PETR 4301

Theory of Reservoir Modeling

Submitted on: 10th December 2015

Prepared by

Team A1

Englert, Brandon (0881854) Gong, Matt (0658215)

Zaitsev, Vladimir (0601327) Troppe, Aron (1100036)

Fall 2015

Assignment #3

Assignment 3

PETR 5325 (Fall 2015) 2

TABLE OF CONTENTS

LIST OF FIGURES ........................................................................................................................ 3

LIST OF TABLES .......................................................................................................................... 4

INTRODUCTION .......................................................................................................................... 5

TASK 1 ........................................................................................................................................... 8

Subtask 1 ..................................................................................................................................... 8

Subtask 2 ................................................................................................................................... 11

Subtask 3 ................................................................................................................................... 13

Subtask 4 ................................................................................................................................... 14

Subtask 5: .................................................................................................................................. 17

Analytical Solution ............................................................................................................... 17

Subtask 6 ................................................................................................................................... 21

TASK 2 ......................................................................................................................................... 25

TASK 3 ......................................................................................................................................... 32

Dirichlet boundary condition ................................................................................................ 32

Neumann condition ............................................................................................................... 36

TASK 4 ......................................................................................................................................... 38

TASK 5 ......................................................................................................................................... 40

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PETR 5325 (Fall 2015) 3

LIST OF FIGURES

Figure 1: Linear Flow ..................................................................................................................... 8

Figure 2: Parameter Values ............................................................................................................. 9

Figure 3: Section of Numerical Grid ............................................................................................ 10

Figure 4: Pressure vs Lenth, 20 Step Discretizatoin ..................................................................... 11

Figure 5: Pressure vs Time ........................................................................................................... 12

Figure 6: Pressure vs Distance and Time ...................................................................................... 13

Figure 7: Pressure vs distance with 200 ft intervals ..................................................................... 15

Figure 8: Pressure vs Time with 200 ft interval ............................................................................ 16

Figure 9: Average Reletive Error .................................................................................................. 22

Figure 10: Average Relative Error vs Distance ............................................................................ 23

Figure 11: Average Relative Error vs. Time ................................................................................. 24

Figure 12: Radial Continuity Equation ......................................................................................... 26

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PETR 5325 (Fall 2015) 4

LIST OF TABLES

Table 1: Maximum Solution Stability Limits ............................................................................... 14

Table 2: Comparison of Maximum Discretization Ratio .............................................................. 17

Table 3: Black-Oil vs Compositional Simulator ........................................................................... 39

Table 4: Commercial Simulators .................................................................................................. 40

Table 5: Public Domain Simulators .............................................................................................. 45

Assignment 3

PETR 5325 (Fall 2015) 5

INTRODUCTION

The diffusivity, or “heat equation”, describes the flow of fluid in porous media. In its linear form

(1), the diffusivity equation states that the acceleration of the pressure change with distance is

proportional to the rate of its change with time. For a slightly compressible fluid:

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= �𝜙𝜙𝜙𝜙𝜙𝜙

0.002637𝑘𝑘�𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

(1)

𝑃𝑃 is fluid pressure (psi), 𝑥𝑥 is linear distance (ft), and 𝜕𝜕 is time (days). The constants of

proportionality include 𝜙𝜙, porosity, 𝜙𝜙, viscosity, (cP), 𝜙𝜙, pore volume compressibility, (sip), and

𝑘𝑘, permeability (mD). The multiplier 0.002637 is needed to maintain dimensional consistency.

The group 𝜙𝜙𝜙𝜙𝜙𝜙0.002637𝑘𝑘

delineated in the brackets in (1) must have the units of 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 /𝐴𝐴𝐴𝐴𝑇𝑇𝐴𝐴, as

shown (2):

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= �𝜙𝜙𝜙𝜙𝜙𝜙

0.002637𝑘𝑘�𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

(2a)

(𝑝𝑝𝑝𝑝𝑇𝑇)(𝑓𝑓𝜕𝜕2) = �

𝜙𝜙𝜙𝜙𝜙𝜙0.002637𝑘𝑘

�(𝑝𝑝𝑝𝑝𝑇𝑇)

(𝑑𝑑𝐴𝐴𝑑𝑑𝑝𝑝) (2b)

(𝑑𝑑𝐴𝐴𝑑𝑑𝑝𝑝)(𝑝𝑝𝑝𝑝𝑇𝑇)

(𝑝𝑝𝑝𝑝𝑇𝑇)(𝑓𝑓𝜕𝜕2) = �

𝜙𝜙𝜙𝜙𝜙𝜙0.002637𝑘𝑘

� (2c)

(𝑑𝑑𝐴𝐴𝑑𝑑𝑝𝑝)(𝑓𝑓𝜕𝜕2) = �

𝜙𝜙𝜙𝜙𝜙𝜙0.002637𝑘𝑘

� (2d)

A numerical prediction of pressure at various times and distances may be obtained by

discretizing time and space into numbered segments and deriving the numerical solution of (1):

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PETR 5325 (Fall 2015) 6

𝑃𝑃𝑖𝑖𝑛𝑛+1 = 𝑃𝑃𝑖𝑖𝑛𝑛 +∆𝜕𝜕

𝛼𝛼2(∆𝑥𝑥)2(𝑃𝑃𝑖𝑖+1𝑛𝑛 − 2𝑃𝑃𝑖𝑖𝑛𝑛 + 𝑃𝑃𝑖𝑖−1𝑛𝑛 ) (3a)

𝑃𝑃𝑖𝑖𝑛𝑛+1 = 𝑃𝑃𝑖𝑖𝑛𝑛 + 𝐾𝐾𝐷𝐷(𝑃𝑃𝑖𝑖+1𝑛𝑛 − 2𝑃𝑃𝑖𝑖𝑛𝑛 + 𝑃𝑃𝑖𝑖−1𝑛𝑛 ) (3b)

In (3a), 𝑇𝑇 represents the numeration of the distance interval ∆𝑥𝑥, and 𝑛𝑛 represents that of the time

interval ∆𝜕𝜕. The expression 𝛼𝛼2 represents �𝜙𝜙𝜙𝜙𝜙𝜙𝑘𝑘� the hydraulic diffusivity. This expression will

allow the iterative estimation of pressure values, assuming all boundary conditions are known.

In (3b), 𝐾𝐾𝐷𝐷 is a dimensionless multiplier that expresses the combined effect of the hydraulic

diffusivity, and the sizes of the length and time steps. To show how the discretization step

lengths affect the value of 𝐾𝐾𝐷𝐷, we resolve 𝐾𝐾𝐷𝐷 into two components:

𝑚𝑚𝐾𝐾

= 𝐾𝐾𝐷𝐷 ; ≡ 𝑑𝑑𝑇𝑇𝑇𝑇𝑇𝑇𝑛𝑛𝑝𝑝𝑇𝑇𝑑𝑑𝑛𝑛𝑑𝑑𝑇𝑇𝑝𝑝𝑝𝑝 (4a)

𝐾𝐾 = 𝐶𝐶𝛼𝛼2 ; �𝑑𝑑𝐴𝐴𝑑𝑑𝑝𝑝𝑓𝑓𝜕𝜕2

� (4b)

𝑇𝑇 = ∆𝑡𝑡(∆𝑥𝑥)2 ; �𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

𝑓𝑓𝑡𝑡2� (4c)

𝐶𝐶 is a conversion factor necessary to render 𝐾𝐾 into units of 1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑓𝑓𝑡𝑡2

. As shown (5):

𝜙𝜙𝜙𝜙𝜙𝜙𝑘𝑘

=𝜙𝜙𝑃𝑃

𝑇𝑇𝑑𝑑 × 𝑝𝑝𝑝𝑝𝑇𝑇�

1000 𝑇𝑇𝑑𝑑𝐷𝐷

��𝐷𝐷 × 𝐴𝐴𝜕𝜕𝑇𝑇 × 𝑝𝑝 × 𝜙𝜙𝑇𝑇2

𝜙𝜙𝑃𝑃 × 𝜙𝜙𝑇𝑇 × 𝜙𝜙𝑇𝑇3 � �14.6959 𝑝𝑝𝑝𝑝𝑇𝑇

𝐴𝐴𝜕𝜕𝑇𝑇�

(30.48 𝜙𝜙𝑇𝑇)2

(𝑓𝑓𝜕𝜕)2 (5a)

𝜙𝜙𝜙𝜙𝜙𝜙𝑘𝑘

= �1000

1� �𝑝𝑝1� �

14.6959 1

�(30.48 )2

(𝑓𝑓𝜕𝜕)2�

𝑑𝑑𝐴𝐴𝑑𝑑86400 𝑝𝑝

� (5b)

Assignment 3

PETR 5325 (Fall 2015) 7

𝜙𝜙𝜙𝜙𝜙𝜙𝑘𝑘

= �1000

1� �𝑝𝑝1� �

14.6959 1

�(30.48 )2

(𝑓𝑓𝜕𝜕)2�

𝑑𝑑𝐴𝐴𝑑𝑑86400 𝑝𝑝

� = 158 𝑑𝑑𝐴𝐴𝑑𝑑𝑝𝑝𝑓𝑓𝜕𝜕2

(5c)

In other words:

• There are 158 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑓𝑓𝑡𝑡2

per 1 𝜙𝜙𝑐𝑐𝑚𝑚𝑑𝑑 × 𝑝𝑝𝑑𝑑𝑖𝑖

• We are given 𝛼𝛼2 in terms of 𝜙𝜙𝑐𝑐𝑚𝑚𝑑𝑑 × 𝑝𝑝𝑑𝑑𝑖𝑖

• We need to convert 𝛼𝛼2 to 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑓𝑓𝑡𝑡2

to fit into (4a).

• So therefore we need a 𝐶𝐶 factor of 158 in (4b).

The value of 𝐾𝐾 does not depend on the length of discretization intervals; rather on the reservoir

and fluid properties implied in 𝛼𝛼2. The value of 𝑇𝑇, by contrast, expresses the impact of the

discretization. Together, their quotient 𝐾𝐾𝐷𝐷 acts as the constant of proportionality for (3b).

