Dake L.P. Fundamentals of reservoir engineering

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(a) TIME LEVEL n

(b) TIME LEVEL n + 1

Fig. 10.46 Example of water saturation instability (oscillation) resulting from the

application of the

IMPES solution technique (−

−−

− −

−−

− −

−−

− correct, and 



 incorrect

saturations)

Consider, as an example, a water drive from left to right in the two blocks of a linear

model, shown in fig. 10.46(a) - (c), in which the average saturations depicted by the

dashed lines are correct, and by the solid lines, incorrect. During the time step from n

to n+1 the water saturation in block i must increase due to the advancing water from

block i − 1 (not shown). Since the water relative permeability in block i is held constant,

commensurate with the saturation shown in fig. 10.46(a), then not enough water will

flow out of block i during the time step, and consequently the water saturation will be

artificially high in the block at time n+1. For the time step from n+1 to n+2, the water

relative permeability is evaluated for this latter saturation and too much water will leave

the block resulting, at time n+2, in too low a saturation in block i and too high a value in

block i+1. The overall result will be an oscillation in solution for the water saturation.

Instabilities of this sort, which are inherent in the

IMPES solution method, can be

overcome by drastically reducing the length of the time step, but this results in

excessively long computing times.

Blair and Weinaug

developed a numerical simulator which employed implicitly

evaluated permeabilities. This so-called fully implicit procedure greatly improved the

stability of solution of the finite difference equations and consequently permitted the

use of much larger time steps in the simulation. Unfortunately, this benefit was largely

offset by the increased computational work per time step due to the difficulty of solving

the sets of fully implicit finite difference equations. An alternative and computationally

simpler method is that of semi-implicit formulation of the finite difference

equations

23,24,25

, in which the saturation dependent relative permeabilities and pseudo

IMMISCIBLE DISPLACEMENT 412

capillary pressures are evaluated at time level n+1 using the first order Taylor series

expansion, i.e.

n1 n

rr t

kk S

æö

=+ ∆

ç÷

èø

and

,n 1 ,n

rc t

PP S

°°

æö

=+ ∆

ç÷

èø

where ∆

S is the time change in saturation,

n1 n

SS,

− and the first order derivatives are

evaluated at time level n. Peaceman has demonstrated

that this approach produces

perfectly acceptable stability in the majority of numerical simulations, while

Chappelear

has illustrated some of the practical difficulties associated with its use

and how they can be overcome. On substituting such expressions for

n1 ,n1

kandP,

++°

in the finite difference equations (10.83) and (10.84); the latter equations can be

algebraically manipulated in such a way that the saturation changes again disappear

on addition since

SS0∆+∆=

The resulting water pressure equation, which is similar although more complex than

equ. (10.86), can then be solved, following which the saturations can be determined as

described previously for the

IMPES method. The time required to solve the finite

difference equations using the semi-implicit method is longer than when using explicit

saturation dependent parameters, but this is more than compensated by the additional

length of the time step which can be tolerated using the method.

In describing the solution of equ. (10.86) the fact that a grid block could contain an

injection or production well was temporarily overlooked. In such a case there will be an

additional term, Q

+ Q

, on the right hand side of the equation, which is the total oil

and water rate per unit of reservoir bulk volume and is positive for injection and

negative for production. Precisely how these rates are incorporated into the solution of

equations such as equ. (10.86) depends on the way the simulator is being operated.

There is a method of running the simulator in what is described as the material balance

mode. When using this, one attempts to match the observed reservoir pressure history

with the numerical simulation. In this mode the desired time steps are usually specified

as input to the simulation and the observed injection and production rates of each fluid

are assigned to each grid block, containing a well, as a function of time. In this case,

since all the rates are specified they do not add any complication to the method of

solution of equ. (10.86), described previously. As the outcome, the pressures

calculated by the simulator, for each grid block containing a well, should be compared

with the pressures measured in the field, using the dynamic grid block pressure

method described in Chapter 7, sec. 7. This aspect of history matching is tantamount

to calibrating the simulation model and will be recognised as being nothing more than a

IMMISCIBLE DISPLACEMENT 413

sophisticated version of the history matching, using the volumetric material balance

equation, described in Chapters 1, 3 and 9.

