickness.

<strong>Figure 3 </strong>  Radial model of oil with bottom water.

Figure 3 Radial model of oil with bottom water.

We compare the ultimate water-cut values from Eq. (2) and Eq. (5) with the the design of simulated experiments shown in Table 2 representing wide variety of reservoir/bottom-water systems. For creating matrix of experiments, we use the 3-level Box-Behnken design8,9 to consider any non-linearity of the factors in the design. Three-levels (low, intermediate and high) of the reservoir parameters are chosed based on the practical field range values of reservoir properties: Mobility, horizontal permeability, aquifer thickness, penetration ratio and anisotropy ratio, as shown in Table 1. For 5 parameters chosen in this study, the design stipulates 46 number of runs (reservoir systems). Critical-rate values, q cr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGXbWaaSbaaS qaaiaadogacaWGYbaakeqaaaaa@39FE@ , for different reservoir systems used in Eq. (5) are estimated using Eq. A-12.

Levels

Mobility
( M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@37C5@ )

Aquifer thickness ( h w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaadEhaaOqabaaaaa@3912@ )

Horizontal permeability ( k h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaadIgaaOqabaaaaa@3906@ )

Penetration ratio
( h op h o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaadI gadaWgaaWcbaGaam4BaiaadchaaeqaaaGcbaGaamiAamaaBaaaleaa caWGVbaabeaaaaaaaa@3C1C@ )

Anisotropy ratio
( k v k h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaadU gadaWgaaWcbaGaamODaaqabaaakeaacaWGRbWaaSbaaSqaaiaadIga aeqaaaaaaaa@3B2D@ )

Low (-1)

1

20

50

0.2

0.01

Intermediate (0)

3

75

100

0.5

0.1

High (+1)

10

500

500

0.8

1

Table 1 Three-level values of different reservoir/aquifer system parameters

Using the pressure drawdown simulation data for different runs, oil and water production-rates were calculated using Eqs. (3) and (4) as shown in Table 2, which were then subsequently compared with their simulated data (from Table 2) shown in Figures 4 and 5. Near unit-slope correlation plot and high R2 value close to 1, approve the validity of underlying assumptions of these proposed models (Eqs. (3) and (4)) to a larger extent. The slight discrepancy is due to the assumptions of 1) piston-like displacement process and 2) displaced water completion as shown in Figure 2 that neglects the additional skin due to water inflow from aquifer to the oil-zone. Further, the comparison plot between the predicted values of WCult from Eqs. (2) and (5) and the simulated values (from Table 2) is shown in Figure 6.

<strong>Figure 4 </strong>  Simulated vs. predicted oil production rate (Eq. 3).

Figure 4 Simulated vs. predicted oil production rate (Eq. 3).

<strong>Figure 5 </strong>  Simulated vs. predicted water production rate (Eq. 4).

Figure 5 Simulated vs. predicted water production rate (Eq. 4).

<strong>Figure 6 </strong>  Simulated vs predicted ultimate water-cut with Eq. (2) and Eq. (5).

Figure 6 Simulated vs predicted ultimate water-cut with Eq. (2) and Eq. (5).

Reservoir-system #

Mobility( M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@37C5@ (

Aquifer thickness, ( h w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaadEhaaeqaaaaa@3908@ )

Horizontal perm. ( k h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaadIgaaeqaaaaa@38FC@ )

Penetration ratio ( h op h o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaadI gadaWgaaWcbaGaam4BaiaadchaaeqaaaGcbaGaamiAamaaBaaaleaa caWGVbaabeaaaaaaaa@3C1C@ )

Anisotropy ratio, k v k h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaadU gadaWgaaWcbaGaamODaaqabaaakeaacaWGRbWaaSbaaSqaaiaadIga aeqaaaaaaaa@3B2D@

Simulated WCult

WCult (From Eq. 2)

WCult (From Eq. 5)

Abs. Discrepancy (Eq. 2 and 5)

Pressure drawdown ( p e p w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaSbaaS qaaiaadwgaaeqaaOGaeyOeI0IaamiCamaaBaaaleaacaWG3baabeaa aaa@3C12@ )

Simulated oil-rate

Simulated water-rate

Predicted Oil-rate (From Eq. 3)

Predicted water-rate (From Eq. 4)

