SPESPE 20742

General Coning Correlations Based on Mechanistic StudiesS.H. Lee" and W.B. Tung, Chevron Oil Field Research CO.

·SPE Member

Copyright 1990, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the 65th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in New Orleans, LA, September 23-26, 1990.

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

ABSTRACT

The coning of water can impact well productivity and in­crease water treatment requirements. Coning correlationsare often used to model water-breakthrough time and wa­ter cut due to coning. Most existing coning correlations arebased on a steady state approximation so that the predic­tion of water-breakthrough time and initial water cut devel­opment are generally unreliable. By first determining thekey reservoir, production, and well completion parametersthrough a theoretical analysis, improved correlations weredeveloped. They have been extensively compared with nu­merical simulation results to validate their general applica­bility. The correlations simplify the procedure in the studyof the effects of production rate, vertical and horizontal per­meabilities, well completion location, physical properties offluids, relative permeabilities, aquifer thickness, completioninterval size, and drainage radius on coning dynamics.

The new correlations can be used in a stand-alone programor in reservoir simulation. In field-scale reservoir simula­tion, correlations are often employed to model sub-scaleflow behavior near a well. Hence, technical issues in imple­menting correlations into a finite difference simulator arealso discussed. In addition, the new correlations are ex­tended to multi-layered, nonhomogeneous reservoir models.

1. INTRODUCTION

The coning of water (i.e., the production of water from anunderlying aquifer) can impact well productivity and in-

References and illustrations at end of paper

269

crease water treatment requirements. Engineers usecorrelations or numerical simulation to model water­breakthrough time and water cut due to coning. Althoughnumerical simulation is a more rigorous means to modelwater coning, engineers often use simple correlations for apreliminary esitmate of coning phenomena because, com­pared with numerical simulation, they are simple and easyto use.

In addition, coning correlations are employed in field-scalereservoir simulation. In many reservoir simulation stud­ies, the grid size is so large that the local coning phenom­ena around a wellbore cannot be resolved in finite differ­ence simulation. Many approximate methods have beendevised to overcome this shortcoming. For instance, spe­cial pseudo-relative permeability curves for well blocks, lo­cal grid refinements around wells, or coning correlations areoften employed in field-scale modeling. Though local gridrefinement is the most rigorous method of the three, it isalso the most expensive for a large model with many wells.A pseudo-relative function method is cumbersome to im­plement and requires pre-simulation to develop the curves.As a result, coning correlations are still commonly used inreservoir simulation.

Coning correlations have been studied by many investiga­tors ( e.g., Muskat1 , Arthur2 , Meyer and Garder 3, Schols 4,

Chappelear and Hirasakis , Wheatley 6, Hoyland, Papatza­cos, & Skajaeveland 7, and Abass 8). Nevertheless, all cur­rent methods include some crude assumptions, which limittheir applicability in a wide range of problems. Many cor­relations employ a steady state approximation, and adoptthe concept of critical rates. Because the steady state ap­proximation cannot hold for cone development in generaloperating conditions, especially in an early stage of produc-

2 General Coning Correlations Based on Mechanistic Studies SPE 20742

(1)

(3)

(2)

(4)

In domains I, II, and III of Figure 1, fluid flow is governedby Darcy's law. Hence, the governing equations become

3. FORMULATION

An extensive analysis of governing equations and boundaryconditions provides a logical approach to determine char­acteristic variables and important independent variables.

The relative permeability of water at Sw = 1 - Sor andthat of oil at So = 1 - Swe are included in equations (2)and (3), respectively. Note that kro = 1 for most cases.The incompressibility condition of fluids can be expressedas follows:

The matching boundary conditions between domains I andII are given by

In Section 2, the problem to be modeled is specified. Be­cause of the many controlling parameters in this problem,we modeled a simple case and then extended the corre­lations to more complex and general cases. Basically wemodel coning phenomena in a homogeneous and axisym­metric reservoir with a bottom aquifer. Furthermore, thetotal liquid production rate was held constant. In Sections3 and 4, the governing equations and boundary conditionsare derived. From these equations characteristic variablesand important controlling parameters are determined. Thecorrelations are derived in a variable domain nondimension­alized with respect to characteristic variables.

tion, most correlations are very poor in estimating water­breakthrough time.

We also studied coning phenomena with a boundary in­tegral method to understand their detailed dynamics bet­ter. These results are briefly summarized in Section 5. Adetailed study of cone dynamics by a boundary integralmethod will be reported in a forthcoming paper. Section6 describes the procedure to construct new coning correla­tions based on finite difference reservoir simulation results.The correlations and simulation results are compared inSection 7. In Section 8 technical issues in implementing,coning correlations in a finite difference simulator are dis­cussed.

cP = cPo = Po -Pwgz, for r = r e and -ha.:::; z < 0 (9)

and the no-flow boundary condition at the drainage bound­ary in the oil zone becomes

(7)

(8)

(5)

(6)

(11)q = 21lTwl Un dzhp

for r = r e and 0:::; z :::; (ho + hp + hI) (10)Un = 0,

and between domains II and III by

U(2) = U(3)w 0

The constant aquifer potential, cP, at the drainage boundarycan be described by

Here, Un is the normal component of fluid velocity at theboundary. Furthermore, the boundary condition at thecompletion becomes

Here, fe is the cone shape function which describes theheight of the cone. Velocity is always continuous at domaininterfaces, whereas pressure tends to jump due to densitydifferences. In equation (8), the pressure jump at the conefront is linearly proportional to the cone height and thedensity difference between oil and water.

Since the Darcy velocity is an apparent (filter) velocity, theactual cone front velocity is larger than the Darcy velocity.

2. PROBLEM DESCRIPTION

A schematic diagram of water coning around a well is shownin Figure 1. Water coning in an axisymmetric geometry of ahomgeneous reservoir was selected. Fluids are incompress­ible. Because the force of gravity is mostly dominant inwater coning, it was assumed that the oil zone encroachedby water is uniform with residual oil saturation, Sor. Thisassumption is reasonable if the transition zone is negligi­ble, and it also simplifies the problem so that only the endpoints of relative permeabilty curves are needed to developthe correlations. It entails that, in domain II of Figure1, only water is mobile. The water coning is driven by abottom aquifer in which water is supplied at the drainageboundary. The aquifer pressure at the drainage radius re­mains constant and the oil zone has a no-flow boundary atthe drainage radius, r e' In this model, an equilibrium shapeto the water cone does not exist because oil is continouslyreplaced by water.

