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Based on Buckley-Leverett theory, one inverse problem model of the oil-water relative permeability was modeled and proved when the oil-water relative permeability equations obey the exponential form expression, and under the condition of the formation permeability that natural logarithm distribution always obey normal distribution, the other inverse problem model on the formation permeability was proved. These inverse problem models have been assumed in up-scaling cases to achieve the equations by minimization of objective function different between calculation water cut and real water cut, which can provide a reference for researching oil-water two-phase flow theory and reservoir numerical simulation technology.

Reservoir numerical simulation technology is a growing new discipline with the emergence and development of the computer technology and computational mathematics, which has achieved rapid development and wide application all over the world, for example, the study of reservoir numerical simulation based on formation parameters [

In order to get the answers which will be able to apply in large-scaling cases and solve the corresponding problems in reservoir scaling, inverse problem models have to be assumed to achieve the equations by minimization of objective function differently between real water cut and calculation water cut. The theoretical grid model is shown in Figure

Reservoir profile model.

If we know the distribution

The first question: The optimization distribution

The inverse problem mathematical model on oil-water relative permeability is as follows:

If we know the equations

The second question: an optimization distribution problem

The inverse problem mathematical model of the formation permeability is as follows:

Here,

And thus

The inverse models (

Without considering these factors of gravity and capillary pressure, through the Buckley-Leveret theory of two-phase flow one established the oil-water two-phase plane radial flow mathematical model:

We suppose the liquid flow rule is a plane radial flow from reservoir limit to well and choose a volume element in the vertical direction of streamline, as shown in Figure

According to the seepage principle [

In the

We can derive from (

Meanwhile, the inflow volume of the volume element is

In the

From

Then deriving from (

If

We obtain

Radial flow unit model.

A special saturation definition about “

In the reservoir engineering, the different lithological character has the different corresponding oil-water relative permeability curve equations [

If the oil-water relative permeability equations

Relying on oil-water relative permeability equations (

We can derive from (

If

If we obtain the value of the water saturation [

From (

If the formation permeability logarithmic function distribution

According to

According to the area superposition principle of normal distribution curve, Select

According to the area superposition principle of normal distribution curve and giving a initial value

Relying on the distribution

We can derive from (

If

If we obtain the value of the water saturation

Equation (

The different distribution

According to the above inverse problem mathematical model, based on the definition of the normal distribution, different

Finally, the idea of constructing inverse problem models, according to the historical dynamic production data, can be realized, which attaches great importance to formation heterogeneity, observation of water flooding front position, and prediction of dynamic producing performance.

The authors gratefully thank the support of Canada CMG Foundation and the referees for valuable suggestions.