It’s worthwhile noting that we could dispense with the need for both 𝐶𝐶 and 𝐾𝐾 if we had just

replaced the 0.002637 divisor in (1) with the reciprocal of our 𝐶𝐶 factor: 1158

= 0.0063285. That

would allow us to state:

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= �𝜙𝜙𝜙𝜙𝜙𝜙

0.0063285𝑘𝑘�𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

(6a)

𝛼𝛼2 = �𝜙𝜙𝜙𝜙𝜙𝜙

0.0063285𝑘𝑘� (6b)

𝑇𝑇𝛼𝛼2

= 𝐾𝐾 (6c)

In fact, this is an indication that the value of 0.002637 is wrong for the given units of (1), and

reflects some other set of units. We will therefore use (6) to work Task 1.

As an aside, it appears that we can guess the units for which the divisor of 0.002637 was

intended. If you add another zero to 0.002637 to transform it into 0.0002637 and take the

Assignment 3

PETR 5325 (Fall 2015) 8

reciprocal, you get 10.0002637

= 3792.188. Dividing by our 𝐶𝐶 factor of 158, you get 3792.188158

=

24.00. Meaning: If you were using units of hours instead of days in (1) and in (4b), then you

would use a factor of 𝐶𝐶 = 3792.188 in (4b), or a divisor of 0.0002637 in (1), to correctly

convert 𝜙𝜙𝑐𝑐𝑚𝑚𝑑𝑑 × 𝑝𝑝𝑑𝑑𝑖𝑖

into ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑑𝑑𝑓𝑓𝑡𝑡2

.

TASK 1

Subtask 1

Consider the volumetric flow of fluid, 𝑄𝑄, through the depicted control volume.

Figure 1: Linear Flow

Our units of measurement and their given values are shown:

Assignment 3

PETR 5325 (Fall 2015) 9

Figure 2: Parameter Values

The boundary conditions are displayed next:

𝑃𝑃0 = 𝑃𝑃(0, 𝜕𝜕) = 1800 𝑝𝑝𝑝𝑝𝑇𝑇, 𝑓𝑓𝑑𝑑𝐴𝐴 𝜕𝜕 > 0 (6a)

𝑃𝑃1 = 𝑃𝑃(𝐿𝐿, 𝜕𝜕) = 1800 𝑝𝑝𝑝𝑝𝑇𝑇, 𝑓𝑓𝑑𝑑𝐴𝐴 𝜕𝜕 > 0 (6b)

𝑃𝑃(𝑥𝑥, 0) = 1800 �1 + sin �𝜋𝜋𝑥𝑥𝐿𝐿�� 𝑝𝑝𝑝𝑝𝑇𝑇, 𝑓𝑓𝑑𝑑𝐴𝐴 0 < 𝑥𝑥 < 𝐿𝐿 (6c)

We wish to use (3) and (6) to develop a grid-based numerical discretization of (1) with twenty

steps in ∆𝑥𝑥 and one hundred 1-day steps in ∆𝜕𝜕. We will then graph 𝑃𝑃 𝑣𝑣𝑝𝑝 𝑥𝑥 and examine the

isoline curves at 25 step time intervals.

First we need to calculate 𝐾𝐾 based on (6):

1𝛼𝛼2

=1

� 𝜙𝜙𝜙𝜙𝜙𝜙0.0063285𝑘𝑘�

=1

�0.16 × 2 × 0.000220.0063285 × 32 �

= 2876.6 (7a)

𝑇𝑇 =∆𝜕𝜕

(∆𝑥𝑥)2 =1

1002= 0.0001 (7b)

Assignment 3

PETR 5325 (Fall 2015) 10

𝐾𝐾𝐷𝐷 =𝑇𝑇𝛼𝛼2

= 0.0001(2876.6) = 0.28766 (7c)

Next, we use (3b) with (6) and (7c) to develop a numerical approximation grid:

Figure 3: Section of Numerical Grid

As can be seen, (6a) was used for the rightmost row. The top row has been calculated via (6c).

The rest is done with (3b). Here is the isoline chart:

0 100 200

0 1800 2081.58 2356.231 1800 2079.59 2352.292 1800 2077.61 2348.383 1800 2075.64 2344.494 1800 2073.69 2340.645 1800 2071.75 2336.816 1800 2069.83 2333.017 1800 2067.91 2329.23

Assignment 3

PETR 5325 (Fall 2015) 11

Figure 4: Pressure vs Lenth, 20 Step Discretizatoin

Pressure is highest in the middle of the block, at length x = 1000 ft. Then it smoothly drops to

boundary condition pressure as it reaches the end of the block at x = 2000 ft. (this is true for all

times). Pressure is symmetrical about the center of the block.

Pressure drops with time. At t = 0, the pressure is at its maximum, and as time increases it

smoothly drops. The change in pressure is lowest at t = 100 days.

Subtask 2

Our next look at the data will investigate the shape of the pressure – time relationship.

The figure below is P vs. t for the 100 time steps, 1 day each, and 20 distance steps, 100 ft. each.

Assignment 3

PETR 5325 (Fall 2015) 12

Figure 5: Pressure vs Time

Pressure is highest at the beginning, at time = 0 days, and drops slowly as time goes on (this is

true for all locations). Pressure is highest at the center (1000 ft.).

Note that the gradient of the pressure drop is larger at high pressures. This makes sense, as

higher pressure regions would tend to expel their fluid faster. This would result in a steeper

pressure drop.

We do not analyze isolines for values of x between 1000 and 2000 because they would be the

mirror image of the 0 to 1000 foot interval, as established in previous subtask, due to pressure

symmetry around the center of the block at x=1000 ft. So, for example, the pressure curve for x =

800 ft. would be identical to pressure curve for x = 1200 ft. on the above chart, and x = 200 ft.

would be identical to x = 1800 ft. This symmetry can be seen well in Figure 6:

Assignment 3

PETR 5325 (Fall 2015) 13

Figure 6: Pressure vs Distance and Time

In this figure, positon discretization number is displayed on the depth axis. It is seen that the

pressure drop is symmetric with respect to the middle positon, around the 10th discretization.

This corresponds to the 1000 foot interval; this is the reflective axis (Figure 4).

Subtask 3

Accurate numerical solutions depend on keeping the discretization error small. See Task 3 for a

discussion of discretization error. By increasing the size of ∆𝑥𝑥 and ∆𝜕𝜕, we can experimentally

determine the limits on these steps, within which (3a) continues to provide stable

approximations. “Stability” here means that small increments in distance or time should not

generate large or oscillating values of pressure.

With the original time step of 1 day and distance step of 100 ft., the ∆𝑡𝑡(∆𝑥𝑥)2 ratio is 1.00E-4 days

per square foot, and both of the P vs. x and P vs. t plots are stable. When we increase time step

Assignment 3

PETR 5325 (Fall 2015) 14

to 2 days, having a 2.00E-4 ratio, this behavior continues. Our solution stability problems begin

at t = 2.1 days, when the t = 100 day curve on P vs x plot starts to oscillate slightly. The ratio

here is 2.10E-4. Increasing to t = 2.14, the 100 day curve oscillates wildly, but the rest of the

time curves are still stable on P vs x plot. The ratio here is 2.14E-4.

So, the tolerable ratio would depend on the maximum time that we want to simulate. With that,

below is the table for allowable ratios provided we are using twenty distance steps.

Table 1: Maximum Solution Stability Limits

Time (days) Time step, ∆𝒕𝒕 (days) Ratio of ∆𝒕𝒕(∆𝒙𝒙)𝟐𝟐 , (days/sq. ft)

100 2.07 2.07E-4 75 2.22 2.22E-4 50 2.55 2.55E-4 25 4.00 4.00E-4

Note that decreasing time steps to below 1 day does not affect solution stability. It simply moves

curves closer to each other, and in P vs x chart all the pressure curves are grouped much closer to

the t = 0 day curve. Decreasing time to below 1 day does not seem to shift curves on P vs x chart.

Subtask 4

Having seen the effect of changing the length of the time step, we now investigate the result of

changing the distance step.

Assignment 3

PETR 5325 (Fall 2015) 15

Figure 7: Pressure vs distance with 200 ft. intervals

Here, (Figures 8 & 9), we have reduced our 20 distance steps of 100 ft to just 10 steps with 200

ft intervals.

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PETR 5325 (Fall 2015) 16

Figure 8: Pressure vs Time with 200 ft. interval

Note that with 10 distance steps, the pressure declines more steeply in P vs. t plot than in the

same plot with 20 time steps. In the 10 distance step P vs. x chart, the time isolines have

significantly lower maximum pressures at the peak of 1000 ft. (other than the pressure at initial

time, of course).

Let us explore the maximum ratios with distance steps of 200 ft. The initial, stable scenario is

200 ft. and one day; the ratio is 2.50E-5.

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PETR 5325 (Fall 2015) 17

Table 2: Comparison of Maximum Discretization Ratio

∆𝒕𝒕 (days) ∆𝒕𝒕/(∆𝒙𝒙)^𝟐𝟐 (days / sq. ft)

Time 10 x-steps 20 x-steps 10 x-steps 20 x-steps

100 days 8.4 2.07 2.10E-4 2.07E-4 75 days 8.8 2.22 2.20E-4 2.22E-4 50 days 10.0 2.55 2.50E-4 2.55E-4 25 days 15.4 4.0 3.85E-4 4.00E-4

We can see that if we simulate a long period of time, the ratio is about the same, at 2.1E-4.

However if we want to simulate a shorter period of time, say 25 days or 50 days, then the ratio is

higher for the 20 step scenario. In general, the 10 distance step scenario allows for longer time

steps before becoming unstable. Even though the distance discretization is doubled, the

reduction in repetitive calculation error improves the stability of the result.

