A Homotopy Method for the Constrained Inverse Problem in the Multiphase Porous Media Flow

Abstract

This paper considers the constrained inverse problem based on the nonlinear convection-diffusion equation in the multiphase porous media flow. To solve this problem, a widely convergent homotopy method is introduced and proposed. To evaluate the performance of the mentioned method, two numerical examples are presented. This method turns out to have wide convergence region and strong anti-noise ability.

1. Introduction

The multiphase flow in porous media is of great importance in biology [1], chemistry [2], civil engineering [3], mechanics [4], and geosciences [5]. The porous media flow in many engineering problems, including reservoir engineering and reservoir simulation engineering, is often used to simulate the gas and oil flow within the reservoir. The authors in [6] proposed the fractional flow formulation of the multiphase flow equations in porous media, which leads to a nonlinear convection-diffusion saturation equation. Recently, Saeed et al. exactly analyzed the second grade fluid [7] and viscoelastic liquid with single slip assumption [8], and studied the natural convection flow [9] from in mathematical perspective. Abdeljawad et al. [10] and Firdous et al. [11], respectively, considered the MHD Maxwell fluid and Powell-Eyring fluid. Riaz et al. [12,13] mainly researched on the MHD Oldroyd-B fluid. Nowadays there are many different studies that aim to solve the inverse problem in multiphase porous media flow. Hazra et al. [14] solved the direct and inverse modeling in multiphase porous media flow using numerical simulation techniques, and Wang and Zabaras [15] identified the contamination source in multiphase porous media flow. Nilssen et al. [16] recovered the diffusion parameters in multiphase porous media flow with the augmented Lagrangian method. With the development of artificial intelligence, several recent studies focused on the application of deep learning models to the inverse problem in multiphase porous media flow [17,18,19,20].

This paper considers the identification of space-dependent permeability $\mathbf{k}$ for the nonlinear convection-diffusion equation in the multiphase porous media flow:

$$u_t+\nabla ·(\varphi ,\phi )-\nabla ·(\mathbf{k}·N\left(u\right)\nabla u)=f(\mathbf{x},t),(\mathbf{x},t)\in \Omega \times (0,T),$$

(1)

under the boundary and initial conditions

$$u(\mathbf{x},t)=\chi (\mathbf{x},t),(\mathbf{x},t)\in \partial \Omega \times (0,T),$$

(2)

$$u(\mathbf{x},0)=\psi \left(\mathbf{x}\right),\mathbf{x}\in \Omega .$$

(3)

An additional condition is

$$u({\mathbf{x}}^s,t)=\gamma ({\mathbf{x}}^s,t),s=1,2,\dots ,S,t\in (0,T),$$

(4)

where $\varphi$ and $\phi$ are both S-shaped Buckley–Leverett flux functions, N is a nonlinear diffusion function, f is a source function that is piecewise smooth, $\Omega$ is set to be an unit square for simplicity, and $\gamma$ is the measured data (seepage velocity data).

Equation (1) correlates with the multiphase porous media flow. The partial differential equation (PDE) system describing the immiscible displacement of oil by water in porous media under no gravity condition is as follows [6]:

$$\left\{\begin{array}{c}\nabla ·\mathbf{V}=f_1(\mathbf{x},t),\hfill \\ \mathbf{V}=-\mathbf{k}·\varsigma (\mathbf{x},u)(\nabla p-\vartheta (u)\nabla h),\hfill \\ \beta \left(\mathbf{x}\right)u_t+\nabla ·(\varphi \left(u\right)\mathbf{V}+\mathbf{k}·\phi \left(u\right)\nabla h)-\nabla ·(\mathbf{k}·N\left(u\right)\nabla u)=f_2(\mathbf{x},t),\hfill \end{array}\right.$$

(5)

where $\varphi =\frac{{\varsigma }_{ra}}{\varsigma }$, ${\varsigma }_{ra}=\frac{{\mathbf{k}}_{ra}}{\mu }$, $\phi =(\vartheta -\rho )\varphi ϵ$, and the definitions of model parameters are listed in

Table 1. Parameter definitions.