1

10

75

100

0.5

1.0

0.968

0.967

0.958

0.010

609

64

1936

70

1940

2

1

75

100

0.5

0.0

0.720

0.745

0.713

0.046

680

560

1440

480

1460

3

10

20

100

0.5

0.1

0.902

0.888

0.891

0.003

1178

196

1804

182

1800

4

10

75

500

0.5

0.1

0.959

0.966

0.958

0.007

152

82

1918

67

1950

5

1

75

50

0.5

0.1

0.720

0.748

0.708

0.057

1147

560

1440

501

1470

6

3

75

50

0.8

0.1

0.905

0.900

0.884

0.018

962

190

1810

196

1790

7

3

75

100

0.5

0.1

0.903

0.899

0.879

0.023

702

194

1806

205

1800

8

1

20

100

0.5

0.1

0.465

0.442

0.450

0.017

629

1070

930

970

960

9

10

500

100

0.5

0.1

0.974

0.995

0.989

0.006

625

52

1948

19

1990

10

3

500

100

0.5

1.0

0.965

0.982

0.946

0.038

480

70

1930

90

1940

11

3

75

100

0.5

0.1

0.903

0.899

0.879

0.023

710

194

1806

207

1820

12

3

75

100

0.8

1.0

0.909

0.899

0.880

0.022

410

182

1818

206

1820

13

3

75

50

0.5

1.0

0.916

0.899

0.873

0.030

1137

168

1832

218

1810

14

10

75

100

0.2

0.1

0.968

0.967

0.957

0.011

1535

64

1936

72

1940

15

3

20

500

0.5

0.1

0.726

0.701

0.707

0.009

194

548

1452

498

1480

16

10

75

100

0.8

0.1

0.968

0.968

0.962

0.006

524

64

1936

64

1950

17

3

500

100

0.5

0.0

0.920

0.982

0.968

0.014

716

160

1840

48

1880

18

10

75

50

0.5

0.1

0.963

0.968

0.960

0.007

1490

74

1926

65

1910

19

3

75

500

0.5

1.0

0.898

0.894

0.868

0.030

114

204

1796

218

1810

20

3

75

50

0.5

0.0

0.908

0.899

0.883

0.018

1696

184

1816

200

1820

21

3

20

100

0.8

0.1

0.753

0.705

0.710

0.006

731

494

1506

522

1535

22

3

75

100

0.2

0.0

0.887

0.898

0.876

0.025

1805

226

1774

209

1810

23

3

75

100

0.8

0.0

0.887

0.899

0.886

0.015

565

226

1774

193

1810

24

3

20

50

0.5

0.1

0.768

0.705

0.712

0.009

2043

464

1536

525

1560

25

1

500

100

0.5

0.1

0.904

0.948

0.895

0.059

575

192

1808

170

1830

26

3

75

500

0.5

0.0

0.865

0.891

0.874

0.018

166

270

1730

195

1780

27

3

75

500

0.8

0.1

0.891

0.897

0.881

0.018

97

218

1782

198

1810

28

1

75

100

0.8

0.1

0.720

0.748

0.716

0.045

395

560

1440

483

1470

29

3

75

100

0.5

0.1

0.903

0.899

0.879

0.023

714

194

1806

208

1830

30

3

20

100

0.5

1.0

0.755

0.705

0.712

0.011

846

490

1510

515

1540

31

10

75

100

0.5

0.0

0.944

0.967

0.961

0.006

899

112

1888

63

1930

32

3

20

100

0.2

0.1

0.753

0.705

0.715

0.014

1890

494

1506

507

1535

33

3

75

50

0.2

0.1

0.905

0.899

0.870

0.033

2921

190

1810

227

1845

34

3

500

100

0.2

0.1

0.946

0.982

0.949

0.034

1244

108

1892

82

1910

35

1

75

100

0.2

0.1

0.700

0.746

0.689

0.083

1132

600

1400

528

1430

36

3

75

100

0.5

0.1

0.903

0.899

0.879

0.023

714

194

1806

208

1830

37

3

75

100

0.5

0.1

0.903

0.899

0.879

0.023

718

194

1806

209

1840

38

3

500

50

0.5

0.1

0.947

0.983

0.963

0.020

1218

106

1894

60

1940

39

1

75

100

0.5

1.0

0.710

0.747

0.695

0.075

456

580

1420

523

1450

40

3

75

100

0.5

0.1

0.903

0.899

0.879

0.023

714

194

1806

208

1830

41

3

500

500

0.5

0.1

0.938

0.976

0.957

0.020

121

124

1876

60

1920

42

3

20

100

0.5

0.0

0.755

0.704

0.710

0.008

1163

490

1510

517

1530

43

3

500

100

0.8

0.1

0.946

0.983

0.967

0.016

413

108

1892

54

1940

44

1

75

500

0.5

0.1

0.700

0.733

0.693

0.057

112

600

1400

488

1430

45

3

75

100

0.2

1.0

0.909

0.898

0.855

0.051

1077

182

1818

256

1840

46

3

75

500

0.2

0.1

0.891

0.891

0.863

0.033

287

218

1782

224

1815

Table 2Simulated and predicted data (WCult, oil-rate and water-rate) for an experimental matrix: h o =25ft MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaad+gaaOqabaGaeyypa0JaaGOmaiaaiwdacaWGMbGaamiDaaaa @3D6F@ ; Q=2000 bbl day MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGrbGaeyypa0 JaaGOmaiaaicdacaaIWaGaaGimamaaliaabaGaamOyaiaadkgacaWG SbaabaGaamizaiaadggacaWG5baaaaaa@4157@  

It is clear from the unit-slope correlation plot (Figure 6) that both the formulas give practically the same result. This infers that though the formula 2 ignores the inevitable non-radial flow to a partially penetration well, it still manages to conform to a more realistic physics-based formula 5 and hence predict the simulated WCult value.

Figure 7a shows the average absolute discrepancy (error), in percentage between the presently-used formula 2 and the proposed formula 5 using the data from Table 2. Also, Figure 7b shows the discrepancy between the formulas Eq. (2) and Eq. (5) for light oil reservoirs (M<3). From these two figures, it can be inferred that for the light oil reservoirs (when the mobility ratio is <3), the theoretical formula 2 may significantly deviate from the better (physically accurate) formula 5 for some cases (Figure 7a) with discrepancy as high as 8% (Figure 7b), which may not be reflected in Figure 6 due to considerable wide variety of sample size.  In this study, any discrepancy exceeding the limit of 5% would be considered significant. This implies that for the light oil reservoir, the simplified assumptions of formula 2 may no longer allow it to better predict the actual WCult values, for which the formula 5 can serve better. This can be also be justified by the mathematical proof in Appendix B. So, in practice, formula 5 should be preferred for general use.

<strong>Figure 7a </strong>  Average absolute discrepancy, in % between formulas 5 and 2.

Figure 7a Average absolute discrepancy, in % between formulas 5 and 2.

<strong>Figure 7b </strong>  Absolute Discrepancy, in % between formulas 5 and 2 for runs having M<3.

Figure 7b Absolute Discrepancy, in % between formulas 5 and 2 for runs having M<3.

On the other hand, for moderate to high mobility ratio reservoirs (M 3), Figure 7a shows that the average discrepancy between the formulas is less than 5%, which is insignificant. This implies that in those conditions, formula (5) can be reduced to formula (2), which is also shown mathematically in Appendix B. So, Eq. (2), being simpler than Eq. (5), suffices to predict WCult for viscous oil reservoirs (M≥3).

Conclusions

Results of the study are summarized in the following conclusions:

  1. A new analytical formula for WCult has been proposed including the physical effect ignored in the presently-used formula: partial penetration of oil zone, and aquifer. The formula utilizes the new models of oil and water production-rates during the ultimate water-cut stage. The derivation of models considers the piston-like displacement process and the inflow of oil and water into separate completions at the top of oil-zone and aquifer respectively.
  2. The proposed formulas are systematically verified for wide variety of reservoir systems using design of simulated experiments (IMEX). High R2 value for the plot between the simulated and the predicted oil and water production-rates approves the validity of the proposed model’s underlying assumptions to a large extent. However, sight discrepancy can be attributed to the above assumptions.
  3. In general, both the formulas (proposed and presently-used) of WCult predicts almost the same results which matches the simulated WCult values. However, for the light oil reservoirs (mobility ratio<3), simulations showed that the theoretical presently used-formula may significantly deviate from the (physically accurate) proposed formula. This is also confirmed by mathematical proof, so in practice, proposed formula should be preferred for the possible avoidance of errors.
  4. On the other hand, for viscous oil reservoirs (Mobility ratio≥3), comparison of the simulations with the predicted values showed that the presently-used formula suffices to predict the WCult values. This fact that the proposed formula reduces to presently-used formula for the above reservoirs, can be justified mathematically.

Nomenclature

μ o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaWgaa WcbaGaam4Baaqabaaaaa@39C9@ = viscosity of oil, cp
μ w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaWgaa WcbaGaam4Daaqabaaaaa@39D1@ = viscosity of water, cp
Δρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarcqaHbp GCaaa@3A19@ = density difference between water and oil, lb/ft3
B o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbWaaSbaaS qaaiaad+gaaeqaaaaa@38DA@ = oil formation volume factor, bbl/stb
B w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbWaaSbaaS qaaiaadEhaaeqaaaaa@38E2@ = water formation volume factor, bbl/stb
BOR = balanced-oil-rate
h o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaad+gaaeqaaaaa@3900@ = oil-zone thickness, ft
h op MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaad+gacaWGWbaabeaaaaa@39F5@ = perforated length, ft
h opo MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaad+gacaWGWbGaam4Baaqabaaaaa@3AE9@ = length of well-completion occupied by oil during WCult stage, ft
h opw MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaad+gacaWGWbGaam4Daaqabaaaaa@3AF1@ = length of well-completion occupied by water during WCult stage, ft
h w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaadEhaaeqaaaaa@3908@ = aquifer thickness, ft
k h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaadIgaaeqaaaaa@38FC@ = horizontal permeability, md
k o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaad+gaaeqaaaaa@3903@ = effective permeability of oil, md
k ro MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaadkhacaWGVbaabeaaaaa@39FA@ = relative permeability of oil
k rw MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaadkhacaWG3baabeaaaaa@3A02@ = relative permeability of water
k v k h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWccaqaaiaadU gadaWgaaWcbaGaamODaaqabaaakeaacaWGRbWaaSbaaSqaaiaadIga aeqaaaaaaaa@3B2F@ =Anisotropy ratio, fraction
k w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaadEhaaeqaaaaa@390B@ = effective permeability of water, md
M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad2eaaa a@3854@ = mobility ratio between water and oil, fraction
p e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaSbaaS qaaiaadwgaaeqaaaaa@38FE@ = reservoir pressure, psi
p w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaSbaaS qaaiaadEhaaeqaaaaa@3910@ = well-bottomhole pressure, psi
q cr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghakm aaBaaaleaacaWGJbGaamOCaaqabaaaaa@3A8D@ =critical oil rate, bbl/day
q o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghakm aaBaaaleaacaWGVbaakeqaaaaa@39AC@ = oil flow rate, bbl/day
q w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghakm aaBaaaleaacaWG3baabeaaaaa@39AA@ = water flow rate, bbl/day
Q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGrbaaaa@37C9@ = Total production rate, bbl/day
r w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaadEhaaeqaaaaa@3912@ = wellbore radius, ft
r e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaadwgaaeqaaaaa@3900@ = reservoir radius, ft
S o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaad+gaaeqaaaaa@38EB@ = Partial penetration skin due to oil-inflow
S w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaadEhaaeqaaaaa@38F3@ = Partial penetration skin due to water-inflow
T = Ratio of aquifer thickness to oil-zone thickness
WC = water-cut, fraction
WCult MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadEfaca WGdbGaamyDaiaadYgacaWG0baaaa@3C0A@ = Ultimate water cut, fraction