Coning phenomena in a reservoir are so complex that rea­sonably simple, practical, and useful correlations have beendifficult to derive. Hence, we adopted a step-by-step ap­proach to this problem, instead of including all the com­plexities of the physical phenomena at once. A well-defined,simple, but representative, case is first modeled, and thenadditional complexities are gradually added to the correla­tions. The physical phenomena to be modeled are specifiedin this section.

270

SPE 20742 S. H. Lee and W. B. Tung 3

The actual front velocity can be readily calculated from thematerial balance at the front.

The boundary conditions at the drainage radius, r = re ,

are expressed as

(27)

(29)

The velocity components and coordinates were rescaledwith nondimensionalized permeabilities as in equations (14)and (15). Fortunately, in the rescaled and nondimensionaldomain the permeability, K, does not directly appear inequations (16) to (18). It is imbedded in the geometri­cal description of the problem. Furthermore, porosity onlychanges the front velocity as shown in equation (12).

and

4. CHARACTERISTIC VARIABLES AND IMPOR­TANT INDEPENDENT VARIABLES

In addition, the boundary condition at the completion isgiven by

Even in the simplified problem described in Section 2,there are many independent physical variables: directionalpermeabilities (kv , kh), relative permeabilities (kro , krw ),por.osity (¢», densities (Po, Pw), viscosities (Po, Pw), residualoil saturation (Sor), and connate water saturation (Swc). Inaddition, the geometrical description involves drainage ra­dius (re ), wellbore radius (rw ), aquifer thickness (h,,), for­mation thickness (h t + hp + ho + h,,), and top and bottomposition of the completion (ho, ho+hp ). The boundary con­ditions include production rate (q), wellbore pressure (Pw),and aquifer pressure (Po) at the drainage radius. Becausethere are so many independent variables, it is important toidentify a small set of variables that mainly govern coningdynamics.

In Section 3 the governing equations and boundary" condi­tions were nondimensionalized with respect to characteris­tic variables: lc = ho, Pc = 100 psia, U c = Pckckro/1cPo,kc = kv , t c = lc/uc, and qc = l~/tc. Here, the distance­between the bottom of the well completion and the origi­nal oil-water contact line was considered the most relevantlength scale for water breakthrough. The vertical perme­ability was also chosen as a characteristic variable, whichcontrols vertical motion of the water cone. We had the free­dom to choose either a characteristic pressure or a charac­teristic velocity from operating conditions. For simplicity,we chose a characteristic pressure from a typical pressuredrop between the wellbore and reservoir pressure. All othercharacteristic variables were then determined from the gov­erning equations and boundary conditions.

(12)

(13)

(22)

(23)

(19)

(24)

(14)

(25)

(20)

(15)

(16)

(17)

(18)

(2)Uw

U front = (2) (3)¢>(Sw - Sw )

liZ) = -m2VP(2)

li~3) = _ VP(3)

Here,

~ (khK= oFurther, when the velocity components are rescaled as

poml=-­

kroPw

krwPom2=--

kroPw

The equation of continuity becomes

We chose as characteristic variables lc = ho, Pc = 100 psia,U c = Pckckro/1cPo, kc = kv, t c = lc/uc and qc = l~/tc, andnondimensionalized the governing equations and boundaryconditions with respect to these characteristic variables.Henceforth, a tilde, ~, denotes a dimensionless variable.

Without loss of generality, a coordinates system is chosenin the princij>al directions of~:K so that the off-diagonalelements of K vanish. Then K has two non-zero diagonalelements.

P<l) = P(2)

and between domains II and III by

The matching conditions between domains are given by,between domains I and II,

_ Xi

Xi == ../ki

the governing equations (1)-(3) can be simplified as

liZ) = -ml VP<l)

_ 11.i

Ui == ../ki

and the components of the position vector are rescaled as

where

C_ gf)"p1c

g-Pc

(26)By examining equations (16) to (29), one can easily iden­tify independent variables in the rescaled nondimensional

271

4 General Coning Correlations Based on Mechanistic Studies SPE 20742

domain that characterize coning development. In the gov­erning equations the controlling parameters are the mobil­ity ratios, m1 and m2, as defined in equations (19) and (20).In the boundary conditions the production rate, q, the grav­itational force due to density difference, Cg , and the ratio of

actual front velocity and Darcy velocity, rl = </>(S~) -S~»),are independent variables. The geometrical description isdetermined by wellbore radius, r w , drainage radius, r e ,

aquifer thickness, ha , perforation interval, hp , and the dis­tance between the top of the formation and the top of thecompletion, ht .

These variables are, however, too numerous to develop rea­sonably simple correlations. We further reduced the num­ber of controlling variables based on coning dynamics. Themobility ratio m1 is generally much larger than 1. It is nota controlling parameter because it is not specific for givenreservoir conditions. The ratio of actual front velocity andDarcy velocity, r I, only adjusts the front location based onmaterial balance, which is independent of coning dynam­ics. A coning correlation, developed at a specific value ofr I, can be easily generalized by adjusting the time scale fora reservoir with a different rl'

Since rw « 1 and re » 1 in most reservoirs, the well boreradius and the drainage radius can be excluded in studyingconing dynamics. In a gravitationally stabilized produc­tion, the oil production from the zone above the comple­tion is minimal, which implies that ht is not important in

this study. We first assume that ha >> 1 and hp << 1and develop correlations based on this simple geometricaldescription. Later the correlations are adjusted to includethe effects of small ha and large hp • Given these condi­tions, three important controlling parameters are identified:mobility ratio, m = m2; gravitational force, Cgj and pro­duction rate, q. Correlations are developed based on thesethree parameters.

5. PRELIMINARY STUDY OF CONING DYNAMICSBY A BOUNDARY INTEGRAL METHOD

Water coning phenomena are dominated by the water frontmotion, which carries a sharp saturation change as in aBuckley-Leverett displacement process. In a conventionalfinite difference method, the location of water fronts canbe indirectly tracked from the dispersed saturation distri­bution. Due to its numerical dispersity and finite differ­ence approximation, it is rather cumbersome to study thedetailed dynamics of moving fronts by a finite differencemethod. It is always helpful to have a numerical methodthat can conveniently describe the detailed dynamics of awater front near a well completion. Consequently we devel­oped a boundary integral method to study the dynamics ofwater fronts.