Parameter	Definition
$\mathbf{k}$	absolute permeability
$\mathbf{V}$	total Darcy velocity
$\varsigma$	total mobility of phases
p	global pressure
$\vartheta$	density of wetting phase
h	height
$\beta$	porosity
$f_1$	injection well
$f_2$	production well
${\varsigma }_{ra}$	mobility ratio
${\mathbf{k}}_{ra}$	relative permeability
$\mu$	viscosity
$\rho$	density
$ϵ$	phase mobility of nonwetting phase

. Units for $\mathbf{k}$, $\mathbf{V}$, p, h, $\rho$ are, respectively, $m^2$, $m/s$, N, m, $kg/m^3$, and other parameters are dimensionless.

Equation (1) closely resembles Equation (5) if the time derivative and convection terms are not considered. The coefficients of time derivative terms in Equations (1) and (5) are respectively constant 1 and $\beta \left(\mathbf{x}\right)$; the convection term in Equation (5) has varying coefficient and permeability dependence, and the one in Equation (1) does not.

Oil reservoir simulation on the basis of this inverse problem is an effective tool, which can provide help for petroleum reservoir management, such as the choices of the fluid production and injection rates, well locations, imaging method (see Figure 1). Generally, the permeability model has problems with equivalence, non-uniqueness, hidden or suppressed layers and lack of model resolution, because the measured data are insufficient, inconsistent and inaccurate [21]. Moreover, the primary difficulties of the traditional methods (e.g., Levenberg–Marquardt, Gauss–Newton) are the local convergence property and that the cost function has numerous local minima. To cope with these problems, a homotopy method is developed associated with the constraints of well logs to identify the permeability coefficient.

The homotopy method has global convergence under certain weak assumptions [22], and so has been successfully used to solve various nonlinear problems [23,24,25,26,27,28]. Recently, some authors also used it to solve inverse problems. For instance, Mallick et al. [29,30] used this method to estimate the thermal parameters within annular fins. Biswal et al. [31] applied the homotopy method for the inverse analysis of Jeffery–Hamel flow problem. Sattari Shajari and Shidfar [32] studied the homotopy solution of the wave equation inverse source problem. Liu [33,34] combined the homotopy method with multiscale ideas to develop the hybrid algorithm, and applied it to nonlinear inverse problems. Hu et al. [35] identified the parameters of a cracked beam using the homotopy algorithm. Courbot and Colicchio [36] analyzed the solution of the gridless sparse recovery problem using the homotopy method. The homotopy method has also been applied to the fractional inverse Stefan problem [37], the backward heat conduction problem [38], the porosity reconstruction on the basis of Biot elastic model [39], etc.

Usually, the measured data have a low signal-to-noise ratio. With the aim of restraining noise and improving identified model quality, the constraint condition has a wide application in many inversion fields [40,41,42,43]. The reason lies in that the constraint data are recorded from the interior of the model to be identified, and are less noisy than the measured data.

In previous works [44,45], we have verified the effectiveness of the homotopy and multigrid-homotopy methods for the inverse problem of the nonlinear diffusion equation:

$$u_t-\nabla ·(\mathbf{k}·N(u,\nabla u)\nabla u)=f(\mathbf{x},t),$$

(6)

which is an intermediate step of permeability identification in multiphase porous media flow. Different from [44,45], this paper not only considers the permeability identification based on the nonlinear convection-diffusion Equation (1), which can more accurately describe the multiphase flow process in porous media than the nonlinear diffusion Equation (6), but also introduces a well-log constraint to this inverse problem. The resulted constrained inverse problem can be transformed into a nonlinear constrained optimization problem. Because of the ill-posedness of problem and the local convergence property of traditional methods, we first impose Tikhonov regularization, and then use a widely convergent homotopy method to solve the normal equation of the regularized cost function. The final part of the paper consists of a report of numerical experiments that demonstrates the performance of the method.

