[PDF] A Computer Method to Calculate Two-Phase Flow in Any Irregularly

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A Computer Method to Calculate Two-Phase Flow In Any

Irregularly Bounded

Porous Medium

ABSTRACT

R. V. HIGGINS

MEMBER AIME

A. J. LEIGHTON

JUNIOR MEMBER AIME

A fast method is needed for calculating thoroughly the performance of two-phase flow in reservoir rock with com plex geometry. The authors present such a method and show its accuracy by comparison with laboratory perform ance of a jive-spot pattern. In making the performance calculations, the quadrant of the jive-spot pattern was divided into four channels. The computer time required for the calculation was one minute. This is roughly about minute per channel.

INTRODUCTION

By using an electronic computer it is now possible to make reservoir performance calculations in which fewer assumptions and more significant variables are included than was possible with a desk calculator. These perform ance calculations include such variables as the properties of the reservoir rock and its contained fluids, properties of the rock-fluid system and even different well-spacing patterns.

Usually, the more variables

that are included the higher the computer cost-and time on the computer is expensive.

As a result, a short procedure

is needed so the many vari ables will be included and accurate performance calcula tions will result.

If a reservoir is composed of alternate lay

ers of permeable and impervious rock, and if each conduct ing layer has a different relative permeability, the combined computer cost for calculating each layer by the present long computer programs'" (which includes different well spacing patterns and other variables) would be expensive.

The performances

of complex spacing patterns in a field often are studied by a potentiometric model, but in these model studies the analogy is to the flow of only a single phase. This paper presents a procedure to calculate the per formance of a five-spot pattern with two phases flowing using a potentiometric model as a guide for the dimensions of channels and other related data.

The time required to

calculate the complete performance is about one minute on the IBM 7090 or four minutes on the IBM 704. The procedure can be extended to calculate the performance of more complex well spacings or an entire field with little modification. The procedure has been tested on the performance of the laboratory water floods of a five-spot pattern reported Original manuscript received in Society of Petroleum Engineers office Nov. 24, 1961. Revised manuscript received April 25, 1962. Paper pre sented at 32nd Annual California Regional Meeting of SPE, Nov 2-3 1961, in Bakersfield, Calif. . • given at end of paper. SPE 243

JU]\'E, 1962

U. S. BUREAU OF MINES

SAN FRANCISCO, CALIF.

by Douglas, et al: The range of viscosity ratios in the laboratory experiments is from 0.083 to 754, which is ex tensive.

The accuracy of the performance calculations is

excellent as shown by the plotted points in Fig. 1. The recovery as influenced by the geometry of the five spot, taking into account the deviation from true radial character, has been predicted by Aronofsky and Ramey.' The authors used a potentiometric model to determine the sweep efficiency to breakthrough of the flooding agent.

Dyes, Caudle and

Erickson: by means of a laboratory

model, show the influence of mobility ratio on recovery after breakthrough of the injected material. Dyes, et ai, used the X-ray shadowgraph technique and miscible phases in their studies. Craig, Geffen and

Morse" from the analysis

of the performance of their laboratory model correlate many variables, including average saturation of a flooding agent determined by permeability relationship. Craig, et aI, used X-ray shadowgraphs to determine the sweep efficien cies in sandstone models.

In this report the sweep efficiency

is not needed, and the important effect of continuously changing saturation gradients before and after breakthrough .8 0.083 .7 o c w u .6 c 0 0: Il. .5 0 IIJ .4 ::Ii 0

IIJ .3

0: 754
0 Il. .2 --Experimental

A00. Calculation by Douglas et 01 (REF. 2)

+ Higgins -Leighton long method ( REF. II .. Higgins -Leighton short method

0.5 1.0 1.5 2.0 2.5 3.0

PORE VOLUMES WATER INJECTED

FIG. I-COMPARISON OF LABORATORY AND COMPUTER PERFOR:'.I

ANCE CALCULATIONS FOR THE FIVE:SPOT PATTERN.

679 Downloaded from http://onepetro.org/JPT/article-pdf/14/06/679/2213588/spe-243-pa.pdf by guest on 01 July 2023

is used in determining recovery, rates, cumulative injection and other data that are functions of time.

