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PAPER 2005­114

An Improved Pseudo­Time for Gas Reservoirs With Significant Transient Flow

D.M. ANDERSON Fekete Associates Inc.

L. MATTAR Fekete Associates Inc.

This paper is to be presented at the Petroleum Society’s 6 th Canadian International Petroleum Conference (56 th Annual Technical Meeting), Calgary, Alberta, Canada, June 7 – 9, 2005. Discussion of this paper is invited and may be presented at the meeting if filed in writing with the technical program chairman prior to the conclusion of the meeting. This paper and any discussion filed will be considered for publication in Petroleum Society journals. Publication rights are reserved. This is a pre­print and subject to correction.

Abstract The use of semi­analytic methods for correcting flow equations to accommodate changing gas properties with pressure, has become increasingly common. It is a mainstay of modern production decline analysis as well as gas deliverability forecasting. The use of pseudo­time is one method which enables a time­based correction of gas properties, honoring the gas material balance within the time­based flow equation. By using pseudo­time, the analytical well / reservoir models, derived for the liquid case (slightly compressible fluid) can be modified for gas by re­evaluating the gas properties as the reservoir pressure depletes. These gas correction procedures are well documented in the literature. Also well documented is the iterative nature of the gas properties correction methods, as original gas­in­place is a required input into the equations.

The pseudo­time correction is based on the average reservoir pressure and works very well for boundary dominated flow. However, when transient flow prevails, the pseudo­time concept is not valid and its use can create anomalous responses. This will occur in low permeability systems or in reservoirs with

irregular shapes, especially where some of the boundaries are very distant from the well.

The semi­analytic gas correction has a “representative pressure” at its root, which, in the existing models, is always the average reservoir pressure. We propose a straight­forward modification to the determination of this pressure as follows. The representative pressure ought to be based on a “radius of investigation” or “region of influence” (in the case of non­ radial systems), rather than the average reservoir pressure. In the case of a depleting system, the representative pressure would be the same as the average reservoir pressure. The following paper outlines the proposed procedure and illustrates its advantages over the existing method, by using synthetic and field data examples.

PETROLEUM SOCIETY CANADIAN INSTITUTE OF MINING, METALLURGY & PETROLEUM

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Introduction

Background Literature on the derivation and usage of pseudo­time is

prevalent (2)(3) . The definition that will be used in this paper is shown below.

∫ = t

t i t a

c dt c t

0 ) (

µ µ 1

The above is used in the pseudo­steady­state equation for gas, which is at the core of most modern production decline analysis methods. It is also used in analytical well / reservoir models, whose conventional formulations are only valid for slightly compressible fluids with constant properties over a given pressure range. These models enjoy widespread usage for both history matching and forecasting, and their inclusion of pseudo­time for gas reservoirs is vital.

To illustrate the value of pseudo­time, let us take the simple case of a vertical well in the center of a circular gas reservoir. We will assume constant rate production and pseudo­steady state conditions. Thus, the model that describes the pressure response at the well can be reduced to the pseudo­steady­state equation for gas (5) .

− +

= −

4 3 ln * 6 417 .

) ( ) (

2

wa

e

i i g

i pwf pi

r r

kh Tq e 1

t f G Z c

qp p p µ 2

The “f(t)” in equation (2) is the chosen time function. Figure 1 shows the pressure response plotted against time for two cases­ f(t) = time (t) and f(t) = pseudo­time (ta). Also compared is the numerical solution using the same input parameters. Upon comparison of the solutions, it is clear that pseudo­time has a significant impact on the flow equation for gas. We also note that the pseudo­time solution appears to be identical to the numerical solution. Thus, without using pseudo­time, the pseudo­steady­state equation would be incorrect for gas reservoirs.

