Numerical Simulation of Naturally Fractured Reservoirs F. Sonier, SPE, Simulation and Modelling Consultancy Ltd. P. Souillard, Elf Aquitaine F.T. Blaskovich, SPE, Simulation and Modelling Consultancy Ltd.

Summary. The most important and difficult aspect of modeling a naturally fractured reservoir is the correct calculation of the exchange of fluids between the matrix rock and the surrounding fractures. Several authors have published alternative techniques for handling this problem over the past few years. However, because each of these alternatives has some limitations, a new and more general technique has been developed. This new technique is used to simulate matrix/fracture exchange with special emphasis on the gravity forces included in the exchange terms. The exchange terms and the gravity forces within the exchange terms simulate the behavior of a single matrix block surrounded .by fractures that may contain several different fluids. The gravity forces are internally calculated as functions of saturation. This technique has been incorporated into a new three-dimensional (3D), three-phase, fully implicit model for simulating fluid flow in a naturally fractured reservoir. The description of the porous medium might include highly fractured, microfractured, and nonfractured regions. Several examples explain the use of a new naturally fractured reservoir model and the essential differences between the new approach and those used in earlier naturally fractured reservoir models.

Introduction The numerical simulation of naturally fractured reservoirs is a sub-ject that has been described extensively in the literature of the oil industry for the last 25 years.

In the initial work done, the primary goal of the simulation of fractured systems was to study pressure behavior during well tests. This was based primarily on the analytical solutions obtained by Barenblatt et at. l and Warren and RootZ for describing single-phase flow near a wellbore in a naturally fractured reservoir. Their work was based on the concept of a fractured continuum filled with noncontinuous matrix blocks. Warren and Root also delivered a one-dimensional (ID) radial model, which Kazemi3 later extend-ed to study two-dimensional (2D) flow during a well test. Further developments were made by Iffly et at.,4 Yamamoto et at.,5 Kleppe and Morse, 6 and others, but these models were used either to match laboratory results or to study the behavior of a single matrix block.

More recently, more complex problems have been studied with full-field reservoir models. Our interest in this paper is the study of fractured reservoirs with full-field models.

Examples are presented that show some of the differences be-tween a new fractured reservoir model and other models described in the literature. Comparisons are also made that point out some of the complexities of fractured reservoir simulation and the potential problems with the use of a single-porosity approach to study vari-ous types of fractured reservoir behavior.

One of the first papers to describe a fractured reservoir simula-tor that could be applied to full-field model studies was by Saidi7 and was concerned with the simulation of the very highly fractured reservoirs in Iran. This model represented the fracture system as a 2D network because the fracture conductivity was so large that vertical equilibrium approximations were adequate. In addition, be-cause of the extremely low matrix-rock permeabilities, flow be-tween grid cells in the model occurred only in the fractures. The matrix rock in each grid cell acted only as a source or sink term in the simulator flow equations. This model worked well for the particular applications that it was designed for. However, the ap-proach is not general enough to model effectively the many differ-ent types of naturally fractured reservoirs that exist in different parts of the world.

Another approach to modeling fractured reservoirs was devel-oped by Reiss et at. 8 and Rossen. 9 In both of these models, the exchange of fluids between the matrix and surrounding fractures

'Now at Standard Oil Co. Copyright 1988 Society of Petroleum Engineers

1114

was controlled by describing the recovery from the matrix as a func-tion oftime. These "transfer functions" had to be determined out-side the simulator (analytically, experimentally, or from the simulation of a matrix block). The basic problem with this tech-nique is that it did not account for changes in reservoir conditions and operating practices with time. It was a "static" approach to describing a "dynamic" process in the reservoir.

In this paper, we define a "static" approach to be one that is strongly dependent on input from outside the model, such as the transfer functions described above. In contrast, a "dynamic" ap-proach is a fully integrated one in which the model internally cal-culates the results, which can change as a result of changes in conditions (pressure, saturation, fluid composition, and field oper-ating practices) during the simulation.

The first attempt at a more general approach to fractured reser-voir simulation was described by Kazemi et at. to In this paper, a 3D model for water/oil applications was described in which matrix/fracture exchange terms were calculated as functions of reser-voir conditions. The exchange terms controlled the movement of fluids between the matrix and fractures within a grid cell and were a function of viscous and capillary forces.

As in the earlier models, the Kazemi et at. model treated the matrix as a discontinuous series of sources and sinks that was sur-rounded by a continuous fractured medium. The flow between grid cells occurred only in the fractures. For the flow of either phase (oil or water) between the matrix and fractures in a grid cell, Kazemi proposed that the exchange term for Phase IX should have the fol-lowing form:

( kk Vb) E =0 001127 _,_a_ a . a( af- am)' ............... (1) JJ.aBa m

The phase potential difference in Eq. 1 is a combination of viscous and capillary forces but does not include any gravity forces.

