A Parallel, Multiscale Approach to Reservoir

Modeling

Omer Inanc Tureyen and Jef Caers

Department of Petroleum Engineering

Stanford University

1

Abstract

With the advance of CPU power, numerical reservoir models have

become an essential part of most reservoir engineering applications.

These models are used for predicting future performances or deter-

mining optimal locations of infill wells. Hence in order to accurately

predict, these reservoir models must be conditioned to all available

data. The challenge in data integration for numerical reservoir mod-

els lies in the fact that each data has its own resolution and area of

coverage. The most common data for reservoir characterization are;

well-log/core data, seismic data and production data. The challenge

is that each data set has its own resolution and area of coverage.

Most current approaches to data integration are hierarchical. Fine

scale models are used for integrating well-log/core and seismic data

while coarse models are used to integrate mostly production data. The

drawback of such a hierarchical approach is such that once the scale is

changed, data conditioning, maintained in the previous scale, is lost.

In this paper, we review a general algorithm as a solution to the

multi-scale data integration. Instead of proceeding in a hierarchical

fashion, a fine model and a coarse model is kept in parallel through

out the entire characterization process. The link between the fine

scale and the coarse scale is provided by non-uniform upscaling. An

optimization procedure determines the optimal gridding parameters

that provide the smallest possible mismatch between fine and coarse

scale reservoir models.

A synthetic example application is given and demonstration of

the methodology. The upgridding is accomplish by a static gridding

algorithm, 3DDEGA. This algorithm aims at preserving geology by

1

minimizing heterogeneity within a coarse grid block. The coarse grids

are provided in a corner-point geometry fashion, hence this allows for

accurate description of the reservoir with fewer number of grid blocks.

1 Introduction

Reservoir modelling and prediction calls for the integration of various data

sources into a single reservoir model. Such data sources can be divided into

several groups of which the most important ones are:

• Geological interpretation of reservoir architecture at all scales ranging

from major faults to facies and bedding configurations. Such infor-

mation is often qualitative in nature, yet may constrain the reservoir

model at all scales. In geostatistics such information can be quantified

through variogram or through 3D training images.

• Well-log and core measurements. Such information is often the most

direct type of information, however is only telling of the near well-bore

reservoir heterogeneity and provides information at a foot scale. In

geostatistics, this type of information is often treated as hard data.

• 3D seismic surveys. This information is probably most exhaustive, yet

often at a scale larger than the reservoir modelling scale. Seismic is

known to act as a low-pass filter, resulting in seismic images that lack

important fine-scale heterogeneities. In geostatistics this type of data

is treated as soft data.

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• Reservoir dynamic data, most particularly from pressure and flow mea-

surements, or increasingly common, 4D seismic. The scale at which

this information informs the reservoir is largely unknown beyond the

fact that is quite coarse, its scale varies spatially, depending on well

configurations and time, depending on depletion strategies.

In building reservoir models, it should be recognized that each piece of

information has its own characteristic and scale, yet no single source of in-

formation may determine the reservoir model uniquely. While some redun-

dancy of information may be present, it is generally considered that each of

the above four data sources will contribute to modelling the reservoir.

The current practice of reservoir modelling consists of modelling the reser-

voir first using the static data (sources one to three), then only using dynamic

information (source four), see Figure-1. In the static reservoir modelling

stage, the reservoir is modelled on the ”geostatistical” scale. In the vertical

direction, the size of the geostatistical grid cell is typically equal to the scale

of the well-log and core data (about 1 ft), while in the horizontal direction

the grid cell size is often related to the scale at which seismic surveys (100ft

typically).

Two important problems exist with this approach:

• A missing scale problem occurs: the core data at a 0.1ft×0.1ft×1ft

scale is represented by the geostatistical cell of 100ft×100ft×1ft which

implies an intrinsic upscaling of cell properties. This may constitute a

serious problem in strongly heterogeneous system.

• Flow simulation cannot be performed at this scale as flow simulators

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cannot be evaluated on grid dimensions larger than a few 100.000 grid

blocks. Hence the million cell geostatistical model needs to be upscaled

if any production data needs to be integrated.

