spe 38290 linear transient flow solution for primary oil recovery

Copyright 1997, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the 1997 SPE Western Regional Meeting held inLong Beach, California, 25–27 June 1997.

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AbstractWe present analytical solutions for primary production,producer infills and early response to waterflood in a lowpermeability, compressible, layered reservoir filled with oil,water and gas. The sample calculations are for the CaliforniaDiatomites, but the equations apply to other tight rock systems.Primary oil recovery from rows of hydrofractured wells in alow permeability reservoir is described by linear transient flowof oil, water and gas with the concomitant pressure decline.During primary, it may be desirable to drill infill wells toaccelerate oil production. At some later time, the infill wellsmay be converted into waterflood injectors for pressuresupport and incremental oil recovery. In this paper, we analyzethe pressure response and fluid flow rates due to the originalwells and infill wells drilled halfway between the originalwells, and - finally - due to water injection at the infill wells.All of the formation and fluid properties are described by asingle hydraulic diffusivity, α, assumed to be independent oftime and production or injection. We solve the one-dimensional pressure diffusion equation analytically usingpressure boundary conditions at the original and infill wellsand use superposition to account for the water injection. Wegive solutions for the pressure in the formation and discusshow to calculate oil, water and gas rates and cumulatives asfunctions of time at both the original wells and infill wells.Finally, we present a computational example of oil productionfrom a stack of seven diatomite layers with different propertiesand show the effects of infill wells and water injection on thetotal oil production. We show that results of this analyticalsolution and a compositional numerical simulation for primaryproduction in the diatomite agree well. Our analysis can

predict the onset of pressure depletion and quantify how longto produce from the infill wells before injecting water. Itshows that producing from the infill well for a few yearssignificantly increases the production from the field and canminimize the lost production at the infill well due toconversion to a waterflood injector. Our analysis alsogenerates very reliable, well-by-well, field-wise forecasts offluid production and water injection.

IntroductionThe late and middle Miocene diatomaceous oil fields in theSan Joaquin Valley, California, are located in Kern County,some forty miles west of Bakersfield. An estimated original-oil-in-place in the Monterey diatomaceous fields exceeds 10billion barrels and is comparable to that in Prudoe Bay inAlaska.

Cyclic bedding of the diatomite1 is a well documentedphenomenon, attributed to alternating deposition of detritusbeds, clay, and biogenic beds. The cycles span length-scalesthat range from a fraction of an inch to tens of feet, reflectingthe duration of depositional phases from semi-annual tothousands of years. On a large scale, there are at least sevendistinct oil producing layers with good lateral continuity withineach layer, but little vertical continuity between adjacentlayers. The diatomaceous rocks are very porous (25-65percent), rich in oil (35-70 percent), and nearly impermeable(0.1-10 millidarcy). The high porosity and oil saturation,together with large thickness (up to 1000 feet) and area (up toa few square miles per field) translate into the gigantic oil-in-place estimates.

To compensate for the low reservoir permeability, all wellsin the diatomite must be hydrofractured. A typical well has 3-8fractures with an average fracture half-length of 150 feet.Wells are usually spaced along lines following the maximumin-situ stress every 330 feet (2-1/2 acre), 165 feet (1-1/4 acre)or even 82 feet (5/8 acre). Thousands of hydrofractures havebeen already induced and thousands more may be created asnew recovery processes, such as waterflood2-4 or steam drive5-6

on 5/8 acre spacing, become commercially viable.Primary oil production on 2-1/2 acre spacing, followed by

infill to 1-1/4 acre and subsequent conversion to waterflood isof great interest to the producers of the diatomaceous oil

SPE 38290

Linear Transient Flow Solution for Primary Oil Recovery with Infill and Conversion toWater InjectionEric D. Zwahlen, Lawrence Berkeley National Laboratory, and Tad W. Patzek, SPE, University of California, Berkeley

SPE 38290 LINEAR TRANSIENT FLOW SOLUTION FOR PRIMARY OIL RECOVERY WITH INFILL AND CONVERSION TO WATER INJECTION 2

fields. Fig. 1 shows a schematic of the production and infillprocess. We start from the mathematical formulation of theproblem and present analytical solutions. We then present acomputational example of a seven-layer diatomite reservoir.We also compare this analytical solution for primaryproduction to a 1-D compositional reservoir simulation.Finally we match, well-by-well, the oil production and waterinjection in a waterflood project and perform rank orderstatistics.

Problem StatementIn a compressible, homogeneous porous medium, the pressuredistribution follows a simple diffusion equation. With suitableboundary conditions and an initial condition, the pressure andfluid velocity in the medium can be calculated analytically. Inthis paper we present the governing equations and somegraphical results for primary production, infill production, andfinally water injection in the infill wells. See ref. 7 for a morecomplete derivation and listing of the solution equations.

We begin with a reservoir at some uniform initial pressurepi at t = 0. First, we drill a series of wells with spacing 2L andwellbore flowing pressure pwell as shown in Fig. 2. All wellsare hydrofractured, and all of the fractures are rectangular andhave permeabilities that are much higher than the formationpermeability. Therefore, we can assume that the uniformpressure pwell is imposed throughout the entire hydrofracture.Second, at some time tinf, we drill infill wells halfway betweenthe original wells and also produce these wells at pwell. Third,at time tinj, we inject water into the infill wells at the downholeinjection pressure pinject and continue producing at the originalwells. This final step is to quantify the effect of re-pressurization of the formation.

This statement of primary production, followed by infilland injection, is a simplification of what actually occurs inpractice. Here we assume uniform and constant properties ineach layer, constant pressures in the wells, and we neglect theeffect production and injection may have on fluid and rockproperties. We assume each of the layers in the reservoir isindependent and solve the problem for each layer separately.These assumptions allow us to derive an analytical solutionthat can teach us the effect of each of the parameters on theproduction.

Original Primary Production. The one-dimensional pressurediffusion equation is

∂∂

α ∂∂

p

t

p

xx L t= ≤ ≤ >

2

2 0 0, , .....................................(1)

where

α λφ

= t

tc............................................................................(2)

is the hydraulic diffusivity which accounts for the totalcompressibility of the formation, ct , and the total fluid

mobility, λt. All of the formation and fluid properties are

combined into the single constant parameter α. For largervalues of α, the pressure response will be faster. Asmentioned, we assume α remains constant during productionand injection.