The boundary integral method has been successfully ap­plied in a variety of fields: solid mechanics, low-Reynolds-

272

number fluid motion, ground water flow, and well produc­tivity calculation (cf. Brebbia9 , geller et al. 10 , Ligget &Liull , and Lee12 ). The boundary integral method hastwo distinct advantages over the conventional finite dif­ference or finite element method in calculating satura­tion fronts. First, the potential strengths at the bound­ary elements are directly calculated. Therefore, only theboundaries must be discretized, which renders great flex­ibility and convenience in describing the boundary sur­faces. Secondly, the order of spatial dimensionality forthe flow problem is reducedj three-dimensional problemsbecome two-dimensional (surface) integral equations, andtwo-dimensional problems become one-dimensional (line)integral equations. The boundary integral method is moreefficient in computation and requires less memory storagethan a finite difference or finite element method for linearequations. Because a detailed description of the methodand results will be reported in a forthcoming paper, we willonly briefly summarize the preliminary numerical results tohelp understand the general dynamics of water coning.

The shape of the cone front is shown for several time stepsin Figure 2, and the height of the cone apex is plottedin Figure 3. These figures clearly indicate that the conedevelops through four distinct steps:

(1) Initially the cone develops rapidly to produce a shapein gravitational equilibrium.

(2) Later, the cone maintains a relatively steady develop­ment speed.

(3) As the cone tip approaches the completion interval, thenarrow region around the cone center grows rapidly.

(4) After water breaks through, the cone growth slowsdown substantially.

These four steps are typical for all coning phenomena. Thispreliminary study of cone dynamics clearly indicates thatthe water-interface motion is different before and after wa­ter breakthrough. It entails that these two phenomenashould be modeled separately. For practical purposes, es­timates of water breakthrough time and water cut projec­tions are needed, rather than the actual shape function ofthe water cone. In the following section, we model the aver­age cone development velocity before water breakthrough,and water cut projections after water breakthrough.

6. CONSTRUCTION OF CONING CORRELATIONS

As discussed in the previous section, water coning phenom­ena were modeled separately before and after water break­through due to different coning dynamics. Breakthroughtime and water cut projections after breakthrough were di­rectly modeled with important physical and operational pa­rameters. Correlations were first developed based on threekey controlling parameters: q (flow rate), Cg (gravitationalforce due to density difference), and m (mobility ratio).Then the effects of aquifer thickness, ha , and perforation

SPE 20742 S. H. Lee and W. B. Tung 5

interval, hp , were added in the correlations. A base modelwas constructed with the physical properties in Table 1 andstraight-line relative permeability curves. With a finite dif­ference simulator, coning phenomena were systematicallyinvestigated to determine their dependencies on physicalparameters and geometrical and boundary conditions. Inaddition, the effects of relative permeability, oil viscosityvariation, and a small drainage radius were also investi­gated. The correlations are expressed with nondimensionaland rescaled variables, as defined in Section 2. For simplic­ity, the tilde and hat for dimensionless and rescaled vari­ables are omitted in this section.

that the functional dependencies of C9 and m are separa­ble, and f and 9 are developed with a constant mobilityratio ( m = 4.146). Subsequently the mobility effect onwater-breakthrough time was examined by changing mo­bility and production rate for the model. The results aredepicted in Figure 6. Because the effect of mobility ratio onwater breakthrough time is strongly coupled with produc­tion rate, the following equation is proposed to representthe curves in Figure 6,

6.1 Water-Breakthrough Timeand the equation of u is also modified as

q, Cg, and m dependency h(m,q)u = f(q)g(Cgr )h(4.146,q) (36)

Instead of modeling water-breakthrough time (twb), wechose to model the average cone development velocity,which is the reciprocal of water-breakthrough time.

For extreme. values of q, h(m,q) is limited by

hem, q) = h(m,0.2074), for q < 0.2074

h(m,q) = h(m,3.), for q > 3.

hp and ha dependency

From regression of the curves, we obtain

h1(q) = 1.4103 - 2.0228q, for 0.2064::; q ::; 1.1144h1(q) = -0.000012085 _161.9204e-4.71624q,

for 1.1144 < q ::; 3.

h2(q) = 0.49724 + 0.78129q, for 0.2074::; q ::; 0.4179

h2(q) = 1.22017 - 0.94544q, for 0.4179 < q ::; 3.h3(q) = 0.54307 - 0.68135e-l.09375q,

for 0.2074::; q ::; 1.1144

h3 (q) = 0.02545 +1.130e-1.l43q , for 1.144 < q::; 3.

(32)

(30)

(31)u = u(q,Cg,m)

u = f(q)g(Cgr)h(m,q)

1u=­

twb

The average cone development velocity can be expressed asa general functional.

. Using a finite difference simulator, breakthtough time iscomputed for models with various q and Cg • The mobility

ratio, m, is kept constant (rn = 4.146). The simulationresults are plotted in Figure 4. The figure clearly showssimilarity in shape among the curves with different produc­tion rates. Hence, a similarity transformation is proposed.

Velocity at Cg = Cgm is plotted in Figure 5. From theseresults, functions f(q) and g(Cgr ) are determined.

f(q) = 0.61490q - 2.32377q2 + 6.93017q3,

for 0::; q ::; 0.426

f(q) = 1.1977q - 0.13422, for 0.426 < q

g(Cgr ) = 26.8977 - 69.9435Cgr + 47.8545C;r,

for Cgr ::; 0.58

g(Cgr ) = 9.0600 -15.9050Cgr + 7.8366C;r,

for 0.58 < Cgr ::; 1

g(Cgr ) = 0.01309 + 1.0033e-3.3016(Cgr-1.), for 1. < Cgr

whereC gr = Cg/Cgm

Cgm = h(q)

Regression of the curves in Figure 4 yields

Cgm = h(q) = 0.597q + 0.0484

(33)

(34)

The average cone development velocity is also dependent onthe perforation interval and aquifer thickness. A large com­pletion interval for a given production rate delays break­through time. When a line sink is distributed over a largeinterval, some portion of the sink is far from the oil-watercontact line. Therefore, the total pulling force (singularitystrength) of the sink becomes smaller, compared with a wellwith a small completion interval and the same productionrate. A thin aquifer restricts water flow to prolong water­breakthrough time. These effects of perforation intervaland aquifer thickness were added to the correlations.