Subtask 5:

Analytical Solution

Recall our diffusivity equation and its boundary conditions:

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= �𝜙𝜙𝜙𝜙𝜙𝜙

0.002637𝑘𝑘�𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

(1a)

𝑃𝑃0 = 𝑃𝑃(0, 𝜕𝜕) = 1800 𝑝𝑝𝑝𝑝𝑇𝑇, 𝑓𝑓𝑑𝑑𝐴𝐴 𝜕𝜕 > 0 (1b)

𝑃𝑃1 = 𝑃𝑃(𝐿𝐿, 𝜕𝜕) = 1800 𝑝𝑝𝑝𝑝𝑇𝑇, 𝑓𝑓𝑑𝑑𝐴𝐴 𝜕𝜕 > 0 (1c)

𝑃𝑃(𝑥𝑥, 0) = 1800 �1 + sin �𝜋𝜋𝑥𝑥𝐿𝐿�� 𝑝𝑝𝑝𝑝𝑇𝑇, 𝑓𝑓𝑑𝑑𝐴𝐴 0 < 𝑥𝑥 < 𝐿𝐿 (1d)

Assignment 3

PETR 5325 (Fall 2015) 18

Consider this proposed solution (2a) to our linear diffusivity equation. For simplicity we

substitute 𝛼𝛼2 = � 𝜙𝜙𝜙𝜙𝜙𝜙0.002637𝑘𝑘

�:

𝑃𝑃(𝑥𝑥, 𝜕𝜕) = 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

0.002637𝑘𝑘𝜋𝜋2

𝜙𝜙𝜙𝜙𝜙𝜙𝐿𝐿2𝜕𝜕�� (2a)

𝑃𝑃(𝑥𝑥, 𝜕𝜕) = 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (2b)

We will first verify (1b), (1c), and (1d) by their subsequent substitution into (2b):

𝑃𝑃0 = 𝑃𝑃(0, 𝜕𝜕) = 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋(0)𝐿𝐿

�𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (3a)

= 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛(0)𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (3b)

= 1800 �1 + (0)𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (3c)

= 1800 (3d)

𝑃𝑃1 = 𝑃𝑃(𝐿𝐿, 𝜕𝜕) = 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋(𝐿𝐿)𝐿𝐿

�𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (4a)

= 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛(𝜋𝜋)𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (4b)

= 1800 �1 + (0)𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (4c)

= 1800 (4d)

Assignment 3

PETR 5325 (Fall 2015) 19

𝑃𝑃(𝑥𝑥, 0) = 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝�−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2(0)�� (5a)

= 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝(0)� (5b)

= 1800 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� (1)� (5c)

= 1800 �1 + sin �𝜋𝜋𝑥𝑥𝐿𝐿�� (5d)

Now that we know the boundary conditions check out, let’s generate our partial derivative for

time. Using (2b):

𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

= 1800𝜕𝜕𝜕𝜕𝜕𝜕�1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �

𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (6a)

𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

= 1800 �𝜕𝜕𝜕𝜕𝜕𝜕

(1) +𝜕𝜕𝜕𝜕𝜕𝜕�𝑝𝑝𝑇𝑇𝑛𝑛 �

𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕��� (6b)

𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

= 1800 �0 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿�𝜕𝜕𝜕𝜕𝜕𝜕�𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕��� (6c)

𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

= 1800 �𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�

𝜕𝜕𝜕𝜕𝜕𝜕�−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (6d)

𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

= 1800 �𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2�𝜕𝜕(𝜕𝜕)𝜕𝜕𝜕𝜕

� (6e)

𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

= 1800 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2� �𝑝𝑝𝑇𝑇𝑛𝑛 �

𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� (1)� (6f)

𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕

= 1800 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2� 𝑝𝑝𝑇𝑇𝑛𝑛 �

𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� (6g)

Subbing into (1a), we have:

Assignment 3

PETR 5325 (Fall 2015) 20

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= 𝛼𝛼2 �1800 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2� 𝑝𝑝𝑇𝑇𝑛𝑛 �

𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (7a)

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= 1800 �−𝜋𝜋2

𝐿𝐿2� 𝑝𝑝𝑇𝑇𝑛𝑛 �

𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� (7b)

The uniformly negative value of 𝜕𝜕2𝑐𝑐

𝜕𝜕𝑥𝑥2 for 0 < 𝑥𝑥 < 𝐿𝐿 implies that 𝑃𝑃 will be curved downward for

the solution domain. This curvature reaches a maximum at (𝑥𝑥, 𝜕𝜕) = �𝐿𝐿2

, 0�, and approaches zero

as 0 ← 𝑥𝑥 → 𝐿𝐿 or as 𝜕𝜕 → ∞.

The 1st partial derivative of 𝑃𝑃 with respect to 𝑥𝑥 is also developed from (2b):

𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

= 1800𝜕𝜕𝜕𝜕𝑥𝑥

�1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�� (8a)

𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

= 1800 �𝜕𝜕𝜕𝜕𝑥𝑥

(1) +𝜕𝜕𝜕𝜕𝑥𝑥

�𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕��� (8b)

𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

= 1800 �(0) + 𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕�

𝜕𝜕𝜕𝜕𝑥𝑥

�𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿��� (8c)

𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

= 1800 �𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� 𝜙𝜙𝑑𝑑𝑝𝑝 �

𝜋𝜋𝑥𝑥𝐿𝐿�𝜕𝜕𝜕𝜕𝑥𝑥

�𝜋𝜋𝑥𝑥𝐿𝐿�� (8d)

𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

= 1800 �𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� 𝜙𝜙𝑑𝑑𝑝𝑝 �

𝜋𝜋𝑥𝑥𝐿𝐿� �𝜋𝜋𝐿𝐿�𝜕𝜕𝜕𝜕𝑥𝑥

(𝑥𝑥)� (8e)

𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

= 1800 �𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� 𝜙𝜙𝑑𝑑𝑝𝑝 �

𝜋𝜋𝑥𝑥𝐿𝐿� �𝜋𝜋𝐿𝐿� (1)� (8f)

𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥

= �1800𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� �

𝜋𝜋𝐿𝐿�� 𝜙𝜙𝑑𝑑𝑝𝑝 �

𝜋𝜋𝑥𝑥𝐿𝐿� (8g)

Now, we use (8g) to find the 2nd partial:

Assignment 3

PETR 5325 (Fall 2015) 21

𝜕𝜕𝜕𝜕𝑥𝑥

�𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥� = �1800𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� �

𝜋𝜋𝐿𝐿��

𝜕𝜕𝜕𝜕𝑥𝑥

�𝜙𝜙𝑑𝑑𝑝𝑝 �𝜋𝜋𝑥𝑥𝐿𝐿�� (9a)

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= �1800𝑇𝑇𝑥𝑥𝑝𝑝 �−𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� �

𝜋𝜋𝐿𝐿�� �−𝑝𝑝𝑇𝑇𝑛𝑛 �

𝜋𝜋𝑥𝑥𝐿𝐿��

𝜕𝜕𝜕𝜕𝑥𝑥

�𝜋𝜋𝑥𝑥𝐿𝐿� (9b)

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= �1800𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� �−

𝜋𝜋𝐿𝐿�� �

𝜋𝜋𝐿𝐿�𝜕𝜕𝜕𝜕𝑥𝑥

(𝑥𝑥) (9c)

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= �1800𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� �−

𝜋𝜋2

𝐿𝐿2�� (1) (9d)

𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

= 1800 �−𝜋𝜋2

𝐿𝐿2� 𝑝𝑝𝑇𝑇𝑛𝑛 �

𝜋𝜋𝑥𝑥𝐿𝐿� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝐿𝐿2𝜕𝜕� (9e)

Since (9e) is identical to (7b), we conclude that (2a) is a solution to (1a). It is one of a 2-

parameter family of solutions on 0 < 𝑥𝑥 < 𝐿𝐿:

𝑃𝑃(𝑥𝑥, 𝜕𝜕) = 𝜸𝜸 �1 + 𝑝𝑝𝑇𝑇𝑛𝑛 �𝜋𝜋𝑥𝑥𝑳𝑳� 𝑇𝑇𝑥𝑥𝑝𝑝 �−

𝜋𝜋2

𝛼𝛼2𝑳𝑳2𝜕𝜕�� (10)

The 𝛾𝛾 parameter determines 𝑃𝑃(0,1) and 𝑃𝑃(0, 𝐿𝐿). The 𝐿𝐿 parameter is the upper domain boundary

in 𝑥𝑥. Both parameters are needed to determine 𝑃𝑃(𝑥𝑥, 0).

Subtask 6

It’s always good to check on the accuracy of a numerical approximation process. By

numerically approximating a known analytical function, we can relate the relative accuracy of

the approximation process with time and distance. For this subtask we used the numerical

method with 100 ft. long distance steps (delta x = 100 ft.) and 1 day time steps (delta t = 1 day)

to arrive at matrix which had 0 to 99 day columns, and 0 to 20th distance steps (0 to 2000 ft.).

Assignment 3

PETR 5325 (Fall 2015) 22

The analytical matrix, which uses analytical solution (equation directly in terms of x and t), had

the same dimensions: 0 to 2000 feet distance in 0 to 20th time steps and 0 to 99th day in 1 day

time steps.

Average relative error was calculated (Figure 11):

Figure 9: Average Reletive Error

Here, (x0-x) is the absolute of difference (positive value) between the numerical and analytical

solutions, and x is the magnitude (positive value) of the analytical solution.

For the actual values to compare, we used the average pressure value for each row and column in

both numerical and analytical matrices. For example, average pressure was calculated for each

day step (each column), an array of those average pressures was created, and compared to array

of average pressure values in each column of second matrix. The same process was followed for

the average pressure for each row (time step) in both matrices.

Assignment 3

PETR 5325 (Fall 2015) 23

Figure 10: Average Relative Error vs Distance

The figure above shows average relative error with respect to length of the simulated block. We

can see that error is greatest in the middle of the block at x = 1000 ft., and the error smoothly

decreases toward zero at the ends of the block (x = 0 and x = 2000), where we are given the

value of pressure as a boundary condition.

Assignment 3

PETR 5325 (Fall 2015) 24

Figure 11: Average Relative Error vs. Time

This figure shows the average relative error with respect to simulation time. As the time grows,

so does the error. From the appearance of the curve, the relative error appears to grow without

bound as simulation time increases. This is likely because the numerical method becomes less

and less valid when we move further away from boundary/initial condition in both distance and

time.

However, an inspection of the fitting equation, displayed on the graph, indicates something

different. The quadratic equation will reach a maximum value for average relative error.

Afterwards, we might expect error to decrease as time grows without bound.