Later, Gilman and Kazemi 11 presented an updated and improved version of Kazemi et at.'s original model in which they attempted to handle three-phase flow and to model gravity effects in the matrix/fracture exchange terms that were not included in the origi-nal model. This was done by defining different depths (Dm and Df ) for the matrix and fractures within a gridblock. The resulting matrix/fracture exchange term was essentially the same as in Eq. 1, but the potential difference was redefined as

af-rtm =(Pa.r'YafDf)-(Pam -'YamDm) . ........... (2) SPE Reservoir Engineering, November 1988

TABLE 1-MOLE FRACTIONS OF COMPONENTS 1 THROUGH 3 AVAILABLE IN THE THREE PHASES

Matrix Medium Fracture Medium Xl Yl xwd=O) Xl Yl Xwl (=O) X2 Y2 Xw2 X2 Y2 Xw2 X3(=0) Y3( =0) XW3 X3(=0) Y3(=0) Xw3

The pressure terms Pam and Pa! in Eq. 2 mayor may not include a capillary pressure, depending on the phase being defined.

The gravity term included in the flow equations shown above was another static, instead of dynamic, approach to the problem. The use of two different depths (D! and Dm) in the matrix/fracture ex-change implied that the gravity force would re~~in c~n~tant.regardless of saturation changes with time. In addition, It Implied that the fractures are located in one portion of the grid cell while the matrix rock is located in another portion of the same grid cell. In reality, gravity effects should change as a function ofthe fluid dis-tribution in the matrix and fractures, and we usually do not know the actual distribution of the matrix and fractures within a grid cell. We normally make the assumption that both matrix and fractures are distributed evenly across the entire grid cell.

The problems with the Gilman gravity terms are demonstrated in an example described later in this paper.

Thomas et al. 12 also developed a three-phase fractured reservoir simulation model. Their equation for describing matrix/fracture ex-change was similar to that used by Kazemi et al. shown in Eq .. 1.

However, Thomas et al. recognized some of the problems In-herent in the Kazemi et al. model. They tried, for example, to in-clude the gravity effect in the matrix/fracture exchange by using pseudocapillary pressure and relative permeability cur~es for b~th the fractures and the matrix. This approach to handlIng gravity forces is also static instead of dynamic. The use of pseudofunc-tions implies that the gravity term has to be generated by simulat-ing laboratory experiments and creating a function based on matching the experimental results. In addition, Tho~nas et al. used the same pseudo functions to describe both the flow In the fractures and the exchange between the matrix and fractures, which should normally require different functions because of the different types of behavior exhibited.

The pseudo functions used by Thomas et al. resulted in a verti~al equilibrium system in their model in both the fractures and matrIx. This reduced the flow from three to two dimensions and was based on the assumption that the vertical matrix permeability is so ~arge that the vertical eqUilibrium concept is valid and that the predorrunant recovery mechanism is vertical flow caused by gravity.

In addition, Thomas et al. found that the shape factor, a, had to be adjusted depending on whether the matrix blocks were sur-rounded by water or gas. In effect, this type of fitting procedu:e is similar to the transfer function approach that had been used In earlier models.

Again, as with the Kazemi et al. system, the Thomas et al. model only allowed for flow in the fractures. The matrix acted only as a source or sink term.

Bossie-Codreanu et al.13 presented a 3D, three-phase black-oil model that simulated naturally fractured reservoirs by using in-dividual grid cells to represent the matrix and fractures. This model was a conventional single-porosity simulator in which individual fractures and matrix blocks were gridded. The matrix/fracture ex-change was controlled by including scaling coefficients that could be adjusted by the user to match the reservoir performance ..

There were several problems with this approach; the most Im-portant ones were memory and computer-time requirements. As a matter of fact, to model a full reservoir would require many more grid cells than would be required by using a real naturally frac-tured reservoir simulator. To simulate the Kazemi et al. five-spot problem required as many as 4 to 14 times the number of grid cells required with a normal fractured reservoir simulator.

Blaskovich et al. 14 and, later, Hill and Thomas 15 presented a simulator that incorporated a much more general treatment of the flow phenomena that can occur in a naturally fractured reservoir. In contrast to all previous simulators described here, this model

SPE Reservoir Engineering, November 1988

accounted for flow within both the matrix and fracture systems, as well as flow between the matrix and fractures. In this case, both the matrix and fractures were considered to be continuous media. However, the exchange terms used were based on an approach simi-lar to that of Thomas et al., 12 which relied on pseudofunctions to include a gravity effect.

Litvak16 described another 3D, three-phase fractured reservoir simulator that could be used in full-field model studies. He described a further refinement of the matrix/fracture exchange-term calcula-tion in which he attempted to account for gravity forces in a differ-ent way than had been used in the previous models.

Essentially, the matrix/fracture exchange term was based on the Kazemi et al. formulation (Eq. I), but Litvak also included an ad-ditional gravity term, PG, that he expressed for the case of water/oil exchange as

In this expression, the terms 2! and 2m represented the water lev-els in the fractures and matrix, respectively, for a single matrix block. A similar expression was used for the gas/oil case.