From symmetry, we write the equations only for 0 ≤ x ≤ L,while the original wells are at x = ±L. The initial condition isuniform pressure everywhere in the layer,

p x p x Li( , ) ,0 0= ≤ ≤ ................................................... (3)

The boundary conditions for primary production are

∂∂p

xt

p L t p

t t

well

inf

0 0,

,,

a fa f

=

=

UV|W|

< ................................................. (4)

Before the infill time, the symmetry halfway between theoriginal wells at x = 0 requires a no-flow boundary condition,which is specified by the gradient of the pressure being equalto zero. The pressure at the primary production well at x = L isspecified as the well flowing pressure.

This system of equations can be solved by separation ofvariables.8 The zero gradient condition at x = 0 leads to acosine expansion of the initial condition. The pressure beforeinfill is

p x t p

p px L

e

x L t t

well

i welln n

n

t L

n

inf

n

,

( )cos /

,

,

/

a f

b g b g=

+ − −

≤ ≤ < <

−

=

∞

∑2 1

0 0

2 2

0

λλ

λ α ........... (5)

where

λ πn n n= + =2 1

20 1 2a f , , , ,K ...................................... (6)

This cosine series clearly shows that the boundaryconditions before the infill time are satisfied. The timedependence of the pressure is controlled by α divided by L2.

We note that solving the problem by Laplace Transformleads easily to an error function solution better suited for earlytimes

p x t p p p

n x L

t L

n x L

t L

x L t t

well i well

n

n

inf

,

/

/

/

/

,

a f b g

a f a f a f= + − ×

− − + − + + +RS|T|

UV|W|

L

NMM

O

QPP

≤ ≤ <

=

∞

∑1 12 1

2

2 1

2

0

2 20

erfc erfcα α

(7)

Primary After Infill. At the infill time, an infill well is drilledat x = 0, and the pressure is specified at both boundaries of thesystem. This set of boundary conditions leads to a sine seriesexpansion of the original cosine series. We set t = tinf in (5) anduse the result as the initial condition for a new set of sideconditions to solve (1). The initial condition is


p x t p p px L

e x Linf well i welln n

n

t L

n

n inf, ( )cos /

,/d i b g b g

= + − − ≤ ≤−

=

∞

∑2 1 02 2

0

λλ

λ α(8)

The boundary conditions are specified as constant pressureconditions by

p t p

p L t pt t

well

wellinf

0,

,,

a fa f

=

=UVW

≥ ..................................................(9)

Again by separation of variables, the solution to (1) isgiven by

p x t p

p p x Lee

x L t t

well

i well m mm

t t Ln t L

n m nn

inf

m infn inf

,

sin /( )

,

,

//

a f

b g b gd i

d i

=

+ − −−

≤ ≤ >=

∞ − −−

=

∞

∑ ∑41

0

12 2

0

2 22 2

β βλ β λ

β αλ α

(10)

where

β πm m m= =, , , ,1 2 3 K .........................................(11)

Oil Flow Rate Before Infill. The oil flow rate is proportionalto the pressure gradient in the reservoir. The oil flow rate atthe original primary wells qo

( )1 is given by

q Akk p

xoro

o x L

( )1 = −=µ

∂∂

.....................................................(12)

where A is twice the area of the production wellhydrofractures, with the factor of 2 coming from symmetry(there are actually two production wells or, alternatively, twosides of a hydrofracture). The superscript (1) refers to theoriginal wells. Differentiating the pressure solution in (5) andusing the definition of the flow rate gives the flow rate beforeinfill

q Akk p p

Le t to

ro

o

i well t L

ninf

n( ) / ,1

0

2 02 2

=−

≤ ≤−

=

∞

∑µλ αb g

......(13)

Differentiating (7) gives a form of the flow rate equationwhich is better suited for early times and clearly shows thesquare root of time dependence

q Akk p p

te

x L t t

oro

o

i well n

n

t L

n

inf

( ) / ,

,

1

1

1 2 1

0

2

2

=−

+ −

RS||

T||

UV||

W||

≤ ≤ <

−FHG

IKJ

=

∞

∑µ πααb g a f

......(14)

For early times, the summation is negligible because theexponential terms are approximately zero.

Oil Flow Rate After Infill. After the infill well is drilled, oilis produced from both the original primary well and the newinfill well. After infill, the oil flow rate can be obtained bydifferentiating (10) and using (12) at x = 0 for the infill well

and x = L for the original well.

Cumulative Oil Production. We are interested in the totalamount of oil that is produced from the original wells and theinfill wells. The cumulative oil production up to some time t isgiven by

Q q dt Akk p

xdto o

t ro

oxor

x L

t= ′ = − ′z z =

=

0 00µ∂∂

............................. (15)

In the integration, we use (5) up until the infill time and(10) after the infill time. The pressure gradient from (5) is zeroat x = 0 until the infill time so there is no contribution to theproduction at x = 0 until the infill well is drilled.

Water Injection at Infill Well. Finally, we investigate theeffect of water injection at the infill well in order to re-pressurize the formation. We admit this approach is onlyapproximate for water injection as it neglects the effect ofincompressible Buckley-Leverett displacement of the oil bythe injected water, as well as water imbibition. We assume therock and fluid properties continue to be the same as originallypresent in the formation, and will use the same α. We thencalculate the pressure in the system, the rate of injection andcumulative injection of water, and the rate and cumulativeproduction of oil at the original well.

Because of the linearity of the equations, this waterinjection problem can be solved by superposition. We continueto calculate the pressure, flow rate, and cumulative productionat the infill well using the equations previously discussed. Thetotal pressure, injection, and production will be the sum of theprevious infill problem and the following injection problem.The equations for the water injection calculation are

∂∂

α∂∂

p

t

p

xx L t tinj inj

inj= ≤ ≤ >2

2 0, , ........................ (16)

For simplicity, we use the same hydraulic diffusivity α inthis injection problem as in the previous infill problem todescribe the compressibility of the formation and the fluids.The initial condition is

p x t p x Linj inj well( , ) ,= ≤ ≤0 ..................................... (17)

This states that the initial pressure for this injectionsuperposition calculation is the well flowing pressure.

The boundary conditions are constant pressure:

p t p

p L t pt t

inj inject

inj wellinj

0,

,,

a fa f

=

=

UV|W|

> .......................................... (18)

The pressure at the infill well (now an injector) isprescribed as pinject and the pressure at the original productionwell is prescribed as pwell.