The Trouble With Transient Flow In the conventional definition of pseudo­time the

compressibility and viscosity terms in equation (1) are evaluated at average reservoir pressure conditions. Clearly, the average reservoir pressure is a function only of total pore volume and cumulative produced fluids. If the well of interest is producing the reservoir under boundary dominated conditions (depletion mode), the average reservoir pressure is a very reasonable datum at which to establish bulk fluid properties. However, if the well production is still transient and no reservoir boundaries have been observed, average reservoir pressure based on total reservoir volume may be a poor datum to use. Consequently, pseudo­time can cause anomalous model responses, under certain conditions.

To illustrate the above, consider two gas reservoirs of very different size but having identical properties (permeability, skin etc). If we allow reservoir fluids to flow from identical vertical wells in both reservoirs, we would expect identical pressure responses in both cases, until the boundaries of the smallest

reservoir have been observed. Yet, when we use the analytical models with pseudotime, we observe with a slightly greater pressure drop in the larger reservoir. This discrepancy is clearly not based on physical reality, but rather, is caused by pseudo­ time. The error is usually negligible. However, in cases where the pressure drawdown is severe, not only at the sandface, but over a significant areal extent in the vicinity of the well, the error can be significant.

One such case is that of a long and narrow reservoir, with the well located near one bounded end. Figure 2 illustrates the significant localized pressure drawdown that occurs in this type of configuration.

To illustrate the pseudo­time issue, let us plot the pressure response at the well for two scenarios.

1. Bounded Narrow Rectangular System

2. Semi­Infinite Narrow Rectangular System

Both reservoirs are the same width. The bounded system has its end boundaries located beyond what we will call the “region of influence”. The region of influence is defined as the region outside of which there is no measurable pressure disturbance. Therefore, we would expect the pressure responses for cases 1 and 2 to be identical. In fact, we can see from Figure 3 that they are quite different. The pressure response for scenario 2 falls significantly below that of scenario 1. This is contrary to physics.

The pressure responses ought to be identical, but because of pseudo­time, there is an artificially induced difference between them. The error arises because the pseudo­time for the semi­ infinite reservoir (case 2) is based on an average reservoir pressure that remains constant at initial conditions through time (infinite reservoirs do no deplete), while the finite case has a falling average reservoir pressure. Which one is correct? The answer is that they are both wrong. However, since scenario 1 allows for some reduction in the pressure at which the bulk properties are evaluated, its pressure response will be closer to the correct answer.

Additional Sources of Error

It is worth noting that there is a secondary source of error in the pseudo­time procedure that has nothing to do with transient or boundary dominated flow. Rather, the error results from the assumption that bulk properties evaluated at a single average pressure can adequately represent the range of fluid properties observed throughout the reservoir. This error, usually negligible, can be significant if there are large pressure gradients. Such would be the case when the well produces at very high drawdown from a very low permeability reservoir. The error would be present during both transient and boundary dominated flow. The solution suggested below does not address this error.

Solution Method The proposed solution involves altering the definition of

conventional pseudo­time, such that the gas properties are evaluated at the average pressure of the region of influence, rather than the average pressure of the total pore volume. Evaluating this pressure mathematically would involve first defining the region of influence and then integrating the

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pressure response over the domain and dividing by the area of the region (assuming no pressure variation in the vertical direction).

A

pdA p A

∫ = ~ 3

where p ~ is the average pressure inside the region of influence.

For practical purposes, a much more convenient way to evaluate the pressure is to use a material balance approach within the region of influence.

− = ir

p

G G

Z p

Z p 1 ~ ~

4

where

g

w r ir

B s h A G ) 1 ( −

= φ 5

and

r A is the area of the region of influence

Once in boundary dominated flow, the average reservoir pressure is established using precisely this method, but with the area in equation (3) being defined as the total reservoir area.

Establishing the Region of Influence

Determining the region of influence is not a trivial matter. There are (at least) two reasonable methods for establishing an affected reservoir area. One would be to calculate the pressure distribution through the entire reservoir area and delineate the region, outside of which, there is less than some minimum percentage of pressure drops (say 1%). This procedure, although analytically sound, would be very costly computationally, as a large array of solved pressures would be required at every time­step in the computation of pseudo­time. In addition, some established analytical solutions for pressure at the wellbore (such as that of a bounded reservoir containing an infinite conductivity fracture) do not include the pressure solution for all points inside the reservoir domain.