Using the vertical equilibrium concept in the fractures, Litvak determined the number of matrix blocks surrounded by gas or by water within a grid cell. He then applied the water/oil term (Eq. 3) to those matrix blocks below the water/oil contact and the simi-lar gas/oil equation to those matrix blocks above the gas/oil c~ntact and in this way determined the total exchange caused by gravity within a grid cell. Litvak did not explain how the contact levels in the matrix and fractures were calculated.

Because of the need to determine the contact levels within the grid cell at any timestep, Litvak's gravity terms could not be solved implicitly and this could normally cause stability problems. In his formulation, however, this technique does not add any significant instability because his is strictly an implicit pressure, explicit satu-ration (IMPES) model. Although his approach incorporates an in-ternally calculated gravity term, the concept of counting matrix blocks above and below contacts is very similar to the technIques used in the earlier transfer-function models.

Model Description On the basis of all the models described above, we decided to use a different approach to modeling the fluid-flow phenomena in natur-ally fractured reservoirs. The new model includes the ability to han-dle both dual-porosity and dual-permeability reservoirs, as well as reservoirs that contain some combination of microfractured, mac-rofractured, and unfractured areas. Although this paper concen-trates primarily on the development of matrix/fracture exchange terms, we first need to define the total flow equations used in the new model.

Flow Equations and Solution Techniques. In the model, we use a compositional formulation. The water phase is treated. in ~he same way as the two hydrocarbon phases, and the molar contInUity equa-tions for Component k are expressed in finite-difference form as follows.

Fracture Flow Equation.

Matrix Flow Equation.

1115

w w w w w w a: a: a: a: a: a: :::I :::I :::I :::I :::I :::I I- MATRIX l- I-0 MATRIX l- I- MATRIX

I-0 0 0 0 0 < < < < < < a: a: a: a: a: a: II. II. II. II. II. II.

CASE A CASE B CASE C

NO GRAVITY EFFECT MAXIMUM GRAVITY EFFECT NO GRAVITY EFFECT

Zwm = 0.0 Zwm = 0.0

Fig. 1-Minlmum and maximum gravity effects.

The variables Eot. Egb and Ewk represent the matrix/fracture ex-change terms per component, which will be described in more de-tail below. The flow terms qo, qg, and qw are expressed in moles per day, and ~o' ~g, and ~w are the molar densities of oil, gas, and water, respectively.

The model has the capability for handling a fully extended black-oil system in which the variables x, y, and x w in Eqs. 4 and 5 rep-resent the mole fractions of Component k contained in the oil, gas, and water phases, respectively. At the present time, only three com-ponents are included in the model, but because of the composition-al formulation used, it could be extended to n components in the future.

Table 1 shows the mole fractions of the three components (Com-ponents 1, 2, and 3) available in the three phases.

In the extended black-oil system, the following rules apply. 1. The oil component can exist in the oil and gas phases. (The

oil in the gas phase is y 1 and is equivalent to an oil vaporization ratio, or rs term.)

2. The gas component can exist in the oil, gas, or water phases. (The gas in the water phase is xw2 and is equivalent to a gas solu-bility ratio, or Rsw term.)

3. The water component can only exist in the water phase. Phase behavior is handled through the use of K values in the stan-

dard equilibrium equations. The above flow equations are completed with classic constraint

equations and capillary pressure relations. In a dual-permeability system, a choice of six primary unknowns is made in each grid, depending on the types of fluids present. All terms in the equa-tions are expanded into linear combinations of the primary unknowns and the equations are solved with the Newton-Raphson method.

To avoid stability problems, especially when modeling dual-porosity or dual-permeability systems, the fully implicit solution technique has been developed as the basic version. Sequential so-lutions and the IMPES solution are obtained as simplified versions of the fully implicit one.

Exchange Terms. For ease of understanding, we will develop the matrix/fracture exchange terms Eo, Eg, and Ew in terms of black-oil properties so that they can be compared with the previous authors' models.

For a Phase 01, let

Ta =O.()()1127(AX~Y~)u(_k_) .................. (6) JLaBa m

Then, we can write

X[Poj-Pom-d'Ygo(Zgf-Zgm)-d'Ywo(Zlif-Zwm)], ....... (7)

1116

Ew=Tw[wwkrwm +(l-ww)krnf]

x[Poj-Pom-(Pcwoj-Pcwom)+d'Ywo(Zwj-Zwm)], ....... (8)

and

Eg = Tg[wgkrgm +(l-wg)krgf]

X [Poj-Pom-(Pcgoj-Pcgom)+d'Ygo(Zgf-Zgm)], ........ (9)

where d'Y go = half the oil and gas gradient difference, d'Ywo = half the water and oil gradient difference,

W O' wg , Ww = upstream weighting coefficients for the oil,

gas, and water relative permeabilities, respectively, and

Zlif' Zwm, Zgf, Zgm = water and gas levels in the fracture and

matrix within a grid cell, respectively.

A derivation of these matrix/fracture exchange terms is given in the Appendix.