We will not deal with the missing scale problem but with the latter prob-

lem only. The common approach to history matching is to history match

on the coarse or upscaled reservoir model, either manually or using some

gradient-based method. Performing history matching in such fashion has the

following problems:

• Any fine scale reservoir information (core or well-log) may be lost when

the coarse scale model is perturbed.

• Important fine and coarse scale geological information may be de-

stroyed while history matching. Particularly when the history match-

ing method does not take into account statistics such as variogram or

multiple-point statistics that are imported in the fine scale geological

model in order to honor the prior geological data.

In many practical settings however, a history match can still be achieved,

even at the cost of destruction of any seismic data conditioning or geological

realism. The cost however is often predictivity of the resulting reservoir mod-

els. For this reason, presuming geology while history matching has deserved

little attention, since often obtaining history match has become a goal on its

own.

This shortcoming in the work flow of building the reservoir model first

static, then dynamic has been recognized by several authors (amongst which

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the paper by Tran T.T. and Behrens (1999) is pioneering). An improved work

flow has been proposed, shown in Figure-2. Instead of simply history match-

ing the coarse scale model, a posterior downscaling of the history matched

coarse model is performed. The downscaling step allows to re-integrate the

fine-scale information lost during the history matched procedure. The down-

scaled realization can then be upscaled to any desired grid dimensions. The

main problem with this approach is that the final upscaled reservoir model

(Figure-2) need not match the production history due to the fact that the

downscaling procedure introduces noise/error in the match. The downscal-

ing procedure is purely geostatistical, production data is not utilized at this

stage. In fact one may argue that the approach in Figure-2 is still a sequen-

tial hierarchical approach to reservoir modelling. Upscaling, downscaling

and history matching ”operations” are performed in a sequential fashion,

with the last performed ”operation” potentially destroying the achievements

of its predecessors.

In this paper we propose a ”parallel” or ”joint” approach to the problem

of reservoir model building. Instead of going through a sequential procedure

of scale-change (down or up) followed by history matching, we propose to

integrate upscaling into the history matching loop. Instead of perturbing the

coarse scale model, we propose to perturb the fine scale reservoir model. A

simple loop is created of first creating a fine scale realization, then upscaling

it, then evaluating flow on the coarse scale realization. The flow results of the

coarse model are used to perturb the fine scale realization. This sequence of

steps is repeated until a history match is achieved. This approach is not new

and has been successfully applied by the authors Caers (2001), Mezghani and

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Roggero (2001) and Tureyen and Caers (2002) and also others.

In this paper we show however that this approach has a fundamental and

potentially dangerous flaw. When perturbing a fine scale geostatistical model

using flow results from another, namely coarse scale realization, it is implic-

itly implied that the flow results evaluated on coarse and fine are similar,

hence no or few upscaling errors occur. Indeed, any perturbation of reservoir

parameters linked to the fine scale should be based on the calculation of a

mismatch function that is correctly reflecting fine scale parameter changes.

This is not necessarily true in the presence of upscaling errors. The danger

therefore lies in the fact that the user may assume no upscaling errors exist

(while in reality upscaling errors could be quite severe) and still achieve a

successful history match. Recall that production data may not necessarily

provide a strong constraint to the reservoir model when few wells are avail-

able, hence history matching on poorly upscaled reservoir models is feasible.

The same argument was used above to explain the current state-of-the-art

approach in history matching by perturbing coarse scale realizations. The

result is, as above, a model that matches history but has lost important fine

geological or seismic data, hence lost prediction power or accuracy.

The main contribution of this paper is to correct this flaw by not only

closing the loop between history matching and upscaling but also by reducing

upscaling errors through gridding optimization, while history matching. This

will amount to reducing the upscaling errors between fine and coarse such

that all relevant fine scale geological data is maintained in the coarse scale

model throughout the history matching procedure.

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The further outline of this paper can now be defined clearly: we will first

review the work of Tureyen and Caers (2002) and show a counter example of

what can go wrong in such a procedure. Next we introduce a fairly general

optimization framework for reducing upscaling errors between fine and coarse

scale realizations. Our methodology or work flow is generic in many ways.

We will therefore use various existing geostatistical, upscaling and history

matching procedures and refrain from using a case study to demonstrate our

point of generality.

