Inverse Problem Models of Oil-Water Two-Phase Flow Based on Buckley-Leverett Theory

Abstract

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.

1. Introduction

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 [1], the well models and impacts [2], a numerical simulator for low-permeability reservoirs [3], and so on. However, the majority of methods have a great amount of calculation. It will waste a lot of time and energy when we research the oil-water relative permeability equations or the formation permeability distribution. Therefore, according to the inverse problem modeling based on oil-water two-phase flow, we propose a method to obtain the relevant results for our research. It is important to define the formation permeability distribution and oil-water relative permeability equations, which provide necessary information for oil reservoir evaluation, for example, large-scaling evaluation, which could be applied in reservoir numerical simulation during evaluation.

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 1.

Figure 1 

Reservoir profile model.

If we know the distribution {𝐾₁,𝐾₂,…,𝐾_𝑛}, then the inverse problem model will be as shown in (1).

The first question: The optimization distribution {𝑎_𝑤,𝑏_𝑤,𝑎_𝑜,𝑏_𝑜} and the relative permeability equations 𝐾_𝑟𝑤(𝑆_𝑤), 𝐾_𝑟𝑜(𝑆_𝑤).

The inverse problem mathematical model on oil-water relative permeability is as follows: objectivefunction:𝐸=min^𝑛𝑡_𝑡=1𝑓_𝑤(𝑡)−𝑓^(history)_𝑤(𝑡)²,𝑆where𝑛𝑡istimesteps;initialcondition:_{𝑤𝑗𝑖(0)}=𝑆_𝑗𝐼𝑤,𝑄⁰_𝑗=00≤𝐾𝑟𝑜^𝑖_𝑗<1,0≤𝐾𝑟𝑤^𝑖_𝑗𝐾<1_𝑟𝑤𝑆_𝑤=𝑎_𝑤𝑆_𝑤−𝑆_𝑤𝑐1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑏𝑤𝐾_𝑟𝑜𝑆_𝑤=𝑎_𝑜1−𝑆_𝑜𝑟−𝑆_𝑤1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑏𝑜if𝑟_𝑒>𝑟^𝑖_𝑗𝑤𝑓,then𝑆_𝑤𝑗^𝑖(𝑡)=𝑆_𝑗𝐼𝑤if𝑟_𝑒≤𝑟^𝑖_𝑗𝑤𝑓,then𝑆_𝑤𝑗^𝑖(𝑡)=𝑓_𝑤^′−1𝑆_𝑤𝑓_𝑤=𝑓_𝑤𝑆_𝑤;mathematicalmodel:𝐾𝑟𝑜^𝑖_𝑗=𝐾𝑟𝑜𝑆_𝑤𝑗^𝑖(𝑡),𝐾𝑟𝑤^𝑖_𝑗=𝐾𝑟𝑤𝑆_𝑤𝑗^𝑖(𝑡)𝑇^𝑖_𝑗=2𝜋ℎ_𝑗𝐾_𝑗ln𝑟/𝑟^𝑖_𝑗𝐾^𝑖_𝑗𝑟𝑜𝜇_𝑜+𝐾^𝑖_𝑗𝑟𝑤𝜇_𝑤𝑇_𝑗=1∑^𝑚_𝑖=11/𝑇^𝑖_𝑗,𝑇=^𝑛_𝑗=1𝑇_𝑗,𝑄^𝑡_𝑗=𝑇_𝑗𝑇⋅𝑄_𝑡𝑟²_𝑒−𝑟^𝑖_𝑗²=𝑓^𝑡_𝑤𝑆_𝑤^𝑖_𝑗^′𝑄^𝑡_𝑗𝜙_𝑗⋅𝜋ℎ_𝑗,𝑟²_𝑗𝑤𝑓=𝑟²_𝑒−𝑓^𝑡_𝑤𝑆^𝑖_𝑗𝑤𝑓^′𝑄^𝑡_𝑗𝜙_𝑗⋅𝜋ℎ_𝑗𝑄^𝑡_𝑤𝑗=𝑄^𝑡_𝑗⋅𝑓^𝑡_𝑤𝑆¹_𝑤𝑗,𝑄^𝑡_𝑜𝑗=𝑄^𝑡_𝑗⋅1−𝑓^𝑡_𝑤𝑆¹_𝑤𝑗𝑄^𝑡_𝑤=^𝑛_𝑗=1𝑄^𝑡_𝑤𝑗,𝑄^𝑡_𝑜=^𝑛_𝑗=1𝑄^𝑡_𝑜𝑗,𝑓_𝑤(𝑡)=𝑄^𝑡_𝑤𝑄^𝑡_𝑤+𝑄^𝑡_𝑜. (1)

If we know the equations 𝐾_𝑟𝑤(𝑆_𝑤), 𝐾_𝑟𝑜(𝑆_𝑤), another inverse problem model will be obtained as shown in (2).