This makes sense, too. As the pressure throughout the reservoir heads to the boundary pressures

at 𝑥𝑥 = {0, 𝐿𝐿} as time increases without bound, both the numerical and analytic expressions tend

asymptotically toward that boundary pressure. The difference between those expressions

Assignment 3

PETR 5325 (Fall 2015) 25

becomes smaller. So, the relative error, which is that difference divided by the analytic answer,

must go down, too.

Using the first derivative test on the trend line equation, we can estimate the time of the

maximum relative error:

2(−8𝐸𝐸 − 08)𝑥𝑥 + 2𝐸𝐸 − 05 = 0 (8a)

𝑥𝑥 =−2𝐸𝐸 − 05

2(−8𝐸𝐸 − 08) = 0.125𝐸𝐸3 = 125 𝑑𝑑𝐴𝐴𝑑𝑑𝑝𝑝 (8b)

So then we would expect the relative error to reach its maximum at 125 days.

TASK 2

The continuity equation is the basis for modeling fluid flow in porous media. In its radial form,

it may be combined with Darcy’s law and an equation of state to generate the radial diffusivity

equation.

Consider the volume element of thickness ∆𝐴𝐴 (below) located at distance 𝐴𝐴 from a well at the

center of a considered circular system. Suppose there is radial fluid flow through the element

towards the well at the center. The reservoir system is assumed to have constant thickness ℎ, and

constant fluid and rock properties. We will now derive the continuity equation in its radial form

(ibid), which represents the conservation of mass for flow in radial systems (in the equation 𝜌𝜌

represents fluid density, 𝑢𝑢 represents the fluid flow velocity, and 𝜙𝜙 the porosity of the system).

Assignment 3

PETR 5325 (Fall 2015) 26

Figure 12: Radial Continuity Equation

We start off by assuming that the annular volume element depicted above is fixed in space with

the positive radial direction defined as outward from the wellbore. The Continuity Principle

requires the following axiom:

𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐴𝐴𝜙𝜙𝜙𝜙𝑢𝑢𝑇𝑇𝑢𝑢𝑑𝑑𝐴𝐴𝜕𝜕𝑇𝑇𝑑𝑑𝑛𝑛 = 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐼𝐼𝑛𝑛 + 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐺𝐺𝑇𝑇𝑛𝑛𝑇𝑇𝐴𝐴𝐴𝐴𝜕𝜕𝑇𝑇𝑑𝑑𝑛𝑛 − 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝑂𝑂𝑢𝑢𝜕𝜕 (1)

The Law of Conservation of Mass requires that matter is neither generated, nor destroyed.

Therefore, we simplify to:

𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐴𝐴𝜙𝜙𝜙𝜙𝑢𝑢𝑇𝑇𝑢𝑢𝑑𝑑𝐴𝐴𝜕𝜕𝑇𝑇𝑑𝑑𝑛𝑛 = 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐼𝐼𝑛𝑛 − 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝑂𝑂𝑢𝑢𝜕𝜕 (2)

Dividing both sides of the equation by time, we restate this as a balance of mass rates:

𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐴𝐴𝜙𝜙𝜙𝜙𝑢𝑢𝑇𝑇𝑢𝑢𝑑𝑑𝐴𝐴𝜕𝜕𝑇𝑇𝑑𝑑𝑛𝑛 𝑅𝑅𝐴𝐴𝜕𝜕𝑇𝑇 = 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐹𝐹𝑑𝑑𝑑𝑑𝐹𝐹 𝑅𝑅𝐴𝐴𝜕𝜕𝑇𝑇 𝐼𝐼𝑛𝑛 − 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐹𝐹𝑑𝑑𝑑𝑑𝐹𝐹 𝑅𝑅𝐴𝐴𝜕𝜕𝑇𝑇 𝑂𝑂𝑢𝑢𝜕𝜕 (3)

We now need to define the geometry of our control volume. The inner side of the annulus is

located at distance 𝐴𝐴 from the wellbore, and has a thickness ℎ. Its area 𝐴𝐴|𝑜𝑜 is given by:

Assignment 3

PETR 5325 (Fall 2015) 27

𝐴𝐴|𝑜𝑜 = 2𝜋𝜋𝐴𝐴ℎ (4)

The outer side of the annulus is located at (𝐴𝐴 + ∆𝐴𝐴). Its area 𝐴𝐴|𝑜𝑜+∆𝑜𝑜 is given by:

𝐴𝐴|𝑜𝑜+∆𝑜𝑜 = 2𝜋𝜋(𝐴𝐴 + ∆𝐴𝐴)ℎ (5)

The volume of the annulus is the difference between the smaller and larger cylinders:

𝑉𝑉 = 𝜋𝜋(𝐴𝐴 + ∆𝐴𝐴)2ℎ − 𝜋𝜋𝐴𝐴2ℎ (6)

This simplifies to:

𝑉𝑉 = 𝜋𝜋ℎ[2𝐴𝐴∆𝐴𝐴 + (∆𝐴𝐴)2] (7)

By making ∆𝐴𝐴 arbitrarily small in comparison to 𝐴𝐴, we further simplify to:

𝑉𝑉 ≈ 2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ (8)

The pore volume 𝑉𝑉𝑝𝑝, is simply the control volume 𝑉𝑉 multiplied by its porosity 𝜙𝜙:

𝑉𝑉𝑝𝑝 ≈ 2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ𝜙𝜙 (9)

To find the mass 𝑀𝑀𝑝𝑝 within the pore volume, we multiply by the density 𝜌𝜌. Even though we are

assuming constant fluid properties, we are not assuming constant density. The value of 𝜌𝜌 will be

dependent on the radial distance from the wellbore. Our expressions will be general enough to

describe a fluid whose density varies with 𝐴𝐴, such as a gas. Therefore, 𝜌𝜌 must be considered

separately in each of our expressions, and we may not cancel a 𝜌𝜌 that appears in one expression

Assignment 3

PETR 5325 (Fall 2015) 28

with a 𝜌𝜌 appearing in a different expression. However, as ∆𝐴𝐴 becomes arbitrarily small, 𝜌𝜌

becomes constant within the pore volume. So we may state:

𝑀𝑀𝑝𝑝 ≈ 2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ𝜙𝜙𝜌𝜌 (10)

The mass in the pore volume at any time 𝜕𝜕 may be designated as 𝑀𝑀𝑝𝑝�𝑡𝑡:

𝑀𝑀𝑝𝑝�𝑡𝑡 ≈ (2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ𝜙𝜙𝜌𝜌)|𝑡𝑡 (11)

At time 𝜕𝜕 + ∆𝜕𝜕 we similarly state:

𝑀𝑀𝑝𝑝�𝑡𝑡+∆𝑡𝑡 ≈ (2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ𝜙𝜙𝜌𝜌)|𝑡𝑡+∆𝑡𝑡 (12)

So the accumulation of mass ∆𝑀𝑀𝑝𝑝 over the interval ∆𝜕𝜕 is simply:

∆𝑀𝑀𝑝𝑝 = 𝑀𝑀𝑝𝑝�𝑡𝑡+∆𝑡𝑡 − 𝑀𝑀𝑝𝑝�𝑡𝑡 (13)

We have justly replaced the “approximately equal ≈” sign, as the volumetric error of (∆𝐴𝐴)2

from (7) does not change over ∆𝜕𝜕, and must therefore disappear in the difference.

The rate of mass accumulation in the pore volume is:

∆𝑀𝑀𝑝𝑝

∆𝜕𝜕=𝑀𝑀𝑝𝑝�𝑡𝑡+∆𝑡𝑡 − 𝑀𝑀𝑝𝑝�𝑡𝑡

∆𝜕𝜕 (14)

Plugging in (11) and (12) we obtain:

∆𝑀𝑀𝑝𝑝

∆𝜕𝜕=

(2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ𝜙𝜙𝜌𝜌)|𝑡𝑡+∆𝑡𝑡 − (2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ𝜙𝜙𝜌𝜌)|𝑡𝑡∆𝜕𝜕

(15)

Assignment 3

PETR 5325 (Fall 2015) 29

Since the geometrical boundaries of the pore volume do not change over ∆𝜕𝜕, we pull out the

geometric factors:

∆𝑀𝑀𝑝𝑝

∆𝜕𝜕= 2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ

(𝜙𝜙𝜌𝜌)|𝑡𝑡+∆𝑡𝑡 − (𝜙𝜙𝜌𝜌)|𝑡𝑡∆𝜕𝜕

(16)

The expression of (16) indicates that the quantity (𝜙𝜙𝜌𝜌) varies over ∆𝜕𝜕. The increase in density 𝜌𝜌

of the pore volume is obviously due to the accumulation of mass within it. In addition, the

porosity 𝜙𝜙 of the control volume can also increase, though the geometrical boundaries of the

control volume remain constant. This increase in porosity is due to the increase of pore pressure

with mass accumulation. The increasing pressure compresses the rock matrix, generating an

increasing pore volume – therefore a higher porosity for the control volume.

Taking the limit as ∆𝜕𝜕 → 0, we may state the mass accumulation rate as:

𝜕𝜕𝑀𝑀𝑝𝑝

𝜕𝜕𝜕𝜕= 2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ

𝜕𝜕(𝜙𝜙𝜌𝜌)𝜕𝜕𝜕𝜕

(17)

We substitute (17) into (3) to obtain:

2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ𝜕𝜕(𝜙𝜙𝜌𝜌)𝜕𝜕𝜕𝜕

= 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐹𝐹𝑑𝑑𝑑𝑑𝐹𝐹 𝑅𝑅𝐴𝐴𝜕𝜕𝑇𝑇 𝐼𝐼𝑛𝑛 − 𝑀𝑀𝐴𝐴𝑝𝑝𝑝𝑝 𝐹𝐹𝑑𝑑𝑑𝑑𝐹𝐹 𝑅𝑅𝐴𝐴𝜕𝜕𝑇𝑇 𝑂𝑂𝑢𝑢𝜕𝜕 (18)

To find the mass flow rate, we first designate 𝑢𝑢, the average apparent fluid velocity. The actual

velocity of the fluid is higher, due to the fact that the fluid must move through the pore throats,

and can’t move through the rock matrix. However, 𝑢𝑢 is the average velocity over one of the

curved annular surfaces of the control volume as fluid appears to move radially toward the

wellbore.