2 Review: Basic Parallel Modelling

The objective of the parallel modelling approach for reservoir characterization

is basically to avoid the problem of “choosing a scale” by working on multiple

scales jointly. Unlike the hierarchical modelling approach given in section-

1, the parallel modelling scheme gives the flexibility to update each scale

throughout the entire characterization process.

To provide a clear understanding some of the notation that will be used

through out this paper is first introduced:

z : {z (u), ∀ u∈ Reservoir} the reservoir property at grid block

u=(x,y,z) e.g. permeability;

z (r) : a perturbation of the fine scale reservoir model z . The

magnitude of perturbation is parameterized using

some parameters r ;

z up : the uniquely determined upscaled reservoir model,

upscaled from z ;

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z up(r) : the model upscaled from z (r);

FSM (z ) : the flow simulation model evaluated on z ;

RP(z ) : the flow response when FSM is performed on z ;

S θ(z ) : The upscaling method S evaluated on the fine scale model

z . θ are upscaling or upgridding parameters belonging

to the upscaling method S ;

D : reservoir production data to be matched.

Note at this point that we make full abstraction of the parametrization of

the problem z (r). r can be a single parameter such as in, the gradual defor-

mation method (see Roggero and Hu (1998)), the probability perturbation

method (see Caers (2003)), or r may be parametrization using sensitivity co-

efficients (see Landa (1997) and Wen and Cullik (1998)). Figure-3 schemat-

ically illustrates the main algorithm of the parallel approach on a history

matching problem. First a fine scale geostatistical model (z (u) that honors

the seismic and hard data) is generated. In history matching, some initial

starting model is perturbed using a perturbation scheme. Perturbations are

often parametrized by a set of parameters r that change the reservoir model

z into a perturbed model z (r). Note that we attach such perturbation to

the fine scale, not the coarse scale model. The next step is to upscale z (r)

to a coarser model z up(r), non-uniformly through the following relationship:

z up(r) = Sθ(z (r)) (1)

Here S represents the upscaling technique applied on z (r) and θ repre-

sent the upgridding parameters regarding the upscaling technique S. In the

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method of Tureyen and Caers (2002), Figure-3, both S and θ are determined

prior to the history matching and remain fixed while history matching.

Once the coarse model is determined, the flow response is obtained by

evaluating the full flow simulation (FSM ) on the coarse model through the

following relationship:

RPz up(r) = FSM(z up(r)) (2)

Finally the r parameters are optimized so that the following objective

function is minimized.

minr

O(r) = min ‖ RPz up(r)−D ‖ (3)

At this point it should be noted that the fine scale model is perturbed

(with the r parameters) with respect to the flow response of the coarse scale

model. Such an approach makes upscaling a part of the history matching

process and is actually the basis of parallel modelling. Some of the advantages

of such an approach can be listed as follows:

• Fine scale and coarse scale data are integrated at the same time at their

relative scales. The well log/core and the seismic data are honored at

the fine scale and the production data is honored on the coarse scale,

• Relevant geology is maintained at all scales. This is accomplished on

the coarse grid through the use of either non-uniform or unstructured

gridding,

• No posterior upscaling is required since it is the non-uniformly gridded

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model that matches the history. This is a result of the fact that full

flow simulations are performed on the coarse model,

• It is a generic method and serves as an alternative approach to reservoir

characterization.

2.1 Example

To make a better understanding of the proposed approach by Tureyen and

Caers (2002), an example will be briefly reviewed in this section. Figure-4

gives the reference permeability field along with its corresponding water cut

curve and sample statistics regarding the generation of the reference field.

This field represents a cross-section of a reservoir which is modelled with 50

grid blocks in the horizontal and 50 grid blocks in the vertical directions. The

flow scheme takes place with an injector and a producer under fixed bottom

hole pressures (5500psi for the injector and 4500psi for the producer). The

production history is for 500 days.

In this example the perturbation was accomplished through a single pa-

rameter gradual deformation method, Roggero and Hu (1998). Sequential

Gaussian Simulation (Deutsch and Journel (1998)) was used to generate the

reference permeability field and to generate realizations for the gradual defor-

mation method. The upgridding technique was adopted from Durlofsky and

Milliken (1997), a single phase upscaling technique (Tran (1995)) was used

to upscale the models once they were upgrided and finally a two-phase fi-

nite difference simulation model was chosen as the full flow simulation model

(FSM ).