In more normal application, however, it is necessary for the simulator to forecast the

injection or production rate of each well. Considering a single well situated at the centre

of a grid block the simulator must be able to determine, for the currently evaluated

average fluid pressures and saturations, how much of each fluid can flow into or out of

the grid block, through the well. Since pressure and saturation gradients do not exist

within each individual grid block, but only between blocks, the physical modelling of

well performance presents difficulties. Pressure-gradients can be very high in the

proximity of the well which should require the subdivision of the grid block into a fine

mesh of blocks close to the well in order to correctly account for these gradients. Such

an approach would result in very high computer costs due to the large number of grid

blocks involved in the simulation.

An alternative, approximate method of modelling well performance is to design an

analytical well model to account for either injection or production from a grid block. For

instance, the total production of water from a block, q

= Q

x (net bulk volume of

block), could be predicted as

– p

) (10.87)

in which p

and p

are the current bottom hole flowing pressure and average grid block

pressure for the water, respectively . The order of subtraction of these pressures

ensures that water production is negative and injection Positive, in accordance with the

normal convention. The term a is the productivity index of the well and is normally

evaluated for steady state flow (Chapter 6, table 1) as

2kkh

(Darcy units)

ln S

æö

−+

ç÷

èø

7.08 10 kk h

(field units)

ln S

−

æö

−+

ç÷

èø

where r

is the equivalent grid block radius

∆∆

for a well draining from the centre of a block. The choice of a steady state well model is

usually justified by the fact that transients, within a grid block, normally die out in a

relatively short time compared to the average time step used by the simulator; and the

condition at the block boundary is probably closer to steady state (open boundary —

constant pressure) than semi-steady state (closed boundary — pressure depletion).

IMMISCIBLE DISPLACEMENT 414

It has been found that, even for a simulation which can be satisfactorily run using the

IMPES solution technique, the well model will be unstable unless extremely small time

steps are employed. This is because of the saturation dependence of the relative

permeability term in the productivity index

. For this reason the PI is always evaluated

semi implicitly as

n1 n

αα

æö

=+ ∆

ç÷

èø

for each liquid. Thus the equation for the water rate, (10.87), at time n+1, in block i,

which contains a well, is

w,i

n1 n n1

w,i

w,i w ,i t wf ,i w,i

w,i

q ( S )(p p )

æö

=+ ∆ −

ç÷

èø

(10.88)

and a similar expression can be formulated for the oil phase. Such expressions are

added to the right hand side of equs. (10.83) and (10.84), in grid blocks containing

wells and addition of the equations, again with much algebraic manipulation, will lead

to a modified form of the pressure equation, (10.86), in which the

S∆ terms cancel on

addition. The subsequent solution for phase saturations and pressures permits the

calculation of the rates by back substitution of these values in the well model equation,

(10.88), for both oil and water.

It is intuitively obvious that since the term

kk /

controls the flow of fluids from one

grid block to the next, it should be evaluated at the grid block faces. What is not so

obvious, however, is precisely how the values of the three parameters in this

expression are determined at the block faces since they are only defined at the grid

block nodes.

The viscosities, which are pressure dependent, are considered to be slowly varying

functions of position, especially in the case of incompressible displacement, and are

therefore evaluated as the average of their values at adjacent grid block nodes; thus

for regular grid block spacing

i+½

= (

i+1

) / 2

The absolute permeability, which normally remains constant throughout the simulation,

is. determined as the harmonic mean of its value at adjacent grid nodes. Consider the

general case of flow between grid blocks of unequal length, as shown in fig. 10.47.