Assignment 3

PETR 5325 (Fall 2015) 30

Although 𝐴𝐴 has already been defined as positive in the outward radial direction, we will define 𝑢𝑢

as positive towards the wellbore; 𝑢𝑢 is always a positive number that is dependent on the radial

distance from the wellbore. We designate 𝑢𝑢|𝑜𝑜 as the fluid velocity at position 𝐴𝐴. The lower

velocity at (𝐴𝐴 + ∆𝐴𝐴) is then 𝑢𝑢|𝑜𝑜+∆𝑜𝑜.

The volumetric outflow 𝑞𝑞𝑜𝑜𝑜𝑜𝑡𝑡 through the inner annular surface located at 𝐴𝐴 is simply the average

velocity multiplied by the surface area 𝐴𝐴|𝑜𝑜:

𝑞𝑞𝑜𝑜𝑜𝑜𝑡𝑡 = 𝑢𝑢|𝑜𝑜𝐴𝐴|𝑜𝑜 (19)

Subbing in (4), we obtain:

𝑞𝑞𝑜𝑜𝑜𝑜𝑡𝑡 = 𝑢𝑢|𝑜𝑜(2𝜋𝜋𝐴𝐴ℎ) (20)

Since 𝑢𝑢 is being evaluated at position 𝐴𝐴, it is convenient to group those variables together:

𝑞𝑞𝑜𝑜𝑜𝑜𝑡𝑡 = 2𝜋𝜋ℎ(𝑢𝑢𝐴𝐴)|𝑜𝑜 (21)

The mass outflow rate 𝜕𝜕𝑀𝑀𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝑡𝑡

is then found by multiplying by fluid density evaluated at 𝐴𝐴:

𝜕𝜕𝑀𝑀𝑜𝑜𝑜𝑜𝑡𝑡

𝜕𝜕𝜕𝜕 = 2𝜋𝜋ℎ(𝜌𝜌𝑢𝑢𝐴𝐴)|𝑜𝑜 (22)

The mass inflow rate may be similarly derived through the following steps:

𝑞𝑞𝑖𝑖𝑛𝑛 = 𝑢𝑢|𝑜𝑜+∆𝑜𝑜𝐴𝐴|𝑜𝑜+∆𝑜𝑜 (23)

𝑞𝑞𝑖𝑖𝑛𝑛 = 𝑢𝑢|𝑜𝑜+∆𝑜𝑜2𝜋𝜋(𝐴𝐴 + ∆𝐴𝐴)ℎ (24)

𝑞𝑞𝑖𝑖𝑛𝑛 = 2𝜋𝜋ℎ(𝑢𝑢𝐴𝐴)|𝑜𝑜+∆𝑜𝑜 (25)

Assignment 3

PETR 5325 (Fall 2015) 31

𝜕𝜕𝑀𝑀𝑖𝑖𝑛𝑛

𝜕𝜕𝜕𝜕 = 2𝜋𝜋ℎ(𝜌𝜌𝑢𝑢𝐴𝐴)|𝑜𝑜+∆𝑜𝑜 (26)

Subbing (22) and (26) into (18), we have:

2𝜋𝜋𝐴𝐴∆𝐴𝐴ℎ𝜕𝜕(𝜙𝜙𝜌𝜌)𝜕𝜕𝜕𝜕

= 2𝜋𝜋ℎ(𝜌𝜌𝑢𝑢𝐴𝐴)|𝑜𝑜+∆𝑜𝑜 − 2𝜋𝜋ℎ(𝜌𝜌𝑢𝑢𝐴𝐴)|𝑜𝑜 (27)

Rearranging and simplifying:

1𝐴𝐴

(𝜌𝜌𝑢𝑢𝐴𝐴)|𝑜𝑜+∆𝑜𝑜 − (𝜌𝜌𝑢𝑢𝐴𝐴)|𝑜𝑜∆𝐴𝐴

=𝜕𝜕(𝜙𝜙𝜌𝜌)𝜕𝜕𝜕𝜕

(28)

It’s important to understand what this equation implies. Over the positive distance interval ∆𝐴𝐴,

as one moves a small increment away from the well-bore, the product 𝜌𝜌𝑢𝑢𝐴𝐴 will increase if mass

in accumulating in the control volume. If the control volume is losing mass, then 𝜌𝜌𝑢𝑢𝐴𝐴 will

decrease over ∆𝐴𝐴. In a black oil reservoir above the bubble point experiencing steady state flow,

𝜌𝜌 is approximately constant, 𝜕𝜕(𝜙𝜙𝜌𝜌)𝜕𝜕𝑡𝑡

is zero; therefore 𝑢𝑢 ∝ 1𝑜𝑜.

The Taking the limit as ∆𝐴𝐴 → 0, we state:

1𝐴𝐴𝜕𝜕(𝜌𝜌𝑢𝑢𝐴𝐴)𝜕𝜕𝐴𝐴

=𝜕𝜕(𝜙𝜙𝜌𝜌)𝜕𝜕𝜕𝜕

(29)

This is the radial continuity equation. Q.E.D.

Assignment 3

PETR 5325 (Fall 2015) 32

TASK 3

Dirichlet boundary condition

Suppose it is known that 𝑃𝑃 is a function of (𝑥𝑥, 𝜕𝜕) with the conditions:

𝜕𝜕2𝑐𝑐

𝜕𝜕𝑥𝑥2= 𝛼𝛼2 𝜕𝜕𝑐𝑐

𝜕𝜕𝑡𝑡 ; 𝛼𝛼 is a constant (1𝐴𝐴)

𝑥𝑥| 0 ≤ 𝑥𝑥 ≤ 𝐿𝐿 (1𝑏𝑏)

𝜕𝜕| 𝜕𝜕 ≥ 0 (1𝜙𝜙)

𝑃𝑃(𝑥𝑥, 0) = 𝑓𝑓(𝑥𝑥) 𝑓𝑓𝑑𝑑𝐴𝐴 𝑥𝑥| 0 < 𝑥𝑥 < 𝐿𝐿 ; 𝑓𝑓(𝑥𝑥) is an explicit function (1𝑑𝑑)

𝑃𝑃(0, 𝜕𝜕) = 𝑃𝑃(𝐿𝐿, 𝜕𝜕) = 0 for 𝜕𝜕| 𝜕𝜕 ≥ 0 (1𝑇𝑇)

We wish to derive a numerical approximation of 𝑃𝑃 for discretized values of 𝑥𝑥| 0 < 𝑥𝑥 < 𝐿𝐿 and

of 𝜕𝜕 > 0. The step lengths are denoted ∆𝑥𝑥 and ∆𝜕𝜕 respectively. The step number in 𝑥𝑥 will be

noted by the subscript 𝑇𝑇, and the step number in 𝜕𝜕 will be noted by the superscript 𝑛𝑛. Hence, we

seek an approximation function 𝜓𝜓 of the equivalent forms:

𝑃𝑃(𝑇𝑇∆𝑥𝑥, (𝑛𝑛 + 1)∆𝜕𝜕) = 𝜓𝜓�𝑃𝑃�(𝑇𝑇 + 1)∆𝑥𝑥,𝑛𝑛∆𝜕𝜕�,𝑃𝑃(𝑇𝑇∆𝑥𝑥,𝑛𝑛∆𝜕𝜕),𝑃𝑃�(𝑇𝑇 − 1)∆𝑥𝑥,𝑛𝑛∆𝜕𝜕�� (2𝐴𝐴)

𝑃𝑃(𝑥𝑥𝑖𝑖, 𝜕𝜕𝑛𝑛 + ∆𝜕𝜕) = 𝜓𝜓{𝑃𝑃(𝑥𝑥𝑖𝑖 + ∆𝑥𝑥, 𝜕𝜕𝑛𝑛),𝑃𝑃(𝑥𝑥𝑖𝑖, 𝜕𝜕𝑛𝑛),𝑃𝑃(𝑥𝑥𝑖𝑖 − ∆𝑥𝑥, 𝜕𝜕𝑛𝑛)} (2𝑏𝑏)

𝑃𝑃(𝑥𝑥𝑖𝑖, 𝜕𝜕𝑛𝑛+1) = 𝜓𝜓{𝑃𝑃(𝑥𝑥𝑖𝑖+1, 𝜕𝜕𝑛𝑛),𝑃𝑃(𝑥𝑥𝑖𝑖, 𝜕𝜕𝑛𝑛),𝑃𝑃(𝑥𝑥𝑖𝑖−1, 𝜕𝜕𝑛𝑛)} (2𝜙𝜙)

𝑃𝑃𝑖𝑖𝑛𝑛+1 = 𝜓𝜓{𝑃𝑃𝑖𝑖+1𝑛𝑛 ,𝑃𝑃𝑖𝑖𝑛𝑛,𝑃𝑃𝑖𝑖−1𝑛𝑛 } (2𝑑𝑑)

Our explicit function for 𝑃𝑃𝑖𝑖0, and our given values for 𝑃𝑃0𝑛𝑛 and 𝑃𝑃𝐿𝐿𝑛𝑛 form the boundary conditions

that allow us to calculate all 𝑃𝑃𝑖𝑖𝑛𝑛 in an iterative fashion. For example:

Assignment 3

PETR 5325 (Fall 2015) 33

𝑃𝑃11 = 𝜓𝜓{𝑃𝑃20,𝑃𝑃10,𝑃𝑃00} (3)

We know from (1𝑑𝑑) that 𝑃𝑃20 = 𝑓𝑓(2) and that 𝑃𝑃10 = 𝑓𝑓(1). We also know from (1𝑇𝑇) that 𝑃𝑃00 = 0.

If we just knew 𝜓𝜓, then we could find 𝑃𝑃11. Similarly:

𝑃𝑃21 = 𝜓𝜓{𝑃𝑃30,𝑃𝑃20,𝑃𝑃10} (4)

In this case, all the arguments of 𝜓𝜓 are known from 𝑓𝑓(𝑥𝑥).