The differences,

for oil,

d'Ywo(Zwj-Zwm) .................................. (11)

for water, and

dr go (Zgf-Zgm) .................................. (12)

for gas, will represent the gravity effects in the exchange terms. The gravity term in the matrix/fracture exchange is based on the

height of the matrix blocks, Lz, and the change of saturation within the matrix and surrounding fractures. The saturations are convert-ed to hypothetical fluid levels by assuming that all matrix blocks in the grid cell exist at the same saturation and that the saturation within the fractures (and the matrix) is the same in any portion of the grid cell.

Therefore, the following expressions can be used to express the levels:

SPE Reservoir Engineering, November 1988

...

Ol Ol :; Ol Ol

0 ....

< a:

= 0 , a: w ....

< ;t

4

0 0

I I I I I I I I

// I ~ __

-'

'" '" -'

'" '"

e ....

c: -'

? II: W ....

< ~

o 200 400

.-0 .-Af/

600 800

I /

TIME - DAYS

KAZEMI CASE 1 (NO GRAVITY) KAZEMI CASE 2 (NO GRAVITY)

I I

I

1000

~

I I

I

I

I

I

1200 1400

- -e - KAZEMI CASE 1 WITH FRACTURES AT TOP OF BLOCK

- -B - KAZEMI CASE 1 WITH FRACTURES AT BOTTOM OF BLOCK

----+ - NEW MODEL WITH GRAVITY CL z = 30ft)

Fig. 4-A comparison with gravity effects proposed in other models.

of the differences between the behavior of naturally fractured reser-voir simulators and more conventional single-porosity simulators.

WOR Performance. Kazemi et aI. Five-Spot Problem. The Kazemi et al. five-spot problem (described in Ref. 10) was used to com-pare the results of the new model with previously published models. Note that Kazemi et al. did not include gravity effects. They showed the results of three cases with this model: Case 1, a fractured reser-voir with capillary pressure; Case 2, a fractured reservoir with zero capillary pressure; and Case 3, an unfractured reservoir (with ef-fective permeability from the fracture system and all other reser-voir description data from the matrix system).

We simulated the above three cases with our model by setting the gravity terms to zero and found that our results matched the results shown by Kazemi et ai. Good model stability and material balances made it possible to simulate the problem with average timestep sizes of 60 to 90 days under automatic timestep cont~ol.

Gravity Effects Without Capillary Pressure. To show the Im-portance of the gravity effects, we simulated Kazemi et ai.'s sec-ond case with our model but included the new gravity terms in the potential difference.

Fig. 2 shows the results from Kazemi et ai. without gravity ~ffects (Case 2 described above) and the results from our model WIth gravity effects and assuming two different matrix block heights (Lz= 10 and 30 ft [3 and 9.1 m]). Note that, in each case, the value of the shape factor, a, used was the same. This was done to dem-onstrate the importance of the matrix block height, Lz, on the results.

The results show the dramatic importance of the gravity terms when capillary forces are negligible. The difference in results occurs because the gravity force in the water-exchange term acts in the same direction as a capillary pressure to displace oil from the matrix with water. Referring to Eqs. 10 and 11 and Fig. 1, we can see that water will tend to move from the fractures to the matrix be-cause of gravity as long as Zwj>Zwm' At the same time, oil will tend to move from the matrix to the fractures as long as Zwm

TABLE 2-GOR PERFORMANCE MODEL BASIC RESERVOIR DATA

Reservoir grid 9x3x3 M=.dY, ft 500 Thickness, ft 20 to 100 U, ft- 2 0.09 Lx =L y, ft 10 Lz,ft 20 Fracture porosity, % 0.005 Matrix porosity, % 0.25 Fracture permeability (effective)

kXf=k yf , md 100 kZf' md 5

Matrix permeability k xm =kYm =kZm' md

Rock compressibility Matrix = fracture, psi -1 3 x 10-6

Datum depth, ft 9,290 Datum pressure, pSia 4,014.7 Fluid properties

Oil compressibility, psi- 1 1.47x10-5 Oil specific gravity 0.787 Gas specific gravity 0.792 Bubblepoint pressure, psia 1,614.7

Pressure Rs Bo iJ.o Bg iJ.g Bw iJ.w (psia) (scflSTB) (RB/STB) (cp) (RB/Mcf) (cp) (RB/STB) (cp) 14.7 1.0 1.062 1.040 226.28 0.008 1.0410 0.31

264.7 90.5 1.150 0.975 12.09 0.010 1.0403 0.31 514.7 180.0 1.207 0.910 6.01 0.011 1.0395 0.31

1,014.7 371.0 1.295 0.830 2.84 0.014 1.0380 0.31 2,014.7 636.0 1.435 0.695 1.34 0.019 1.0350 0.31 3,014.7 930.0 1.565 0.594 0.91 0.023 1.0320 0.31 4,014.7 1,270.0 1.695 0.510 0.74 0.027 1.0290 0.31 5,014.7 1,618.0 1.827 0.449 0.65 0.031 1.0258 0.31 9,014.7 2,984.0 2.357 0.203 0.51 0.047 1.0130 0.31