THEORY

The material-balance equation

(I) relating variables when a single displacing fluid enters a small sand element, has been stated by Buckley and Lever ett." They show that, in terms of QT the total amount of displacing fluid entering the system, Eq. 1 may be expressed

Lll = ) (2)

cf>A dSDi, Treating many conduits as approximately one-dimensional, and neglecting pressure gradients transverse to the main flow, Eq. 2 may be expressed approximately as Eq. 3a. The integral term, a volume, has properties similar to those that Buckley and Leverett found applicable to "1, distance in arbitrary units". This property, combined with tech niques discussed later, makes it possible to calculate rapidly the performance of reservoirs whose flood pattern is non linear. f'.p (x) A(x) dx = Q,I' x, (3a) The reservoir was divided into sand elements, or cells. The cell boundaries need not be parallel to one another. For convenience in making the performance calculations, the channel volume was divided into cells of equal volume. The use of equal volumes also has the advantage of main taining a parallelism to the linear model. In addition, by a few changes, the nonlinear program can calculate readily the performance of the linear model. In the procedure described in this paper, the stream channels in the five-spot were divided into cells of equal volume; the length of each cell is fixed by the equal-volume requirement.

The sum of the length of the cells in a chan

nel is the total distance between the inlet and outlet. Dur ing flow, the saturation distribution changes take place in the equal volumes, as they do in the equal volumes in the linear model. Because of the different lengths and mean effective cross-sectional areas, the shape factor G (geome tric resistivity) of each cell of equal volume is different. The shape factor was determined from the potentiometric model in Fig. 2, whereas in the linear model this is readily calculated for one cell and is the same for the remaining cells. In a true radial model, the shape factor and equal volumes can be calculated analytically. The section of the saturation profile through a cell was used to determine the average permeability to water and oil of the cell. The average permeability' of each cell at the end of a time interval and the shape factor determine the resistance to flow. From the sum of the resistances and the pressure drop between the inlet and outlet wells, the in stantaneous rate of flow was obtained. The flood performance to breakthrough was computed at the successive invasion of each cell by water. When beginning in the primary stage, the entire saturation distri bution at breakthrough from inlet to outlet was used for the first cell, then the first two cells and, correspondingly, for the remainder of the cells until breakthrough. The quantity of oil produced during each step is that displaced by the change in water saturation of the newly invaded cell and the small change in the upstream cells. This is a 680

Equipotential lines

----I>' --Streamlines FIG. 2-EQUIPOTENTIAL LINES AND STREAMLINES USED TO CALCl;' LATE PERFORMANCE OF A FIVE-SPOT WATER FLOOD (AFTER VAUGHN AND WATKINS, REF. ll). CHANNEL 1 WAS CHOSEN BETWEEN STREAMLINES 1 AND 2, CHANNEL 2 BETWEEN STREAMLINES 2 AND

3, ETC. THE QUADRANT BOUNDARY WAS USED INSTEAD OF

STREAMLINE 5 FOR CHANNEL 4.

constant quantity until breakthrough when cells of equal volume are used, as each step-wise calculation uses the saturation distribution after the water-oil interface reaches the end of a cell. After breakthrough from the last cell, the oil produced is equal to the saturation change in all cells as the satura tion profile changes with water throughput QT. The elapsed time required to produce the oil is obtained by dividing the volume of oil produced between two consecutive instan taneous rates by the average of the two rates. The production from all the channels in the five-spot pattern was summed for the same time to obtain the over all performance.

DESCRIPTION OF PROCEDURES USED

The computer calculated first the f w curve from the per meability-saturation curves, the water saturation at the out let face at breakthrough and the average water saturation at breakthrough. Then the saturation interval from break through to the irreducible water saturation was divided into equal parts to use the Stirling' equation to obtain the derivatives

1'. The I' and the permeability to water at the

equal water saturations were calculated.

With these data,

the computer's equivalent look-up table for the curve shown in Fig.