A more practical method for establishing a region of influence is to use the well established concept of the radius of investigation. The radius of investigation is defined as follows (1) .

c kt rinv φµ 948

= 6

The radius of investigation is an accepted measure of the speed of propagation of the pressure transient through diffusive media, and is a function only of permeability, porosity, fluid properties and time. Thus, the radius of investigation propagates through any unbounded diffusive media at precisely the same velocity, regardless of reservoir geometry or changes in pressure drawdown. Appendix A presents two synthetic model

responses, clearly demonstrating that this is true for an arbitrary reservoir shape and well location.

It is important to note that the radius of investigation does not account for the magnitude of pressure depletion over time, only the propagation of the leading edge of the region of influence.

Given the above, delineation of the region of influence in an isotropic reservoir, containing a single vertical well, involves evaluating the intersection of the outward progressing circle prescribed by the radius of investigation, with the reservoir area itself.

A r A inv r I 2 π = 7

where A is the area of the reservoir (defined by the model).

The formula given in (7) applies to any arbitrary reservoir shape and well location, but is restricted to vertical well geometry and isotropic and homogeneous reservoirs.

More advanced well / reservoir geometries such as fractured wells, anisotropic media, heterogeneities and/or layers would follow much the same procedure. However, the definition of the velocity of transient propagation would have to be defined separately for all directions and possibly as a function of space. The additional solution complexity and computational expense would likely preclude the practical application of a pseudo­time correction in such cases, in favor of using a numerical model (which explicitly evaluates fluid properties at each grid block). In order for analytical models to remain favorable, they must be convenient and efficient to solve.

Case Studies

Synthetic Example 1: Vertical Well in a Rectangular Reservoir The corrected pseudo­time solution for a vertical well arbitrarily located in a rectangular shaped reservoir of arbitrary dimensions is constructed as follows.

Area of the Region of Influence

The area of the region of influence is defined as 2 inv r r A π = in

an unbounded system. As the region of investigation expands beyond system boundaries, area elements are subtracted from the infinite acting “circle”. Thus, the solution for Ar changes with time, and is different for different well / reservoir geometries.

To simplify the geometry of the solution significantly, we can approximate the region of influence as a rectangle (rather than a circle with subtracted segments), with negligible impact on the result. The solution for area is divided into four different geometries, any combination of which may be part of a prescribed Ar (t) function for a given pseudo­time calculation (Figure 4).

For example, well / reservoir geometry such as that shown in Figure 4(C) would have a region of investigation whose area would follow the formulas, in succession, for the single boundary, perpendicular boundaries and three boundaries solutions.

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From Figure 4, it appears that the description of time dependent areas of influence is somewhat cumbersome, as it requires adjustments on the fly that depend on the particular geometry of the model. Since the pseudo­time correction is negligible in all but the most “severe” well / reservoir geometries, we may further approximate the area of the region of influence by identifying the dominant geometry factor (single boundary, perpendicular boundaries, three boundaries or parallel boundaries) and applying the appropriate area formula.

For instance, if d1, d2 and d3 are all of similar magnitude in the three boundary case, we may apply that area formula from time zero until the radius of investigation reaches the far­field boundary. Likewise, if d1 and d2 are much closer than d3, and d3 and the far­field boundary are on the same order of magnitude, then we may apply the area formula for the perpendicular boundaries case from time zero.

Average Pressure in the Area of Influence and Corrected Pseudo­time

The solution for the area developed above is used in conjunction with equations (4) and (5), describing respectively, average pressure (p/z) within the region of influence and gas­in­ place within the region of influence. The corrected pseudo­time is calculated using the following equation.