TABLE 3-GOR PERFORMANCE MODEL RELATIVE PERMEABILITY AND CAPILLARY PRESSURE DATA

Matrix Data Sw krw krow Pcwod Pcwoi (fraction) (fraction) (fraction) (psi) (psi)

0.250 0.0000 0.7000 3.6000 3.6000 0.300 0.0128 0.5180 2.1300 2.0000 0.400 0.0520 0.2975 1.0800 0.9200 0.500 0.1000 0.1750 0.5375 0.3750 0.600 0.1700 0.0910 0.2785 0.0000 0.700 0.2960 0.0224 0.1420 -0.4000 0.750 0.4000 0.0000 0.1000 -1.0000 1.000 1.0000 0.0000 0.0000 -1.0000

Sg krg krog P Cgod (fraction) (fraction) (fraction) (psi)

0.000 0.0000 0.7000 0.0000 0.030 0.0000 0.6500 0.2000 0.100 0.0277 0.4529 0.3970 0.200 0.0820 0.2541 0.5539 0.300 0.1529 0.1262 0.7126 0.400 0.2692 0.0525 1.0266 0.500 0.5000 0.0000 2.0000

Fracture Data Sw krw krow Pcwod Pcwol (fraction) (fraction) (fraction) (psi) (psi)

0.000 0.0000 1.0000 0.0000 0.0000 0.500 0.5000 0.5000 0.0000 0.0000 1.000 1.0000 0.0000 0.0000 0.0000

Sg krg krog Pcgod (fraction) (fraction) (fraction) (psi)

0.000 0.0000 1.0000 0.0000 0.500 0.5000 0.5000 0.0000 1.000 1.0000 0.0000 0.0000

SPE Reservoir Engineering, November 1988 1119

1000

8000 i ! If I RUN 4 ! J / 5000 RUN 3; ;

/ / " RUN 2 4000 i jv 3000 ! // RUN 1 2000 I i / 1 1/

-"-~ ~ u ~~

tOOO

0 1 2 3 7 10

TIME - YEARS

Fig. 6-GOR performance during pressure depletion.

at its maximum value. As a result, water easily goes from the frac-tures to the matrix and rapidly displaces all the movable oil into the fractures. Water breakthrough at the producer is very late and oil recovery at the end of 1,500 days is even better than in the single-porosity case. At the end of the simulation, the WOR is higher than in the other cases because there is less oil remaining to be produced.

In the case with the fractures at the bottom of the grid cell, Gil-man and Kazemi's gravity term works against the capillary pres-sure and prevents water from entering the matrix blocks. As shown in Fig. 4, the results appear to be even worse than those obtained by Kazemi et al. 10 in his Case 2, with no capillary pressure or gravity forces. This occurs because, on the basis ofthe Gilman and Kazemi formulation, we have included a negative gravity term in the matrix/fracture exchange equation.

Note that no confusion should be made here; we do not agree with Gilman and Kazemi's gravity term, which uses two different depths, DI and Dm, in the exchange terms. Our model computes gravity potentials for the exchange terms on the basis of satura-tions and matrix block heights with no explicit reference to depths (see the Appendix for details).

GOR Perfonnance. Three-Phase, 3D Problem. A simple 3D reser-voir model was constructed to demonstrate the importance of gravity and capillary effects on GOR performance in a three-phase frac-tured reservoir. The reservoir has a constant 5 dip and thins rapidly from a maximum thickness of 300 ft [91 m] at the deepest point to 60 ft [18 m] at the highest point.

To represent this structure, the model grid consists of 81 grid cells. A schematic showing a typical X-Z cross-section through the model grid is shown in Fig. 5. The matrix and fracture properties are constant for all grid cells. The reservoir is initially undersatu-rated and there is no aquifer. All important model properties are shown in Tables 2 and 3.

To describe the matrix blocks, representative block dimensions were used and internally converted into a shape factor in the model. For all cases run, a constant shape factor of 0.09 ft -2 [0.98 m -2] was used. An oil producer was located in the center of the reser-voir in Grid Cell (5,2,2). In all model runs, the well produced with a maximum oil rate of 500 STBID [79 stock-tank m3 /d] and a minImum bottornhole pressure limit of 2,500 psia [17.2 MPa].

In all cases described below, the model ran with no stability prob-lems. The material balances were perfect. A typical IO-year run as a fractured reservoir required about 90 timesteps. However, no attempt was made to optimize the performance of the model dur-ing these runs, and on the basis of experience with other similar problems, we are confident that much larger timesteps could have been used here.

Gas/Oil Depletion. The first series of model runs shows the be-havior of the reservoir during a lO-year period of pressure deple-tion. The following sections describe the results obtained from adjusting various parameters.

Single vs. Dual Porosity. To compare the performance of a frac-tured reservoir with a nonfractured reservoir, Run I was made with

1120

1000

8000

5000 ~ " I ,

4000 I \ I , , I \ / RUN e \

3000 \ ;' , I \

I \ 2000 I

---

\ j RUN 5 ~ , 1000 "--*--fII_

0 1 2 3 7 10

TIME - YEARS

Fig. 7-GOR performance after water Injection.

a single-porosity approach. In Run 1, the total PV of matrix and fractures and the effective permeability of the fracture system were used to describe the reservoir. The relative permeability and capil-lary pressure curves used for the matrix in the subsequent cases, as shown in Table 3, were used.