For simplicity of modeling, simple functional dependenciesof u on ha and hp are proposed.

h(m,q)u = 01(hp )02(ha)f(q)g(Cgr ) h(4.146, q) (37)

The functionals of 01 and 02 are determined from simula­tion results with various aquifer thickness and perforationintervals.

The next step was to model the mobility dependency ofthe correlation. In the above equation of u, it is assumed

The simulation results for hp dependency are modeled bythe following equations.

ol(hp ) = 1, for hp ::; 0.1428

273

6 General Coning Correlations Based on Mechanistic Studies SPE 20742

a1 (hp

) = 0.1965 - 0.1175hp + 2.0752e-6.4807hp,

for 0.1428 < hp :::; 0.857

a1 (hp

) = 0.01315 +0.803ge-2.5602hp, for 0.857 < hp

Water flow restriction due to the aquifer size is modeled by

a2(ha) = 1, for ha > 1.428

a2(ha) = 0.97979 - 0.7901e-4.2025h., for 0:::; ha :::; 1.428

6.2 Water Cut Prediciton

the water cut performance was regressed. The results areplotted in Figures 7,8, and 9. In these figures the curves ap­pear to have a similiar shape for different production rates.So it seeems feasible to represent the curves with a singlecurve by normalizing Cg with Cgm as in modeling water­breakthrough time. Using this conjecture, we propose thefollowing functional forms for the parameters:

(45)

t' = t - twb

Note that time, t, is a nondimensionalized time w.r.t. t e ·

The three coefficients a, b, and c are dependent on the con­trolling parameters.

Next, correlations to predict water cut projections after wa­ter breakthrough were developed. Upon examining watercut performance for various cases, a single functional formwith an independent variable, time, and three coefficientswas devised to represent water cut performance.

The Cg dependency of the coefficients are obtained as fol­lows. For Cgr :::; 1.0,

(47)

(46)()b2(Cgr )

b ex: b1 q b (C )b3 (m)2 gr,base

() C2(Cgr ) ()c ex: C1 q (C ) C3 m

C2 gr,base

0.426Cgr,base =-C

gm

Since at> b1 , and C1 are determined with constant Cg( =0.426),

a2(Cgr ) = -0.01446 + 1.0942e-O.90527C••

b2 (Cgr ) = 45.85469 - 45.520ge-O.101826C••

C2 (Cgr) = 1.18127 - 2.77058Cgr + 4.35034C;r - 2.56542C;r

(39)

(40)

(41)

(38)

b = b(q, Cg, m, hp , ha )

c = c(q, Cg, m, hp , ha )

I ct')Water Cut = (a + bt )(1. - e-

q, Cg, and m dependency For Cgr > 1.0,

We first examine the coefficient dependency on q with theassumption of

aex:a1(q) (42)

b ex: b1 (q) (43)

c ex: C1 (q) (44)

while other variables are fixed (m = 4.146, Cg = 0.426,hp= 0.142, and ha = 2.143). Coefficients a, b, and c versusproduction rates are tabulated in Table 2, and the resultsare regressed with polynomials and exponential functions.

a1(q) = (1. + 0.058245q)(0.755788 - 0.49752 e-S.16409q)b1(q) = 3.9975.10-4 -1.42251·1O-3e-21.0453q,

for q :::; 0.8527

b1(q) = 3.9975.10-4, for q> 0.8527

C1(q) = 4.29719.10-3 -1.25198 .1O-1q + 2.400953q2,

for q :::; 0.0974

C1(q) = 2.93319.10-1 - 5.45488q + 26.6589q2,

for 0.0974 < q :::; 0.162

C1(q) = -7.72884.10-2 + 1.32738q - 2.39082 '1O-1q2,

for 0.162 < q :::; 2.776

C1(q) = 1.76507, for 2.776 < q

a2(Cgr ) = 1.21555 - 0.80422e-4.1248C••

b2(Cgr ) = 0.077663 + 4.7436e-6.6002C••

C2(Cgr ) = 0.012916 + 0.18722e-19.5342C••

Thirdly the mobility ratio dependent terms, a3, b3, and C3were successively determined from simulation results.For m > 4.146,

a3(m) = 1.097858 - 0.33261ge-O.31419m

b3 (m) = 0.859829 +0.256246e-O.12558m

c3(m) = 1.65041-1.26192 e-O.15555m

For m :::; 4.146,

a3(m) = 1 - (0.3785q _1.565)(e-l.036q+3.57 _ e<-l.036q+3.57)m.)

b3(m) = 1 + (-0.59618 + 1.57235q2)(mr _1)2, for q :::; 0.663

b3(m) = 1. + (1.62q - 0.979)(m r -I?, for 0.663 < q :::; 1.14

b3(m) = 1. + 0.867(mr _1)2, for 1.14 < q

c3(m) = 0.6mr + 0.4

where m r = m/4.146

Secondly the coefficient dependencies on Cq were investi­: gated. Simulations with various q and Cg were made and

hp and ha dependency

(a) perforation interval

274

SPE 20742 S. H. Lee and W. B. Tung 7

As in the previous subsection 6.1- of water-breakthroughtime, a modified production rate was introduced to modelthe effect of perforation interval.

(48)

The function "Ip is given by

"IP = 0.32364 + 1.03274e-2.8S048hp, for 0.1429 ~ hp

"IP = 1, for hp < 0.1429

Numerical simulation results reveal that the coefficient C

can be modeled with the modified production rate as

Here,

13 1 + -c.h.b = . Cae

f3c = 1. + cse-ceh•

Cl = -0.3948 - 0.0789(hp - 0.14286)

C2 = 3.0318 - 0.16797(hp - 0.14286)

Ca = 2.2014 - 1.4429(hp - 0.14286)

C4 = 5.46755 - 2.07882(hp - 0.14286)

Cs = -0.16578 - 0.58342(hp - 0.14286)

C6 = 6.2606 + 0.93565(hp - 0.14286)

Coefficient a cannot be modeled only with q* , especially forlarge hp , so that a modifier Q a is introduced.