Appendix A: Derivation of new analytical WCult formula

Assuming piston-like displacement process, the rise of water cone before final stabilization covers larger area of oil completion. Eventually, the ratio of well completion producing oil and water becomes equal to the ratio of oil and water zone thickness, when ultimate water-cut is reached.3 So, the length of well-completion occupied by oil during WCult stage:

               h opo = h o h o + h w × h op MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaad+gacaWGWbGaam4BaaqabaGccqGH9aqpdaWcaaqaaiaadIga daWgaaWcbaGaam4BaaqabaaakeaacaWGObWaaSbaaSqaaiaad+gaae qaaOGaey4kaSIaamiAamaaBaaaleaacaWG3baabeaaaaGccqGHxdaT caWGObWaaSbaaSqaaiaad+gacaWGWbaabeaaaaa@4851@                                                                                                     (A-1) And, the length of well-completion occupied by water during WCult stage:                                                                     h opw = h w h o + h w × h op MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaad+gacaWGWbGaam4DaaqabaGccqGH9aqpdaWcaaqaaiaadIga daWgaaWcbaGaam4DaaqabaaakeaacaWGObWaaSbaaSqaaiaad+gaae qaaOGaey4kaSIaamiAamaaBaaaleaacaWG3baabeaaaaGccqGHxdaT caWGObWaaSbaaSqaaiaad+gacaWGWbaabeaaaaa@4861@                                               (A-2)
This follows that the well completion system during water cone stabilization stage can be assumed to be the combination of the oil completion (producing only oil) at the top of oil-zone and the displaced water completion (producing only water) at the top of aquifer (Figure 2). So, oil inflow rate due to partial penetration in oil-zone (producing only oil) is given by,
                                                                        q o = 2π k o h o ( p e p w ) μ o (ln r e r w +so) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaaaaaa aapeGaamyCaKqbaoaaBaaabaqcLbmacaWGVbaajuaGbeaajugibiab g2da9OWaaSaaaeaajugibiaaikdacqaHapaCcaWGRbWcdaWgaaqaaK qzadGaam4BaaWcbeaajugibiaadIgalmaaBaaabaqcLbmacaWGVbaa leqaaKqzGeGaaiikaiaadchakmaaBaaaleaacaWGLbaabeaajugibi abgkHiTiaadchakmaaBaaaleaajugWaiaadEhaaSqabaqcLbsacaGG PaaakeaajugibiabeY7aTTWaaSbaaeaajugWaiaad+gaaSqabaqcLb sacaGGOaGaciiBaiaac6gakmaaliaabaqcLbsacaWGYbGcdaWgaaWc baGaamyzaaqabaaakeaajugibiaadkhakmaaBaaameaacaWG3baale qaaaaajugibiabgUcaRiaadohaoiaad+gajugibiaacMcaaaaaaa@6332@                                                        
Since,k_o=k_h k_ro , we get:
                                                                                                           q o = 2π k h k ro h o ( p e p w ) μ o (ln r e r w +so) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaaaaaa aapeGaamyCaKqbaoaaBaaabaqcLbmacaWGVbaajuaGbeaajugibiab g2da9OWaaSaaaeaajugibiaaikdacqaHapaCcaWGRbGcdaWgaaWcba qcLbmacaWGObaaleqaaKqzGeGaam4AaSWaaSbaaeaajugWaiaadkha caWGVbaaleqaaKqzGeGaamiAaSWaaSbaaeaajugWaiaad+gaaSqaba qcLbsacaGGOaGaamiCaOWaaSbaaSqaaiaadwgaaeqaaKqzGeGaeyOe I0IaamiCaOWaaSbaaSqaaKqzadGaam4DaaWcbeaajugibiaacMcaaO qaaKqzGeGaeqiVd02cdaWgaaqaaKqzadGaam4BaaWcbeaajugibiaa cIcaciGGSbGaaiOBaOWaaSGaaeaajugibiaadkhakmaaBaaaleaaca WGLbaabeaaaOqaaKqzGeGaamOCaOWaaSbaaWqaaiaadEhaaSqabaaa aKqzGeGaey4kaSIaam4Ca4Gaam4BaKqzGeGaaiykaaaaaaa@6804@    (A-3)
Where, s o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaSbaaS qaaiaad+gaaeqaaaaa@390B@ is the skin factor10 due to oil-inflow and is given by,
             S o =( 1 h opD 1)ln π 2 r oD + 1 h opD ln[ h opD 2+ h opD ( A1 B1 ) 1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaad+gaaeqaaOGaeyypa0JaaiikamaalaaabaGaaGymaaqaaiaa dIgadaWgaaWcbaGaam4BaiaadchacaWGebaabeaaaaGccqGHsislca aIXaGaaiykaiGacYgacaGGUbWaaSaaaeaacqaHapaCaeaacaaIYaGa amOCamaaBaaaleaacaWGVbGaamiraaqabaaaaOGaey4kaSYaaSaaae aacaaIXaaabaGaamiAamaaBaaaleaacaWGVbGaamiCaiaadseaaeqa aaaakiGacYgacaGGUbWaamWaaeaadaWcaaqaaiaadIgadaWgaaWcba Gaam4BaiaadchacaWGebaabeaaaOqaaiaaikdacqGHRaWkcaWGObWa aSbaaSqaaiaad+gacaWGWbGaamiraaqabaaaaOWaaeWaaeaadaWcaa qaaiaadgeacqGHsislcaaIXaaabaGaamOqaiabgkHiTiaaigdaaaaa caGLOaGaayzkaaWaaWbaaSqabeaadaWccaqaaiaaigdaaeaacaaIYa aaaaaaaOGaay5waiaaw2faaaaa@63D9@                                                  (A-4)
          h opD = h opo h o = h op h o + h w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaad+gacaWGWbGaamiraaqabaGccqGH9aqpdaWccaqaaiaadIga daWgaaWcbaGaam4BaiaadchacaWGVbaabeaaaOqaaiaadIgadaWgaa WcbaGaam4BaaqabaaaaOGaeyypa0ZaaSaaaeaacaWGObWaaSbaaSqa aiaad+gacaWGWbaabeaaaOqaaiaadIgadaWgaaWcbaGaam4Baaqaba GccqGHRaWkcaWGObWaaSbaaSqaaiaadEhaaeqaaaaaaaa@4B27@ (From Eq. (A-1))                                                                   (A-5)