We define a normalized pressure that scales the injectionpressure to the original formation pressure


pp p

p pinject well

i well

* =−

−.........................................................(19)

The solution for the pressure in the formation due only towater injection is a sine series

p x t p

p p px

L

x Le

x L t t

inj well

i wellm

m

t t L

m

inj

m inj

,

sin /,

,

* /

a f

b g b g d i

=

+ − − −RST|

UVW|

≤ ≤ >

− −

=

∞

∑1 2

0

2 2

1

ββ

β α(20)

We differentiate this pressure solution with respect to x toget the water flow rate in the absence of oil pressure as in (12).The flow rate at x = 0 is the water injection rate, and the flowrate at x = L is production at the original well. We can thenintegrate this injection rate solution over time using (15) to getthe cumulative water injection, again in the absence of theflowing oil.

Total Pressure, Flow Rates, and Cumulative Production bySuperposition. We are actually interested in the total (or net)pressure, flow rates and cumulative production including theoriginal infill solution and this water injection solution. Wenow present the superposition equations necessary to calculatethe net pressure, flow rates, and cumulative production afterwater injection begins. The linearity of the equations allows usto add the results of the infill problem to the results of theinjection problem to get the total. The following equations arevalid for t > tinf. The superscripts (1) and (2) refer the originaland infill wells, respectively.

The total pressure in the formation is sum of the pressurecalculated by the original solution for p and the solution for pinj

p p p p t tnetinj well inj= + − >, ..................................(21)

The net oil production rate at the original well qonet( ),1 is the

sum of the oil flow rates from the infill problem and theinjection problem

q q q t tonet

o o inj inj( ), ( )

,( ) ,1 1 1= + > ..................................(22)

Similarly, the net cumulative oil production at the originalwell is

Q Q Q t tonet

o o inj inj( ), ( )

,( ) ,1 1 1= + > ..................................(23)

The net rate of water injection is given by the waterinjection rate from the injection problem minus the oilproduction rate calculated from the original infill problem,

q q q t tw injnet

w inj o inj,( ),

,( ) ( ),2 2 2= − > ..................................(24)

The net cumulative production at the infill well is slightlymore complicated. After water injection begins at the infillwell, there is no more oil production from this well. We letQo inj,

( )2 be the cumulative oil production at the infill well up

until the time at which water injection begins. Then the netcumulative water injection is given by

Q Q Q Q t twnet

w inj o o inj inj( ),

,( ) ( )

,( ) ,2 2 2 2= − − >d i ................... (25)

Computational Example – Single WellAs an example, we model an average well in Section 33 of theSouth Belridge diatomite field. This region can be divided intoseven separate layers (diatomite cycles), each with its ownmaterial and fluid properties. For each layer, we assume thatthe initial producer spacing is 330 feet (2-1/2 acre), and thetip-to-tip length of the hydrofracture is also 330 feet. Theimportant properties of each layer are summarized inFigs. 3-5. These data are averages of well logs taken at 1-footintervals through the reservoir column.

Fig. 3 shows the amount of oil in each layer calculated asS hoφ , which when multiplied by the pattern area gives theOOIP. The depths and thicknesses of each layer are shown onthe vertical axis. Layer G is the thickest and has a high oilsaturation giving it a large volume of oil. Layer K is very thin,so it has little oil to produce. The remaining layers are betweenthose extremes.

Fig. 4 shows the formation absolute permeability for eachlayer, again as a function of depth. The permeability starts at0.15 md in the upper layers, is slightly higher towards thebottom, and jumps to 0.85 md in layer M.

More important than the absolute permeability is theabsolute permeability times the oil relative permeability, kkro

shown in Fig. 5. We use the Stone II model with reasonablecoefficients and exponents to calculate the oil relativepermeability. Large values of kkro are desirable so layers G andK are good, while layer I is obviously poor.

To get a quantitative estimate of the productivity of a layer,we must calculate its hydraulic diffusivity, α. We use generalcorrelations9-11 for the viscosity, mobility, and compressibilityand give the results in Fig. 6.

Layer K has the highest α and will react the fastest andproduce well. However, it is a thin layer with little oil in placeso the total volume will be small. Layers G, J, L, and M haveintermediate values of α and good amounts of oil, so they willbe good producers. Layers H and I have the lowest values of αand will produce poorly and show little effect of infill andwater injection.

We assume that the layer pressure corresponds to the oilbubble-point pressure for a particular layer. The bubble pointpressure as a function of depth is given by

pbp = + ×29 7 0. .438 depth(ft), psia ............................. (26)

The layer temperature is calculated from the averagethermal gradient for the diatomite which is given as

T = + × °72 0 0 024. . depth(ft), F .................................. (27)

For the calculations we impose a back-pressure of 50 psiaon the producers. The original wells produce for 10 years(3650 days) after which an infill well is drilled between theoriginal wells. After one year of production from the infillwell, at 11 years (4015 days), water is injected at pinject into theinfill well, where


pinject = ×0 7. depth(ft), psia ........................................(28)

These parameter values are used to calculate the pressure,oil flow rate at the wells, and cumulative oil production fromeach layer as a function of time for 50 years (18250 days =135 day1/2).

Fig. 7 shows the cumulative primary production inthousands of barrels in each layer and the total for 50 yearswithout infill or injection. The slope of the total primaryproduction is approximately 3000 barrel/day1/2, which isrepresentative of a very good well in the diatomite. A poorerwell might not have the entire hydrofracture area open and theproduction would be less.

Layer G, which has the most oil in place and high α,produces the most oil. The next two layers, H and I, have lowoil saturations and low α values, and are poor producers. Thedeeper layers have high α values and are good producers.Layer K has a low cumulative production because it has littleoil in place.

Fig. 8 shows the percent oil recovery of all of the layersand the total for 50 years. The total is recovery is about 10percent with the deeper layers J through M better, and the toplayers G through I worse.

Fig. 9 shows the pressure profiles for infill at 10 years andwater injection at 11 years for layers G, I, K, and M. Theselayers show representative values of α with layer I being thelowest and layer K the largest. Large values of α give fasterpressure response. In these figures, the position labeled 0 feetis the position of the infill well, and the position at 165 feet isthe original well. The pressure at the original well is constantat 50 psia.

In Fig. 9a, the initial pressure in layer G is 350 psia and thepressure at the “original” well is maintained at 50 psia. Thepressure profile at x = 0 is horizontal until 10 years indicatingthe no-flow (or symmetry) boundary condition. At 10 yearswhen the infill well is drilled, the pressure at x = 0 has notdecreased significantly. The 11th year profile is just beforeinjection at 513 psia begins at the infill well. Finally, thepressure distribution becomes a linear, steady-state profilewith injection.