2. Discretization

Equations (1)–(4) can be discretized by using the finite-difference scheme as follows:

$$\left\{\begin{array}{c}\frac{u_{i,j}^k-u_{i,j}^{k-1}}{h_t}+\nabla ·(\varphi (u_{i,j}^{k-1}),\phi (u_{i,j}^{k-1}))-\nabla ·({\mathbf{k}}_{i,j}{N}_{i,j}^k\nabla u_{i,j}^k)=f_{i,j}^k,\hfill \\ i=1,\dots ,I-1;j=1,\dots ,J-1;k=1,\dots ,K,\hfill \\ u_{0,j}^k={\chi }_{0,j}^k,j=0,\dots ,J;k=1,\dots ,K,\hfill \\ u_{1,j}^k={\chi }_{1,j}^k,j=0,\dots ,J;k=1,\dots ,K,\hfill \\ u_{i,0}^k={\chi }_{i,0}^k,i=0,\dots ,I;k=1,\dots ,K,\hfill \\ u_{i,1}^k={\chi }_{i,1}^k,i=0,\dots ,I;k=1,\dots ,K,\hfill \\ u_{i,j}^0={\psi }_{i,j},i=0,\dots ,I;j=0,\dots ,J,\hfill \\ u_{{\mathbf{x}}^s}^k={\gamma }_{{\mathbf{x}}^s}^k,s=1,\dots ,S;k=1,\dots ,K,\hfill \end{array}\right.$$

(7)

where

$$\begin{array}{cc}& u_{i,j}^k=u(i{h}_x,j{h}_y,k{h}_t),{\mathbf{k}}_{i,j}=\mathbf{k}(i{h}_x,j{h}_y),f_{i,j}^k=f(i{h}_x,j{h}_y,k{h}_t),\hfill \\ & {\psi }_{i,j}=\psi (i{h}_x,j{h}_y),{\chi }_{i,j}^k=\chi (i{h}_x,j{h}_y,k{h}_t),{\gamma }_{{\mathbf{x}}^s}^k=\gamma ({\mathbf{x}}^s,k{h}_t),\hfill \\ & I=1/h_x,J=1/h_y,K=T/h_t,\hfill \end{array}$$

$h_t$, $h_x$, $h_y$ are, respectively, the time and spatial step lengths. $\nabla ·(\varphi (u_{i,j}^{k-1}),\phi (u_{i,j}^{k-1}))$ and $\nabla ·({\mathbf{k}}_{i,j}{N}_{i,j}^k\nabla u_{i,j}^k)$ are, respectively, the discretizations of convection and diffusion terms, which will be described in the Appendix A and Appendix B.

By Equation (7), this inverse problem can be formulated as the following nonlinear operator equation:

$$R\left(\mathbf{K}\right)=\Gamma ,$$

(8)

where

$$\begin{array}{cc}& {\mathbf{K}}^{\top }=({\mathbf{k}}_{1,1},{\mathbf{k}}_{1,2},\dots ,{\mathbf{k}}_{1,J},{\mathbf{k}}_{2,1},{\mathbf{k}}_{2,2},\dots ,{\mathbf{k}}_{2,J},\dots ,{\mathbf{k}}_{I,1},{\mathbf{k}}_{I,2},\dots ,{\mathbf{k}}_{I,J}),\hfill \\ & {\Gamma }^{\top }=({\gamma }_{{\mathbf{x}}^1}^1,{\gamma }_{{\mathbf{x}}^2}^1,\dots ,{\gamma }_{{\mathbf{x}}^S}^1,{\gamma }_{{\mathbf{x}}^1}^2,{\gamma }_{{\mathbf{x}}^2}^2,\dots ,{\gamma }_{{\mathbf{x}}^S}^2,\dots ,{\gamma }_{{\mathbf{x}}^1}^K,{\gamma }_{{\mathbf{x}}^2}^K,\dots ,{\gamma }_{{\mathbf{x}}^S}^K).\hfill \end{array}$$