r oD =( r w h o ) ( k v k h ) 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaad+gacaWGebaabeaakiabg2da9maabmaabaWaaSGaaeaacaWG YbWaaSbaaSqaaiaadEhaaeqaaaGcbaGaamiAamaaBaaaleaacaWGVb aabeaaaaaakiaawIcacaGLPaaadaqadaqaamaaliaabaGaam4Aamaa BaaaleaacaWG2baabeaaaOqaaiaadUgadaWgaaWcbaGaamiAaaqaba aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaadaWccaqaaiaaigdaaeaa caaIYaaaaaaaaaa@4843@ ;   A=4/ h opD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbGaeyypa0 ZaaSGbaeaacaaI0aaabaGaamiAamaaBaaaleaacaWGVbGaamiCaiaa dseaaeqaaaaaaaa@3D5E@ ;       B=4/ 3 h opD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbGaeyypa0 ZaaSGbaeaacaaI0aaabaGaaG4maiaadIgadaWgaaWcbaGaam4Baiaa dchacaWGebaabeaaaaaaaa@3E1C@          
Now, again water inflow rate due to partial penetration in an aquifer (producing only water) is given by,
q w = 2π k w h w ( p e p w ) μ w (ln r e r w + s w ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaaaaaa aapeGaamyCaKqbaoaaBaaabaGaam4DaaqabaqcLbsacqGH9aqpkmaa laaabaqcLbsacaaIYaGaeqiWdaNaam4AaOWaaSbaaSqaaiaadEhaae qaaKqzGeGaamiAaOWaaSbaaSqaaiaadEhaaeqaaKqzGeGaaiikaiaa dchakmaaBaaaleaacaWGLbaabeaajugibiabgkHiTiaadchakmaaBa aaleaajugWaiaadEhaaSqabaqcLbsacaGGPaaakeaajugibiabeY7a TTWaaSbaaeaacaWG3baabeaajugibiaacIcaciGGSbGaaiOBaOWaaS GaaeaajugibiaadkhakmaaBaaaleaacaWGLbaabeaaaOqaaKqzGeGa amOCaOWaaSbaaWqaaiaadEhaaSqabaaaaKqzGeGaey4kaSIaam4CaO WaaSbaaSqaaiaadEhaaeqaaKqzGeGaaiykaaaaaaa@5E2F@ Since,k_w=k_h k_rw , we get:
                                                                                                          q w = 2π k h k rw h w ( p e p w ) μ w (ln r e r w + s w ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaaaaaa aapeGaamyCaKqbaoaaBaaabaGaam4DaaqabaqcLbsacqGH9aqpkmaa laaabaqcLbsacaaIYaGaeqiWdaNaam4AaOWaaSbaaSqaaKqzadGaam iAaaWcbeaajugibiaadUgalmaaBaaabaqcLbmacaWGYbGaam4DaaWc beaajugibiaadIgakmaaBaaaleaacaWG3baabeaajugibiaacIcaca WGWbGcdaWgaaWcbaGaamyzaaqabaqcLbsacqGHsislcaWGWbGcdaWg aaWcbaqcLbmacaWG3baaleqaaKqzGeGaaiykaaGcbaqcLbsacqaH8o qBlmaaBaaabaGaam4DaaqabaqcLbsacaGGOaGaciiBaiaac6gakmaa liaabaqcLbsacaWGYbGcdaWgaaWcbaGaamyzaaqabaaakeaajugibi aadkhakmaaBaaameaacaWG3baaleqaaaaajugibiabgUcaRiaadoha kmaaBaaaleaacaWG3baabeaajugibiaacMcaaaaaaa@6430@   (A-6)
So, the skin factor, S w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaadEhaaeqaaaaa@38F3@ due to water-inflow can be represented by10:
                                                        S w =( 1 h wpD 1)ln π 2 r wD + 1 h wpD ln[ h wpD 2+ h wpD ( Aw1 Bw1 ) 1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaSbaaS qaaiaadEhaaeqaaOGaeyypa0JaaiikamaalaaabaGaaGymaaqaaiaa dIgadaWgaaWcbaGaam4DaiaadchacaWGebaabeaaaaGccqGHsislca aIXaGaaiykaiGacYgacaGGUbWaaSaaaeaacqaHapaCaeaacaaIYaGa amOCamaaBaaaleaacaWG3bGaamiraaqabaaaaOGaey4kaSYaaSaaae aacaaIXaaabaGaamiAamaaBaaaleaacaWG3bGaamiCaiaadseaaeqa aaaakiGacYgacaGGUbWaamWaaeaadaWcaaqaaiaadIgadaWgaaWcba Gaam4DaiaadchacaWGebaabeaaaOqaaiaaikdacqGHRaWkcaWGObWa aSbaaSqaaiaadEhacaWGWbGaamiraaqabaaaaOWaaeWaaeaadaWcaa qaaiaadgeacaWG3bGaeyOeI0IaaGymaaqaaiaadkeacaWG3bGaeyOe I0IaaGymaaaaaiaawIcacaGLPaaadaahaaWcbeqaamaaliaabaGaaG ymaaqaaiaaikdaaaaaaaGccaGLBbGaayzxaaaaaa@6601@   (A-7)
          h wpD = h opw h w = h op h o + h w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaadEhacaWGWbGaamiraaqabaGccqGH9aqpdaWccaqaaiaadIga daWgaaWcbaGaam4BaiaadchacaWG3baabeaaaOqaaiaadIgadaWgaa WcbaGaam4DaaqabaaaaOGaeyypa0ZaaSaaaeaacaWGObWaaSbaaSqa aiaad+gacaWGWbaabeaaaOqaaiaadIgadaWgaaWcbaGaam4Baaqaba GccqGHRaWkcaWGObWaaSbaaSqaaiaadEhaaeqaaaaaaaa@4B3F@ (From Eq. (A-2))                                                                   (A-8)
r wD =( r w h w ) ( k v k h ) 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaadEhacaWGebaabeaakiabg2da9maabmaabaWaaSGaaeaacaWG YbWaaSbaaSqaaiaadEhaaeqaaaGcbaGaamiAamaaBaaaleaacaWG3b aabeaaaaaakiaawIcacaGLPaaadaqadaqaamaaliaabaGaam4Aamaa BaaaleaacaWG2baabeaaaOqaaiaadUgadaWgaaWcbaGaamiAaaqaba aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaadaWccaqaaiaaigdaaeaa caaIYaaaaaaaaaa@4853@ ;   Aw=4/ h wpD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbGaam4Dai abg2da9maalyaabaGaaGinaaqaaiaadIgadaWgaaWcbaGaam4Daiaa dchacaWGebaabeaaaaaaaa@3E62@ ;    Bw=4/ 3 h wpD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbGaam4Dai abg2da9maalyaabaGaaGinaaqaaiaaiodacaWGObWaaSbaaSqaaiaa dEhacaWGWbGaamiraaqabaaaaaaa@3F20@                                  
From Eqs. (A-5) and (A-8), we get:
                                                                     h wpD = h opD = h pD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaadEhacaWGWbGaamiraaqabaGccqGH9aqpcaWGObWaaSbaaSqa aiaad+gacaWGWbGaamiraaqabaGccqGH9aqpcaWGObWaaSbaaSqaai aadchacaWGebaabeaaaaa@4388@                                            (A-9)