Layer I, shown in Fig. 9b, reacts much more slowly thanthe other layers. The initial pressure is 480 psia. By 10 yearswhen the infill well is drilled, the pressure wave is only about aquarter of the way in from the original well. Even at 50 years,the steady-state is not yet reached. The infill and injection doesnot significantly affect the original well.

Layer K, shown in Fig. 9c, has the highest α, and thepressure wave reaches the infill well position at 5 years. By 10years the pressure has dropped enough so that oil production atthe original well is also falling. After injection begins, thesteady-state is nearly reached in 15 to 20 years.

Layer M, shown in Fig. 9d, appears very similar to layer G.Even though layers G and M have different properties, their αvalues are similar which results in similar pressure responsetimes.

Fig. 10 shows the percent oil recovery or water injectionfor infill at 10 years and water injection at 11 years for eachlayer in the formation calculated as the volume produced orinjected divided by the original oil in place times 100 percent.The final total increase in recovery due to injection is about3 percent. The scale of each figure, except layer K, is the sameso that the layers can be easily compared.

Each of the plots has five separate curves. The top curve,shown as a bold solid line, is the percent oil recovery from theoriginal well, including the effect of the infill well and waterinjection. The upward trend at late times is due to the extra oilproduced by water injection. The normal solid line is thepercent oil recovery at the original well with infill but withoutinjection at the infill well. In most of the layers, the differencebetween recovery with and without injection is about 3percent. In layer K the effect is rapid and large, because of thehigh α. However, we stress that much of the increasedrecovery in this layer at late times may be just the injectedwater recirculated through the producer. Layers H and I showalmost no effect of the water injection because of low oilsaturation and low α. This calculation shows that in theabsence of Buckley-Leverett banking of oil, the incrementaloil recovery from pressure support by water injection will besmall. Hence in layers with a low oil saturation or unfavorablemobility ratio, one cannot expect a big waterflood response.

The first dotted curves show the percent oil recovery at theinfill well where production starts from zero at the infill time.The “Infill (no injection)” curves show the production if therewere no injection, and the “Infill (with injection)” curves showthe production with injection. These latter recovery curves(including water injection) become flat after water injectionbegins because no more oil is produced at the infill well. Formost of the layers, a significant amount of oil production fromthe infill well is lost because of injection. However, injectionmay be necessary to preserve the integrity of the formation anddecrease well failure. Thus, producing from the infill well fortwo to five years before injection begins instead of just onemay be better.

The water injection curve is the volume of water injecteddivided by the original oil in place in the layer times 100percent to keep the units consistent. Even though the injectionbegins at 11 years, the effect at the original wells is not seenuntil almost 20 years. Layer K shows the effect sooner, butlayers H and I show almost no effect of injection even at 50years. The rate of water injection becomes constant at latetimes so the percent water injection rises linearly with time forlate times. However, it appears to bend upward when plottedhere versus the square root of time.

To further explain the figures, we specifically considerlayer G. We see from Fig. 7 that the total volume of oilproduced in 50 years is approximately 100 MSTBO with noinfill or injection, which is about 8 percent recovery. Withinfill and injection, the original well produces just under 9percent or 110 MSTBO, and the infill well produces anadditional 15 MSTBO in the one year of production. The 15


MSTBO could be increased to about 30 MSTBO, if waterinjection began at 13 years instead of 11.

Comparison with THERM SimulatorWe now compare the results of this analytical solution with a1-D simulation done with a reservoir simulator, THERM. Thenumerical simulation is of primary production on 2-1/2 acres.As shown in Figs. 11 and 12, the two analyses give nearly thesame results. The data for these simulations are for a deeplayer, e.g. layer M, but having a moderate permeability andhigh oil saturation. The depth in the analytical solution waschosen to match the initial pressure in the numericalsimulation. The well flowing pressure is fixed at 100 psia inboth cases. The only parameters that are different in the twocalculations are kkro and µo, where kkro = 0.7 md and 0.8 mdand µo = 4.5 cp and 5-6 cp in the analytical and numericalsimulations, respectively.

Fig. 11 compares the percent oil recovery. The numericalsimulation is carried to 36 years and the analytical solution to50 years. Until the very end of the simulation the results aresimilar. The numerical simulation predicts a lower ultimaterecovery, probably because depletion changes the properties ofthe system. Thus α is not actually constant for the entireproduction time. However, the differences are not significantuntil after 30 years.

Fig. 12 compares the average pressure in the layer. Theaverage pressure predicted by the analytical solution fallsslightly slower than that predicted by THERM. This is alsoevidenced by the slightly higher production shown in Fig. 11.Also, near the end of the simulation, the pressure flattens outmore quickly, and the production slows down.

We see that our simple model can accurately predict therecovery and pressures for primary production. The PVTproperties of the general correlations and relativepermeabilities agree well with those predicted by THERM.

Computational Example – Multiple WellsAs demonstrated above, our analytical model can be used tocapture the detailed, layer-by-layer behavior of a single“typical” producer in the diatomite. On the other hand, Fig. 8shows that while some reservoir layers suffer from pressureinterference, the total cumulative production from this well is aremarkably straight line when plotted versus the square root oftime for up to 50 years. A downward deviation from this lineforeshadows pressure depletion, and an upward deviation is asign of reservoir re-pressuring by water injection and, perhaps,of Buckley-Leverett displacement of the oil by the injectedwater. The latter mechanism is not modeled here.

Hence, Eq. (14) can be used to match and then forecast theoil, water and gas production, as well as water injection, for anentire field project. Here, we will simplify this equation, sothat only the square-root-of-time behavior of individual wellsis retained. In other words, we neglect the late-time wellinterference.

As an illustration, we use the Crutcher-Tufts waterflood

project in Sections 1 and 12 of the Dow-Chanslor Fee in theMiddle Belridge Diatomite, Kern County, CA (Fig. 13).Because of the page limit, we focus here on Section 1 alone,but we perform rank order statistics on all 121 producers and37 water injectors in both sections.

The input data are ASCII-format queries from a ProductionAnalyst database for Dow-Chanslor. The monthly wellrecords for each producer include the current calendar time,the number of days on production in a given month, and themonthly produced oil, gas and water. For injectors, themonthly records consist of the calendar time, the number ofdays on injection (if available), the injected water, and theaverage wellhead pressure (WHP). These data are processedby our custom software package which performs theproduction and injection forecasts.