The measured data are denoted by ${\overline{\gamma }}_{{\mathbf{x}}^s}^k$, which can form the vector $\overline{\Gamma }$ in the same sequence as $\Gamma$:

$${\overline{\Gamma }}^{\top }=({\overline{\gamma }}_{{\mathbf{x}}^1}^1,{\overline{\gamma }}_{{\mathbf{x}}^2}^1,\dots ,{\overline{\gamma }}_{{\mathbf{x}}^S}^1,{\overline{\gamma }}_{{\mathbf{x}}^1}^2,{\overline{\gamma }}_{{\mathbf{x}}^2}^2,\dots ,{\overline{\gamma }}_{{\mathbf{x}}^S}^2,\dots ,{\overline{\gamma }}_{{\mathbf{x}}^1}^K,{\overline{\gamma }}_{{\mathbf{x}}^2}^K,\dots ,{\overline{\gamma }}_{{\mathbf{x}}^S}^K),$$

and the permeability, known from the well logs of a well located at point $i_0$ in the $x$-direction, is denoted by

$${\overline{\mathbf{K}}}_{i_0}^{\top }=({\overline{\mathbf{k}}}_{i_0,1},{\overline{\mathbf{k}}}_{i_0,2},\dots ,{\overline{\mathbf{k}}}_{i_0,J}).$$

Let Y denote a matrix

$$Y={\left(\begin{array}{cccccccccccc}0& 0& \dots & 0& 1& 0& \dots & 0& 0& 0& \dots & 0\\ 0& 0& \dots & 0& 0& 1& \dots & 0& 0& 0& \dots & 0\\ ⋮& ⋮& & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& & ⋮\\ 0& 0& \dots & 0& 0& 0& \dots & 1& 0& 0& \dots & 0\end{array}\right)}_{J\times (I\times J)},$$

such that $Y\mathbf{K}=({\mathbf{k}}_{i_0,1},{\mathbf{k}}_{i_0,2},\dots ,{\mathbf{k}}_{i_0,J})$, then Equation (8) turns into a nonlinear constrained optimization problem

$$\underset{\mathbf{K}}{min}{\parallel R\left(\mathbf{K}\right)-\overline{\Gamma }\parallel }^2,\mathrm{subject} \mathrm{to} Y\mathbf{K}={\overline{\mathbf{K}}}_{i_0}.$$

(9)

By combining the objective function and constraint into a penalty function, we can attack Equation (9) by solving an unconstrained problem. For instance, a penalty function can be defined as

$$\parallel R\left(\mathbf{K}\right)-\overline{\Gamma }{\parallel }^2+\omega {\parallel Y\mathbf{K}-{\overline{\mathbf{K}}}_{i_0}\parallel }^2,$$

(10)

where $\omega$ is the penalty parameter. After that, the solution of Equation (9) can be obtained by minimizing this unconstrained function Equation (10) for a large value of $\omega$.

3. Identification Method

In the first part of this section, a basic iterative method is given by the successive linearization technique. Then, in the second part, the strategy used for the development of a homotopy method is presented.