The GOR performance of the oil producer is shown in Fig. 6. The reservoir pressure declined below the bubblepoint during the first year. At that time, the GOR began to decrease because all the gas in the reservoir remained immobile because of the critical gas saturation. The GOR declined steadily until the end of the third year, when the critical gas saturation in the wellblock exceeded the critical saturation. After that time, the GOR rose slowly until the end of the run.

In Run 2, the reservoir was simulated with a fractured approach. To describe the matrix, a block height, Lz, of 20 ft [6 m] was used in every grid cell. The GOR behavior of the producer is shown in Fig. 6. The reservoir pressure declined rapidly and reached the bubblepoint in the first year. At that time, the GOR of the producer rose slightly because gas was liberated in the fractures, where the critical gas saturation was zero. After a small increase, the GOR began to decline as gas migrated updip through the fractures. Gas in the matrix remained immobile until the critical saturation of 3 % was reached. The fracture system at the top of the reservoir quick-ly filled with gas and, because of viscous and gravity forces, be-gan to enter the matrix blocks. By the end of 3 years, gas in the fractures began to reach the well. At that time, the GOR began to increase much more rapidly than it had in Run 1 and exceeded 6,000 scf/STB [1080 std m3/stock-tank m3] by the eighth year.

Gravity Effects. Run 3 was similar to Run 2, with the exception that the effective matrix block height in each grid cell was changed from 20 to 10 ft [6 to 3 m]. The shape factor was held constant at the same value as was used in the earlier cases (0.09 ft -2 [0.98

m~2]). The GOR performance of the oil producer is shown in Fig. 6.

Compared with Run 2, the GOR began to increase sooner in Run 3 because of the reduced influence of gravity. As a result, there was less exchange of gas between the fractures and matrix, and the gas in the fractures reached the producer faster.

In Run 4, the reservoir was modeled as a naturally fractured sys-tem, as in Run 2, but no gravity forces were included in the matrix/fracture exchange. Therefore, gas movement from the frac-tures to the matrix was not possible because the pressure in the matrix (i.e., the viscous and capillary forces) was almost always larger than in the fractures. The GOR behavior in Run 4 is shown in Fig. 6. Without gravity forces, the GOR in Run 4 rose very rapid-ly after 3.5 years, when the gas in the fractures reached the well-block.

The results of Runs 3 and 4 point out the very large difference in results with and without the inclusion of a gravity term in the matrix/fracture exchange of oil and gas.

Depletion Followed by Water Injection. A second series of model runs was made to study the behavior of the reservoir if, after a period of pressure depletion, water injection is begun. As in the first series

SPE Reservoir Engineering, November 1988

of cases, a total of 10 years was simulated in each case. Water in-jection was begun after 7 years of pressure depletion, when a sig-nificant amount of free gas was being produced.

In all cases, the water injector was completed in the deepest por-tion of the reservoir in Grid Cell (9,2,3). To match the voidage from the oil producer approximately, a constant water-injection rate of 1,000 STBID [159 stock-tank m3/d] was used.

As in the pressure depletion cases described above, a compari-son was made between the behavior of a nonfractured reservoir and a fractured reservoir in the case of depletion followed by water injection. To do this, Run 5 was made with a single-porosity ap-proach and was based on Run 1. The GOR behavior of the producer in Run 5 is shown in Fig. 7. The GOR began to decline slowly after water injection was begun. By the end ofthe lO-year period, the GOR was still declining but was still greater than the solution GOR.

Run 6, a fractured reservoir approach, was based on the pressure-depletion scenario described in Run 2. At the end of7 years, water injection was started. At that time, the producing GOR was nearly 5,000 scf/STB [900 std m3/stock-tank m3]. The GOR behavior, shown in Fig. 7, indicates a rapid, steady decline over a period of 18 months. The producing GOR after 8.5 years reached a level near the normal solution ratio and continued at that level until the end of the lO-year period.

Threshold Capillary Effects. A final set of model runs was made to determine the sensitivity of results to the matrix gas/oil threshold capillary pressure and its relationship to the matrix block height. A modified capillary pressure curve was used in which the threshold capillary pressure was set at 0.20 psi [1.4 kPa]. Several runs were made with various matrix block heights. In each case, the model was first run with a zero threshold capillary pressure and then with a value of 0.2 psi [1.4 kPa]. The matrix block heights used were 2, 4, and 20 ft [0.6, 1.2, and 6.1 m].

The GOR behavior of the producer during these sensitivity runs is shown in Fig. 8.

The inclusion of a threshold capillary pressure always resulted in a more rapid gas breakthrough at the producer. As the matrix block height increased, the overall importance of the threshold capil-lary pressure became less important because the effective capillary height with a threshold pressure of 0.20 psi [1.4 kPa] is equivalent to 1.5 ft [0.5 m], which is large compared with the 2- or 4-ft [0.6-or 1.2-m] matrix block heights, but very small compared with the 20-ft [6.1-m] matrix block. This behavior agrees well with our understanding of the physical phenomenon.