∫ = t

t c dt

i t a c t 0

~ ~ ) ( µ

µ 8

In Figure 5, the “pseudo­time corrected” solution for the semi­ infinite reservoir case presented previously is compared against the numerical solution, showing reasonably good agreement (the remaining error cannot be reduced within the confines of the pseudo time approximation. Note, since the three boundary case causes the most severe localized pressure drawdown of any rectangular geometry investigated, the approximations are expected to hold up as well (or better) for any general rectangular reservoir / vertical well geometry.

Field Example 2: Fractured well in a Rectangular Reservoir

In this field study, an analytical reservoir model has been used to simulate the measured pressure response from a hydraulically fractured well. The production response, analyzed using the Agarwal and Gardener type curves, suggests a bounded system with OGIP of about 1 bcf (see Figure 6). The objective of the modeling exercise is to verify the interpreted gas­in­place, and improve upon the type curve model by fine­tuning.

A satisfactory history match is obtained using a narrow, rectangular reservoir with the well located near a corner (see Figure 7). The simulated OGIP is more than four times higher than that interpreted using the type curve analysis. As the far­ field boundary is moved closer to the well (and thus, OGIP is reduced) the simulated flowing pressures increase! Common sense tells us that the pressure response ought to decrease upon reduction in total reservoir volume in a bounded, depleting system. The anomalous synthesized pressure from the model is a result of the pseudo­time error discussed previously. Figure 7 compares the results from three different boundary locations in the model. It also displays the measured data, illustrating the original history match. Because of the pseudo­time error, the

model does not properly (or uniquely) indicate the OGIP or bounded nature of the reservoir.

Corrected Pseudo­time Model

The area formulas discussed in the previous section can be extended to hydraulically fractured wells. In this example, the “three boundary” case is relevant. For the three boundary system, the modified area formula and geometry is shown in Figure 4(E).

Upon correcting the pressure response using the new definition of pseudo­time, the previous model match is no longer valid, indicating too large an OGIP (see Figure 8). As the far­field boundary is reduced, the pressure response (properly) remains unchanged until its location coincides with the radius of investigation. At that point, the synthetic pressures decrease.

This case illustrates the value of the corrected pseudo­time function when history matching using analytical models with extreme well / boundary geometry. The conventional definition of pseudo­time, in this case generated history matches that did not properly illustrate the boundary dominated response observed from the type curve analysis. In addition, the directionality of the results did not make sense. Only with the corrected pseudo­time, was the proper synthetic pressure response created.

To complete the exercise, the history match is now refined, using the corrected pseudo­time model. The results indicate an OGIP of about 1 bcf (see Figure 9).

Conclusions 1. Without pseudo­time (or similar gas property

correction routines), the analysis and modeling of gas reservoirs would not be possible when using analytical solutions designed for slightly compressible fluids.

2. The conventional definition of pseudo­time is satisfactory under conditions of boundary dominated flow.

3. During transient / transitional flow, the standard definition of pseudo­time may cause incorrect and misleading results, when there are localized regions of high pressure drawdown. These localized regions are caused primarily by severe well / reservoir geometry (such as a narrow reservoir with the well located near a corner). Additional errors may result from large pressure gradients in very low permeability systems.

4. A new “corrected” pseudo­time function is presented that evaluates gas properties at the average pressure of the region of influence, during transient or transitional flow, rather than the average reservoir pressure (based on external no­ flow boundaries). A general formula (equation (7)) is given that describes the region of influence for an arbitrarily located well in a reservoir with an arbitrary shape. Specific solutions are also derived for some simple (common) well / reservoir geometries.

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5. The new pseudo­time definition solves the problems associated with transient flow with severe geometry and yields results that are more consistent with those from a numerical simulator. Its primary limitation is the complexity involved in calculating the area of the region of influence (which changes with time). However, with suitable approximations, this can be simplified greatly.