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The results of the example given by Tureyen and Caers (2002) are given

in Figure-5. Figure-5a gives the flow responses of 20 realizations conditioned

only to hard data (no history matching has been performed). The relatively

wide scatter of the flow responses are obvious. Most importantly it should be

noted that the flow responses for these 20 realizations were obtained by eval-

uating the full flow simulation model on the fine scale. Figure-5b illustrates

the results in which the proposed algorithm has been applied to 20 realiza-

tions. It is clear in this case that all realizations match the history up to 500

days along with relatively accurate future predictions. Most importantly all

full flow simulations were conducted on coarse models. As mentioned ear-

lier the immediate end result of the proposed algorithm are models that are

coarse, non-uniformly gridded and match the history. Hence the realizations

in Figure-5b are coarse, non-uniformly gridded and match the history.

3 Problems With Parallel Modelling

Although the parallel modelling proposed by Tureyen and Caers (2002) offers

an effective method to reservoir characterization, there are some limitations

to this approach. These can be summarized as follows:

• As mentioned in Section-2, during the entire characterization process

there are two models that are kept in parallel; the fine scale and the

coarse scale. Geostatistics is performed on the fine scale whereas full

flow simulations are performed on the coarse scale. Because of this

reason it is guaranteed that the coarse model will match the history

(as shown in the example given in Section-2.1). However at the end

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of the process the corresponding fine scale model, where all hard data

and seismic data are honored, might not match the history.

• The level of coarsening can play an important role during the process.

Tureyen and Caers (2002) upscaled from a 50×50 fine scale model to

a 25×25 coarse scale model. However the decision regarding the level

of coarsening is fixed prior to characterization. Taking into account

that the fine scale reservoir is perturbed and may significantly change

during the history matching phase, the level of coarsening may not be

known prior to history matching.

3.1 Example Case

To demonstrate the importance of the upscaling level, we present an example

similar to that given in Section-2.1. Most of the outline of the example is

similar to that given in the previous section, hence only the results will be

discussed.

In this example the 50×50 fine grid is upscaled to non-realistic grid di-

mensions of 5×5 on the coarse scale. Figure-6a illustrates this case. It is

clear that the permeability field (Figure-6a) is not representative of the fine

scale geology. However when we consider Figure-6b (which shows the flow

responses of 5 realizations with grid dimensions of 5×5 at the end of the

history matching process) it is clear that models with such grid dimensions

can match the history equally well as the ones shown in the example given

in Section-2.1. Recall that the history matching procedure reduces the mis-

match between field and simulated data on the coarse grid, hence it is possible

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that the reservoir can be perturbed till history match on the coarse scale,

regardless of the upscaling errors.

As a result, when performing flow simulation on Figure-7a, a serious

mismatch between field and simulated data is observed. This might have the

following consequences:

• The fine scale models are the closest representation of the actual geo-

logical variability. Despite history matching the coarse scale model, the

fine scale model still show considerable variability in the flow response,

Figure-(7b), hence geological variability has not been drastically re-

duced. In other words, a ”poor” history matching procedure does not

necessarily reduce geological uncertainty, despite the fact that a history

match is obtained.

• Fine scale models are often used for well placement optimization prob-

lems (Guyaguler (2001)) hence a history match on the fine scale may

be desirable.

4 Proposed Solution: Gridding Optimization

The essential shortcoming in the method of Tureyen and Caers (2002) lies in

the fact that upscaling errors are not accounted for. The idea of perturbing

the fine scale geological model based on the flow results of the coarse model

would only make sense if the upscaling errors are minimal. Otherwise, if

upscaling errors are not negligible, the fine scale models will not match the

history. In this paper we propose a method that aims at improving the

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correction between the fine scale and the coarse scale response by reducing

the upscaling errors, while history matching.

First, we proceed by defining additional notation similar to the ones given

in Section-2.