Q a = 1.0, for hp ~ 0.7667

Q a = 0.966 - 0.3766(hp - 0.857), for hp > 0.7667

Finally, as coefficient b is not successfully modeled with q*,b is directly modeled with a modifier Qb without q*.

c = c(q*,Cg,m)

a = Q a 'a(q*,Cg,m)

Here,

b = Qb' b(q,Cg,m)

For hp ~ 0.1429,Qb = 1.0

and for hp > 0.1429,

Qb = 6.1872 - 5.2816e-O.16129hp

(49)

(50)

(51)

6.3 Relative Permeability and Oil Viscosity Variation

The correlations have been developed with the assumptionof straight-line relative permeability curves and constantfluid properties. In some reservoirs, relative permeabilitycurves can be very different from straight lines, and oilviscosity can also change as pressure changes. Numericalsimulation results showed that relative permeability andoil viscosity variations have substantial impact on break­through time, but minimal impact on water cut develop­ment after water breakthrough. We thus investigated theeffect of nonstraight-line relative permeability curves andoil viscosity variation on water-breakthrough time.

In general, if relative permeabilities do not differ too muchfrom a straight line, the end-point relative permeabilitiesat Sw = Swc and Sw = 1 :..- Sor determine coning phenom­ena. The correlations were developed with the assumptionthat the actual shape of relative permeabilities are not im­portant in coning dynamics. Nevertheless, when oil rel­ative permeability becomes very small, almost practicallyimmobile at Sw << 1 - Sor, the shape effect of relativepermeability curves cannot be neglected.

Since the effect of relative permeability is mostly due to thechange in oil mobility, a modifier for the mobility ratio isdevised to include this effect.

This modified mobility ratio is used only in the correlationsof breakthrough time, not in those of water cut projections.When the mobility ratio is reduced according to the the­oretical front saturation of the fractional flow curve, theagreement between the correlations and numerical simula­tion results improves significantly.

(b) aquifer size

The effect of aquifer size is modeled with a modifier 13. Itis noted that the modifier is dependent on the perforationinterval as well as aquifer thickness. The following function­als are thus proposed to characterize the effect of aquiferthickness on water cut development.

(52)

*m =Qmm (55)

( ) ( *) C2(C;r) ()C= f3c ha, hp Cl q (C* ) Ca m

C2 gr,base

For ha > 1.714,

f3a = f3b = f3c = 1.0

For ha ~ 1.714,

(53)

(54)

275

In some reservoir fluids, oil viscosity varies as pressure isreduced. Since there is a noticeable pressure drop arounda wellbore, oil viscosity variation can affect breakthroughtime as well. This can also be modeled with equation (55)by adjusting the mobility ratio.

6.4 Boundary Conditions at the Drainage Radius

The effect of the drainage radius on coning dynamics was

8 General Coning Correlations Based on Mechanistic Studies SPE 20742

7.1 Breakthrough Time

7. COMPARISON WITH FINITE DIFFERENCE SIMU­LATION RESULTS

To include the effect of the drainage radius in the correla­tions, the breakthrough time twb and the coefficients b andc in equation (36) are modified as

In Tables 3 and 4, the correlations can predict water­breakthrough time for models with a wide range of physicaland geometrical parameters. It proves that the scaling andnondimensionalization processes adopted in the develop­ment of the correlations are theoretically consistent, whichensures the general applicability of the correlations.

A base model was constructed with the physical propertiesin Table 1, the production rate, q = 500 bbls/day, andthe reservoir geometry oj ho = 35 ft, hp = 5 ft, andha = 75 ft. By using the base model, the prediction ofthe new correlations were compared with simulation results.The results are shown in Figure 10 together with the pre­diction of the Chappelear-Hirasaki correlation. Since thedistance between the original oil-water contact line and theperforation interval is small, water breaks through early,followed by a very slow increase in water-cut for a longperiod. The new correlations represent the simulation re­sults extremely well, whereas the Chappelear-Hirasaki for-

7.2 Water Cut Projection

The correlations were also tested with the base reservoirmodel whose physical properties are summarized in Table1. Straight line relative permeability curves and constantoil viscosity were used in this study. In Table 4 break­through time calculated by the correlations are comparedwith simulation results for various production rates, dis­tance between the original oil-water contact line and thebottom of perforation interval, perforation interval size, andaquifer thickness. The correlations provided very satisfac­tory results in which the discrepancy error is less than 15percent.

The dimensional analysis in Section 3 shows that as thepermeability of an isotropic reservoir decreases, the char­acteristic production rate, qe, decreases. The rescaled pro­duction rate, q, increases as a result, which warrants earlywater-breakthrough. In the second case, when only the ver­tical permeability increases, the rescaled production rate qdoes not change, but the characteristic time increases. Sothe breakthrough time becomes linear to the chracteristic.time scale. In the last case, as the distance between the oil­water contact line and the bottom of perforation intervalincreases, the characteristic length increases accordingly.This creates a smaller rescaled production rate and alsodecreases the rescaled aquifer thickness and perforation in­terval. This combined effect prolongs water-breakthroughtime. In Table 3, the estimate of water-breakthrough timefrom the correlations agrees very well with finite differencesimulation results for all the different cases tested.

md, and in the second group the vertical permeability ischanged, while the horizontal permeability is kept constantat 750 md. In the last group the characteristic length (ho)is changed from 20 ft to 80 ft. These conditions representa wide range of physical properties in a reservoir.

(57)

(58)

(56)

b* = 'T/b b

c* = 'T/e C

Here,

The first example is the model reported in Chappelear andHirasaki5 . Since the oil and water viscosities are identi­cal in this model, it is a favorable displacement process(m = 0.25). In Table 3, the predictions of the correlationsare compared with simulation results. In the first group,the isotropic permeability is varied from 100 md to 750

'T/twb = 1, for q > 0.441 or r. > 10

'T/twb = 1- wI(I- (~~)2), for q ~ 0.441 and r. ~ 10

'T/b = W2, for r. ~ 14 or q ~ 1

'T/b = w2(1 - W3) + W3, for r. > 14 and q> 1

'T/e = 1, for q > 0.441

'T/e = 1 - WI (1 - W2), for q ~ 0.441

WI = 5.12(0.1945 _ q2)

14)2W2= (­

r.