The second question: an optimization distribution problem {𝐾₁,𝐾₂,…,𝐾_𝑛} and the standard deviation 𝜎.

The inverse problem mathematical model of the formation permeability is as follows: objectivefunction:𝐸=min^𝑛𝑡_𝑡=1𝑓_𝑤(𝑡)−𝑓^(history)_𝑤(𝑡)²,𝑆where𝑛𝑡istimesteps;initialcondition:_{𝑤𝑗𝑖(0)}=𝑆_𝑗𝑤𝑐,𝑄⁰_𝑗𝑓𝐾=0_𝑗=1√𝑒2𝜋⋅𝜎^{−(ln𝐾𝑗−𝜇)2/2𝜎2}𝜇=Ln𝐾if𝑟_𝑒>𝑟^𝑖_𝑗𝑤𝑓,then𝑆_𝑤𝑗^𝑖(𝑡)=𝑆_𝑗𝑤𝑐if𝑟_𝑒≤𝑟^𝑖_𝑗𝑤𝑓,then𝑆_𝑤𝑗^𝑖(𝑡)=𝑓_𝑤^′−1𝑆_𝑤𝑓_𝑤=𝑓_𝑤𝑆_𝑤mathematicalmodel:𝐾𝑟𝑜^𝑖_𝑗=𝐾𝑟𝑜𝑆_𝑤𝑗^𝑖(𝑡),𝐾𝑟𝑤^𝑖_𝑗=𝐾𝑟𝑤𝑆_𝑤𝑗^𝑖(𝑡)𝑇^𝑖_𝑗=2𝜋ℎ_𝑗𝐾_𝑗ln𝑟/𝑟^𝑖_𝑗𝐾^𝑖_𝑗𝑟𝑜𝜇_𝑜+𝐾^𝑖_𝑗𝑟𝑤𝜇_𝑤𝑇_𝑗=1∑^𝑚_𝑖=11/𝑇^𝑖_𝑗,𝑇=^𝑛_𝑗=1𝑇_𝑗,𝑄^𝑡_𝑗=𝑇_𝑗𝑇⋅𝑄_𝑡𝑟²_𝑒−𝑟^𝑖_𝑗²=𝑓^𝑡_𝑤𝑆_𝑤^𝑖_𝑗^′𝑄^𝑡_𝑗𝜙_𝑗⋅𝜋ℎ_𝑗,𝑟²_𝑗𝑤𝑓=𝑟²_𝑒−𝑓^𝑡_𝑤𝑆^𝑖_𝑗𝑤𝑓^′𝑄^𝑡_𝑗𝜙_𝑗⋅𝜋ℎ_𝑗𝑄^𝑡_𝑤𝑗=𝑄^𝑡_𝑗⋅𝑓^𝑡_𝑤𝑆¹_𝑤𝑗,𝑄^𝑡_𝑜𝑗=𝑄^𝑡_𝑗⋅1−𝑓^𝑡_𝑤𝑆¹_𝑤𝑗𝑄^𝑡_𝑤=^𝑛_𝑗=1𝑄^𝑡_𝑤𝑗,𝑄^𝑡_𝑜=^𝑛_𝑗=1𝑄^𝑡_𝑜𝑗,𝑓_𝑤(𝑡)=𝑄^𝑡_𝑤𝑄^𝑡_𝑤+𝑄^𝑡_𝑜. (2)

Here, 𝑓^(history)_𝑤(𝑡) is history water cut and 𝑓_𝑤(𝑡) is calculation water cut; the first water cut is a known amount from production and another is calculated by our inverse problem models. Water cut means water production rate is a key parameter in reservoir engineering, which can represent water production capacity in reservoir development. If water cut is too high, it may bring negative effects to reservoir production. Therefore, the history matching for water production rate plays an important role in reservoir dynamic analysis and numerical simulation [4–6].