6. The pseudo­time modifications could be easily incorporated into any software, by simply changing the pressure at which gas properties are evaluated. The new pressure would be based on the region of influence, rather than the average reservoir pressure. Note that the region of influence pressure and average reservoir pressure become identical during boundary dominated flow.

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NOMENCLATURE A = Area of the reservoir, ft 2 B g = Formation Volume Factor of gas, rcf/scf cg = Gas Compressibility, psi ­1 ct = Total Compressibility, psi ­1

t c ~ = Compressibility evaluated at the average

reservoir pressure within the region of influence, psi ­1

t c = Compressibility evaluated at the average reservoir pressure, psi ­1

d = distance to boundary, ft G = Gas in Place, MMcf Gp = Cumulative Gas Produced, MMcf h = Net Pay, ft k = Permeability, md q = Production rate, Mcfd p = Pressure, psi pp = Pseudo pressure, psi 2 /cp ppwf = Pseudo pressure at the sandface, psi 2 /cp p ~ = Average pressure within the region of

influence, psi re = Exterior radius, ft rinv = Radius of investigation, ft rwa = Apparent wellbore radius, ft sw = Water saturation t = time, days or hours ta = Pseudo­time, days or hours T = Temperature, °R Xf = Fracture half length, ft Z = Gas compressibility factor Z ~

= Average Gas Compressibility factor within the region of influence

µ ~ = Viscosity at the average reservoir pressure within the region of influence, cp

µ = Viscosity at the average reservoir pressure, cp

µ = viscosity, cp ϕ = porosity

Subscripts i = Initial r = Within the region of influence

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REFERENCES

1. E.R.C.B. Gas Well Testing – Theory and Practice; Energy and Resource Conservation Board, Alberta, Canada, 1975, Third Edition.

2. Agarwal, R.G., Gardner, D.C., Kleinsteiber, S.W., Fussell, D.D., Analyzing Well Production Data Using Combined Type­ Curve and Decline­Curve Analysis Concepts; SPE Reservoir Evaluation and Engineering, October, 1999.

3. Fraim, M.L., Wattenbarger R.A., Gas Reservoir Decline­ Curve Analysis Using Type Curves with Real Gas Pseudopressure and Normalized Time; SPE Formation Evaluation, December, 1987.

4. Palacio, J.C., Blasingame, T.A., Decline­Curve Analysis Using Type Curves – Analysis of Gas Well Production Data; Paper SPE 25909 presented at the Joint Rocky Mountain Regional and Low Permeability Reservoirs Symposium, Denver, CO, April 26­28, 1993.

5. Blasingame, T.A., Lee, W.J., Variable­Rate Reservoir Limits Testing; Paper SPE 15028 presented at the Permian Basin Oil and Gas Recovery Conference, Midland, TX, March 13­14, 1986

Appendices

Appendix A:

The purpose of the following is to illustrate that the propagation of the radius of investigation along any unobstructed direction is not affected by the influence of boundaries in other directions.

The radius of investigation is defined as

c kt rinv φµ 948

= (A­1)

As shown, the radius of investigation is a function of the permeability, time, porosity, and the fluid properties.

Let us consider two reservoirs with identical properties but of different size and shape. Reservoir A is square shaped with the well in the center, while Reservoir B is long and narrow. Reservoir B has a much smaller total volume than Reservoir A. However, the distance from the well to the "x" boundary is the same in both reservoirs. A schematic is shown in Figure 10. For simplicity we will assume that both reservoirs produce single phase oil.

Data for Reservoirs A and B are as follows:

k = 20 md XeA = YeA = 10000 ft φ = 10% XeB = 10000 ft μ = 1.4194 cp YeB = 100 ft c = 1.02e­5 psi­1

The time to reach the outer boundary X of both reservoirs is calculated by applying equation A­1. A time of 1715 hours is calculated when the radius of investigation is 5000 ft (distance to the boundary from a well positioned in the centre of each reservoir).