Similarly: If 𝐼𝐼 = 𝐿𝐿∆𝑥𝑥

, then 𝐼𝐼 represents the total number of steps in 𝑥𝑥. Then:

𝑃𝑃𝑖𝑖𝑛𝑛+1 = 𝜓𝜓{𝑃𝑃𝑖𝑖+1𝑛𝑛 ,𝑃𝑃𝑖𝑖𝑛𝑛,𝑃𝑃𝑖𝑖−1𝑛𝑛 } (5𝐴𝐴)

𝑃𝑃𝐼𝐼−11 = 𝜓𝜓{𝑃𝑃𝐼𝐼0,𝑃𝑃𝐼𝐼−10 ,𝑃𝑃𝐼𝐼−20 } (5𝑏𝑏)

From (1𝑇𝑇), the Dirichlet boundary condition, we know that 𝑃𝑃𝐼𝐼0 = 0. The other arguments,

again, are from (1𝑑𝑑). Obviously, then, we can find any 𝑃𝑃𝑖𝑖1. Once we find all 𝑃𝑃𝑖𝑖1, we can use

(2𝑑𝑑) to find any 𝑃𝑃𝑖𝑖2 for 𝑥𝑥| 0 ≤ 𝑥𝑥 ≤ 𝐿𝐿 :

𝑃𝑃𝑖𝑖𝑛𝑛+1 = 𝜓𝜓{𝑃𝑃𝑖𝑖+1𝑛𝑛 ,𝑃𝑃𝑖𝑖𝑛𝑛,𝑃𝑃𝑖𝑖−1𝑛𝑛 } (6𝐴𝐴)

𝑃𝑃𝑖𝑖2 = 𝜓𝜓{𝑃𝑃𝑖𝑖+11 ,𝑃𝑃𝑖𝑖1,𝑃𝑃𝑖𝑖−11 } (6𝑏𝑏)

It’s clear, then, that we will be able to find any 𝑃𝑃𝑖𝑖𝑛𝑛. This, then, is our motivation for seeking the

approximation function 𝜓𝜓. If 𝑃𝑃(𝑥𝑥, 𝜕𝜕) represents the pressure transient of a depleting reservoir,

then 𝜓𝜓 would allow the generation of a numerical solution for 𝑃𝑃 in the typical case where an

analytical solution is not available.

Assignment 3

PETR 5325 (Fall 2015) 34

To begin, we take the Taylor series forward expansion of 𝑃𝑃 with regard to 𝑥𝑥 (equivalent forms):

𝑃𝑃(𝑥𝑥 + ∆𝑥𝑥, 𝜕𝜕) = 𝑃𝑃(𝑥𝑥, 𝜕𝜕) +∆𝑥𝑥1!�𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝑥𝑥

𝑡𝑡

+(∆𝑥𝑥)2

2!�𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

�𝑥𝑥

𝑡𝑡

+(∆𝑥𝑥)3

3!�𝜕𝜕3𝑃𝑃𝜕𝜕𝑥𝑥3

�𝑥𝑥

𝑡𝑡

+ ⋯ (7𝐴𝐴)

𝑃𝑃𝑖𝑖+1𝑛𝑛 = 𝑃𝑃𝑖𝑖𝑛𝑛 +∆𝑥𝑥1!�𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝑖𝑖

𝑛𝑛

+(∆𝑥𝑥)2

2!�𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

�𝑖𝑖

𝑛𝑛

+(∆𝑥𝑥)3

3!�𝜕𝜕3𝑃𝑃𝜕𝜕𝑥𝑥3

�𝑖𝑖

𝑛𝑛

+ ⋯ (7𝑏𝑏)

Next, we take the backwards expansion:

𝑃𝑃𝑖𝑖−1𝑛𝑛 = 𝑃𝑃𝑖𝑖𝑛𝑛 +(−∆𝑥𝑥)

1!�𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝑖𝑖

𝑛𝑛

+(−∆𝑥𝑥)2

2!�𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

�𝑖𝑖

𝑛𝑛

+(−∆𝑥𝑥)3

3!�𝜕𝜕3𝑃𝑃𝜕𝜕𝑥𝑥3

�𝑖𝑖

𝑛𝑛

+ ⋯ (8)

We add (7𝑏𝑏) and (8):

𝑃𝑃𝑖𝑖+1𝑛𝑛 + 𝑃𝑃𝑖𝑖−1𝑛𝑛 = 2𝑃𝑃𝑖𝑖𝑛𝑛 + (∆𝑥𝑥)2 �𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

�𝑖𝑖

𝑛𝑛

+ ⋯𝑂𝑂(∆𝑥𝑥)4 (9)

The term 𝑂𝑂(∆𝑥𝑥)4 represents the summation of the remaining terms not shown. The term 𝑂𝑂

means, “of order.” In other words, the remaining terms are at most of order, or proportional to

(∆𝑥𝑥)4. Assuming ∆𝑥𝑥 ≪ 1, then 𝑂𝑂(∆𝑥𝑥)4 is negligible and may be safely ignored. This error

introduced to our approximation due to ignoring the remaining terms of the Taylor series

expansion is called the discretization error.

Rearranging (9) and simplifying:

𝑃𝑃𝑖𝑖+1𝑛𝑛 − 2𝑃𝑃𝑖𝑖𝑛𝑛 + 𝑃𝑃𝑖𝑖−1𝑛𝑛

(∆𝑥𝑥)2 = �𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

�𝑖𝑖

𝑛𝑛

(10)

Assignment 3

PETR 5325 (Fall 2015) 35

To make this equation more intuitively meaningful, consider the equivalent form:

�𝑃𝑃𝑖𝑖+1𝑛𝑛 − 𝑃𝑃𝑖𝑖𝑛𝑛∆𝑥𝑥 � − �𝑃𝑃𝑖𝑖

𝑛𝑛 − 𝑃𝑃𝑖𝑖−1𝑛𝑛

∆𝑥𝑥 �

∆𝑥𝑥=�𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝑖𝑖+12

𝑛𝑛− �𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝑖𝑖−12

𝑛𝑛

∆𝑥𝑥= �

𝜕𝜕 �𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝜕𝜕𝑥𝑥

�

𝑖𝑖

𝑛𝑛

(11)

The group �𝑐𝑐𝑖𝑖+1𝑛𝑛 −𝑐𝑐𝑖𝑖

𝑛𝑛

∆𝑥𝑥� represents the estimate of �𝜕𝜕𝑐𝑐

𝜕𝜕𝑥𝑥�𝑖𝑖

𝑛𝑛 at a forward half-step, ∆𝑥𝑥

2. The central

approximation of change in �𝜕𝜕𝑐𝑐𝜕𝜕𝑥𝑥�𝑖𝑖

𝑛𝑛 over ∆𝑥𝑥 is expressed by the middle of (11).

Plugging (10) into (1a), we obtain:

𝑃𝑃𝑖𝑖+1𝑛𝑛 − 2𝑃𝑃𝑖𝑖𝑛𝑛 + 𝑃𝑃𝑖𝑖−1𝑛𝑛

(∆𝑥𝑥)2 = 𝛼𝛼2 �𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕�𝑖𝑖

𝑛𝑛

(12)

The right side of (1a) has become defined in (12) at the particular discretization intervals of 𝑇𝑇 and

𝑛𝑛. This is the consequence of estimating the left side of (1a) in (12) at those intervals.

We now take the Taylor forward expansion-based approximation of 𝑃𝑃𝑖𝑖𝑛𝑛+1:

𝑃𝑃𝑖𝑖𝑛𝑛+1 = 𝑃𝑃𝑖𝑖𝑛𝑛 +∆𝜕𝜕1!�𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕�𝑖𝑖

𝑛𝑛

+(∆𝜕𝜕)2

2!�𝜕𝜕2𝑃𝑃𝜕𝜕𝜕𝜕2

�𝑖𝑖

𝑛𝑛

+(∆𝜕𝜕)3

3!�𝜕𝜕3𝑃𝑃𝜕𝜕𝜕𝜕3

�𝑖𝑖

𝑛𝑛

+ ⋯ (13)

𝑃𝑃𝑖𝑖𝑛𝑛+1 = 𝑃𝑃𝑖𝑖𝑛𝑛 +∆𝜕𝜕1!�𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕�𝑖𝑖

𝑛𝑛

+ ⋯𝑂𝑂(∆𝜕𝜕)2 (14)

𝑃𝑃𝑖𝑖𝑛𝑛+1 − 𝑃𝑃𝑖𝑖𝑛𝑛

∆𝜕𝜕= �

𝜕𝜕𝑃𝑃𝜕𝜕𝜕𝜕�𝑖𝑖

𝑛𝑛

(15)

Note that this approximation has a larger discretization error, which is proportional to (∆𝜕𝜕)2.

The left hand side of (15) would actually approximate �𝜕𝜕𝑐𝑐𝜕𝜕𝑡𝑡�𝑖𝑖

𝑛𝑛+12 more closely. However, by we

Assignment 3

PETR 5325 (Fall 2015) 36

can minimize this error as much as we want by making ∆𝜕𝜕 smaller. Subbing (15) into (12), we

obtain:

𝑃𝑃𝑖𝑖+1𝑛𝑛 − 2𝑃𝑃𝑖𝑖𝑛𝑛 + 𝑃𝑃𝑖𝑖−1𝑛𝑛

(∆𝑥𝑥)2 = 𝛼𝛼2𝑃𝑃𝑖𝑖𝑛𝑛+1 − 𝑃𝑃𝑖𝑖𝑛𝑛

∆𝜕𝜕 (16)

Rearranging and solving for 𝑃𝑃𝑖𝑖𝑛𝑛+1:

𝑃𝑃𝑖𝑖𝑛𝑛+1 = 𝑃𝑃𝑖𝑖𝑛𝑛 +∆𝜕𝜕

𝛼𝛼2(∆𝑥𝑥)2(𝑃𝑃𝑖𝑖+1𝑛𝑛 − 2𝑃𝑃𝑖𝑖𝑛𝑛 + 𝑃𝑃𝑖𝑖−1𝑛𝑛 ) (16)

This is our approximation function 𝜓𝜓 for 𝑃𝑃𝑖𝑖𝑛𝑛+1.