And thus 𝑖=1,2,3,…,𝑚, 𝑗=1,2,3,…,𝑛, 𝑚 is the average partition total, 𝑛 is the partition total, 𝑇 is grid conductivity, ℎ_𝑗 is each longitudinal layer stratum thickness, 𝐾_𝑗 is each longitudinal layer permeability, 𝜙_𝑗 is each longitudinal layer porosity, 𝑆^𝑖_𝑤𝑗 is grid water saturation at different time, 𝑆_𝑤𝑐 is the initial water saturation, 𝐸 is the objective function, 𝑟_𝑒 is reservoir radius, 𝐾 is average permeability, 𝐾_𝑟𝑜 is oil relative permeability, 𝐾_𝑟𝑤 is water relative permeability, 𝑄_𝑡 is the total fluid production, 𝑄^𝑡_𝑤 is the total water yield of fluid producing edge at different time, 𝑄^𝑡_𝑜 is the total oil production of fluid producing edge at different time, 𝑄^𝑡_𝑗 is the total fluid production of each longitudinal layer at different time, 𝑆_𝑤𝑓 is the frontier saturation, 𝑟_𝑤𝑓 is the corresponding displacement of frontier saturation.

The inverse models (1) and (2) contain Buckley-Leverett theory of two-phase flow [7] and establish the oil-water two-phase plane radial flow function: 𝑟²_𝑒−𝑟²=𝑓^′_𝑤⋅𝑄^𝑡/𝜙⋅𝜋⋅ℎ without considering these factors of gravity and capillary pressure; in addition, that shows the movement rules of isosaturation surface. As said above, most researchers always rely on forward problems [8] of core sampling experiment and mathematical statistics for an answer, As a result, there is always a large error between the calculation water cut and the history water cut in reservoir water cut analysis. In this paper, according to historical water cut and calculation water cut to establish the objective function, 𝐸=min∑^𝑛𝑡_𝑡=1(𝑓_𝑤(𝑡)−𝑓^(history)_𝑤(𝑡))². By the end an inverse problem model on the optimal solution distribution {𝑎_𝑤,𝑏_𝑤,𝑎_𝑜,𝑏_𝑜} and oil-water relative permeability equations 𝐾_𝑟𝑤(𝑆_𝑤), 𝐾_𝑟𝑜(𝑆_𝑤) based on the Buckley-Leveret theory of two-phase flow can be realized, and another inverse problem model was modeled under the condition of the formation permeability logarithmic function are always obey normal distribution, Finally, it can provide a key information for reservoir numerical simulation studies.

2. Mathematical Model of Oil-Water Two-Phase Plane Radial Flow

Theorem 1. 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: 𝑟²_𝑒−𝑟²=𝑓^′_𝑤⋅𝑄^𝑡.𝜙⋅𝜋ℎ (3)

Proof. 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 2.
According to the seepage principle [9], we can get a flow equation of the volume element: 𝑞_𝑤=𝑞⋅𝑓_𝑤. (4)
In the 𝑑𝑡 time, the flow volume of the volume element is 𝑄_out=𝑑𝑞_𝑤⋅𝑑𝑡. (5)
We can derive from (4) and (5) that 𝑄_out=𝑞⋅𝑑𝑓_𝑤⋅𝑑𝑡. (6)
Meanwhile, the inflow volume of the volume element is 𝑄_in=𝜙⋅2𝜋ℎ⋅𝑟𝑑𝑟⋅𝑑𝑠_𝑤. (7)
In the 𝑑𝑡 time, relying on the Buckley-Leveret theory of two-phase flow, 𝑄_in=𝑄_out. from (6) and (7), we have that 𝑞⋅𝑑𝑓_𝑤⋅𝑑𝑡=𝜙⋅2𝜋ℎ⋅𝑟𝑑𝑟⋅𝑑𝑆𝑤 as follows: 𝑞⋅𝑑𝑓_𝑤𝑑𝑆𝑤⋅𝑑𝑡=𝜙⋅2𝜋ℎ⋅𝑟𝑑𝑟. (8)
From 𝑓_𝑤=𝑓_𝑤(𝑆_𝑤), we can get that 𝑓^′_𝑤=𝑑𝑓_𝑤/𝑑𝑆_𝑤.
Then deriving from (8) that 𝑞⋅𝑑𝑓_𝑤(𝑆𝑤)⋅𝑑𝑡=𝜙⋅2𝜋ℎ⋅𝑟𝑑𝑟, after infinitesimal calculus, we obtain 𝑓^′_𝑤^𝑡₀𝑞⋅𝑑𝑡=𝜙⋅2𝜋ℎ⋅^𝑟𝑒_𝑟𝑟𝑑𝑟. (9)
If 𝑄^𝑡=∫^𝑡₀𝑞⋅𝑑𝑡, then (9) turns to 𝑓^′_𝑤⋅𝑄^𝑡=𝜙⋅𝜋ℎ⋅(𝑟²_𝑒−𝑟²).
We obtain 𝑟²_𝑒−𝑟²=𝑓^′_𝑤⋅𝑄^𝑡.𝜙⋅𝜋ℎ (10) From the reservoir profile grid model, the plane radial flow model of the grids in the longitudinal each layer can shows that 𝑟²_𝑒−𝑟^𝑖_𝑗²=𝑓^𝑡_𝑤𝑆_𝑤^𝑖_𝑗^′𝑄^𝑡_𝑗𝜙_𝑗⋅𝜋ℎ_𝑗. (11)