Ultimate Water-cut, during water-cut stabilization stage3 is given by:
                                                                       WCult=( 1 q cr Q ) q w q w + q o =( 1 q cr Q ) 1 1+ q o q w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaam4qai aadwhacaWGSbGaamiDaiabg2da9maabmaabaGaaGymaiabgkHiTmaa laaabaGaamyCamaaBaaaleaacaWGJbGaamOCaaqabaaakeaacaWGrb aaaaGaayjkaiaawMcaamaalaaabaGaamyCamaaBaaaleaacaWG3baa beaaaOqaaiaadghadaWgaaWcbaGaam4DaaqabaGccqGHRaWkcaWGXb WaaSbaaSqaaiaad+gaaeqaaaaakiabg2da9maabmaabaGaaGymaiab gkHiTmaalaaabaGaamyCamaaBaaaleaacaWGJbGaamOCaaqabaaake aacaWGrbaaaaGaayjkaiaawMcaamaalaaabaGaaGymaaqaaiaaigda cqGHRaWkdaWcaaqaaiaadghadaWgaaWcbaGaam4Baaqabaaakeaaca WGXbWaaSbaaSqaaiaadEhaaeqaaaaaaaaaaa@59E3@   (A-10)
Substituting q o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGabaaXliaadghada WgaaWcbaGaam4Baaqabaaaaa@39DB@  and q w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGXbWaaSbaaS qaaiaadEhaaeqaaaaa@3911@  from Eqs. (A-3) and (A-6) in (A-10), we get:
                      WCult=( 1 q cr Q ) 1 1+ 2π k h k ro h o ( p e p w ) μ o (ln r e r w +so) 2π k w h w ( p e p w ) μ w [ ln r e r w + s w ] =( 1 q cr Q ) M h w ln r e r w + s w M h w ln r e r w + s w + h o ( ln r e r w + s o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaam4qai aadwhacaWGSbGaamiDaiabg2da9maabmaabaGaaGymaiabgkHiTmaa laaabaGaamyCamaaBaaaleaacaWGJbGaamOCaaqabaaakeaacaWGrb aaaaGaayjkaiaawMcaamaalaaabaGaaGymaaqaaabaaaaaaaaapeGa aGymaiabgUcaRmaalaaabaWaaSaaaeaajugibiaaikdacqaHapaCca WGRbGcdaWgaaWcbaqcLbmacaWGObaaleqaaKqzGeGaam4AaSWaaSba aeaajugWaiaadkhacaWGVbaaleqaaKqzGeGaamiAaSWaaSbaaeaaju gWaiaad+gaaSqabaqcLbsacaGGOaGaamiCaOWaaSbaaSqaaiaadwga aeqaaKqzGeGaeyOeI0IaamiCaOWaaSbaaSqaaKqzadGaam4DaaWcbe aajugibiaacMcaaOqaaKqzGeGaeqiVd02cdaWgaaqaaKqzadGaam4B aaWcbeaajugibiaacIcaciGGSbGaaiOBaOWaaSGaaeaajugibiaadk hakmaaBaaaleaacaWGLbaabeaaaOqaaKqzGeGaamOCaOWaaSbaaWqa aiaadEhaaSqabaaaaKqzGeGaey4kaSIaam4Ca4Gaam4BaKqzGeGaai ykaaaaaOqaamaalaaabaqcLbsacaaIYaGaeqiWdaNaam4AaOWaaSba aSqaaiaadEhaaeqaaKqzGeGaamiAaOWaaSbaaSqaaiaadEhaaeqaaK qzGeGaaiikaiaadchakmaaBaaaleaacaWGLbaabeaajugibiabgkHi TiaadchakmaaBaaaleaajugWaiaadEhaaSqabaqcLbsacaGGPaaake aajugibiabeY7aTTWaaSbaaeaacaWG3baabeaakmaadmaabaqcLbsa ciGGSbGaaiOBaOWaaSGaaeaajugibiaadkhakmaaBaaaleaacaWGLb aabeaaaOqaaKqzGeGaamOCaOWaaSbaaWqaaiaadEhaaSqabaaaaKqz GeGaey4kaSIaam4CaOWaaSbaaSqaaiaadEhaaeqaaaGccaGLBbGaay zxaaaaaaaaaaWdaiabg2da9maabmaabaGaaGymaiabgkHiTmaalaaa baGaamyCamaaBaaaleaacaWGJbGaamOCaaqabaaakeaacaWGrbaaaa GaayjkaiaawMcaamaalaaabaWaaSaaaeaajugibiaad2eacaWGObqc fa4aaSbaaSqaaKqzadGaam4DaaWcbeaaaOqaaiGacYgacaGGUbWaaS GaaeaacaWGYbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaamOCamaaBaaa leaacaWG3baabeaaaaGccqGHRaWkcaWGZbWaaSbaaSqaaiaadEhaae qaaaaaaOqaamaalaaabaqcLbsacaWGnbGaamiAaKqbaoaaBaaaleaa jugWaiaadEhaaSqabaaakeaaciGGSbGaaiOBamaaliaabaGaamOCam aaBaaaleaacaWGLbaabeaaaOqaaiaadkhadaWgaaWcbaGaam4Daaqa baaaaOGaey4kaSIaam4CamaaBaaaleaacaWG3baabeaaaaGccqGHRa WkdaWcaaqaaiaadIgadaWgaaWcbaGaam4Baaqabaaakeaadaqadaqa aiGacYgacaGGUbWaaSGaaeaacaWGYbWaaSbaaSqaaiaadwgaaeqaaa GcbaGaamOCamaaBaaaleaacaWG3baabeaaaaGccqGHRaWkcaWGZbWa aSbaaSqaaiaad+gaaeqaaaGccaGLOaGaayzkaaaaaaaaaaa@C663@       (A-11)
Where, M= k rw μ w k ro μ o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaeyypa0 ZaaSGaaeaadaWcaaqaaiaadUgadaWgaaWcbaGaamOCaiaadEhaaeqa aaGcbaGaeqiVd02aaSbaaSqaaiaadEhaaeqaaaaaaOqaamaalaaaba Gaam4AamaaBaaaleaacaWGYbGaam4BaaqabaaakeaacqaH8oqBdaWg aaWcbaGaam4Baaqabaaaaaaaaaa@44E5@
Critical rate, q cr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGXbWaaSbaaS qaaiaadogacaWGYbaabeaaaaa@39F4@ in above Eq. (A-11) can be substituted by the following formula11:
                                                        q cr =0.0783× 10 4 [ Δρ k o ( h o 2 h op 2 ) μ o B o ][ 0.7311+ 1.943 r e h o k v k h ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGXbWaaSbaaS qaaiaadogacaWGYbaabeaakiabg2da9iaaicdacaGGUaGaaGimaiaa iEdacaaI4aGaaG4maiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacq GHsislcaaI0aaaaOWaamWaaeaadaWcaaqaaiabfs5aejabeg8aYjaa dUgadaWgaaWcbaGaam4BaaqabaGcdaqadaqaaiaadIgadaWgaaWcba Gaam4BaaqabaGcdaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGObWa aSbaaSqaaiaad+gacaWGWbaabeaakmaaCaaaleqabaGaaGOmaaaaaO GaayjkaiaawMcaaaqaaiabeY7aTnaaBaaaleaacaWGVbaabeaakiaa dkeadaWgaaWcbaGaam4BaaqabaaaaaGccaGLBbGaayzxaaWaamWaae aacaaIWaGaaiOlaiaaiEdacaaIZaGaaGymaiaaigdacqGHRaWkdaWc aaqaaiaaigdacaGGUaGaaGyoaiaaisdacaaIZaaabaWaaSaaaeaaca WGYbWaaSbaaSqaaiaadwgaaeqaaaGcbaGaamiAamaaBaaaleaacaWG VbaabeaaaaGcdaGcaaqaamaalaaabaGaam4AamaaBaaaleaacaWG2b aabeaaaOqaaiaadUgadaWgaaWcbaGaamiAaaqabaaaaaqabaaaaaGc caGLBbGaayzxaaaaaa@6DEB@  (A-12)
Where, all the parameters are in field units.