FSM ∗ : a flow simulator that is a fast approximation of FSM ;

RP* : the flow response when FSM ∗ is performed on z ;

Figure-8 illustrates the proposed method schematically. The work flow

starts by constructing a fine scale geostatistical model. However, instead of

directly proceeding with the upscaling and upgridding phase (proposed in

Section-2) a gridding optimization is performed to ensure that the upscaling

errors between the flow responses of the fine scale and the coarse scale models

are minimized. In other words we can define an ε parameter such that;

ε = ‖FSM(zup(r))− FSM(z (r))‖ (4)

or

ε = ‖FSM(Sθ(z (r)))− FSM(z (r))‖ (5)

at the end of the gridding optimization the ε parameter is minimized. It is

clear that in all cases we will not have access to the information FSM (z (r))

(which is the flow response of the fine scale model when the full flow simu-

lation is performed), if we had, there would not be a necessity for upscaling

in the first place. Hence the challenge is to reduce the upscaling errors ε

without knowing FSM (z (r)).

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To achieve this we introduce a fast approximation to the flow model,

FSM∗. The proposed approach for the optimization is such that, instead of

optimizing on the error ε directly, we introduce an approximate error ε∗, for

any given r , given by:

ε∗(θ, S) = ‖FSM∗(Sθ(z (r)))− FSM∗(z (r))‖ (6)

The error ε∗ is evidently a function of the upscaling method S and its

parameters θ. ε∗ can be minimized by finding an optimal set S and θ. Mini-

mizing ε∗ therefore consists of reducing the mismatch between the production

data FSM∗(z (r)) evaluated on the fine scale using the approximated flow

simulator FSM∗ and the production data. FSM∗(z up(r)), is the same flow

evaluated on the coarse scale. Note two important points:

1. To minimize ε∗ the actual flow response D is not used, neither does

one have to use the same boundary conditions as in the reservoir.

2. In order to find an optimal θ and S we need to evaluate FSM∗ one

single time on the fine scale realization z (r) to obtain a reference fast

flow response. Multiple flow simulations are required on z up(r) to

find the optimal θ and S. Hence the CPU time spent in solving the

optimization problem of finding S and θ will be small compared to

running FSM on z up(r).

The assumption made is the following: Once ε∗ has been made small

enough ε has also been reduced, although probably not by the same amount

as ε∗. In other words the objective function defined on FSM∗ namely ε∗ is

monotonically varying with the objective function defined on FSM namely ε.

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The ranking of models provided by FSM is the same as the ranking provided

by FSM∗: the errors ε and ε∗ need not be the same in absolute magnitude.

A similar approach related to the problem of uncertainty quantification is

taken in Ballin and Journel (1993).

As a result the overall proposed method consists of two optimization

levels: the outer level concerns history matching, the inner level consists of

a grid optimization.

5 Example

In this section we present a synthetic case study for the approach presented in

the previous section. A different upscaling/upgridding approach will be used

in this case. In their example Tureyen and Caers (2002) used the method of

Durlofsky and Milliken (1997), which was a flow based gridding technique.

In this example we use a static based upgridding method described in the

following subsection.

5.1 3DDEGA

3DDEGA (see Garcia (1990)) is an effective and efficient gridding algorithm,

where the objective is to automatically generate coarse grids that fit the ge-

ological heterogeneities. The output of the algorithm are quadrilateral or

hexahedric grids that can be a direct input into commercial numerical reser-

voir simulators through a corner-point geometry description of grid blocks.

The main idea behind the 3DDEGA algorithm is to generate coarse grid

blocks that are as homogeneous as possible in terms of an input variable

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(permeability map, porosity map, facies map, etc.) that is obtained from an

underlying fine scale model. The grid edges are assigned elastic properties

that allow them to expand or shrink to make the coarse blocks homogeneous.

In order to accomplish this, the algorithm goes through three major steps,

which are:

• Determine the grid block heterogeneities, which is also accomplished in

three steps;

– Retrieve the fine scale grid cells within a coarse grid block,

– Compute the same statistics of the fine grid cells within each

coarse grid block

– Determine a heterogeneity index for the coarse block from the

internal block statistics.

• Update the grid-edge elasticity coefficients (that are a function of the

heterogeneity index),

• Compute the new grid-vertex locations that minimize the heterogeneity

which is defined by the heterogeneity index.

Figure-9 illustrates how the 3DDEGA algorithm responds to a unique

example of fine scale heterogeneity. The fine scale contains a box with high

permeability. In the coarse scale the algorithm adjusts itself to refine around

the region of high permeability and coarsen through out in the other regions.