W3 = (1 - e2-

2q)

For simplicity, the tilde and hat for dimensionless andrescaled variables are omitted in the above equations. Notethat coefficient a is independent of r •.

In this section the correlations for water-breakthrough timeand water cut profile after breakthrough are compared withsimulation results from finite difference simulation. As pre­viously mentioned, the development of the correlations areintended to mimic finite difference simulation results. Acomparison with field data may not be meaningful due tolarge uncertainties in field properties and measurements. Itwas thus assumed that a comparison with finite differencesimulation solely determines the accuracy of correlations.

studied with various well models. Numerical simulationresults clearly showed that coning dynamics generally be­comes less dependent on boundary conditions at the drain­age radius as the nondimensinalized and rescaled produc­tion rate, q, increases. However, in a model with a lowproduction rate, the development rate of water-cut afterwater breakthrough is directly dependent on the size of thedrainage domain.

276

SPE 20742 S. H. Lee and W. B. Tung 9

relation predicts an instant water breakthrough, and thepredicted water cut is 10 to 20 percent lower than the sim­ulation results.

In order to validate the scaling and characteristic vari­ables in the correlations, the correlations were tested witha model which is geometrically twice as large as the previ­ous one. The results are drawn in Figure 11. In this en­larged model, water breaks through early as well, but thestabilized water-cut is much smaller than the previous case.This phenomenon is well predicted by the new correlations.The Chappelear-Hirasaki correlation completely misses thetrend of simulation results. This comparison also validatesthe scaling and nondimensionlization techinque employedin Sections 2 and 3.

The correlations were also examined with reservoir modelswith different perforation intervals and aquifer thickness.A model with hp = 40ft and ha = 20ft was constructed.The completion interval (hp ) is larger than the character­istic length (ho) for this model. The comparison betweensimulation results and correlations is depicted in Figure 12.The new correlations represent simulation results very sat­isfactorily, while the Chappelear-Hirasaki correlation con­sistently predicts water cut 20 to 30 lower percent than thesimulation results.

In Figure 13 the water cut performance from the correla­tions and simulation results are plotted for various produc­tion rates. The distance between the bottom of the perfo­ration interval and the original oil-water contact line is 35ft in this model. Because the characteristic length ho is rel­atively small, we expect an immediate water breakthroughand a quick stabilization of water cut. This is confirmedby simulation results in Figure 13. The correlations alsoreveal the same characteristics, and the agreement betweenthe simulation and correlations appears to be very good. Amodel with twice the size of the previous one was also inves­tigated. The comparison is shown in Figure 14. The watercut performance is very similiar to the case in Figure 13,because they are basically identical except the geometricalSIze.

8. IMPLEMENTATION IN A FINITE DIFFERNCE SIM­ULATOR

The coning correlations were developed based on a simplephysical model described in Section 2. In order to applythe correlations in a finite difference simulator, they shouldbe extended to a more general physical model. Fortunatelyit can be accomplished by employing appropriate approx­imations. In implementing the coning correlations into afinite difference simulator, four main technical issues haveto be resolved: (1) the effect of changes in the productionrate, (2) the evaluation of effective physical properties (i.e., permeability, porosity, and saturations) for a multi­layered reservoir model, (3) the distribution of producd flu-

277

ids among layers in a multi-layered reservoir model, and (4)the effect of the well-block size.

Production rate changes

Often the production rate of a well varies, depending onthe operational conditions and market demands of oil. Inreservoir simulation, the production histol'Y is usually incor­porated as realistically as possible. In Section 5, the coningcorrelations were developed based on a constant liquid pro­duction rate. It is necessary to show here how the corre­lations can be utilized in reservoir simulation with varyingproduction rates.

For a fixed rate of total fluid production, the water satura­tion change in the drainage volume can be derived as

LlSwV</> = itq(l- WCUT(t,q))dt (59)

Here, WCUT is the water cut from the correlations; LlSwis the averaged water saturation change in the drainagedomain; V is the total volume; and </> is the porosity.

In most applications, when the production rate is changed,the cone quickly alters its shape in order to attain anew quasi-steady shape under the current production rate.Therefore, we assume that the cone shape is dependent onlyon the current production rate and the fluid saturation inthe drainage domain. This assumption allows us to extendthe coning correlations to a well with varying productionrates.

From current and initial cell saturations, one can readilyobtain the water saturation change in a drainage domainaround a producing well. The time needed to achieve thewater saturation change with the current production ratecan be computed from the above equation. Let us denotethe calculated time by tt. Substituting q and tt into theconing correlations will readily yield the estimate of currentwater-cut.

Effective physical properties

Instead of deriving different coning correlations for a multi­layered, heterogeneous reservoir model, we conjecture thatthe current coning correlations can be still applicable if aheterogeneous model can be mapped into a homogeneousmodel by effective physical properties.

In general, vertical flow goes through each layer sequen­tially, and horizontal flow comes in parallel from all thelayers. From this physical intuition, an arithmetic averagewas chosen as an effective horizontal permeability, and aharmonic average as an effective vertical permeability.

Models with the detailed layer heterogeneities and witheffective permeabilities were studied by a finite differencemethod. The water cut performance generally agreed witheach other to confirm the above simple conjecture. There-

10 General Coning Correlations Based on Mechanistic Studies SPE 20742

fore, the correlations can be adequately utilized for multi­layered models, if effective permeabilities are employed.Other physical properties are not very important as longas they are not too different among layers. An arithmeticaverage appears to be proper for other physical properties.

Distribution of liquids among layers

The coning correlations calculate oil and water productionrates. The completion block in finite difference simulationmay not contain enough water to meet the water produc­tion of the correlations. A mechanism should be devised toproduce water from the grid blocks which contain enoughwater to satisfy the water-oil ratio from the correlations.

This difficulty can be circumvented if production fluids areallocated based on the potential difference and availablityof fluids from the completion cells as well as from all thecells below the completion. The production of each layer isproportional to the potential difference between well-blockand well bore and also the productivity index of the layer.In essence, the simulator produces fluids from all the layersin which the well is located, independent of the actual lo­cation of completion intervals. Because one does not haveenough information in distributing liquids from a coarsegrid simulation, this is the only logical way of fluid distri­bution.