Appendix B: Mathematical convergence of new formula to presently-used formula

Using Eqs. (A-4), (A-7) and (A-9), Eq. 5 can be rewritten as:
                                 ( 1 q cr Q ) M h w ln r e r w + s w M h w ln r e r w + s w + h o ( ln r e r w + s o ) =( 1 q cr Q ) M h w ln r e r w +( 1 h pD 1 )ln π h w 2 r w ( k v k h ) 1 2 + 1 h pD ln[ h opD 2+ h pD ( A1 B1 ) 1 2 ] M h w ( ln r e r w +( 1 h pD 1 )ln π h w 2 r w ( k v k h ) 1 2 + 1 h pD ln[ h opD 2+ h pD ( A1 B1 ) 1 2 ] ) + h o ( ln r e r w +( 1 h pD 1 )ln π h o 2 r w ( k v k h ) 1 2 + 1 h pD ln[ h opD 2+ h pD ( A1 B1 ) 1 2 ] ) =( 1 q cr Q ) M h w M h w + h o × ( ln r e r w +( 1 h pD 1 )ln π h w 2 r w ( k v k h ) 1 2 + 1 h pD ln[ h opD 2+ h pD ( A1 B1 ) 1 2 ] ) ( ln r e r w +( 1 h pD 1 )ln π h o 2 r w ( k v k h ) 1 2 + 1 h pD ln[ h opD 2+ h pD ( A1 B1 ) 1 2 ] ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaaig dacqGHsisldaWcaaqaaiaadghadaWgaaWcbaGaam4yaiaadkhaaeqa aaGcbaGaamyuaaaaaiaawIcacaGLPaaadaWcaaqaamaalaaabaqcLb sacaWGnbGaamiAaKqbaoaaBaaaleaajugWaiaadEhaaSqabaaakeaa ciGGSbGaaiOBamaaliaabaGaamOCamaaBaaaleaacaWGLbaabeaaaO qaaiaadkhadaWgaaWcbaGaam4DaaqabaaaaOGaey4kaSIaam4Camaa BaaaleaacaWG3baabeaaaaaakeaadaWcaaqaaKqzGeGaamytaiaadI gajuaGdaWgaaWcbaqcLbmacaWG3baaleqaaaGcbaGaciiBaiaac6ga daWccaqaaiaadkhadaWgaaWcbaGaamyzaaqabaaakeaacaWGYbWaaS baaSqaaiaadEhaaeqaaaaakiabgUcaRiaadohadaWgaaWcbaGaam4D aaqabaaaaOGaey4kaSYaaSaaaeaacaWGObWaaSbaaSqaaiaad+gaae qaaaGcbaWaaeWaaeaaciGGSbGaaiOBamaaliaabaGaamOCamaaBaaa leaacaWGLbaabeaaaOqaaiaadkhadaWgaaWcbaGaam4DaaqabaaaaO Gaey4kaSIaam4CamaaBaaaleaacaWGVbaabeaaaOGaayjkaiaawMca aaaaaaGaeyypa0ZaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGXb WaaSbaaSqaaiaadogacaWGYbaabeaaaOqaaiaadgfaaaaacaGLOaGa ayzkaaWaaSaaaeaadaWcaaqaaiaad2eacaWGObWaaSbaaSqaaiaadE haaeqaaaGcbaGaciiBaiaac6gadaWccaqaaiaadkhadaWgaaWcbaGa amyzaaqabaaakeaacaWGYbWaaSbaaSqaaiaadEhaaeqaaaaakiabgU 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GaciiBaiaac6gadaWcaaqaaiabec8aWjaadIgadaWgaaWcbaGaam4D aaqabaaakeaacaaIYaGaamOCamaaBaaaleaacaWG3baabeaakmaabm aabaWaaSGaaeaacaWGRbWaaSbaaSqaaiaadAhaaeqaaaGcbaGaam4A amaaBaaaleaacaWGObaabeaaaaaakiaawIcacaGLPaaadaahaaWcbe qaamaaliaabaGaaGymaaqaaiaaikdaaaaaaaaakiabgUcaRmaalaaa baGaaGymaaqaaiaadIgadaWgaaWcbaGaamiCaiaadseaaeqaaaaaki GacYgacaGGUbWaamWaaeaadaWcaaqaaiaadIgadaWgaaWcbaGaam4B aiaadchacaWGebaabeaaaOqaaiaaikdacqGHRaWkcaWGObWaaSbaaS qaaiaadchacaWGebaabeaaaaGcdaqadaqaamaalaaabaGaamyqaiab gkHiTiaaigdaaeaacaWGcbGaeyOeI0IaaGymaaaaaiaawIcacaGLPa aadaahaaWcbeqaamaaliaabaGaaGymaaqaaiaaikdaaaaaaaGccaGL BbGaayzxaaaacaGLOaGaayzkaaaaaiabgUcaRmaalaaabaGaamiAam aaBaaaleaacaWGVbaabeaaaOqaamaabmaabaGaamiBaiaac6gadaWc caqaaiaadkhadaWgaaWcbaGaamyzaaqabaaakeaacaWGYbWaaSbaaS qaaiaadEhaaeqaaaaakiabgUcaRmaabmaabaWaaSaaaeaacaaIXaaa baGaamiAamaaBaaaleaacaWGWbGaamiraaqabaaaaOGaeyOeI0IaaG ymaaGaayjkaiaawMcaaiGacYgacaGGUbWaaSaaaeaacqaHapaCcaWG ObWaaSbaaSqaaiaad+gaaeqaaaGcbaGaaGOmaiaadkhadaWgaaWcba Gaam4DaaqabaGcdaqadaqaamaaliaabaGaam4AamaaBaaaleaacaWG 2baabeaaaOqaaiaadUgadaWgaaWcbaGaamiAaaqabaaaaaGccaGLOa GaayzkaaWaaWbaaSqabeaadaWccaqaaiaaigdaaeaacaaIYaaaaaaa aaGccqGHRaWkdaWcaaqaaiaaigdaaeaacaWGObWaaSbaaSqaaiaadc hacaWGebaabeaaaaGcciGGSbGaaiOBamaadmaabaWaaSaaaeaacaWG ObWaaSbaaSqaaiaad+gacaWGWbGaamiraaqabaaakeaacaaIYaGaey 4kaSIaamiAamaaBaaaleaacaWGWbGaamiraaqabaaaaOWaaeWaaeaa daWcaaqaaiaadgeacqGHsislcaaIXaaabaGaamOqaiabgkHiTiaaig daaaaacaGLOaGaayzkaaWaaWbaaSqabeaadaWccaqaaiaaigdaaeaa caaIYaaaaaaaaOGaay5waiaaw2faaaGaayjkaiaawMcaaaaaaaGaey ypa0ZaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGXbWaaSbaaSqa aiaadogacaWGYbaabeaaaOqaaiaadgfaaaaacaGLOaGaayzkaaWaaS aaaeaacaWGnbGaamiAamaaBaaaleaacaWG3baabeaaaOqaaiaad2ea caWGObWaaSbaaSqaaiaadEhaaeqaaOGaey4kaSIaamiAamaaBaaale aacaWGVbaabeaakiabgEna0oaalaaabaWaaeWaaeaacaWGSbGaaiOB amaaliaabaGaamOCamaaBaaaleaacaWGLbaabeaaaOqaaiaadkhada WgaaWcbaGaam4DaaqabaaaaOGaey4kaSYaaeWaaeaadaWcaaqaaiaa igdaaeaacaWGObWaaSbaaSqaaiaadchacaWGebaabeaaaaGccqGHsi slcaaIXaaacaGLOaGaayzkaaGaciiBaiaac6gadaWcaaqaaiabec8a WjaadIgadaWgaaWcbaGaam4DaaqabaaakeaacaaIYaGaamOCamaaBa aaleaacaWG3baabeaakmaabmaabaWaaSGaaeaacaWGRbWaaSbaaSqa aiaadAhaaeqaaaGcbaGaam4AamaaBaaaleaacaWGObaabeaaaaaaki aawIcacaGLPaaadaahaaWcbeqaamaaliaabaGaaGymaaqaaiaaikda aaaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadIgadaWgaaWcba GaamiCaiaadseaaeqaaaaakiGacYgacaGGUbWaamWaaeaadaWcaaqa aiaadIgadaWgaaWcbaGaam4BaiaadchacaWGebaabeaaaOqaaiaaik dacqGHRaWkcaWGObWaaSbaaSqaaiaadchacaWGebaabeaaaaGcdaqa daqaamaalaaabaGaamyqaiabgkHiTiaaigdaaeaacaWGcbGaeyOeI0 IaaGymaaaaaiaawIcacaGLPaaadaahaaWcbeqaamaaliaabaGaaGym aaqaaiaaikdaaaaaaaGccaGLBbGaayzxaaaacaGLOaGaayzkaaaaba WaaeWaaeaacaWGSbGaaiOBamaaliaabaGaamOCamaaBaaaleaacaWG LbaabeaaaOqaaiaadkhadaWgaaWcbaGaam4DaaqabaaaaOGaey4kaS YaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGObWaaSbaaSqaaiaadcha caWGebaabeaaaaGccqGHsislcaaIXaaacaGLOaGaayzkaaGaciiBai aac6gadaWcaaqaaiabec8aWjaadIgadaWgaaWcbaGaam4Baaqabaaa keaacaaIYaGaamOCamaaBaaaleaacaWG3baabeaakmaabmaabaWaaS GaaeaacaWGRbWaaSbaaSqaaiaadAhaaeqaaaGcbaGaam4AamaaBaaa leaacaWGObaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaamaali aabaGaaGymaaqaaiaaikdaaaaaaaaakiabgUcaRmaalaaabaGaaGym aaqaaiaadIgadaWgaaWcbaGaamiCaiaadseaaeqaaaaakiGacYgaca GGUbWaamWaaeaadaWcaaqaaiaadIgadaWgaaWcbaGaam4Baiaadcha caWGebaabeaaaOqaaiaaikdacqGHRaWkcaWGObWaaSbaaSqaaiaadc hacaWGebaabeaaaaGcdaqadaqaamaalaaabaGaamyqaiabgkHiTiaa igdaaeaacaWGcbGaeyOeI0IaaGymaaaaaiaawIcacaGLPaaadaahaa WcbeqaamaaliaabaGaaGymaaqaaiaaikdaaaaaaaGccaGLBbGaayzx aaaacaGLOaGaayzkaaaaaaaaaaa@A7C3@    (B-1)