Neumann condition

Let’s suppose now that the boundary condition at 𝐿𝐿 is changed:

𝜕𝜕𝑐𝑐(𝐿𝐿,𝑡𝑡)𝜕𝜕𝑥𝑥

= 0 for 𝜕𝜕| 𝜕𝜕 ≥ 0 (17)

Now we are no longer given the value of any 𝑃𝑃𝐼𝐼𝑛𝑛. We will be unable to utilize 𝜓𝜓 to find any 𝑃𝑃𝑖𝑖𝑛𝑛

whose antecedent’s 𝜓𝜓 function depends on a value of 𝑃𝑃𝐼𝐼𝑛𝑛. We need to use (17) to develop an

approximation procedure for 𝑃𝑃𝐼𝐼𝑛𝑛.

Let’s add ∆𝑥𝑥 to 𝐿𝐿 to create an imaginary set of 𝑃𝑃𝐼𝐼+1𝑛𝑛 values. Now we use (7b) and (8) to write the

forward and backward Taylor expansions of 𝑃𝑃𝐼𝐼𝑛𝑛:

𝑃𝑃𝐼𝐼+1𝑛𝑛 = 𝑃𝑃𝐼𝐼𝑛𝑛 +∆𝑥𝑥1!�𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝐼𝐼

𝑛𝑛

+(∆𝑥𝑥)2

2!�𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

�𝐼𝐼

𝑛𝑛

+(∆𝑥𝑥)3

3!�𝜕𝜕3𝑃𝑃𝜕𝜕𝑥𝑥3

�𝐼𝐼

𝑛𝑛

+ ⋯ (18)

𝑃𝑃𝐼𝐼−1𝑛𝑛 = 𝑃𝑃𝐼𝐼𝑛𝑛 +(−∆𝑥𝑥)

1!�𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝐼𝐼

𝑛𝑛

+(−∆𝑥𝑥)2

2!�𝜕𝜕2𝑃𝑃𝜕𝜕𝑥𝑥2

�𝐼𝐼

𝑛𝑛

+(−∆𝑥𝑥)3

3!�𝜕𝜕3𝑃𝑃𝜕𝜕𝑥𝑥3

�𝐼𝐼

𝑛𝑛

+ ⋯ (19)

Assignment 3

PETR 5325 (Fall 2015) 37

Subtract (19) from (18) and simplifying:

𝑃𝑃𝐼𝐼+1𝑛𝑛 − 𝑃𝑃𝐼𝐼−1𝑛𝑛 = 2∆𝑥𝑥 �𝜕𝜕𝑃𝑃𝜕𝜕𝑥𝑥�𝐼𝐼

𝑛𝑛

+(∆𝑥𝑥)3

3�𝜕𝜕3𝑃𝑃𝜕𝜕𝑥𝑥3

�𝐼𝐼

𝑛𝑛

+ ⋯ (20𝐴𝐴)

𝑃𝑃𝐼𝐼+1𝑛𝑛 − 𝑃𝑃𝐼𝐼−1𝑛𝑛 = 2∆𝑥𝑥(0) + 𝑂𝑂(∆𝑥𝑥)3 (20𝑏𝑏)

𝑃𝑃𝐼𝐼+1𝑛𝑛 = 𝑃𝑃𝐼𝐼−1𝑛𝑛 (20𝜙𝜙)

Note that at (20b) we applied (17) to render �𝜕𝜕𝑐𝑐𝜕𝜕𝑥𝑥�𝐼𝐼

𝑛𝑛= 0. Taking (16) at 𝐼𝐼:

𝑃𝑃𝐼𝐼𝑛𝑛+1 = 𝑃𝑃𝐼𝐼𝑛𝑛 +∆𝜕𝜕

𝛼𝛼2(∆𝑥𝑥)2(𝑃𝑃𝐼𝐼+1𝑛𝑛 − 2𝑃𝑃𝐼𝐼𝑛𝑛 + 𝑃𝑃𝐼𝐼−1𝑛𝑛 ) (21)

Subbing (20c) into (21) and simplifying, we have:

𝑃𝑃𝐼𝐼𝑛𝑛+1 = 𝑃𝑃𝐼𝐼𝑛𝑛 +∆𝜕𝜕

𝛼𝛼2(∆𝑥𝑥)2(𝑃𝑃𝐼𝐼−1𝑛𝑛 − 2𝑃𝑃𝐼𝐼𝑛𝑛 + 𝑃𝑃𝐼𝐼−1𝑛𝑛 ) (22𝐴𝐴)

𝑃𝑃𝐼𝐼𝑛𝑛+1 = 𝑃𝑃𝐼𝐼𝑛𝑛 +2∆𝜕𝜕

𝛼𝛼2(∆𝑥𝑥)2(𝑃𝑃𝐼𝐼−1𝑛𝑛 − 𝑃𝑃𝐼𝐼𝑛𝑛) (22𝑏𝑏)

As long as we are given all of 𝑃𝑃𝑖𝑖0, including 𝑃𝑃𝐼𝐼0, we can then use (22b) to find 𝑃𝑃𝐼𝐼1. If we use (16)

to find the rest of 𝑃𝑃𝑖𝑖1, then we can use (22b) again to find 𝑃𝑃𝐼𝐼2. Clearly, we will be able to

approximate the remaining values of 𝑃𝑃𝑖𝑖𝑛𝑛.

Sometimes the Neumann condition is stated in a more general form:

𝜕𝜕𝑐𝑐(𝐿𝐿,𝑡𝑡)𝜕𝜕𝑥𝑥

= 𝐶𝐶 for 𝜕𝜕| 𝜕𝜕 ≥ 0 ; 𝐶𝐶 is a constant (23)

This would require the modification of (20b) through (22b) as follows:

Assignment 3

PETR 5325 (Fall 2015) 38

𝑃𝑃𝐼𝐼+1𝑛𝑛 − 𝑃𝑃𝐼𝐼−1𝑛𝑛 = 2∆𝑥𝑥(𝐶𝐶) + 𝑂𝑂(∆𝑥𝑥)3 (24𝐴𝐴)

𝑃𝑃𝐼𝐼+1𝑛𝑛 = 𝑃𝑃𝐼𝐼−1𝑛𝑛 + 2𝐶𝐶∆𝑥𝑥 (24𝑏𝑏)

𝑃𝑃𝐼𝐼𝑛𝑛+1 = 𝑃𝑃𝐼𝐼𝑛𝑛 +∆𝜕𝜕

𝛼𝛼2(∆𝑥𝑥)2(𝑃𝑃𝐼𝐼−1𝑛𝑛 + 2𝐶𝐶∆𝑥𝑥 − 2𝑃𝑃𝐼𝐼𝑛𝑛 + 𝑃𝑃𝐼𝐼−1𝑛𝑛 ) (25𝐴𝐴)

𝑃𝑃𝐼𝐼𝑛𝑛+1 = 𝑃𝑃𝐼𝐼𝑛𝑛 +2∆𝜕𝜕

𝛼𝛼2(∆𝑥𝑥)2(𝑃𝑃𝐼𝐼−1𝑛𝑛 − 𝑃𝑃𝐼𝐼𝑛𝑛 + 𝐶𝐶∆𝑥𝑥) (25𝑏𝑏)

For these cases, we need to know the value of 𝐶𝐶 to complete our approximations.

TASK 4

Reservoir simulation is a branch of reservoir engineering in which reservoir models are

built to predict fluid flow behavior through porous media. Such reservoir models can be either

physical (such as laboratory sand packs) or more often numerical. Numerical (i.e. mathematical)

models are built on a set of equations that - subject to certain assumptions and constraints -

describes the physical processes active in the reservoir. Reservoir simulation is an invaluable

tool used widely by the upstream petroleum industry to estimate reservoir field performance over

time under various producing schemes, thereby leading to more accurate assessments of risk and

reserves and more cost-effective field development planning.

This section of the report presents a comparative table that highlights key similarities and

differences between the two principal types of numerical reservoir simulators - black-oil and

compositional simulators. The key difference between these two types of models is rooted in the

methods used by the two simulators to characterize the phase behavior of a reservoir fluid.

Assignment 3

PETR 5325 (Fall 2015) 39

Table 3: Black-Oil vs Compositional Simulator

BLACK-OIL SIMULATOR COMPOSITIONAL SIMULATOR

Description • A fluid model which assumes that the stock-tank oil and gas have multiple components, and that all of the resulting fluid PVT behavior is a function of temperature and pressure only

• A fluid model which treats each individual component in the hydrocarbon fluid mixture separately and handles each in terms of moles of the individual components

Capabilities • Characterized by the number of fluid phases present (1, 2, or 3), the direction of flow, and the type of solution used for the complex fluid flow equations

• Concerned with modeling the phase behavior of the individual components (C1, C2, etc.) present in a hydrocarbon fluid mixture

Similarities • Both simulators are built to model reservoir fluid behavior for the purposes of aiding in the design of production processes and equipment

Differences • Governed only by fluid flow mechanisms

• Oil and gas phases are each represented by a single component

• Liquid/gas stock-tank compositions of the separator flash conditions are assumed constant, regardless of pressure

• Equations are written in terms of stock-tank volumes

• Governed by fluid flow and phase composition flow mechanisms

• Oil and gas phases are represented as multi-component mixtures

• Equations are written at reservoir conditions under the assumption that a valid EOS model can be used to represent fluid phase behavior

Advantages • Suitable for most petroleum reservoirs; can be used in ~90% of all petroleum reservoir simulation studies

• Can be used to model most reservoir fluids, including dry gases, wet gases, heavy oils and volatile oils

• Run times normally do not adversely affect the required total study time

• Capable of handling higher degree of complexity in fluid composition and behavior

• Can be used to accurately model lean gas cycling in the presence of oil that will vaporize in the lean gas

Disadvantages • Breaks down in cases where lean gases such as CO2 or CH4 are cycled in the presence of a liquid phase Cannot model variability in oil

vaporization since there’s only 1 oil component

Under-predicts early oil vaporization in the presence of injected lean gas when used to model gas recycling processes

• Does not model the effects of changes in reservoir fluid composition that may occur with depletion over time

• Run times can become long enough to adversely affect total study times and can require significantly more computing (CPU) power to perform

• Software interfaces tend to lag behind those of black oil simulators since it is used only by a small community of experts

• Requires the use of a verified EOS model to represent the fluid properties of the different components

Assignment 3

PETR 5325 (Fall 2015) 40

Process

Applications

• Black-oil simulators can be appropriately used for the vast majority of petroleum reservoir fluid systems

• For wet gas and volatile oil applications, a modified black-oil simulator can be constructed if PVT data for the reservoir includes both the gas dissolved in the oil phase and the oil dissolved in the gas.