Figure 2 

Radial flow unit model.

Theorem 2. A special saturation definition about “𝑆_𝑤𝑓’’: from the water cut function curve 𝑓_𝑤=𝑓_𝑤(𝑆_𝑤), selecting the point of the irreducible water saturation as a fixed point and joining any other point on the curve, and constructing function 𝑘(𝑆_𝑤𝑓)=(𝑓_𝑤(𝑆_𝑤𝑓)−𝑓_𝑤(𝑆_𝑤1))/(𝑆_𝑤𝑓−𝑆_𝑤1), if Max{𝑘(𝑆_𝑤𝑓)}, then one can call “𝑆_𝑤𝑓’’ frontier saturation. Now the corresponding displacement of frontier saturation 𝑟_𝑤𝑓 is 𝑟²_𝑗𝑤𝑓=𝑟²_𝑒−𝑓^𝑡_𝑤𝑆^𝑖_𝑗𝑤𝑓^′𝑄^𝑡_𝑗𝜙_𝑗⋅𝜋ℎ_𝑗. (12)

3. The Inverse Problem Model on Oil-Water Relative Permeability

3.1. The Oil-Water Relative Permeability Equations

In the reservoir engineering, the different lithological character has the different corresponding oil-water relative permeability curve equations [10], but the most commonly form is used as a kind of the exponential form expression, as follows: 𝐾_𝑟𝑤𝑆_𝑤=𝑎_𝑤𝑆_𝑤−𝑆_𝑤𝑐1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑏𝑤,𝐾_𝑟𝑜𝑆_𝑤=𝑎_𝑜1−𝑆_𝑜𝑟−𝑆_𝑤1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑏𝑜. (13) In reservoir profile grid model, the oil-water relative permeability curve equations of each grid can be showed: 𝐾_𝑟𝑤𝑆^𝑖_𝑤𝑗=𝑎_𝑤𝑆^𝑖_𝑤𝑗−𝑆_𝑤𝑐1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑏𝑤,𝐾_𝑟𝑜𝑆^𝑖_𝑤𝑗=𝑎_𝑜1−𝑆_𝑜𝑟−𝑆^𝑖_𝑤𝑗1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑏𝑜. (14) In the equations, we can know different {𝑎_𝑤,𝑏_𝑤,𝑎_𝑜,𝑏_𝑜} corresponding to its own 𝐾_𝑟𝑤(𝑆_𝑤), 𝐾_𝑟𝑜(𝑆_𝑤).

3.2. The Inverse Problem Mathematical Model of Oil-Water Relative Permeability Equations

Theorem 3. If the oil-water relative permeability equations 𝐾_𝑟𝑤(𝑆_𝑤), 𝐾_𝑟𝑜(𝑆_𝑤) always obey (13) and one can calculate the water cut 𝑓_𝑤(𝑡) based on the model of the Theorem 1, if the objective function 𝐸=min∑^𝑛𝑡_𝑡=1(𝑓_𝑤(𝑡)−𝑓_𝑤^{(ℎ𝑖𝑠𝑡𝑜𝑟𝑦)}(𝑡))² can be satisfied, then the inverse problem model (1) will have the optimal solution of the distribution {𝑎_𝑤,𝑏_𝑤,𝑎_𝑜,𝑏_𝑜} and oil-water relative permeability equations 𝐾_𝑟𝑤(𝑆_𝑤), 𝐾_𝑟𝑜(𝑆_𝑤).