The diagnostic plots in Figure 10 illustrate that the propagation of the radius of investigation is identical for A and B. This becomes clear when comparing the derivative plots for each case. The influence of the final (Xe) boundary, and subsequent onset of pseudo­steady­state (unit slope), begins at exactly the same time. Furthermore, the time, as shown in Figure 10 is the same as that calculated using equation A­1. Thus, we confirm that the radius of investigation along a particular direction is unaffected by the influence of boundaries in other directions.

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Figures:

Pressure Response Comparison for time and pseudo time

0

500

1000

1500

2000

2500

3000

0 20 40 60 80 100 120 140

time (days)

Flow

ing sand

face pressure (psia)

Pseudo­time

time

Numerical

Figure 1: Pressure Responses Using Time and Pseudo time

Pressure Response Comparison for time and pseudo time

0

500

1000

1500

2000

2500

3000

0 20 40 60 80 100 120 140

time (days)

Flow

ing sand

face pressure (psia)

Pseudo­time

time

Numerical

Figure 1: Pressure Responses Using Time and Pseudo time

Init ial Pressure

Figure 2: Illustration of High Localized Drawdown Due to Boundaries – same well rates, same flow duration

Well Flow ing Pressure

Init ial Pressure

Figure 2: Illustration of High Localized Drawdown Due to Boundaries – same well rates, same flow duration

Well Flow ing Pressure

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Figure 3: Flowing Pressure Response – Bounded and Semi­Infinite Reservoirs (using conventional) pseudo­time

Pwf vs Time­ "Three Boundary" Reservoir

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250

Time (h)

pwf (psia)

Bounded (slightly beyond region of influence)

Semi­Infinite

Figure 3: Flowing Pressure Response – Bounded and Semi­Infinite Reservoirs (using conventional) pseudo­time

Pwf vs Time­ "Three Boundary" Reservoir

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250

Time (h)

pwf (psia)

Bounded (slightly beyond region of influence)

Semi­Infinite

A. Single Boundary

d 1

r inv

B. Perpendicular Boundaries

d 1

r inv d 2

C. Three Boundaries

( ) ) ( 2 3 1 inv r r d d d A + + =

d 1

r inv d 2

d 3

D. Parallel Boundaries

( ) ) 2 ( 3 1 inv r r d d A + =

d 1

d 3 r inv

d 1

d 2 d 3

r inv

x f

E. Three Boundaries with Hydraulic Fracture

Figure 4: Region of Influence Areas for Common Geometries

) )( ( 2 3 1 inv f r x d d d Ar + + + =

) ( 2 1 d r r A inv inv r + = ) )( ( 2 1 inv inv r r d r d A + + =

A. Single Boundary

d 1

r inv

B. Perpendicular Boundaries

d 1

r inv d 2

C. Three Boundaries

( ) ) ( 2 3 1 inv r r d d d A + + =

d 1

r inv d 2

d 3

D. Parallel Boundaries

( ) ) 2 ( 3 1 inv r r d d A + =

d 1

d 3 r inv

d 1

d 2 d 3

r inv

x f

E. Three Boundaries with Hydraulic Fracture

Figure 4: Region of Influence Areas for Common Geometries

) )( ( 2 3 1 inv f r x d d d Ar + + + =

) ( 2 1 d r r A inv inv r + = ) )( ( 2 1 inv inv r r d r d A + + =

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Figure 5: Pseudo time corrected solution for three boundary case compared with Numerical simulation

Pwf vs Time­ Semi­Infinite "Three Boundary" Reservoir

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250

Time (h)

pwf (psia)

Numerical Corrected pseudo­time

Uncorrected pseudo­time

Figure 5: Pseudo time corrected solution for three boundary case compared with Numerical simulation

Pwf vs Time­ Semi­Infinite "Three Boundary" Reservoir

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250

Time (h)

pwf (psia)

Numerical Corrected pseudo­time

Uncorrected pseudo­time

Figure 6: Production Response analyzed using Agarwal and Gardener Type Curve

OGIP = 1.06 bcf

Figure 6: Production Response analyzed using Agarwal and Gardener Type Curve Figure 6: Production Response analyzed using Agarwal and Gardener Type Curve