Substituting T= h w h o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubGaeyypa0 ZaaSGaaeaacaWGObWaaSbaaSqaaiaadEhaaeqaaaGcbaGaamiAamaa BaaaleaacaWGVbaabeaaaaaaaa@3D10@ , and C=ln r e r w +( 1 h pD 1 )ln π h o 2 r w ( k v k h ) 1 2 + 1 h pD ln[ h opD 2+ h pD ( A1 B1 ) 1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbGaeyypa0 JaciiBaiaac6gadaWccaqaaiaadkhadaWgaaWcbaGaamyzaaqabaaa keaacaWGYbWaaSbaaSqaaiaadEhaaeqaaaaakiabgUcaRmaabmaaba WaaSaaaeaacaaIXaaabaGaamiAamaaBaaaleaacaWGWbGaamiraaqa baaaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaaiGacYgacaGGUbWaaS aaaeaacqaHapaCcaWGObWaaSbaaSqaaiaad+gaaeqaaaGcbaGaaGOm aiaadkhadaWgaaWcbaGaam4DaaqabaGcdaqadaqaamaaliaabaGaam 4AamaaBaaaleaacaWG2baabeaaaOqaaiaadUgadaWgaaWcbaGaamiA aaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaadaWccaqaaiaaig daaeaacaaIYaaaaaaaaaGccqGHRaWkdaWcaaqaaiaaigdaaeaacaWG ObWaaSbaaSqaaiaadchacaWGebaabeaaaaGcciGGSbGaaiOBamaadm aabaWaaSaaaeaacaWGObWaaSbaaSqaaiaad+gacaWGWbGaamiraaqa baaakeaacaaIYaGaey4kaSIaamiAamaaBaaaleaacaWGWbGaamiraa qabaaaaOWaaeWaaeaadaWcaaqaaiaadgeacqGHsislcaaIXaaabaGa amOqaiabgkHiTiaaigdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaada WccaqaaiaaigdaaeaacaaIYaaaaaaaaOGaay5waiaaw2faaaaa@6FF0@  in Eq. (B-1), we get:
                 ( 1 q cr Q ) M h w ln r e r w + s w M h w ln r e r w + s w + h o ( ln r e r w + s o ) =( 1 q cr Q ) MT h o MT h o + h o × ( C+( 1 h pD 1 )lnT ) C =( 1 q cr Q ) M h o h o ×( 1 T +( M+ lnT T C/ ( 1 h pD 1 ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaaig dacqGHsisldaWcaaqaaiaadghadaWgaaWcbaGaam4yaiaadkhaaeqa aaGcbaGaamyuaaaaaiaawIcacaGLPaaadaWcaaqaamaalaaabaqcLb sacaWGnbGaamiAaKqbaoaaBaaaleaajugWaiaadEhaaSqabaaakeaa ciGGSbGaaiOBamaaliaabaGaamOCamaaBaaaleaacaWGLbaabeaaaO qaaiaadkhadaWgaaWcbaGaam4DaaqabaaaaOGaey4kaSIaam4Camaa BaaaleaacaWG3baabeaaaaaakeaadaWcaaqaaKqzGeGaamytaiaadI gajuaGdaWgaaWcbaqcLbmacaWG3baaleqaaaGcbaGaciiBaiaac6ga daWccaqaaiaadkhadaWgaaWcbaGaamyzaaqabaaakeaacaWGYbWaaS baaSqaaiaadEhaaeqaaaaakiabgUcaRiaadohadaWgaaWcbaGaam4D aaqabaaaaOGaey4kaSYaaSaaaeaacaWGObWaaSbaaSqaaiaad+gaae qaaaGcbaWaaeWaaeaaciGGSbGaaiOBamaaliaabaGaamOCamaaBaaa leaacaWGLbaabeaaaOqaaiaadkhadaWgaaWcbaGaam4DaaqabaaaaO Gaey4kaSIaam4CamaaBaaaleaacaWGVbaabeaaaOGaayjkaiaawMca aaaaaaGaeyypa0ZaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGXb WaaSbaaSqaaiaadogacaWGYbaabeaaaOqaaiaadgfaaaaacaGLOaGa ayzkaaWaaSaaaeaacaWGnbGaamivaiaadIgadaWgaaWcbaGaam4Baa qabaaakeaacaWGnbGaamivaiaadIgadaWgaaWcbaGaam4BaaqabaGc cqGHRaWkcaWGObWaaSbaaSqaaiaad+gaaeqaaOGaey41aq7aaSaaae aadaqadaqaaiaadoeacqGHRaWkdaqadaqaamaalaaabaGaaGymaaqa aiaadIgadaWgaaWcbaGaamiCaiaadseaaeqaaaaakiabgkHiTiaaig daaiaawIcacaGLPaaaciGGSbGaaiOBaiaadsfaaiaawIcacaGLPaaa aeaacaWGdbaaaaaacqGH9aqpdaqadaqaaiaaigdacqGHsisldaWcaa qaaiaadghadaWgaaWcbaGaam4yaiaadkhaaeqaaaGcbaGaamyuaaaa aiaawIcacaGLPaaadaWcaaqaaiaad2eacaWGObWaaSbaaSqaaiaad+ gaaeqaaaGcbaGaamiAamaaBaaaleaacaWGVbaabeaakiabgEna0oaa bmaabaWaaSGaaeaacaaIXaaabaGaamivaaaacqGHRaWkdaqadaqaai aad2eacqGHRaWkdaWcaaqaamaalaaabaGaciiBaiaac6gacaWGubaa baGaamivaaaaaeaadaWcgaqaaiaadoeaaeaadaqadaqaamaalaaaba GaaGymaaqaaiaadIgadaWgaaWcbaGaamiCaiaadseaaeqaaaaakiab gkHiTiaaigdaaiaawIcacaGLPaaaaaaaaaGaayjkaiaawMcaaaGaay jkaiaawMcaaaaaaaa@ADA2@                                                                                    (B-2)
Figure B-1 clearly shows the maximum value of lnT T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiGacY gacaGGUbGaamivaaqaaiaadsfaaaaaaa@3A99@  is 0.37. Subsequently, the approximate maximum possible value of lnT T C/ ( 1 h pD 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaamaala aabaGaciiBaiaac6gacaWGubaabaGaamivaaaaaeaadaWcgaqaaiaa doeaaeaadaqadaqaamaalaaabaGaaGymaaqaaiaadIgadaWgaaWcba GaamiCaiaadseaaeqaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaa aaaaaaaa@4264@  is 0.15 for the practical field operating range values of h pD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObWaaSbaaS qaaiaadchacaWGebaabeaaaaa@39CA@  (between 0.1 and 1) and for practical value of T (>0.8). Minimum possible value of 1 r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaaig daaeaacaWGYbaaaaaa@38B5@  tends to 0 for infinite thick aquifers.