3.1. Basic Iterative Method

It is common knowledge that the inverse problem is ill-posed, therefore the regularization technique has to be used for the improvement of numerical stability of the algorithm:

$$\underset{\mathbf{K}}{min}\{\parallel R\left(\mathbf{K}\right)-\overline{\Gamma }{\parallel }^2+\omega \parallel Y\mathbf{K}-{\overline{\mathbf{K}}}_{i_0}{\parallel }^2+{\varpi }_1\parallel W_1{\mathbf{K}\parallel }^2+{\varpi }_2\parallel W_2\mathbf{K}{\parallel }^2\},$$

(11)

where ${\varpi }_1$, ${\varpi }_2$ denote the regularization parameters, $W_1$, $W_2$ denote, respectively, the second-order smooth matrices in the $x$- and $y$-direction.

$$W_1\mathbf{K}=\left[\begin{array}{c}0\\ {\mathbf{k}}_{1,1}-2{\mathbf{k}}_{2,1}+{\mathbf{k}}_{3,1}\\ {\mathbf{k}}_{2,1}-2{\mathbf{k}}_{3,1}+{\mathbf{k}}_{4,1}\\ \cdots \\ {\mathbf{k}}_{I-2,1}-2{\mathbf{k}}_{I-1,1}+{\mathbf{k}}_{I,1}\\ 0\\ 0\\ {\mathbf{k}}_{1,2}-2{\mathbf{k}}_{2,2}+{\mathbf{k}}_{3,2}\\ {\mathbf{k}}_{2,2}-2{\mathbf{k}}_{3,2}+{\mathbf{k}}_{4,2}\\ \cdots \\ {\mathbf{k}}_{I-2,2}-2{\mathbf{k}}_{I-1,2}+{\mathbf{k}}_{I,2}\\ 0\\ \cdots \\ 0\\ {\mathbf{k}}_{1,J}-2{\mathbf{k}}_{2,J}+{\mathbf{k}}_{3,J}\\ {\mathbf{k}}_{2,J}-2{\mathbf{k}}_{3,J}+{\mathbf{k}}_{4,J}\\ \cdots \\ {\mathbf{k}}_{I-2,J}-2{\mathbf{k}}_{I-1,J}+{\mathbf{k}}_{I,J}\\ 0\end{array}\right],W_2\mathbf{K}=\left[\begin{array}{c}0\\ {\mathbf{k}}_{1,1}-2{\mathbf{k}}_{1,2}+{\mathbf{k}}_{1,3}\\ {\mathbf{k}}_{1,2}-2{\mathbf{k}}_{1,3}+{\mathbf{k}}_{1,4}\\ \cdots \\ {\mathbf{k}}_{1,J-2}-2{\mathbf{k}}_{1,J-1}+{\mathbf{k}}_{1,J}\\ 0\\ 0\\ {\mathbf{k}}_{2,1}-2{\mathbf{k}}_{2,2}+{\mathbf{k}}_{2,3}\\ {\mathbf{k}}_{2,2}-2{\mathbf{k}}_{2,3}+{\mathbf{k}}_{2,4}\\ \cdots \\ {\mathbf{k}}_{2,J-2}-2{\mathbf{k}}_{2,J-1}+{\mathbf{k}}_{2,J}\\ 0\\ \cdots \\ 0\\ {\mathbf{k}}_{I,1}-2{\mathbf{k}}_{I,2}+{\mathbf{k}}_{I,3}\\ {\mathbf{k}}_{I,2}-2{\mathbf{k}}_{I,3}+{\mathbf{k}}_{I,4}\\ \cdots \\ {\mathbf{k}}_{I,J-2}-2{\mathbf{k}}_{I,J-1}+{\mathbf{k}}_{I,J}\\ 0\end{array}\right].$$

It is easy to show that Equation (11) is equivalent to (its normal equation):

$$R^{\prime }{\left(\mathbf{K}\right)}^{\top }(R\left(\mathbf{K}\right)-\overline{\Gamma })+\omega Y^{\top }(Y\mathbf{K}-{\overline{\mathbf{K}}}_{i_0})+({\varpi }_1{W}_1^{\top }{W}_1+{\varpi }_2{W}_2^{\top }{W}_2)\mathbf{K}=0.$$

(12)

For the sake of avoiding the influence of second derivative, a successive linearization method can be introduced to construct a basic iterative method.