IMMISCIBLE DISPLACEMENT 415

∆

∆+

∆

Fig. 10.47 Determination of the average, absolute permeability between grid blocks of

unequal size

Assuming, for the calculation of absolute permeability, that the reservoir is completely

saturated with a single fluid and that the flow rate in each block is the same (q

= q

i+l

q). Then applying Darcy's law in each grid block

∆

∆=

and

∆

∆=

But the total pressure drop across the two blocks is

i´ 1 i i i 1

ii1

kxkx

ppp

Akk

∆+∆

∆=∆+∆ =

Also

ii1

(x x )q

∆+∆

∆=

where k

i+½

is the average permeability for flow through the blocks and is the value

assigned at the grid block face i+½. Therefore, equating these two expressions for ∆p

gives

ii1 i i1

i1 i i i1

kk ( x x )

(k x k x )

∆+∆

or, in the more normal case, for equal grid block sizes

ii1

i1 i

2kk

It is more difficult to extrapolate the averaged relative permeabilities from the grid block

nodes, at which they are specified, to the grid block faces. This is due to the saturation

dependence of this parameter which means that it can vary substantially between one

grid block node and the next. Todd et.al.

have demonstrated that a satisfactory

IMMISCIBLE DISPLACEMENT 416

means of extrapolation is by the method of two point upstream weighting. Suppose

flow in the one dimensional reservoir is in the direction of increasing values of i, then

the opposite direction is "upstream" and

r,i r,i

xkr

æö

∆∂

ç÷

∂

èø

in which, for unequal grid block lengths,

k/ x∂∂is established from the two upstream

nodes to give

r,i r,i 1

r,i r,i

ii1

(k k )

kk x

(x x )

−

=+ ∆

∆+∆

or for equal grid block lengths

r,i r,i r,i-1

kkk

=−

Similarly if flow is in the opposite direction, equ. (10.89) becomes

r,i r,i 1 r,i 2

kkk

+++

=−

The direction of flow at the time of evaluation is decided by the potential difference

(pressure difference in a horizontal reservoir) between adjacent grid blocks.

Todd, et.al.

have also shown that a method such as two point upstream weighting

cannot be inserted in a numerical simulator without careful screening. Consider, for

instance, the very simple problem of piston-like displacement of oil by water as shown

in fig. 10.48.

i - 1

−

rw rw

kk kk

∆

rw rw

k,i k

+=−

Fig. 10.48 Overshoot in relative permeability during piston-like displacement

IMMISCIBLE DISPLACEMENT 417

When the sharp water front is about to enter block i the value of the relative

permeability to water at i+½ is − ½ k´

, equ. (10.89). This negative relative permeability

is referred to as overshoot and can obviously occur any time

rw,i 1 rw,i

k3k

−

> . This

situation can arise when a shock front water saturation discontinuity moves through the

reservoir. Such overshoot is forbidden and is overcome by imposing the condition that

k0≥ .

By considering the oil saturation and relative permeabilities corresponding to the

situation shown in fig. 10.48, the reader can verify that another condition which must be

imposed is that

ro,i

k the greater of

≤

The above two rules controlling overshoot are generally applicable for any type of

displacement; that

rw,i

is not allowed to be negative and

ro,i

must be less than or

equal to the greater of the values at adjacent grid nodes. Although more difficult to

code than single point upstream weighting

r,i r,i

(k k )

= , the two point method is more

accurate and, using it, larger grid block sizes can be tolerated. This reduces the

program running time and therefore, simulation costs.

Actual reservoir simulation studies are usually undertaken to determine the areal

distribution of fluids in the reservoir, resulting either from natural water influx or water

injection . As such they require the building of a two or three dimensional model of the

reservoir in which variation in the basic reservoir parameters, such as porosity,

permeability etc., can be allowed for by assigning different values of these parameters

to each grid block. Once a model has been constructed, which closely resembles the

reservoir, it is then necessary to select relative permeabilities and a pseudo capillary

pressure function which characterise the manner in which the water saturation is

distributed in the dip-normal direction, as the flood moves through the reservoir. The

manner in which this can be done is described below.

a) Generate a set of averaged relative permeabilities and the pseudo capillary

pressure for a typical linear cross section through the reservoir, between the

injection and production wells, assuming vertical equilibrium.

b) Set up two simulator models of the same cross section through the reservoir.