• Compositional simulators are built to calculate residual fluid volumes resulting from phase behavior interactions and thus can model fluid saturation as a process variable, rather than requiring it as an input that must be defined by the user in order to perform the simulation

TASK 5

The goal of this task was to research the main commercial and open-source simulators that are

available for use in reservoir simulation studies. The key aspects of each simulator were

organized in the following series of tables.

Table 4: Commercial Simulators

Name ECLIPSE RESERVOIR SIMULATOR

Company Owner(s) Schlumberger

Overview • Arguably the industry’s most complete and robust numerical simulator • Offers fast and accurate prediction of dynamic behavior for all reservoir

types and development schemes

Characteristics

& Key Features

• Covers the entire spectrum of reservoir models – black-oil, compositional, thermal finite-volume, and streamline simulation

• Capable of modeling chemical reactions that occur in the reservoir and can simulate both live and dead oil behaviors

• Ideal for simulating thermal recovery methods like steam-assisted gravity drainage (SAGD) applied to heavy (high viscosity) oil reservoirs Can be used to build low-temperature thermal models suitable for

conducting laboratory experiments • Includes features designed for modeling reservoirs with complex

stratigraphy and well designs Dual completions, horizontal multi-stage fracture treatments, etc.

Applications • Capable of modeling and simulating nearly all processes, including: Steam-assisted gravity drainage (SAGD) Steam flooding In-situ combustion Cyclic steam injection

Assignment 3

PETR 5325 (Fall 2015) 41

Name IMEX RESERVOIR SIMULATOR

Company Owner(s) Computer Modeling Group LTD (CMG)

Overview • Regarded as one of the most efficient conventional three-phase black-oil simulators available on the market

• Used to history match and forecast performance of primary, secondary and tertiary recovery processes

Characteristics

& Key Features

• Generates simulation results faster than with any other black-oil simulator • Allows for quick screening of various recovery mechanisms prior to

moving to more complex simulations • Capable of accurately modeling the matrix-fracture transfer in naturally

and/or hydraulically fractured reservoirs • Allows for easy transition to modeling of EOR processes in sister

applications GEM and STARS • Seamlessly interfaced with CMOST to permit rapid history matching and

efficient development of reservoir management workflows

Applications • Can be used to model shale gas adsorption effects • Useful for a wide variety of applications, including:

Conventional and unconventional reservoirs Enhanced oil recovery (EOR) processes Coupled surface network modeling

Assignment 3

PETR 5325 (Fall 2015) 42

Name JEWELSUITETM SUBSURFACE MODELING

Company Owner(s) Baker-Hughes

Overview • User-friendly subsurface modeling software that can be used to build accurate 3D reservoir models with high structural complexity.

• Models are designed to capture regional and local structural variations and can be easily updated to incorporate new wells and geologic information

Characteristics

& Key Features

• Features a modern intuitive user interface, automatic and semi-automatic workflows and build-in smart rules to accelerate the learning curve for new users and reduce the risk of errors along with project study times

• Offers the most advanced multidisciplinary knowledge and techniques available for quick, accurate and effective data interpretation

• Can be used as a standalone software application or in conjunction with other JewelSuite or third party software to complement other workflows

• Uses a patented grid building technology to generate an advanced “Jewel” grid with faulted cells which are are cut and offset by the faults

Applications • Can be used to build accurate and complex 3D reservoir models • Other areas of application include:

Seismic data visualization and interpretation (time-depth conversion) Well correlation and structural modeling 3D gridding and reservoir property modeling via geostatistics Volumetrics, well planning, and multi-point statistics (MPS)

Assignment 3

PETR 5325 (Fall 2015) 43

Name TNAVIGATOR

Company Owner(s) Rock Flow Dynamics (RFD)

Overview • Interactive black-oil, compositional, and thermal simulator capable of using multi-core, multi-CPU systems

• Designed for performing dynamic black-oil, compositional, and thermal compositional reservoir simulations on laptops, servers and High Performance Computing (HPC) clusters

Characteristics

& Key Features

• Written in C++ • Employs Qt graphics libraries to create true multiplatform system • Offers interactive user control of the simulation run, allowing users to

monitor every step of the simulation at runtime and, with just a simple mouse click, to directly interrupt and modify the simulation configuration

• Incorporates state-of-the-art computing technologies like NUMA and Hyperthreading to exceed the performance of other competitor dynamic simulation tools

• Utilizes a fully implicit time scheme which allows for large time steps based on specific approximation criteria

• Uses a Bi-Conjugate Gradient Stabilized (BCGS) algorithm to solve systems of linear equations

• Handles only state variables of the reservoir model to prevent hard disk operational restrictions during simulation runs

Applications • Useful for a wide variety of applications such as: Injection optimization Waterflooding Sidetrack planning Field development planning and optimum well placement Modeling of natural and hydraulic fracture effects

Assignment 3

PETR 5325 (Fall 2015) 44

Name FLOWSIM STUDIO

Company Owner(s) Plano Research

Overview • Provides a fully implicit, three-phase, three-dimensional black-oil and composite reservoir simulator

• Simulates dynamic responses of petroleum reservoir performance by computing oil, gas and water phase flow behavior within the reservoir

Characteristics

& Key Features

• Capable of handling both single and dual porosity reservoirs • Regarded as being extremely flexible and user-friendly • Features various gridding options like rectangular, radial, corner point, to

handle complex reservoir structures • Local grid refinement and curvilinear grids can be used to model

areas of high activity such as near the well bores • High-level summary of key features:

Well data and simulation controls Real-time reservoir monitoring 2D mapping and plotting and 3D visualization capabilities

Applications • Can be used to simulate surface gas network operations and partial field model linking

• Can also be used for complex well modeling, multi-reservoir simulation, analysis of aquifer influx in water-drive reservoirs, and miscible flooding processes, among others

Assignment 3

PETR 5325 (Fall 2015) 45

Table 5: Public Domain Simulators

Name UTCHEM

Funding / Ownership University of Texas at Austin

Overview • A three-dimensional, multiphase, multicomponent, compositional, variable temperature, finite-difference numerical reservoir simulator

Characteristics

& Key Features

• Code was written to run on most Unix workstations and Windows PCs • Designed for third-order finite difference modeling with a flux limiter • Assumes constant pressure boundaries and can model both horizontal and

vertical wells • Features Cartesian, radial, and curvilinear gridding options • Handles heterogeneous permeability and porosity reservoirs • Also has a variety of groundwater hydrology applications

Applications • Developed by the Department of Energy’s National Petroleum Technology Office for applications such as: Tracer tests for characterizing both single and dual porosity oil reservoirs Polymer and high pH chemical flooding for enhanced oil recovery (EOR) Analysis of Surfactant EOR including the use of polymers and foam Profile control of oil wells with polymer gels Simulating skin damage of oil wells

Name MATLAB RESERVOIR SIMULATION TOOLBOX (MRST)

Funding / Ownership SINTEF Applied Mathematics

Overview • Consists of two main parts: (1) a core that offers basic functionality, and (2) a set of add-on modules that provide discretization, solvers and other workflow tools

Characteristics

& Key Features

• Features a comprehensive set of routines and data structures for reading, representing, processing and visualizing unstructured grids

• Designed with special emphasis on the corner-point gridding format used widely in the oil and gas industry

• Contains mimetic and multiscale flow solvers as well as transport solvers

Applications • Intended mainly as a toolbox for prototyping, testing and demonstration of new simulation techniques and modeling concepts on unstructured grids

Assignment 3

PETR 5325 (Fall 2015) 46

Name PC-GEL SIMULATOR

Funding / Ownership IIT Research Institute

National Institute for Petroleum & Energy Research (NIPER)

Overview • A three-dimensional, three-phase permeability modification simulator developed by incorporating an in-situ gelation model into a black-oil simulator (BOAST)

Characteristics

& Key Features

• Features include: Modeling of transport of each species of polymer/crosslinker system Modeling of gelation reaction kinetics of polymers with crosslinkers Rheology of the gel and polymer Inaccessible pore volume to macromolecules Adsorption of chemical species on rock surfaces Retention of gels on grains in the rock matrix Permeability reductions caused by adsorption of polymers and gels

Applications • Simulation and optimization of primary reservoir field production • Waterflooding and polymer flooding • Permeability modification treatments (acidizing, fracturing, etc.)

Assignment 3

PETR 5325 (Fall 2015) 47

Name GEMINI

(GEO-ENGINEERING MODELING THROUGH INTERNET FORMATICS)

Funding / Ownership U.S. Department of Energy (DOE)

Kansas Geological Survey (KGS)

Overview • A public domain web application centered on analysis & modeling of petroleum reservoirs and resource plays

• Creates projects from on-line data that comes from the KGS server or is uploaded by the user

Characteristics

& Key Features

• Provides optional security features to users (authorized access & passwords) • Features the following standalone analysis toolkit modules: PVT analysis Material balance Gridding and mapping LAS (well log) viewer Lease & field production reporting and analysis

Applications • Designed for small independents and consultants seeking to find, quantitatively characterize and develop bypassed reservoirs by levering the growing base of digital data resource

Assignment 3

PETR 5325 (Fall 2015) 48

Name OPM

(OPEN POROUS MEDIA)

Funding / Ownership Supported by various research groups in different countries

(Suite to be released under General Public Licensing (GPL) agreement)

Overview • The OPM initiative offers a set of tools useful for simulating flow and transport behavior of fluids in porous media

Characteristics

& Key Features

• Notable features of this open-source toolkit include: Maintains open-source code and data sets, so anyone with knowledge or

interest can learn and suggest improvements Built-in functionality to support multiple areas of application Versatile and can be made to support workflows in other scientific or

industrial fields

Applications • Applications are similar to the other software discussed, including: CO2 sequestration processes Enhanced oil recovery methods