It is also important to note how the grid blocks are deviated from orthogo-

nality to better preserve the high permeability region ”as it is” on the coarse

grid.

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To capture heterogeneity quantitatively a block coefficient of heterogene-

ity (heterogeneity index) is defined as follows:

βB =1

∏Nd=1i=1 (1 + µi)− 1

Nd=1∏

i=1

(1 + µi

(σi

B

σimax

)ωi)− 1

(VB

Vnorm

)ωV

(7)

where,

• i = 1,· · ·,Nd, refers to the input variables (fine scale input variables

that are within the limits of the coarse grid block boundaries),

• σimax is the maximum of all internal block variances (or equivalent ex-

pression for a categorical variable) for data variable i,

• µi and ωi (both positive) are a weight and a power assigned to data

variable i,

• VB is block B’s volume,

• ωV is a power assigned to the block volume term,

• Vnorm is such that V ωVnorm=max(V ωV

min,V ωVmax)

However if there is only a single variable then Equation-7 reduces to:

βB =

(σi

B

σimax

)ωi (VB

Vnorm

)ωV

(8)

To obtain a better understanding the procedure will be explained through

Figure-10. For a given coarse grid block (for example grid block B1 in Figure-

10) the inter block variance (σ1B)is calculated (variance of the fine scale input

variable within the limits of coarse grid block B1). Once this is obtained the

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block coefficient of heterogeneity can be determined for both blocks B1 and

B2. The elasticity of the edge that neighbors blocks B1 and B2 can now be

determined from the block coefficients of heterogeneity of both blocks (the

elasticity is a function of the average block coefficients of heterogeneity).

Given the elasticity, the optimal location of the grid vertices can be deter-

mined such that the block coefficient of heterogeneity will be minimized.

The second term in Equation-8 is used for controlling the coarse grid vol-

umes. If a number of very small and very large grid blocks exist neighboring

each other, this term will be large. This term is therefore introduced to

control the quality of the grid by controlling the relative volumes of coarse

grid blocks. ωi and ωV are weights for emphasizing heterogeneity versus em-

phasizing grid conformity. Some more insight into the importance of these

parameters are provided. Figure-11 illustrates the fine scale reference image

and various upscaled coarse models (obtained by evaluating the 3DDEGA

algorithm on the 100×100 reference fine scale image) through which the sen-

sitivities are given. The top row of Figure-11 represent variation in the ωi

parameter, for fixed ωV , and the bottom row represent variation in the ωV

parameter, for fixed ωi.

The ωi parameter emphasizes on geological heterogeneity. As it is clearly

seen from the top row Figure-11 that if ωi=0.0 then no emphasis is given

to geological heterogeneity hence a uniform grid is obtained. As we increase

the ωi parameter more emphasis is given to the regions of higher geological

variability hence the resulting gridding tries to refine these regions as much

as possible. However for large ωi (Figure-11, ωi=1.0 and ωV =0.0) certain

regions are over-refined leading to small grid blocks neighboring large grid

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blocks. This might not be desirable in some cases especially for the stability

of flow calculations. Hence to achieve a control on the block volumes the ωV

parameter is introduced.

The bottom row of Figure-11 studies the effect of the ωV parameter that

has a global control on the grid block volumes. When a negative value of

this parameter is used (such as ωV =-0.5) the algorithm allows very large grid

blocks to neighbor very small grid blocks (Figure-11, ωi=1.0 and ωV =-0.5).

With increasing ωV , more restriction is put on the coarse grid block volumes,

hence for a high ωV (such as ωV =2.0 in the bottom row of Figure-11) the

grid blocks become almost uniform.

5.2 Synthetic Example

In this section we present the enhanced parallel modelling approach with a

synthetic example. Before giving details regarding the example, we present

the work-flow specific to this example, see Figure-12. The work-flow starts

with constructing the fine scale model. Streamline simulation is performed

on the fine scale where the pseudo water cut curve is obtained. Using the

3DDEGA algorithm, the fine scale model is upgridded to a coarse model.