Our software can calculate the oil, gas and waterproduction rates for a typical well in a project. This is done bycalculating the elapsed times on production (the elapsedcalendar time minus the cumulative down time) for allproducers which exist at a given calendar time. In otherwords, at a fixed calendar time, each well has its owncumulative time on production and the correspondingcumulative oil, gas and water production. The number ofactive wells varies with time. For example, at short elapsedtimes on production all wells are counted, and at the longtimes only the oldest wells enter the calculation. The result ofthis time shifting and averaging is shown in Fig. 14. In thisparticular case, the average slope of a producer “prototype” inSection 1 is 900 BO/Days on production . As shown below,

this procedure yields an average well with oil slope close tothe mean of the underlying lognormal distribution of all oilslopes in a project. Thus the calculated average oil slope canbe used to calculate the average oil rate versus time, Fig. 15.Both Figs. 14 and 15 show a short, 100 day, “startup period,”which is excluded from the slope calculations. During thistime period, wells are free-flowing and often must be cleanedup because of back-production of the hydrofracture proppant.

Our software can also fit, in the least-square sense, the oil,gas and water slopes for each producer in a project. Thisautomatic procedure skips the initial start-up period and can bebiased by the user against a late-time waterflood response orpressure depletion. The resulting sample of all possible oilslopes in a project can then be analyzed using the standardstatistical methods. It is well known that any conclusions aboutsample properties, e.g., the mean slope, standard deviation,etc., are predicated on the randomness of the sample.Therefore, we must check if our sample is reasonably random.One way of checking this is to perform order statistics.12 First,we sort the natural logarithms of oil slopes in order ofincreasing magnitude. These sorted values are called orderstatistics. Second, we transform these data to zero mean andunit standard deviation. Third, we calculate the cumulativearea underneath the standard normal distribution that would beassociated with each consecutive data point if it belonged tothis distribution. Fourth, by iterating on the upper limit of the


standard normal distribution, we calculate the value of normalvariable or deviate that corresponds to the cumulative areaassigned to each sample point. Fifth, we plot the orderedsample values versus the calculated normal deviate, Fig, 16. Ifthe sample comes from a single normal distribution, then thisplot results in a straight line with the slope equal to the meanand the intercept equal to the standard deviation of thedistribution. Thus a plot of order statistics is useful indrawing inferences from a sample about the nature of theparent population. If a sample consists of portions ofdistinctly different populations, then a detectable break(“kink”) in the slope of the line results. Therefore, orderstatistics is also useful in judging the homogeneity of thesample (whether it belongs to the same parent population).

Fig. 16 shows that the productivities of all oil wells atDow-Chanslor, expressed through their oil slopes, belong to asingle lognormal population. This is plausible, because allproducer hydrofractures are under compression and similardraw-downs. Hence the oil rate depends only on the productof active hydrofracture area and the oil mobility, modified bythe total hydraulic diffusivity. This product ought to bedistributed lognormally. The underlying population isdepicted in Fig. 17. Its most probable value (mode) is 420, themiddle sample value (median) is 717, and the equal-area value(mean) is 937 BO/Sqrt(D). The mean oil slope from thelognormal distribution is about the same as the average time-shifted slope in Fig. 14. It also follows that 99.7% of allexisting or future producers at Dow-Chanslor should haveslopes lower than 3300 BO/Sqrt(D), unless better and biggerhydrofractures are available.

An identical analysis may be performed for the injectionwells. The time-shifted cumulative water injection versus thesquare root of time on injection is shown in Fig. 18. Anaverage water slope is 34,500 BW/Sqrt(D). Note that the plotof cumulative water is not a straight line (cf. Fig. 14). Thereason for this is shown in Fig. 19, as the history of averageinjector WHP’s. The injection pressures are continuallyincreased with time, thus accelerating the water injectionrelative to Eq. 14. Also, the injector fractures are undertension, have been designed differently over time, and undergoextensions. A more complex analysis, such as in ref. 2, shouldbe used. Nevertheless, with an exception of the initial rateovershoot, the square-root-of-time model of water injectionworks reasonably well.

The rank order statistics of water injection slopes, Fig. 20,reveal at least two, and perhaps four, underlying populations.The old injectors, to the left of the mean, form one or twopopulations, and the new ones, to the right of the mean, alsoform one or two populations. However, the numbers of datapoints representing each sub-population are small. Thus, onaverage, all the water injectors at Dow-Chanslor can be throwninto one lognormal population, shown in Fig. 21. Thepopulation mode is 18,600, its median is 28,700 and the meanis 32,300 BW/Sqrt(D). This population is much wider thanthat shown in Fig. 17; all Dow-Chanslor injectors should have

slopes less than 97,000 BW/Sqrt(D). This large populationwidth reflects the different hydrofracture designs and theinjector operation at different fractions of the overburdenstress. A more detailed statistical analysis of injector behaviorrequires a larger data set and a linear superposition2 approach.Such an analysis can be performed for larger projects, e.g., inthe South Belridge Diatomite.

The ratio of the mean injector and producer slope is35 BW/BO. Hence, from Eq. (14) it follows that

A

A

k

k

k

k hI

P

I

P

rw

ro

o

w

P

I

µµ

αα

0 6

1 11635

.

/ )−≈ …………..…(29)

This ratio cannot be explained by the relativepermeabilities alone. The layer-thickness weighted relativepermeability to oil in the single well example is about 0.2.One should expect the end-point water relative permeability tobe no more than 0.2. The average ratio of the oil and waterviscosity is about 6-10 and the square root of the respectivehydraulic diffusivities is close to 1. Therefore,

A

A

k

kI

P

I

P

≈ −6 10 ,…………………………..…………(30)

i.e., a combination of a larger active hydrofracture area andformation fracturing away from the injection hydrofracturemay be responsible for such large slope ratios. Anotherpossibility is that the producer hydrofractures plug with finesand precipitates.

Finally, our software can forecast the oil, gas and waterproduction, and water injection, for an entire project or field.Two examples for Section 1 are shown in Figs. 22 and 23.Note how well our simple, one-parameter model captures theoverall oil production. This model can be run into the future,resulting in a very reliable forecast of primary production. Anupward deviation from the forecast is the waterflood responseand a downward one is the formation damage and pressureinterference.