Proof. Relying on oil-water relative permeability equations (14) and the conductivity definition of numerical reservoir simulation [11], we get the following: initialvalue:𝑎⁰_𝑤,𝑏⁰_𝑤,𝑎⁰_𝑜,𝑏⁰_𝑜,𝐾_𝑟𝑤𝑆_𝑤=𝑎⁰_𝑤𝑆_𝑤−𝑆_𝑤𝑐1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑏0𝑤𝐾_𝑟𝑜𝑆_𝑤=𝑎⁰_𝑜1−𝑆_𝑜𝑟−𝑆_𝑤1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑏0𝑜𝑇^𝑖_𝑗=2𝜋ℎ_𝑗𝐾_𝑗ln𝑟/𝑟^𝑖_𝑗𝐾^𝑖_𝑗𝑟𝑜𝜇_𝑜+𝐾^𝑖_𝑗𝑟𝑤𝜇_𝑤𝑇_𝑗=1∑^𝑚_𝑖=11/𝑇^𝑖_𝑗,𝑇=^𝑛_𝑗=1𝑇_𝑗,𝑄^𝑡_𝑗=𝑇_𝑗𝑇⋅𝑄_𝑡. (15)
We can derive from (12) and (15): 𝑟_𝑤𝑓.
If 𝑟_𝑒>𝑟_𝑤𝑓, then 𝑆_𝑤𝑗^𝑖(𝑡)=𝑆_𝑗𝑤𝑐. If 𝑟_𝑒≤𝑟_𝑤𝑓 solved inverse function of (11) and calculated water saturation of the fluid producing edge, then 𝑆_𝑤𝑗^𝑖(𝑡)=𝑓_𝑤^′−1(𝑆_𝑤), and so forth 𝑖=1.
If we obtain the value of the water saturation [12], relying on the function 𝑓_𝑤(𝑆_𝑤)∼𝑆_𝑤, we can calculate the value of the longitudinal each layer water cut: 𝑓^𝑡_𝑤(𝑆¹_𝑤𝑗) and the constructing mathematic model as follows 𝑄^𝑡_𝑤𝑗=𝑄^𝑡_𝑗⋅𝑓^𝑡_𝑤𝑆¹_𝑤𝑗,𝑄^𝑡_𝑜𝑗=𝑄^𝑡_𝑗⋅1−𝑓^𝑡_𝑤𝑆¹_𝑤𝑗,𝑄^𝑡_𝑤=^𝑛_𝑗=1𝑄^𝑡_𝑤𝑗,𝑄^𝑡_𝑜=^𝑛_𝑗=1𝑄^𝑡_𝑜𝑗,𝑓_𝑤(𝑡)=𝑄^𝑡_𝑤𝑄^𝑡_𝑤+𝑄^𝑡_𝑜. (16)
From (16) we can calculate the total water cut of the fluid producing edge 𝑓_𝑤(𝑡) and rely on the objective function: 𝐸=min∑^𝑛𝑡_𝑡=1(𝑓_𝑤(𝑡)−𝑓^(history)_𝑤(𝑡))² with numerical optimization calculation in the inverse problem model; an optimization problem of the distribution {𝑎_𝑤,𝑏_𝑤,𝑎_𝑜,𝑏_𝑜} will be obtained, so the 𝐾_𝑟𝑤(𝑆_𝑤) and 𝐾_𝑟𝑜(𝑆_𝑤) can be formed. The proof of Theorem 3 is completed.