We first assume that the kth approximation ${\mathbf{K}}^k$ of Equation (12) has been obtained, and use the linear function

$$E_k\left(\mathbf{K}\right)=R^{\prime }\left({\mathbf{K}}^k\right)(\mathbf{K}-{\mathbf{K}}^k)+R\left({\mathbf{K}}^k\right),$$

instead of $R\left(\mathbf{K}\right)$ in Equation (12):

$$\begin{array}{cc}\hfill & R^{\prime }{\left({\mathbf{K}}^k\right)}^{\top }(R^{\prime }\left({\mathbf{K}}^k\right)(\mathbf{K}-{\mathbf{K}}^k)+R\left({\mathbf{K}}^k\right)-\overline{\Gamma })+\omega Y^{\top }(Y\mathbf{K}-{\overline{\mathbf{K}}}_{i_0})\hfill \\ \hfill & +({\varpi }_1{W}_1^{\top }{W}_1+{\varpi }_2{W}_2^{\top }{W}_2)\mathbf{K}=0.\hfill \end{array}$$

(13)

Then, by denoting the solution of Equation (13) as ${\mathbf{K}}^{k+1}$, we obtain the basic iterative scheme:

$$\left\{\begin{array}{c}{\mathbf{K}}^{k+1}={\mathbf{K}}^k+\Delta {\mathbf{K}}^k,k=0,1,2,\dots \hfill \\ [R^{\prime }{\left({\mathbf{K}}^k\right)}^{\top }{R}^{\prime }\left({\mathbf{K}}^k\right)+\omega Y^{\top }Y+{\varpi }_1{W}_1^{\top }{W}_1+{\varpi }_2{W}_2^{\top }{W}_2]\Delta {\mathbf{K}}^k=\hfill \\ -[R^{\prime }{\left({\mathbf{K}}^k\right)}^{\top }(R\left({\mathbf{K}}^k\right)-\overline{\Gamma })+\omega Y^{\top }(Y{\mathbf{K}}^k-{\overline{\mathbf{K}}}_{i_0})\hfill \\ +({\varpi }_1{W}_1^{\top }{W}_1+{\varpi }_2{W}_2^{\top }{W}_2){\mathbf{K}}^k].\hfill \end{array}\right.$$

(14)

This method is fast and stable, but only has local convergence.

3.2. Homotopy Method

In order to expand the domain of convergence, the homotopy method is introduced to solve Equation (12) by taking into account the fixed-point homotopy equation

$$\begin{array}{cc}\hfill & A(\mathbf{K},a)=a[R^{\prime }{\left(\mathbf{K}\right)}^{\top }(R\left(\mathbf{K}\right)-\overline{\Gamma })+\omega Y^{\top }(Y\mathbf{K}-{\overline{\mathbf{K}}}_{i_0})\hfill \\ \hfill & +({\varpi }_1{W}_1^{\top }{W}_1+{\varpi }_2{W}_2^{\top }{W}_2)\mathbf{K}]+(1-a)(\mathbf{K}-{\mathbf{K}}_0)=0,\hfill \end{array}$$

(15)

where $a\in [0,1]$ and ${\mathbf{K}}_0$ are, respectively, the homotopy parameter and arbitrary initial value. From Equation (15), it is easy to see

$$\begin{array}{cc}\hfill & A(\mathbf{K},0)=\mathbf{K}-{\mathbf{K}}_0,\hfill \\ \hfill & A(\mathbf{K},1)=R^{\prime }{\left(\mathbf{K}\right)}^{\top }(R\left(\mathbf{K}\right)-\overline{\Gamma })+\omega Y^{\top }(Y\mathbf{K}-{\overline{\mathbf{K}}}_{i_0})\hfill \\ \hfill & +({\varpi }_1{W}_1^{\top }{W}_1+{\varpi }_2{W}_2^{\top }{W}_2)\mathbf{K}.\hfill \end{array}$$

(16)

As the homotopy parameter a changed continuously from 0 to 1, the trivial problem $A(\mathbf{K},0)=0$ is continuously deformed to the original problem $A(\mathbf{K},1)=0$, that is, $\mathbf{K}$ is changed from ${\mathbf{K}}_0$ to the solution of Equation (12). In topology, this is called deformation.