Conclusions 1. A new, fully implicit, 3D, three-phase model has been devel-

oped for the study of naturally fractured reservoirs. This model uses a new technique that results in a more physically correct treat-ment of gravity forces in the matrix/fracture exchange than has been used in earlier models.

2. The gravity effect in a naturally fractured reservoir is a dy-namic function of the saturations in the matrix and surrounding frac-tures. It cannot be simulated correctly with a static approach, as has been used by previous authors.

3. From a numerical-solution point of view, the dynamic con-tact levels, Zwj, Zwm, Zgf, and Zgm, are determined in a fully im-plicit way that does not cause any stability or convergence problems.

4. Model results confirm that gravity and capillary forces work together to improve oil recovery by water and work against each other in the recovery of oil by gas. The gravity force itself always works toward improving recovery from the matrix.

5. The use of an incorrect gravity effect, or no gravity effect at all, can lead to very questionable results from a naturally fractured reservoir simulator.

6. The use of a single-porosity simulator to model a naturally fractured reservoir can yield totally different results from those ob-tained with an appropriate fractured reservoir simulator. This ap-proach can often lead to optimistic recoveries from the model, which could be compensated for, but this often requires the use of "artifi-cial" pseudorelative permeability and capillary pressure functions. In addition, pseudos derived from history matching a complex frac-tured reservoir can often be inappropriate for model predictions and can lead to very unrealistic model results.

SPE Reservoir Engineering, November 1988

1000

eooo

15000

3000

2000

1000

10

TIME - YEARS ___ WITH THRESHOLD PRESSURE -- NO THRESHOLD PRESSURE

Fig. 8-Threshold capillary pressure effects for various matrix block heights.

7. Just as the correct physical representation of the matrix/fracture exchange in a reservoir model is important, the use of correct in-put data for describing the capillary and gravity forces (e.g., matrix block height, matrix gas/oil capillary threshold pressure, and wet-tability) is vital in the simulation of the complex processes in a natur-ally fractured reservoir.

Nomenclature B = FVF, RB/STB or RB/Mcf [res m3/stock-tank m3

or res m3/m3] D = depth or depth to grid cell center, ft [m] E = matrix/fracture exchange term, STBID [stock-

tank m3/d] or mol/D k = permeability, md

kr = relative permeability, fraction L = matrix block dimension, ft [m] p = pressure, psi [kPa]

Pc = capillary pressure, psi [kPa] q = molar flow rate, mollD

rs = oil/gas ratio, STB/MMscf [stock-tank m3/std m3]

Rs = GOR, scf/STB [std m3/stock-tank m3] Rsw = gas/water ratio, scf/STB [std m3/stock-tank m3]

S = saturation, fraction I:l.t = time increment, days T = transmissibility, STB/D-psi [stock-tank

m3/d'kPa] V = grid cell volume, RB [res m3] x = mole fraction in the oil phase, fraction

Xw = mole fraction in the water phase, fraction AX = grid cell dimension in the X direction, ft [m]

y = mole fraction in the gas phase, fraction I:l.Y = grid cell dimension in the Y direction, ft [m]

Z = fluid contact level, ft [m] I:J.Z = grid cell dimension in the Z direction, ft [m]

ex = phase (oil, gas, or water) f) = timestep difference, days=on+l-On

I:l.(TI:l.4 = I:l.x(T xl:l.x4 +I:l. y(Tyl:l. y4 +l:l.z(Tzl:l.z4 l' = fluid pressure gradient, psi/ft [kPa/m] p, = viscosity, cp [mPa' s]

~ = molar density, Ibm-mollft3 [kmollm3] a = shape factor = 4(_1_ +_1_ +_1_), ft-2 [m-2] Li Ly Ll

cJ> = porosity, fraction 4> = fluid potential, psi [kPa] w = upstream weighting coefficient, fraction

1I21

Subscripts b = bulk c = capillary f = fracture g = gas i = initial k = fluid component

m = matrix n = timestep level o = oil r = residual

w = water X, Y, Z = grid directions

Acknowledgments We thank the Petronord Group, including Elf Aquitaine Norge A/S, Norsk Hydro A/S, and Total Marine Norsk A/S, as well as SNEA(P) and Total/CFP for their cooperation, assistance, and per-mission to publish this paper.

References 1. Barenblatt, G.I., Zheltov, U.P., and Kochina, G.H.: "Basic Concepts

in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks, " Prikladnaia Matematika i Mehanika, Academii Nauk USSR (1960) 24, No.5, 852-64.

2. Warren, J.E. and Root, P.J.: "The Behavior of Naturally Fractured Reservoirs," SPEJ (Sept. 1963) 245-55; Trans., AIME, 228.

3. Kazemi, H.: "Pressure-Transient Analysis of Naturally Fractured Reser-voirs With Uniform Pressure Distribution," SPEJ (Dec. 1969) 451-62; Trans., AIME, 246.