4. The Inverse Problem Model on the Formation Permeability Logarithmic Function Always Obey Normal Distribution

Theorem 4. If the formation permeability logarithmic function distribution {Ln𝐾₁,Ln𝐾₂,…,Ln𝐾_𝑛} always obey normal distribution 𝑓(𝐾)=(1/√2𝜋⋅𝜎)𝑒^{−(ln𝐾−𝜇)2/2𝜎2}, 𝜇=Ln𝐾, meanwhile, one calculates the value of the 𝑓_𝑤(𝑡) based on the model of Theorem 1, which can be adapted to the objective function 𝐸=min∑^𝑛𝑡_𝑡=1(𝑓_𝑤(𝑡)−𝑓_𝑤^{(ℎ𝑖𝑠𝑡𝑜𝑟𝑦)}(𝑡))²; then the inverse problem model (2) will have the optimal distribution {Ln𝐾₁,Ln𝐾₂,…,Ln𝐾_𝑛} and {𝐾₁,𝐾₂,…,𝐾_𝑛}.

Proof. According to 3𝜎 principle: for normal distribution curve, if 𝑃(𝜇−3𝜎−𝑎<𝑋≤𝜇+3𝜎+𝑎)→1, then there are a left end point value and a right end point value on the curve; the 𝑋 value of the left end point is ln𝐾_min, or 𝜇−3𝜎−𝑎. The 𝑋 value of the right end point is ln𝐾_max, or 𝜇+3𝜎+𝑎. And thus 𝑎>0 and 𝜇=ln𝐾=0.5⋅(ln𝐾_min+ln𝐾_max).
According to the area superposition principle of normal distribution curve, Select 𝑥=ln𝐾 as the starting point, step size Δ𝑥 and make a subdivision for probability curve; if the area summation of these formed closed figures can infinitely approach 1, then we can obtain the ln𝐾_min value of the left end point and the ln𝐾_max value of the right end point, and ln𝐾_max=2⋅ln𝐾−ln𝐾_min.
According to the area superposition principle of normal distribution curve and giving a initial value 𝜎 of normal distribution, then 𝑋₀=ln𝐾_min, and 𝑋_𝑛=ln𝐾_max. 𝐹(𝑋_𝑗) is a cumulative distribution function of the normal distribution [13]. If we use equal step size Δ𝑋 make a subdivision for probability curve, then the area summation of every closed figures 𝑆_𝑗=𝐹(𝑋_𝑗)−𝐹(𝑋_𝑗−1), each longitudinal layer stratum thickness ℎ_𝑗=ℎ⋅𝑆_𝑗, and 𝑋_𝑗=Δ𝑋⋅𝑗+𝑋₀, 𝐾_𝑗=𝑒^𝑋𝑗; if we use equal area Δ𝑆 make a subdivision for probability curve, then the area summation of every closed figures 𝑆_𝑗=𝐹(𝑋_𝑗)−𝐹(𝑋_𝑗−1), 𝑆_𝑗=Δ𝑆=1/𝑛, ℎ_𝑗=ℎ⋅𝑆_𝑗, and 𝑋_𝑗=𝐹⁻¹(𝑆_𝑗), 𝐾_𝑗=𝑒^𝑋𝑗. And thus 𝑗=1,2,3,…,𝑛 (𝑛 is the total number of vertical stratification).
Relying on the distribution {𝐾₁,𝐾₂,…,𝐾_𝑛}, oil-water relative permeability functions: 𝑘𝑟𝑜=𝑘𝑟𝑜(𝑆_𝑤), 𝑘𝑟𝑤=𝑘𝑟𝑤(𝑆_𝑤) [14], the conductivity definition of numerical reservoir simulation, we get the following equations: 𝑘𝑟𝑜^𝑖_𝑗=𝑘𝑟𝑜𝑆_𝑤𝑗^𝑖(𝑡),𝑘𝑟𝑤^𝑖_𝑗=𝑘𝑟𝑤𝑆_𝑤𝑗^𝑖(𝑡)𝑇^𝑖_𝑗=2𝜋ℎ_𝑗𝐾_𝑗ln𝑟/𝑟^𝑖_𝑗𝐾^𝑖_𝑗𝑟𝑜𝜇_𝑜+𝐾^𝑖_𝑗𝑟𝑤𝜇_𝑤𝑇_𝑗=1∑^𝑚_𝑖=11/𝑇^𝑖_𝑗,𝑇=^𝑛_𝑗=1𝑇_𝑗𝑄^𝑡_𝑗=𝑇_𝑗𝑇⋅𝑄_𝑡. (17)
We can derive from (12) and (17) that 𝑟_𝑗𝑤𝑓.
If 𝑟_𝑒≤𝑟_𝑗𝑤𝑓, solving inverse function of (11) and calculating water saturation of the fluid producing edge, then 𝑆_𝑤𝑗^𝑖(𝑡)=𝐹⁻¹(𝑟^𝑖_𝑗,𝜙_𝑗,ℎ_𝑗,𝑄^𝑡_𝑗), and thus 𝑖=1. If 𝑟_𝑒>𝑟_𝑗𝑤𝑓, then 𝑆_𝑤𝑗^𝑖(𝑡)=𝑆_𝑗𝑤𝑐, and thus 𝑖=1.
If we obtain the value of the water saturation (𝑆_𝑤), relying on the function 𝑓_𝑤(𝑆_𝑤)∼𝑆_𝑤, we can calculate the value of the longitudinal each layer water cut: 𝑓^𝑡_𝑤(𝑆¹_𝑤𝑗); then mathematic model can be constructed as follows: 𝑄^𝑡_𝑤𝑗=𝑄^𝑡_𝑗⋅𝑓^𝑡_𝑤𝑆¹_𝑤𝑗,𝑄^𝑡_𝑜𝑗=𝑄^𝑡_𝑗⋅1−𝑓^𝑡_𝑤𝑆¹_𝑤𝑗𝑄^𝑡_𝑤=^𝑛_𝑗=1𝑄^𝑡_𝑤𝑗,𝑄^𝑡_𝑜=^𝑛_𝑗=1𝑄^𝑡_𝑜𝑗𝑓_𝑤(𝑡)=𝑄^𝑡_𝑤𝑄^𝑡_𝑤+𝑄^𝑡_𝑜. (18)
Equation (18) can calculate the total water cut of the fluid producing edge: 𝑓_𝑤(𝑡) and relies on the objective function: 𝐸=min∑^𝑛𝑡_𝑡=1(𝑓_𝑤(𝑡)−𝑓^(history)_𝑤(𝑡))². With numerical optimization calculation in the inverse problem model, the standard deviation 𝜎 and an optimization problem of the distribution {Ln𝐾₁,Ln𝐾₂,…,Ln𝐾_𝑛} will be obtained, so the distribution {𝐾₁,𝐾₂,…,𝐾_𝑛} will be given. The proof of Theorem 4 is completed.