Both models should have only one grid block in the y direction but, in the dip-

normal direction, one of them should be one grid block thick, fig. 10.49(a)

IMMISCIBLE DISPLACEMENT 418

(a) (b)

DEL A M

DEL B

Fig. 10.49 Alternative linear cross sectional models required to confirm the existence of

vertical equilibrium

(model A), using thickness averaged porosity and absolute permeability, while

the second, fig. 10.49(b) (model B) should be constructed with a large number of

layers, irrespective of whether the reservoir is layered or homogeneous.

c) Simulate the displacement of oil by water in both models using the thickness

averaged relative permeabilities and pseudo capillary pressure functions in

model A, and the relevant rock relative permeabilities and actual capillary

pressures, in each separate layer, in model B. If the breakthrough times and

subsequent oil recovery as a function of time are the same in both models then it

means that the assumption of vertical equilibrium, implicit in the generation of the

averaged relative permeability curves, is valid. Since vertical equilibrium is less

likely to occur at high flow rates the comparison between the simulations should

be made using the maximum water injection rate to be used in the study.

d) If the agreement between the cross sectional simulations is poor, so that vertical

equilibrium does not pertain, re-run model A using averaged relative

permeabilities generated using the Stiles method, if the reservoir is layered, or

the rock curves if the reservoir is homogeneous in the dip-normal direction.

Agreement between simulations with models A and B means, in this case, that

there is an extreme lack of vertical equilibrium. For either case c) or d), if

agreement is achieved, the cross sectional model can be reduced to one

dimension and the total three dimensional simulation to an areal, two dimensional

model, with consequent reduction in program running time and considerable cost

saving. In addition, it is worthwhile comparing the results of model A with those

obtained using the analytical method of Buckley-Leverett, using the averaged

relative permeability curves. Favourable comparison, in this latter case, gives the

engineer added confidence in using the simulator.

The comparison between the analytical and simulated results is never exact for,

although both methods are based on the conservation of (water) mass, there are

differences. The Buckley-Leverett theory, for instance, ignores the effect of the

capillary pressure gradient behind the flood front, whereas capillary effects are

included in the simulation. The analytical solution, however, uses continuous

IMMISCIBLE DISPLACEMENT 419

space and time, whereas these assume discrete values in the simulator.

Because of this, shock front displacement is never accurately modelled with a

finite difference simulator. Examples of the comparison between analytical and

simulated shock front displacement have been presented by Todd, et.al.

e) If agreement is not achieved between the cross sectional simulations in either c)

or d) above, then no definite assumption can be made concerning the vertical

equilibrium condition. In this case model A should be re-structured with two

layers of grid blocks in the dip-normal direction, each layer having its own

averaged porosity and permeability. Step a) should be repeated, generating

averaged relative permeabilities and pseudo capillary pressures for each layer.

Simulations with this two layer model should again be compared with the results

from model B. Agreement, in this case, means that while vertical equilibrium may

not apply across the entire thickness of the reservoir, it is appropriate across half

the reservoir. If there is still no agreement between the cross sectional

simulations, model A should be re-structured as a three or four layered model

and the process repeated until satisfactory agreement is achieved. In this manner

it is usually possible to significantly reduce the number of layers.

The procedure outlined above should be undertaken at the commencement of any

large numerical simulation study. It can usually be conducted very quickly and at only a

small fraction of the total study cost. If the reservoir is not areally homogeneous, so

that a cross section between injection and production wells which is representative of

the entire reservoir cannot be selected, then several cross sections must be chosen. It

will then be necessary to repeat step a) — the generation of averaged relative

permeabilities and pseudo capillary pressure for each cross section. If, however,

vertical equilibrium has been clearly validated in one cross section, it is fairly safe to

assume its validity throughout the reservoir, unless there are extremes of thickness or

vertical permeability.

The relative permeability curves obtained from the above study are input to the

reservoir simulation model which has, hopefully, by this stage, been reduced to two

dimensions. Simulations are then conducted to try and match the available history

data. Following a successful history match, the simulator can be used to predict the

future reservoir performance for different proposed production policies, well positions

etc.; and, as Coats

has observed, although a reservoir can only be produced once, its

performance can be simulated in many different ways and at comparatively low cost.

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