4. Iffly, R., Rousselet, D.C., and Vermeulen, J.L.: "Fundamental Study ofImbibition in Fissured Oil Fields," paper SPE 4102 presented at the 1972 SPE Annual Meeting, San Antonio, Oct. 8-11.

5 . Yamamoto, R. H. et al.: "Compositional Reservoir Simulatorfor Fis-sured Systems-The Single-Block Model," SPEJ (June 1971) 113-28.

6. Kleppe, J. and Morse, R.A.: "Oil Production From Fractured Reser-voirs By Water Displacement," paper SPE 5084 presented at the 1974 SPE Annual Meeting, Houston, Oct. 6-9.

7. Saidi, A.M.: "Mathematical Simulation Model Describing Iranian Frac-tured Reservoirs And Its Application To Haft-Kel Field," Proc., Ninth World Pet. Cong., Tokyo (1975) 4, PDI3(3), 209-19.

8. Reiss, L.H., Bossie-Codreanu, D., and Lefebvre du Prey, EJ.: "Flow in Fissured Reservoirs," paper SPE 4343 presented at the 1973 SPE European Petroleum Conference, London, April 2-3.

9. Rossen, R.H.: "Simulation of Naturally Fractured Reservoirs With Semi-Implicit Source Terms," SPEJ (June 1977) 201-10.

10. Kazemi, H. et al.: "Numerical Simulation of Water/Oil Flow in Natur-ally Fractured Reservoirs," SPEJ (Dec. 1976) 317-26; Trans., AIME, 261.

11. Gilman, J .R. and Kazemi, H.: "Improvements in Simulation of Natur-ally Fractured Reservoirs," SPEJ (Aug. 1983) 695-707; Trans., AIME, 275.

12. Thomas, L.K., Dixon, T.N., and Pierson, R.G.: "Fractured Reser-voir Simulation," SPEJ (Feb. 1983) 42-54; Trans., AIME, 275.

13. Bossie-Codreanu, D., Bia, P.R., and Sabathier, J.C.: "The 'Checker Model,' An Improvement in Modeling Naturally Fractured Reservoirs With a Tridimensional, Triphasic, Black-Oil Numerical Model," SPEJ (Oct. 1985) 743-56.

14. Blaskovich, F.T. et al.: "A Multi-Component Isothermal System for Efficient Reservoir Simulation, " paper SPE 11480 presented at the 1983 SPE Middle East Oil Technical Conference, Bahrain, March 14-17.

15. Hill, A.C. and Thomas, G.W.: "A New Approach for Simulating Com-plex Fractured Reservoirs," paper SPE 13537 presented at the 1985 SPE Symposium on Reservoir Simulation, Dallas, Feb. 10-13.

16. Litvak, B.: "Simulation and Characterization of Naturally Fractured Reservoirs," paper presented at the 1985 Reservoir Characterization Technical Conference, Dallas, April.

17. van Golf-Racht, T.D.: Fundamentals of Fractured Reservoir Engineer-ing, Elsevier Scientific Publishing Co., Amsterdam/Oxford/New York City (1982).

Appendix-Expression of Exchange Terms The matrix/fracture exchange terms used in the new model are ex-pressed again in this section in a synthetic form.

1122

The matrix/fracture exchange terms include viscous forces (Term 1), capillary forces (Term 2), and gravity forces (Term 3) and can be expressed by the equations below.

For gas and at Cell J,

or

(Term 1)

+ (P cgofl - P cgoml) (Term 2)

(Term 3)

................................ (A-2)

For water and at Cell J,

or

- (P cwofl - P cwom!)

(Term 1)

(Term 2)

(Term 3)

................................ (A-4)

Similarly, for oil and at Cell J,

E ol = Tolal [(Pofl-Pom!) (Term 1)

-d-y go(Zgfl-Zgml) -d-Ywo(Zwjl-Zwm!)], (Term 3)

................................ (A-5)

where d-y go = half the oil and gas gradient difference, d-y wo = half the water and oil gradient difference,

Tg , To, T w = transmissibilities between matrix and fracture for gas, oil, and water, respectively,

a = shape factor, Pof,Pom = oil pressure in fracture and matrix, respectively,

Pcgo,P cwo = gas/oil and water/oil capillary pressures, Lz = height of matrix block, D = depth to center of grid cell, and

Zwf'Zwm, Zgf,Zgm = dynamic levels reached by water and gas

(functions of saturation) in fracture and matrix media.

51 Metric Conversion Factors bbl x 1.589873 E-Ol cp x 1.0* E-03 ft x 3.048* E-Ol

ft3 x 2.831 685 E-02 psi x 6.894 757 E+OO

Conversion factor is exact SPERE

Original SPE manuscript received for review Oct 5, 1986. Paper accepted for publication Sept. 18, 1987. Revised manuscript received May 2,1988. Paper (SPE 15627) first presented at the 1986 SPE Annual Technical Conference and Exhibition held in New Orleans, Oct. 5-8.

SPE Reservoir Engineering, November 1988