Well-block size

The coning correlations are weakly dependent on the drain­age radius. If the drainage radius is large, the cone canbe established without interference from the presence ofthe drainage boundary. The effect of a small drainage ra­dius is included in the coning correlations through modifiersTJtwb, TJb, and TJe in equations (56) to (58).

eral applicability. For all the models tested, the water­cut projections from the correlations were consistently ingood agreement with simulation results. In comparison,the Chappelear-Hirasaki correlation considerably deviatedfrom the simulation results in many tested cases.

The new correlations can be used in a stand-alone pro­gram or in reservoir simulation. With a stand-alone pro­gram of the correlations, one can study the effects of pro­duction rate, vertical and horizontal permeabilities, wellcompletion location, physical properties of fluids, aquiferthickness, completion interval size, and drainage radius onconing dynamics. The correlations can also be utilized inoptimizing production schemes and planning constructionof water treatment facilities.

The correlations can be incorporated in a reservoir simula­tor for field-scale simulation studies. In field-scale reservoirsimulation, correlations are often employed to model sub­grid scale flow behavior near a well. In regard to imple­menting correlations into a finite difference simulator, fourtechnical issues were resolved: (1) the effect of changes inthe production rate, (2) the evaluation of effective physicalproperties for a multi-layered reservoir model, (3) the dis­tribution of producd fluids among layers, and (4) the effectof the well-block size.

ACKNOWLEDGEMENT

We would like to thank E. Y. Chung, R. M. Toronyi, W.R. Coffelt, S. T. Obut, and D. White for their valuablesuggestions, and D. Teshome for assistance in preparing thefigures used in this memorandum. This work was supportedin part by the Saudi Aramco Petroleum Engineering StudyProgram.

NOMENCLATURE

If a well-block size is larger than twice the drainage radius,the actual drainage radius is of no significance in applyingthe coning correlations. The well block can be consideredthe drainage boundary in computing saturation changes inthe drainage domain. When a well-block size is smaller thantwice the drainage radius, all the neighboring cells withinthe drainage radius should be included in computing thesaturation change in equation (59).

9. CONCLUSIONS

A theoretical analysis was employed to determine the char­acteristic variables and key reservoir, production, and wellcompletion parameters in water coning. By using a system­atic approach to identify functional dependencies of the keyparameters, new coning correlations were developed. Theyconsist of two parts: one for water breakthrough time esti­mation and the other for water-cut development after waterbreakthrough.

The new coning correlations were extensively comparedwith numerical simulation results to validate their gen-

278

Gg :

Ggm :

Ggr :

Ie:g:ho :

ht :

ha :

hp :

K:ke :

kh :k r :

kv :

Ie :m:

p:Pc:

gf:.pIe/Pedefined in equation (33)defined in equation (34)cone shape functiongravitydistance between the original oil-water contactand the bottom of perforation intervalformation thickness above the top of perforationaquifer thicknessperforation interval sizepermeability tensor of the porous mediacharacteristic permeabilityhorizontal permeabilityrelative permeabilityvertical permeabilitycharacteristic lengthmobility ratiopressurecharacteristic pressure

SPE 2074:1 S. H. Lee and W. B. Tung 11

2. Arthur, M. G.: "Fingering and Coning of Water and Gasin Homogeneous Oil Sand", Trans., AIME(1944), vol. 145, ;pp 184-199.

3. Meyer, H. I. and Garder. A. 0.: "Mechanics of Two Im- 'miscible Fluids in Porous Media", J. Appl. Phys. (1954),vol. 25, pp 1400-1406.

4. Schols, R. S.: " An Experimental Formula for the CriticalOil production Rate", Erdoel-Erdgas Z., (January 1972), 'vol. 88, pp 6-11. ;

I

5. Chappelear J. E. & Hirasaki G.J.: "A Model of Oil-WaterConing for Two-Dimensional, Areal Reservoir Simulation" ,Society of Petroleum Engineering Journal( April 1974), pp65-72.

6. Wheatley, M. J.: " An Approximate Theory of Oil/WaterConing", SPE 14210, the SPE 60th Annual Technical Con­ference (1985).

7. Hoyland, L. A., Papatzacos, P., & Skajaeveland, S. M.:"Critical Rate for Water Coning Correlation and Ana­lytical Solution", The SPE European Petroleum Confer­ence(1986), SPE 15855, pp 59-70.

8. Abass, H. H.: "The Critical Production Rate in Water­Connig System", SPE 17311, the SPE Permian Basin Oiland Gas Recovery Conference (1988).

9. Brebbia, C. A. : Progress in Boundary Element Methods,Vol 1, John Wiley & Sons (1981), Vol. 2, Prentech Press(1983).

10. Geller, A. S., Lee, S. H., & Leal, L. G.: "The CreepingMotion of a Spherical Particle Normal to a Deformal Inter­face", J. Fluid Mech.(1986), vol. 169, pp 27-69.

11. Liggett, J. A. & Liu, P. L.-F.: The Boundary Integral Equa-tion Method for Porous Media Flow, George Allen & Unwin(1983).Lee, S. H.: "Analysis of Productivity of Inclined Wells andIts Implication for Finite Difference Reservoir Simulation" ,SPE Production Engineering (May 1989), pp 173-180.

gradient operatordifference operatorfluid viscosityfluid densityporositypotential

production rate.radial coordinatedrainage radiusratio of actual front velocity and Darcy velocitywellbore radiussaturationresidual oil saturationconnate water saturationtimewater-breakthrough timevelocity vectoraverage cone development velocitycharacteristic velocityaxial coordinate

Subscriptn: normalw: water0: oil

Superscriptdimensionless variablesrescaled variables

V:~:

p.:p:<jJ:<1>:

U c :

z:

REFERENCES

r e :

rf :r w :

S:Sor:Swc :t :twb :u:u:

q:r:

1. Muskat, M.: The Flow of Homogeneous Fluids through 12.Porous Media, McGraw-Hill (1937), New York.

Table 1. Physical Properties of the Base Model Table 2. Coefficients a, b, and c vs. Production Rate.

p.o= 6.526 (cp)p.w= 0.57 (cp)po= 0.849pw= 1.122krw= 0.372kro= 1.00<jJ= 0.28Swc= 0.08Sor= 0.14kh= 2100 (md)kv = 1050 (md)

-q- ---!L.. _b_ _c_

0.106 0.476 2.26.10-4 0.06480.162 0.521 3.64.10-4 0.1110.170 0.564 3.76.10-4 0.1460.213 0.611 3.31 .10-4 0.1980.426 0.715 4.33.10-4 0.4720.853 0.788 3.87.10-4 0.8651.705 0.831 3.80.10-4 1.493

279

Table 3. Breakthrough Time Comparison With Simulation Results: Table 4. Breakthrough Time Comparison WitlSflEtion2'O:7 4 2the Model in Chappelear and Hirasaki(1973). the Base Model, k./kh =0.5, kh =2100md.

k. = kh (md) simul.(days) corrl.(days) q ho hp h. tw.(siml) tw.(corrl)

750. 11.4 11.1(bbls/day) (tt) (tt) U!l (days) (days)

300. 8.6 7.1500. 35. 40. 20. 20. 17.