To achieve the specific numerical algorithm, we first divide the interval $[0,1]$ into $0=a_0<a_1<\cdots <a_B=1$, and then solve $A(\mathbf{K},a_b)=0$$(b=1,2,\dots ,B)$ sequentially by some iterative scheme, whose initial value is chosen as the solution ${\mathbf{K}}_{b-1}$ of the previous equation $A(\mathbf{K},a_{b-1})=0$.

For $A(\mathbf{K},a_b)=0$, in the same manner as the construction of Equation (14), we have

$$\left\{\begin{array}{c}{\mathbf{K}}_b^{k+1}={\mathbf{K}}_b^k+\Delta {\mathbf{K}}_b^k,k=0,1,\dots ,b_m,\hfill \\ [a_b{R}^{\prime }{\left({\mathbf{K}}_b^k\right)}^{\top }{R}^{\prime }\left({\mathbf{K}}_b^k\right)+a_b\omega Y^{\top }Y+a_b{\varpi }_1{W}_1^{\top }{W}_1+a_b{\varpi }_2{W}_2^{\top }{W}_2+(1-a_b)I]\Delta {\mathbf{K}}_b^k=\hfill \\ -[a_b{R}^{\prime }{\left({\mathbf{K}}_b^k\right)}^{\top }(R\left({\mathbf{K}}_b^k\right)-\overline{\Gamma })+a_b\omega Y^{\top }(Y{\mathbf{K}}_b^k-{\overline{\mathbf{K}}}_{i_0})\hfill \\ +(a_b{\varpi }_1{W}_1^{\top }{W}_1+a_b{\varpi }_2{W}_2^{\top }{W}_2){\mathbf{K}}_b^k+(1-a_b)({\mathbf{K}}_b^k-{\mathbf{K}}_0)],\hfill \\ {\mathbf{K}}_b^0={\mathbf{K}}_{b-1},{\mathbf{K}}_b={\mathbf{K}}_b^{b_m+1},b=1,2,\dots ,B,\hfill \end{array}\right.$$

(17)

where I refers to the unit matrix. After ${\mathbf{K}}_B$ is obtained, Equation (14) may be used to make correction. Therefore, Equations (14) and (17) are combined into a stabilized method which has a wider domain of convergence for the constrained permeability identification of the nonlinear convection-diffusion equation.

By choosing $a_b=\frac{b}{B}$ and $b_m=0$, Equation (17) can be simplified as

$$\left\{\begin{array}{c}{\mathbf{K}}_{b+1}={\mathbf{K}}_b+\Delta {\mathbf{K}}_b,b=0,1,\dots ,B-1,\hfill \\ [\frac{b}{B}{R}^{\prime }{\left({\mathbf{K}}_b\right)}^{\top }{R}^{\prime }\left({\mathbf{K}}_b\right)+\frac{b}{B}\omega Y^{\top }Y+\frac{b}{B}{\varpi }_1{W}_1^{\top }{W}_1+\frac{b}{B}{\varpi }_2{W}_2^{\top }{W}_2+(1-\frac{b}{B})I]\Delta {\mathbf{K}}_b=\hfill \\ -[\frac{b}{B}{R}^{\prime }{\left({\mathbf{K}}_b\right)}^{\top }(R\left({\mathbf{K}}_b\right)-\overline{\Gamma })+\frac{b}{B}\omega Y^{\top }(Y{\mathbf{K}}_b-{\overline{\mathbf{K}}}_{i_0})\hfill \\ +(\frac{b}{B}{\varpi }_1{W}_1^{\top }{W}_1+\frac{b}{B}{\varpi }_2{W}_2^{\top }{W}_2){\mathbf{K}}_b+(1-\frac{b}{B})({\mathbf{K}}_b-{\mathbf{K}}_0)].\hfill \end{array}\right.$$