In summary, we have demonstrated that an entireprimary/waterflood project in the diatomite, or another lowpermeability rock, can be analyzed with our simple model.Also, production and injection forecasts, including futureinfills and conversions can be performed quite easily. Finally,as a byproduct of fitting the oil, gas and water slopes, aninteresting insight into the reservoir productivity can beobtained. Water injectivity, one the other hand, is moredifficult to analyze, but it also provides hints of thehydrofracture sizes and formation fracturing.

Conclusions1. We have developed a complete analytical model of

transient, linear, single phase flow in a low permeability,layered, and compressible reservoir and have presented resultsfor primary fluid recovery from the original well spacing,followed by an infill program and conversion of the infill wells


to water injectors.2. Our analysis is simplified and has many limiting

assumptions. It is not meant to be a replacement for reservoirsimulations but, being an analytical solution, it separates theeffects of the various parameters on the solution. However, thecalculations presented here for the South Belridge Diatomiteshould give reasonable estimates of the diatomite layerproductivities. This simple solution agrees well with aTHERM reservoir simulation.

3. This analysis can give an estimate of when to drill aninfill well and how long to produce from the infill well beforeconverting it to an injector. Our analysis predicts that about10% of OOIP can be recovered on 2-1/2 acre primary in agood portion of the south Belridge Diatomite. The infill to 1-1/4 acres, followed by a conversion of the infill well to a waterinjector, increases the ultimate recovery by another 3% ofOOIP. Hence, in the absence of a strong Buckley-Leverettbanking of the oil and/or strong capillary imbibition, the effectof pressure support by water injection on the incremental oilrecovery is weak. In lower quality reservoirs (layers or fields)the effect of waterflood may be small.

4. Another important result is the quantification ofreservoir heterogeneity. This analysis helps identify the goodlayers and those layers with fast pressure responses. Thecurrent analysis gives a good estimate for the pressure,production rate, and cumulative production from originalwells, with infill producers drilled at some later time and thenconverted to water injectors. This model can predict the onsetof pressure depletion and quantify the duration of productionfrom the infill wells before injecting water. We have shownthat producing from the infill well for a few years significantlyincreases the production from the field and can minimize thelost production at the infill well due to conversion to awaterflood injector.

5. We have demonstrated that the simplified Eq. (14) alonecan match, well-by-well, the fluid production and injection foran entire primary/waterflood project in the diatomite (oranother low permeability rock). The same simplified modelcan then be used to provide reliable primary production andwater injection forecasts, including future infills andconversions. A positive deviation from the primary forecast isa sign of waterflood response and a negative one suggestspressure depletion and/or producer plugging.

6. As a byproduct of fitting the slopes of cumulative oil,gas and water production versus the square root of time, aninteresting insight can be obtained into the reservoirproductivity and the effective areas of producerhydrofractures. Water injectivity, one the other hand, is moredifficult to analyze, but it also provides hints of thehydrofracture sizes and formation fracturing.

7. Finally, the current analysis can easily be incorporatedinto an economic model of an entire project or field.

AcknowledgmentsThis work was supported by two members of the U.C. Oil®

Consortium, Chevron Petroleum Technology Company, andCalResources, LLC. Partial support was also provided by theAssistant Secretary for Fossil Energy, Office of Gas andPetroleum Technology, under contract No. DE-ACO3-76FS00098 to the Lawrence Berkeley National Laboratory ofthe University of California.

NomenclatureA = twice the area of the production hydrofracture, ft2

kro = relative permeability for oilL =half-spacing of original wells, ftm =summation indexn =summation indexp =pressure in analytical solution, psia

pbp =bubble point pressure, psiapi = initial formation pressure, psia

pinj =pressure in injection superposition equations, psiapinject =water injection pressure, psia

pnet =net (total) pressure with infill and injection, psiapwell =well flowing pressure, psia

Q =cumulative amount, MBq = flow rate, MBPDT = temperature in the formation, °Ft = time after original production begins, days

tinf = time when infill well begins production, daystinj = time when water injection begins, days

x =distance from centerline between original wells, ftα = total fluid mobility, ft2/day

βm =eigenvalueλ =mobility, md/cp

λn =eigenvalueφ =porosityµ =viscosity, cp

Subscriptsf = formationg =gasI = injector

inf = infillinj = injection

o =oilP =producert = total

w =water

Superscripts

(1) =original well(2) = infill wellnet = total including infill and injection

References1. Schwartz, D. E.: “Characterizing the Lithology, Petrophysical

Properties, and Depositional Setting of the Belridge Diatomite,South Belridge Field, Kern County, California,” Studies of the


Geology of the San Joaquin Basin, S. A. Graham and H. C.Olson (eds.), The Pacific Section Society of EconomicPaleontologists and Mineralogists, Los Angeles, CA (1988)281-301.

2. Patzek, T. W.: “Surveillance of South Belridge Diatomite,”paper SPE 24040 presented at the 1992 Soc. Pet. Eng. WesternRegional Meeting, Bakersfield, Ca., Mar. 30 - Apr. 1.

3. Nikravesh, M., A. R. Kovscek, and T. W. Patzek: “Prediction ofFormation Damage During Fluid Injection in to Fractured LowPermeability Reservoirs Via Neural Network,” The SPEFormation Damage Symposium, Lafayette, La., February 14-15,1995.

4. Nikravesh, M., A. R. Kovscek, A. S. Murer, and T. W. Patzek:“Neural Networks for Field-Wise Waterflood Management inLow Permeability, Fractured Oil Reservoirs,” paper 35721presented at the 66th Soc. Pet. Eng. Western Regional Meeting,Anchorage, Alaska, May 22-24, 1996.

5. Kovscek, A. R., R. M. Johnston, and T. W. Patzek:“Interpretation of Hydrofracture Geometry Using TemperatureTransients I: Model Formulation and Verification,” In Situ 9,(September 1996).

6. Kovscek, A. R., R. M. Johnston, and T. W. Patzek:“Interpretation of Hydrofracture Geometry Using TemperatureTransients II: Asymmetric Hydrofractures,” In Situ 9,(September 1996).

7. Zwahlen, E. D., and T. W. Patzek: “Linear Transient FlowSolution for Primary Oil Recovery with Infill and Conversion toWater Injection,” to be published in In Situ.

8. Carslaw, H. S., and J. C. Jeager: Conduction of Heat in Solids,2nd Ed., Clarendon Press, Oxford (1959).

9. Ahmed, T.: Hydrocarbon Phase Behavior, Contributions inPetroleum Geology and Engineering 7, Gulf PublishingCompany, Houston, (1989) 424.