<strong>Figure B-1  </strong>  Pattern graph of log(T)/T vs. T; (T=ratio of aquifer thickness to oil-zone thickness).

Figure B-1 Pattern graph of log(T)/T vs. T; (T=ratio of aquifer thickness to oil-zone thickness).

Now, assuming 5% maximum possible error is permissible in predicted WCult value given by Eq. (B-2); for viscous reservoirs (when mobility ratio ≥ 3), any value of lnT T C/ ( 1 h pD 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaamaala aabaGaciiBaiaac6gacaWGubaabaGaamivaaaaaeaadaWcgaqaaiaa doeaaeaadaqadaqaamaalaaabaGaaGymaaqaaiaadIgadaWgaaWcba GaamiCaiaadseaaeqaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaa aaaaaaaa@4264@  would lie withing this error margin of Eq. (B-2) and hence, the part ‘ lnT T C/ ( 1 h pD 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaamaala aabaGaciiBaiaac6gacaWGubaabaGaamivaaaaaeaadaWcgaqaaiaa doeaaeaadaqadaqaamaalaaabaGaaGymaaqaaiaadIgadaWgaaWcba GaamiCaiaadseaaeqaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaa aaaaaaaa@4264@ ’ can be ignored. So, Eq. (B-2) or Eq. (5) can be rewritten as:
( 1 q cr Q ) M h w ln r e r w + s w M h w ln r e r w + s w + h o ( ln r e r w + s o ) =( 1 q cr Q ) M h o M h o + h o T =( 1 q cr Q ) M h w M h w + h o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdMqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaaig dacqGHsisldaWcaaqaaiaadghadaWgaaWcbaGaam4yaiaadkhaaeqa aaGcbaGaamyuaaaaaiaawIcacaGLPaaadaWcaaqaamaalaaabaqcLb sacaWGnbGaamiAaKqbaoaaBaaaleaajugWaiaadEhaaSqabaaakeaa ciGGSbGaaiOBamaaliaabaGaamOCamaaBaaaleaacaWGLbaabeaaaO qaaiaadkhadaWgaaWcbaGaam4DaaqabaaaaOGaey4kaSIaam4Camaa BaaaleaacaWG3baabeaaaaaakeaadaWcaaqaaKqzGeGaamytaiaadI gajuaGdaWgaaWcbaqcLbmacaWG3baaleqaaaGcbaGaciiBaiaac6ga daWccaqaaiaadkhadaWgaaWcbaGaamyzaaqabaaakeaacaWGYbWaaS baaSqaaiaadEhaaeqaaaaakiabgUcaRiaadohadaWgaaWcbaGaam4D aaqabaaaaOGaey4kaSYaaSaaaeaacaWGObWaaSbaaSqaaiaad+gaae qaaaGcbaWaaeWaaeaaciGGSbGaaiOBamaaliaabaGaamOCamaaBaaa leaacaWGLbaabeaaaOqaaiaadkhadaWgaaWcbaGaam4DaaqabaaaaO Gaey4kaSIaam4CamaaBaaaleaacaWGVbaabeaaaOGaayjkaiaawMca aaaaaaGaeyypa0ZaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGXb WaaSbaaSqaaiaadogacaWGYbaabeaaaOqaaiaadgfaaaaacaGLOaGa ayzkaaWaaSaaaeaacaWGnbGaamiAamaaBaaaleaacaWGVbaabeaaaO qaaiaad2eacaWGObWaaSbaaSqaaiaad+gaaeqaaOGaey4kaSYaaSGa aeaacaWGObWaaSbaaSqaaiaad+gaaeqaaaGcbaGaamivaaaaaaGaey ypa0ZaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGXbWaaSbaaSqa aiaadogacaWGYbaabeaaaOqaaiaadgfaaaaacaGLOaGaayzkaaWaaS aaaeaacaWGnbGaamiAamaaBaaaleaacaWG3baabeaaaOqaaiaad2ea caWGObWaaSbaaSqaaiaadEhaaeqaaOGaey4kaSIaamiAamaaBaaale aacaWGVbaabeaaaaaaaa@8C1A@ (B-3)                                           
Above derivation mathematically proves that Eq. (5) reduces to Eq. (2) in case of viscous oil reservoirs. However, for mobility ratio<3, Eq. (5) may or may not reduce to Eq. (2) depending upon the ratio of aquifer to oil-zone thickness.

Appendix C: Complete Reservoir Simulation Input Data

Parameter

Unit

Value

Datum depth

ft

5000

Thickness of oil zone

ft

25

Depth of WOC

ft

5025

Thickness of water zone

ft

75, varied

Reservoir pressure at datum depth

psi

6000

Position of top completion from formation top

ft

0

Perforated length

ft

12, varied

Horizontal permeability

md

100, varied

Anisotropy ratio

md

0.1, varied

Porosity

fraction

0.3

Well radius

ft

0.25

Outer radius of oil-zone

ft

1000

Outer radius of water zone

ft

1000

Total liquid Production rate

bpd

2000

Table 1 Reservoir and Well Input data

Property

Unit

Value

Reference pressure

psi

6000

Formation oil volume factor

rb/stb

1.2

Relative oil permeability at connate water saturation

fraction

1

Water compressibility

1/psi

3.3202e--6

Oil compressibility

1/psi

1.50E-05

water viscosity

cp

0.5

Oil viscosity

cp

1.5, varied

oil density

lb/cuft

43.65

Water density

lb/cuft

60.55

Bubble point

psi

100

Table C-2 Fluid Properties Input Data

Region

Direction

Grid Number

Oil zone

R

20

Ф

1

Z

25

Water zone

R

29

Ф

1

Z

15

Table C-3 Simulation Grid Data

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