OGIP = 1.06 bcf

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Comparison of Pressure Solutions­ Uncorrected Pseudo­Time

0

1000

2000

3000

4000

5000

6000

7000

0 50 100 150 200 250 300 350 400 450 500

Time (days )

pwf (psia)

xe = 10,000 ft OGIP = 4.3 bcf

xe = 5,000 ft

xe = 2,400 ft

measured data

k = 0.13 md

y e = 1,100 ft x f = 350 ft

x e

Figure 7: Comparison of Pressure Solutions ­ Uncorrected Pseudo Time

Comparison of Pressure Solutions­ Uncorrected Pseudo­Time

0

1000

2000

3000

4000

5000

6000

7000

0 50 100 150 200 250 300 350 400 450 500

Time (days )

pwf (psia)

xe = 10,000 ft OGIP = 4.3 bcf

xe = 5,000 ft

xe = 2,400 ft

measured data

k = 0.13 md

y e = 1,100 ft x f = 350 ft

x e

Figure 7: Comparison of Pressure Solutions ­ Uncorrected Pseudo Time

Comparison of Pressure Solutions­ Corrected Pseudo­Time

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300 350 400 450 500

Time (days)

pwf (psia)

xe = 5,000 ft xe = 10 ,000 ft

xe = 2,400 ft

measured data

xe

k = 0.13 md

ye = 1,100 ft xf = 350 ft

Figure 8: Comparison of Pressure Solutions ­ Corrected Pseudo Time

Comparison of Pressure Solutions­ Corrected Pseudo­Time

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300 350 400 450 500

Time (days)

pwf (psia)

xe = 5,000 ft xe = 10 ,000 ft

xe = 2,400 ft

measured data

xe

k = 0.13 md

ye = 1,100 ft xf = 350 ft

Figure 8: Comparison of Pressure Solutions ­ Corrected Pseudo Time

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Revised His tory Match Using Correc ted Pressures

0

1000

2000

3000

4000

5000

6000

7000

0 50 100 150 200 250 300 350 400 450 500

Time (days)

pwf (psia)

x e = 2,280 ft

k = 0.12 md OGIP = 1 bcf

y e = 1,030 ft x f = 350 ft

Figure 9: History Match using Corrected Pressures

Revised His tory Match Using Correc ted Pressures

0

1000

2000

3000

4000

5000

6000

7000

0 50 100 150 200 250 300 350 400 450 500

Time (days)

pwf (psia)

x e = 2,280 ft

k = 0.12 md OGIP = 1 bcf

y e = 1,030 ft x f = 350 ft

Figure 9: History Match using Corrected Pressures

Figure 10: Comparison of Radius of Investigation between Square and Elongated Reservoir

Diagnostic Plots for Two Reservoir Models: Illustrating the Radius of Investigation

0.1

1

10

100

1000

10000

0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000

Time (hrs)

(Ppi­P

pwf)/q (psi/(bb

l/d)), D

erivative

Outer Boundary (Xe) reached at about 1700 hour s for both r eservoirs .

Reservoir A

XeA = 10000 ft

Reservoir A

Reservoir B

YeB=100 ft

XeA = 10000 ft

YeA = 100000 ft Res ervoir B

Figure 10: Comparison of Radius of Investigation between Square and Elongated Reservoir

Diagnostic Plots for Two Reservoir Models: Illustrating the Radius of Investigation

0.1

1

10

100

1000

10000

0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000

Time (hrs)

(Ppi­P

pwf)/q (psi/(bb

l/d)), D

erivative

Outer Boundary (Xe) reached at about 1700 hour s for both r eservoirs .

Reservoir A

XeA = 10000 ft

Reservoir A

Reservoir B

YeB=100 ft

XeA = 10000 ft

YeA = 100000 ft Res ervoir B