Streamline simulation is performed on the coarse scale model, and the mis-

match between the pseudo water cut curves of the fine scale and the coarse

scale models are then calculated. Streamline simulation is performed multi-

ple times on the coarse scale model until the 3DDEGA gridding parameters

are optimized and the mismatch is minimized. In addition to ωi and ωV

a third parameter is also introduced in the gridding optimization process,

namely the number of grid blocks in the x direction (ncx)for the coarse scale.

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The total number of grid blocks on the coarse scale are fixed (ncx×nc

y remain

fixed). Hence once ncx is varied a value for nc

y can be determined. Once the

gridding optimization is completed, then full flow simulation is performed

on the coarse model and the mismatch is calculated between the calculated

data and the observed field data. This procedure is repeated until a history

match is obtained.

As mentioned earlier the proposed algorithm is generic. In the following

example, the perturbation method (single parameter probability perturba-

tion method) was adopted from Caers (2003). Instead of a variogram based

geostatistical technique, in this case we use a training image based algorithm

for generating realizations, where we model facies instead of continuous per-

meability values. The upgridding is performed by the 3DDEGA algorithm

while the upscaling is achieved by arithmetic averaging of individual values.

The full simulation model and the approximate fast simulation models are

once again taken as a finite difference simulator and a streamline simulator

respectively.

The reference permeability field and its corresponding water cut curve is

given in Figure-13. A constant permeability value of 10000 md is assigned to

the sand facies and 100 md is assigned to the mud facies. The given reference

model is a quarter-five spot pattern with 100×100 grid blocks in the x and

y directions. Each grid block is of 20ft×20ft with a total of 2000 ft in the

x and y directions. The depth is taken as 200 ft. The production history is

taken to be 500 days and the permeability values at the well locations are

treated as conditioning hard data.

Figure-14 illustrates the results for 30 realizations not constrained to

21

water-cut data. As expected the flow responses of the fine scale models con-

ditioned only to hard data give a relatively wide scatter. As a result of the

proposed algorithm, optimally gridded coarse scale models match the history

well, and provide accurate future predictions for the same well configuration.

If grid optimization had not been performed during the characterization

process, then the results given in Figure-15 would be obtained. Even though

a history match is obtained on the coarse scale, the fine scale responses still

present a wide scatter. When performing grid optimization (see Figure-16),

the upscaling errors are minimized and the fine scale models also become

representative of the history.

6 Conclusions

An alternative approach to reservoir characterization is presented in this

paper. The following conclusions are obtained:

• Introducing upscaling/upgridding within the history matching process

is an effective technique for history matching non-uniformly gridded

models directly. However caution needs to be taken such that the

resulting fine scale models need not match history.

• Introducing a second level of optimization for optimizing the gridding

during the upscaling phase results in a better understanding of the

upgridding parameters, furthermore improvements are observed on the

fine scale matches where the full flow simulations are performed only

on the coarse scale models.

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• With the parallel modelling approach, all data are integrated jointly at

their relative scales.

• It is important to note that optimizing gridding parameters while his-

tory matching is different from choosing for an optimal or adequate

set of gridding/upscaling parameters prior to history matching. An

optimal set of parameters for a particular fine scale realization is not

necessarily optimal for another realization, even when both have sim-

ilar geological heterogeneity. The gridding itself should therefore be a

variable of the entire data integration process.

23

References

Ballin, P.R., A. K. and Journel, A.G., Z. L. (1993). Quantifying the Impact

of Geological Uncertainty on Reservoir Performance Forecasts, paper SPE

25238 presented at the 12th SPE Symposium on Reservoir Simulation, New

Orleans, Louisiana, 28 February - 3 March.

Caers, J. (2001). The next frontier in petroleum geostatistics. Keynote ad-

dress, presented at the Annual Meeting of the International Association

for Mathematical Geology, Cancun, Mexico, 9-12 September.

Caers, J. (2003). Geostatistical History Matching Under Training-Image

Based Geological Model Constraints. SPE Journal, pp 218–226.

Deutsch, C. and Journel, A. (1998). GSLIB, Geostatistical Software Library

and Users Guide. Oxford University Press.

Durlofsky, L.J., J. R. and Milliken, W. (1997). A Non-Uniform Coarsening

Approach for the Scale-Up of Displacement Processes in Heterogeneous

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Garcia, M.H., J. A. A. K. (1990). An Automatic Grid Generation and Ad-

justment Method for Modeling Reservoir Heterogeneities. 3rd scrf annual

report, Stanford Center for Reservoir Forecasting, Stanford University,

California, USA.