10. Standing, M. B.: A Pressure-Volume-Temperature Correlationfor Mixtures of California Oils and Gases, SPE Drilling andPractice (1947) 275.

11. Beggs, H. D., and J. R. Robinson: “Estimating the Viscosity ofCrude Oil Systems,” Journal of Petroleum Technology 27(1975) 1140-1141.

12. Mandel, J. “The Statistical Analysis of Experimental Data,”Denver 1984.


P P P P P P

P P P P P P

P

P

I

I2 1/2 acrePrimary

1 1/4 acrePrimary

1 1/4 acreWaterflood

330 ft P=producer I=injector

Fig. 1—Schematic of 2 1/2 acre primary followed by infill to 1 1/4acres followed by conversion to water injection.

-1700

-1600

-1500

-1400

-1300

-1200

-1100

-1000

-900

-800

-700

-600

0 10 20 30 40 50 60 70

So * Phi * h, feet

Dep

th, f

eet

G

H

I

JKL

M

Fig. 3—Amount of oil per unit area of reservoir in each layer.

-1700

-1600

-1500

-1400

-1300

-1200

-1100

-1000

-900

-800

-700

-600

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

kk ro , md

Dep

th, f

eet

G

H

I

JKL

M

Fig. 5—Actual oil permeability in millidarcies for the diatomitelayers.

OriginalWell

OriginalWell

InfillWell

L0 x L0 x

pwell pwell

pwellpinject

Fig. 2—Schematic for mathematical problem describing primaryproduction followed by infill. Note the symmetry in the primaryproduction at x = 0 leads to a no flow boundary. The wells areprescribed pressure boundaries.

-1700

-1600

-1500

-1400

-1300

-1200

-1100

-1000

-900

-800

-700

-600

0 0.2 0.4 0.6 0.8 1

k, md

Dep

th, f

eet

G

H

I

JKL

M

Fig. 4—Average diatomite layer permeability in millidarcies as afunction of depth.

-1700

-1600

-1500

-1400

-1300

-1200

-1100

-1000

-900

-800

-700

-600

0 0.2 0.4 0.6 0.8 1

α, ft2/day

Dep

th, f

eet

G

H

I

JKL

M

Fig. 6—Calculated α in ft 2/day for the diatomite layers.


0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100 120 140

Square Root of Time (days)

Lay

er C

um

ula

tive

Pro

du

ctio

n,

MS

TB

O

0

50

100

150

200

250

300

350

400

450

500

To

tal Cu

mu

lative Pro

du

ction

, M

ST

BO

G

I

J

H

K

L

M

Total

Fig. 7—Cumulative oil production in each layer. No infill orinjection.

0

100

200

300

400

500

600

0 15 30 45 60 75 90 105 120 135 150 165

Position, feet

Pre

ssu

re, p

sia

1 year3 yrs5 yrs10 yrs

11 yrs

12 yrs

15 yrs

20 yrs

50 yrs

Layer G

Fig. 9a—Pressure in layer G, psia, versus the position from infillwell, feet. Infill at 10 years and injection at 11 years.

0

100

200

300

400

500

600

700

800

900

1000

0 15 30 45 60 75 90 105 120 135 150 165

Position, feet

Pre

ssu

re, p

sia

1 year3 yrs5 yrs10 yrs

11 yrs

12 yrs

15 yrs

20 yrs

50 yrs Layer K

Fig. 9c—Pressure in layer K, psia, versus the position from infillwell, feet. Infill at 10 years and injection at 11 years.

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140


Per

cen

t O

il R

eco

very

G

H

I

J

K

LM

Total

Fig. 8—Percent oil recovery in each layer. No infill or injection.

0

100

200

300

400

500

600

700

800

0 15 30 45 60 75 90 105 120 135 150 165

Position, feet

Pre

ssu

re, p

sia

1 year3 yrs

5 yrs

10 yrs

11 yrs

12 yrs

15 yrs

20 yrs

50 yrsLayer I

30 yrs

Fig. 9b—Pressure in layer I, psia, versus the position from infillwell, feet. Infill at 10 years and injection at 11 years.

0

200

400

600

800

1000

1200

0 15 30 45 60 75 90 105 120 135 150 165

Position, feet

Pre

ssu

re, p

sia

1 year

3 yrs5 yrs10 yrs

11 yrs

12 yrs15 yrs

20 yrs

50 yrsLayer M

Fig. 9d—Pressure in layer M, psia, versus the position from infillwell, feet. Infill at 10 years and injection at 11 years.


0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140

Squar e R oot of T im e (d a ys)

Per

cen

t O

il R

eco

very

or

Wat

er In

ject

ion O rig ina l Tota l

In fill (no in jection)

In fill (w ith in jection)

W ater In jection

O rig ina l (no in jection)

La ye r G

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140


Per

cen

t O

il R

eco

very

or

Wat

er In

ject




W ater In jection


La ye r I

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100 120 140


Per

cen

t O

il R

eco

very

or

Wat

er In

ject




W ater In jection


La ye r K

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140


Per

cen

t O

il R

eco

very

or

Wat

er In

ject

ion Original Total

Infill (no injection)

Infill (with injection)

Water Injection

Original (no injection)

Layer M

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140


Per

cen

t O

il R

eco

very

or

Wat

er In

ject




W ater In jection


La ye r H

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140


Per

cen

t O

il R

eco

very

or

Wat

er In

ject




W ater In jection


La ye r J

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140


Per

cen

t O

il R

eco

very

or

Wat

er In

ject




W ater In jection


La ye r L

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140


Per

cen

t O

il R

eco

very

or

Wat

er In

ject

ion Original Total

Infill (no injection)

Infill (with injection)

Water Injection

Original (no injection)

Total

Fig. 10—Percent oil recovery or water injection versus the square root of time, days 1/2, calculated as the amount produced or injected dividedby the original-oil-in-place times 100 percent. Infill at 10 years and injection at 11 years.


0

2

4

6

8

10

12

14

16

0 15 30 45 60 75 90 105 120 135 150


Per

cen

t O

il R

eco

very

1-D THERM Simulation

1-D Analyt ical Solut ion

Fig. 11—Comparison of THERM simulation and analytical solution.Percent oil recovery on primary for 50 years.