100. 6.04 6.1

1000. 70. 10. ISO. 11. 11.

k.(md) [kh = 750J simul.(days) corrl.(days)70. 20. ISO. 21. 18.70. 40. ISO. 54. 48.

750. 11.4 11.1375. 26.8 22.3 1345. 95. 10. 200. 37. 31.75. 101.5 110.9 95. 25. 200. 75. 58.

95. 50. 200. 180. 156.

k. = kh =750. 2000. 95. 10. 200. IS. 12.5ho (tt) simul.(days) corrl.(days) 95. 25. 200. 29. 23.

20. 1.5 0.795. 50. 200. 67. 61.

40. 11.4 11.180. 397. 304.

~..... r .----------tl..~

(DI)

t---__.:z = fc(r)

(D)

cp const

(I)

Figure I. Schematic Diagram of Coning Phenomena

0.8

0.7 TltoESTEP

- 110.6

I 1085." 0.5

~ t 166

8 t 226i.5 1340- t 502

2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

r (radial COOIdinate )

Figure 2. Interface Shape Calculated by a Boundary Integral Method.

1.1.-------------------------------,

q.O.853

q.0.108

q.O.426

q.0.213

10 r-----------------,

0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9 1.0

CgFiguwl. Cone Development Velocity: m=4.146, hp = 0.142, ba = 2.143

800700600400

Time Step

Figure 3. The Position of the Cone Apex Calculated by a Boundary Integral Method

0.1

o.oE---.......~-........-~~ .......~-.......---......-~.......---......-~..Jo 100 200 300

280

1.0r-------------------,2.2.----------------------,.,......--.

2.0 SPE 207 4 20.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 IG._:"~_:_"_--'----J'---'--........-'--:"-....L.----l0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

qm

--- q.O.3115

q.0.4672

q.0.6229

q .1.246

Figure 5. Cone Development Velocity at Cg= Cgm Figure 6: The Effect of Mobility Ratio on Cone Development Velocity

0.80.60.4

Cg

0.2

0.6

\\\

0.4 \" q=O.426

"

':\0.2 ~~=O.213 \, \, .

••••~q=O.106oL_·..::··:,,·._'...J::~_·....=.::·-::b _L.::::::::::o_...,j

o

cq=O.856

"

\\. q=O.426\\

\ \\ .\q=O.213 , .......

" '-q=O.106

00'----'-------:-'-------::'-----'0.2 0.4 0.6 0.8

Cg

0.8

1.6

1.2

0.4

.10-3

2.0r---------------,

b

0.80.6

IIIII q =O.213,,,,"

\\! .:\ '.:\ '.. ,: \ 'Ii ,:\ i\.' /: \ : '-l: \i , ...../: 'i ,~ i\ I~.: \}

0.2 0.4

Cg

1.0

0.8

a 0.6

0.4

0.2

0

Figure 7

Parameter a vs. Cg.

Figure 8Parameter b vs. Cg.

Figure 9Parameter c vs. Cg.

1.0.-----------------------------,

0.9

0.8

0.7 c----===------------------------------0.6

0.5

0.4 Simulation

0.3 New Correlation

C-H Comlation

0.1

Time (days)

Figure 10. Comparison of Cortelations and Simulation Results:q =500bbls/day, ho = 35ft, bp = 5f!, ha = 75ft, Kv = 1050md, Kh =2100md

281

C-H Correlation

Simulation.

New Correlation

--------------------------------------------­

""",

" ..."0.1

0.0O....~~~-,0..0~0-~~~2..0...00~--~3~0.L0~0~-~-40..0-0-~~~5..0...00~~-~8...JOO0

Time (days)

1.0

0.9

0.8

0.7

0.6

8 0.5

~

6000500040003000

Time (days)

20001000

1.0

0.9

0.8 Simulation

0.7 New Correlation

0.8 C-H Correlation

8 0.5

J

Figure II. Comparison of Correlations and Simulation Results:q z 5OObblslday, ho = 70ft, hp =10ft, ha =150ft, Kv =10SOmd, Kh =2100md

Figure 12. Comparison of Correlations and Simulation Resu1ts:q = 5OObblslday, ho = 35ft, hp =4Oft, ha=20ft, Kv =105Omd, Kh = 2100md

1.o,,------------------,

0.9"..,..,,,-__----=..~~"X1j,,-'I .." _

[bbIsIdayJ1.0.-------------------....

[bbIIIday]

0.9

__--...--...-......--.........- ...."W'.-.•..- ...........--.

q = 500, Simulation

q = 250, Simulation

q = 2000, Correlation

q =: 1000, Correlation

q = 500, Correlation

q • 250, Correlation

q =1000. Simulation

q = 2000, Simulation

60005000400030002000

...w ...w~w ~w~._~

,:"/"--------------------_..,

1000

0.8

q =!Ooo, Correlation

q = 500, Correlation

q = 250, Correlation

q =125, Correlation

q = 250, Simulation

q =125, Simulation

-- q =!Ooo, Simulation

-- q = 500, Simulation

600050004000300020001000

Tnne(days)

Figure 13. Comparison of Correlations and Simulation Results:ho =35ft, hp = 5ft, ha =75ft, Kv = 105Omd, Kh = 2100md

Time (days)

Figure 14. Comparison of Correlations and Simulation Results:ho = 70ft, hp = 10ft, ha = 150ft, Kv = 10SOmd, Kh = 2100md.

282