5. Discussions and Conclusions

The different distribution {𝑎_𝑤,𝑏_𝑤,𝑎_𝑜,𝑏_𝑜} corresponds to a group of oil-water relative permeability equations 𝐾_𝑟𝑤(𝑆_𝑤) and 𝐾_𝑟𝑜(𝑆_𝑤). Combining with the objective function 𝐸=min∑^𝑛𝑡_𝑡=1(𝑓_𝑤(𝑡)−𝑓^(history)_𝑤(𝑡))² through the optimization solution method, it will finally bring a set of the optimum distribution {𝑎_𝑤,𝑏_𝑤,𝑎_𝑜,𝑏_𝑜} and the oil-water relative permeability equations 𝐾_𝑟𝑤(𝑆_𝑤) and 𝐾_𝑟𝑜(𝑆_𝑤).

According to the above inverse problem mathematical model, based on the definition of the normal distribution, different (𝜇_𝑖,𝜎²_𝑖) corresponds to its own normal distribution curve with the certain expectation 𝜇_𝑖. If 𝜎_𝑖 goes up, the volatility of its normal distribution will be stronger according to the definition of standard deviation, which will finally lead to the large differential permeability distribution, namely, the strong heterogeneity. With the certain expectation 𝜇_𝑖, each different value of 𝜎_𝑖 corresponds to a group of original values of (𝜇_𝑖,𝜎²_𝑖), which will yield a group of values of {𝐾₁,𝐾₂,…𝐾_𝑛−1,𝐾_𝑛}, namely, the value of permeability of each single formation. And it will also work out liquid production, water production, and integrated water cut of the whole liquid outlet of each single formation. Combining with the objective function 𝐸=min∑^𝑛𝑡_𝑡=1(𝑓_𝑤(𝑡)−𝑓^(history)_𝑤(𝑡))², it will finally bring a set of optimum normal distributions of 𝑋∼(𝜇_𝑖,𝜎²_𝑖) (Etc. 𝑋=ln𝐾) through the optimization solution method. Analysis of the changes of water driving place, oil production, and water production can be done through the related mathematical model.

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.

Acknowledgments

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