0

100

200

300

400

500

600

700

800

0 15 30 45 60 75 90 105 120 135 150


Ave

rag

e P

ress

ure

, psi

a

1-D THERM Simulation

1-D Analytical Solution

Fig. 12—Comparison of THERM simulation and analytical solution.Average layer pressure on primary for 50 years.

0 1 2 3-0.5

0.5

1.5

2.5

B-20

G-19

C-19

A-19

AAA-18

D-18

CC-17

AA-17

25

A-17

27

E-17

G-17

DD-16

BB-16

J-16

H-16

F-16

D-16

AAA-16

EE-15

AA-15

CC-15

24

22

23

I-15

G-15

FF-14

DD-14

BB-14

AAA-14

F-14

H-14

G-14

CC-13

E-14

AA-13

I-13

C-13

A-13

FF-13

L-13

G-13

BB-12

J-13

F-13

H-12

AAA-12

F-12

K-12

AA-11

E-12

CC-11

I-12

G-12

J-11

C-11

G-11

18

L-11

I-11

E-11

B-11

DD-11

GG-10

D-11

GG-11

L-10

CC-10

F-10

KK-10

I-10

AAA-9

FF-10

E-10

C-9A

B-9

AA-9

HH-10

BB-9

L-9

15

H-9

G-9

I-9

JJ-9

AA-8

EE-9

C-9

B-8A

BB-8

G-8A

AAA-8

I-8B

EE-8

II-8

C-8

C-8A

HH-7

KK-8

EE-7

AA-7

12

JJ-8

AAA-7

GG-7

BB-7

DD-6A

II-7

AA-6

B-7

D-7

JJ-7

FF-6

A-6A

II-6

K-5A

LL-5

JJ-5

C-6A

AA-5

GG-5

E-5A

D-5

J-5

Thousands

Tho

usan

ds

Ft East-West

Feet

N-S

Fig. 13—The Crutcher-Tufts waterflood project in Sections 1 (the upper half) and 12 of the Dow-Chanslor Fee, Middle Belridge Di atomite,Kern County, CA. The triangles are water injectors.


0 20 40 60 800

20

40

60

Square Root of Days on Production

Cum

ulat

ive

Oil

MB

O

Fig. 14—Cumulative oil production from an average, time-shiftedCrutcher-Tufts Section 1 well versus the square root of time. Theaverage oil slope for 77 producers is 900 BO per square root ofdays on production. Note that only the oldest wells underperformthe single trend above.

-3 -2 -1 0 1 2 35.0

5.4

5.8

6.2

6.6

7.0

7.4

7.8

8.2

Standard Deviations from Mean

Ln o

f P

rod

ucer

Oil

Slo

pes

BO

/Sq

rt(D

)

Fig. 16—Order statistics for 121 producers in the Dow-ChanslorSection 1 and 12 prove that their “oil slopes” are lognormallydistributed. The intercept of the solid line is the mean and itsslope is the standard deviation of the underlying normalpopulation of the oil slope logarithms. Note the excellent fit.

0 10 20 30 40 50 60 700

20

40

60

80

Square Root of Days On Production

Oil

Rat

e B

OP

D

Fig. 15—Oil production rate from an average, time-shiftedCrutcher-Tufts Section 1 well versus the square root of time. Thisis the oil rate expected from an average well in Section 1. Note a100-day “startup-period.”

0 500 1000 1500 2000 2500 3000 35000.000

0.020

0.040

0.060

0.080

0.100

Oil Slope, BO/Sqrt(D)

Pro

balil

ity

of O

ccur

renc

e, %

Fig. 17—The lognormal distribution of oil production slopes atDow-Chanslor. The distribution mode is 420, median is 717 andmean is 937 BO per square root of days on production. The meanplus three standard deviations (99.7% of all wells) is below 3300BO/Sqrt(D).


0 20 40 60-500

0

500

1000

1500

2000

Square Root of Days on Injection

Cu

mul

ativ

e W

ater

, M

BW

Fig. 18—Cumulative water injection in an average, time-shiftedCrutcher-Tufts Section 1 well versus the square root of time. Theaverage water slope for 21 injectors is 34,500 BW per square rootof days on injection. Note that the fit is not nearly as good as inFig. 14 because of the average WHP increases with time.

-3 -2 -1 0 1 2 39

9.5

10

10.5

11

11.5

Standard Deviations from Mean

Ln o

f In

ject

or S

lope

, B

W/S

qrt(

D)

Fig. 20—Order statistics for 37 water injectors in Section 1 and 12of Dow-Chanslor. The population of “water slopes” is nothomogeneous and there seem to be 2-3 underlying sub-populations. Nevertheless, the water slopes are lognormallydistributed.

0 20 40 600

500

1000

1500

2000

0

200

400

600

800

Square Root of Days On Injection

Wat

er R

ate

BW

PD

Ave

rage

WH

P, p

sig

Fig. 19—Water injection rate and pressure (uppermost curve) inan average, time-shifted Crutcher-Tufts Section 1 well versus thesquare root of time. This is the water injection rate expected froman average well in Section 1, provided that the injection pressureis constant and there is no upper rate limit at early times. Note a100-day “startup-period.”

0 10 20 30 40 50 60 70 80 90 1000.000

1.000

2.000

3.000

Thousands

Tho

usan

dths

Water Injection Slope, BW/Sqrt(D)

Pro

balil

ity

of O

ccur

renc

e, %

Fig. 21—The lognormal distribution of water injection slopes atDow-Chanslor. The distribution mode is 18,600, its median is28,700, and the mean is 32,300 BW per square root of days oninjection. The mean plus three standard deviations (99.7% of allwells) is less than 97,000 BW/Sqrt(D).


1 9 8 1 1 9 8 5 1 9 8 9 1 9 9 3 1 9 9 70

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

2 5 0 0

Year

Tota

l Oil

Rat

e B

OP

D

Fig. 22—The actual (solid line) and predicted (broken line) oil production at the Dow-Chanslor Fee, Section 1. Note that the model pr edictioncan be run into the future and additional infill wells can be added, resulting in a very good oil production forecast.

1 9 9 0 1 9 9 2 1 9 9 4 1 9 9 60

5

1 0

1 5

2 0

2 5

Th

ou

san

ds

Year

Tota

l Wat

er R

ate

BW

PD

Fig. 23—The actual (solid line) and predicted (broken line) water injection at the Dow-Chanslor Fee, Section 1. Note that the model predictioncan be run into the future and additional infill wells can be added, resulting in a reasonably good water injection forecast.

spe 38290 linear transient flow solution for primary oil recovery

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