Guyaguler, B., H. R. (2001). Uncertainty Assesment of Well Placement Opti-

mization, paper SPE 71625 presented at the Annual Technical Conference

and Exhibition, New Orleans, Louisiana, 30 September - 2 October.

24

Landa, J.L., H. R. (1997). A Procedure to Integrate Well Test Data, Reser-

voir Performance History and 4-D Seismic Information into a Reservoir

Description, paper SPE 38653 presented at the Annual Technical Confer-

ence and Exhibition, San Antonio Texas, 5-8 October.

Mezghani, M. and Roggero, F. (2001). Combining Gradual Deformation and

Upscaling Techniques for Direct Conditioning of Fine Scale Reservoir Mod-

els to Dynamic DATA, paper SPE 71334 presented at the Annual Technical

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Roggero, F. and Hu, L. (1998). Gradual deformation of continuous geosta-

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25

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26

7 List of Figures

Figure-1 Hierarchical approach to reservoir characterization.

Figure-2 A traditional approach to history matching.

Figure-3 Basic Parallel Modelling for Reservoir Characterization.

Figure-4 a - Reference field, b - Corresponding water cut curve, c - Sample

statistics regarding the generation of the reference field.

Figure-5 a - Flow responses of 20 realizations (fine scale) conditioned only

to hard data, b - Flow responses of 20 realizations (coarse scale) when

the proposed algorithm is applied.

Figure-6 a - History matched realization with grid dimensions of 5×5, b -

Flow responses of 5 realizations with grid dimensions of 5×5.

Figure-7 a - Fine scale realization corresponding to the history matched

5×5 realization given in Figure-6a, b - Flow responses of 5 fine scale

realizations.

Figure-8 Schematical diagram of the parallel approach with the gridding

optimization.

Figure-9 Typical Gridding Provided by 3DDEGA.

Figure-10 Schematics illustrating various aspects of the block coefficient of

heterogeneity.

Figure-11 Sensitivity results of the ωi and ωV parameters on gridding.

27

Figure-12 The parallel modelling approach specific to the synthetic exam-

ple.

Figure-13 The reference permeability field and its corresponding flow re-

sponse.

Figure-14 Flow responses of 30 non-history matched realizations and 30

history matched realizations.

Figure-15 Comparison of fine scale and coarse scale flow responses in the

case where grid optimization has not been performed.

Figure-16 Comparison of fine scale and coarse scale flow responses in the

case where grid optimization has been performed.

28

8 Figures

Figure 1: Hierarchical approach to reservoir characterization

29

Figure 2: A traditional approach to history matching

30

Figure 3: Basic Parallel Modelling for Reservoir Characterization

31

Figure 4: a - Reference field, b - Corresponding water cut curve, c - Sample

statistics regarding the generation of the reference field.

32

Figure 5: a - Flow responses of 20 realizations (fine scale) conditioned only

to hard data, b - Flow responses of 20 realizations (coarse scale) when the

proposed algorithm is applied.

33

Figure 6: a - History matched realization with grid dimensions of 5×5, b -

Flow responses of 5 realizations with grid dimensions of 5×5.

34

Figure 7: a - Fine scale realization corresponding to the history matched 5×5

realization given in Figure-6a, b - Flow responses of 5 fine scale realizations.

35

Figure 8: Schematical diagram of the parallel approach with the gridding

optimization.

36

Figure 9: Typical Gridding Provided by 3DDEGA

37

Figure 10: Schematics illustrating various aspects of the block coefficient of

heterogeneity

38

Figure 11: Sensitivity results of the ωi and ωV parameters on gridding.

39

Figure 12: The parallel modelling approach specific to the synthetic example.

40

Figure 13: The reference permeability field and its corresponding flow re-

sponse.

41

Figure 14: Flow responses of 30 non-history matched realizations and 30

history matched realizations.

42

Figure 15: Comparison of fine scale and coarse scale flow responses in the

case where grid optimization has not been performed.

43

Figure 16: Comparison of fine scale and coarse scale flow responses in the

case where grid optimization has been performed.

44