3-permeability distribution vs I' or related !:::.(Icf>A)

-was put in storage. The area under Curve B-reciprocal of permeability-in Fig. 3 was divided into sections by the computer, and the area of each section was indexed and stored. The use of sections reduces the time required for the computer to calculate areas under different parts of Curve B shown in Fig. 3. The mean permeability to water of the first cell after the water reaches the end of the first cell is the maximum abscissa, which is the I' at the outlet face at the time vf

JOlJR!"AL OF PETROLEUM TECHNOLOGY Downloaded from http://onepetro.org/JPT/article-pdf/14/06/679/2213588/spe-243-pa.pdf by guest on 01 July 2023

.8 a: .7 U) a: UJ !c.6 .5 _____ ---.J2

DERIVATIVE, f'

1 1 1 o 0.2 0.4 0.6 O.B 1.0

FRACTION OF TOTAL VOLUME, Tot.1 Volume

FIG. 3-WATER SATURATION, WATER PER;\1EABILITY, AND RECIPRO· CAL OF WATER PERMEAlllLITY VS DERIVATIVE OF FRACTIONAL FLOW AC'iD FRACTION OF TOTAL VOLUME BEHIND FLOOD FRONT.

I'bn divided by the area under Curve B in

Fig.

3. When the first two cells have been invaded, the

mean permeability to water for the first cell is one-half of I' br divided by the area under the curve to the abscissa that is 112 I'or. The mean permeability to water in the second cell is 112 I' Or divided by the remaining area. After the interface had reached the end of the third cell, the abscissa length

I'or was divided into three equal parts and

the respective mean permeabilities for the first three cells were then determined. Likewise, the average permeability of the cells was determined as the cells were progressively invaded.

When the last cell was invaded (just at break

through), 1'0" was divided into 40 equal parts to determine the mean permeabilities of the 40 cells. No attempt was made to find the minimum number of cells at which the results would not be reproducible. After breakthrough, the mean permeabilities were ob tained by moving backward' from I' Or' The size of the steps is optional in the computer program and was chosen before a run.

No test has been made to determine the

maximum size possible. Steps of 1/20 1'0" have worked satisfactorily.

The Sw corresponding to I'o,·-m at each in

terval is the water saturation at the outlet face for the instantaneous rate calculation. Also, the curve above the abscissa to

1'"-,,, is the permeability-to-water profile for the

40 cells. The abscissa I' Or-m was divided into 40 equal ele

ments.

The area above each element was divided by 1/40

X I' Or-m to obtain the mean permeability to water for the

40 cells. The abscissa f' Or-m was successively reduced until

the S", corresponding to I' resulted in a water-oil ratio in excess of that previously read into the computer. The division of I'or and I'or-m into equal parts is the result of using cells of equal volume. The corresponding mean permeabilities to oil were determined in a similar manner. The following equations were used in the computation and explain many of the working details. The instantaneous rate at the start in the primary phase,

JUNE, 1962

where the shape factor (geometrical resistances) G

L/A, is

q"i-o initial (primary)

Ka!:::.P

on , .• NCELLS n = 1

The instantaneous rate after the start is

(4) qaj (primary) (K,,!:::.p)/[n ----~k"----) .. G,, n = 1 _ ;:can n + __ ;',:""'.!: + i = 5t cells k1"OIW i = No. Cells

Invaded + 1

where the average water permeability in each cell is f' J i+l dl' f' krwmean = "'i'---____ _ J f'i+ 1 1 k1"W mea n = f' i -o----ccc-d f' k", (f')

All Areas under l/k,,,

vs I' Curve in nth Cell (5) (6) .(6a) where j = number of cells which have been invaded. An equation similar to Eq.

6a was used for the oil permeability.

After the breakthrough

of water,

K,,!:J.P _ . (7)

qu CJ.w j n _ No. Cells (subordmate) -j(-----k .. G" no." 1 (""'m,.an" ----_ .. + fLw fL" where the average permeability to water in each cell IS kt"IFllleallll ~ All Areas underi;k,,,

L.. vs I' Curves in nth Cell

and, similarly, for the oil permeability.

The water-oil ratio

is knr 11-0

WORj=m+scHI,LIS" = k

1 "o fLw for Swat the outlet face, which is S" at I' br-m. (8) (9) In the subordinate phase, the instantaneous oil and water rates are obtained from the instantaneous water-oil ratio and the sum of water and oil rates (Eqs. 7 and 10). q"J (subordinate) qWm+NCELLS

WORm+.\'C;JI.!."

where j = m + NCELLS.

The average oil and water rates are

q"i + qUj', q'j"h = 2 and qll'j-1h ( 10) ( 11 ) (12) If the calculations are started after fill-up, the initial oil ratequotesdbs_dbs14.pdfusesText_20

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