(18)

To test the necessity of introduction of constraints, we also give the homotopy method for the ordinary permeability identification for the nonlinear convection-diffusion equation in the multiphase porous media flow, and compare Equations (14) and (18) with it. Specifically, let $\omega =0$, then Equations (14) and (18) turn into the homotopy method for the ordinary permeability identification problem

$$\left\{\begin{array}{c}{\mathbf{K}}_{b+1}={\mathbf{K}}_b+\Delta {\mathbf{K}}_b,b=0,1,\dots ,B-1,\hfill \\ [\frac{b}{B}{R}^{\prime }{\left({\mathbf{K}}_b\right)}^{\top }{R}^{\prime }\left({\mathbf{K}}_b\right)+\frac{b}{B}{\varpi }_1{W}_1^{\top }{W}_1+\frac{b}{B}{\varpi }_2{W}_2^{\top }{W}_2+(1-\frac{b}{B})I]\Delta {\mathbf{K}}_b=\hfill \\ -[\frac{b}{B}{R}^{\prime }{\left({\mathbf{K}}_b\right)}^{\top }(R\left({\mathbf{K}}_b\right)-\overline{\Gamma })+(\frac{b}{B}{\varpi }_1{W}_1^{\top }{W}_1+\frac{b}{B}{\varpi }_2{W}_2^{\top }{W}_2){\mathbf{K}}_b\hfill \\ +(1-\frac{b}{B})({\mathbf{K}}_b-{\mathbf{K}}_0)],\hfill \end{array}\right.$$

(19)

and

$$\left\{\begin{array}{c}{\mathbf{K}}^{k+1}={\mathbf{K}}^k+\Delta {\mathbf{K}}^k,k=0,1,2,\dots \hfill \\ [R^{\prime }{\left({\mathbf{K}}^k\right)}^{\top }{R}^{\prime }\left({\mathbf{K}}^k\right)+{\varpi }_1{W}_1^{\top }{W}_1+{\varpi }_2{W}_2^{\top }{W}_2]\Delta {\mathbf{K}}^k=\hfill \\ -[R^{\prime }{\left({\mathbf{K}}^k\right)}^{\top }(R\left({\mathbf{K}}^k\right)-\overline{\Gamma })+({\varpi }_1{W}_1^{\top }{W}_1+{\varpi }_2{W}_2^{\top }{W}_2){\mathbf{K}}^k].\hfill \end{array}\right.$$

(20)

4. Numerical Experiments

This section tested two synthetic examples to show the performance of our proposed method. The parameters required in the identification process are given by

$$\begin{array}{cc}& \varphi \left(u\right)=\frac{u^2(1-5(1-u^2))}{u^2+{(1-u)}^2},\phi \left(u\right)=\frac{u^2}{u^2+{(1-u)}^2},N\left(u\right)=u^2-u+1,\hfill \\ & \psi \left(\mathbf{x}\right)=sin\left(\pi x\right)sin\left(\pi y\right),\chi (\mathbf{x},t)=0,\omega ={10}^4,{\varpi }_1={\varpi }_2={10}^{-5},\hfill \\ & {\mathbf{K}}_0\equiv 5,h_x=h_y=\frac{1}{24},i_0=\frac{12}{24},T=0.06,h_t=0.002,B=5.\hfill \end{array}$$

Example 1.

The exact permeability $\mathbf{k}$ in this example is shown in Figure 2. In order to illustrate the noise sensitivity, we add Gaussian noise to the measured data $\overline{\Gamma }$. With 5%, 10%, 15%, and 20% Gaussian noises, the identified permeability results by the homotopy method with constraints are shown in Figure 3, Figure 4, Figure 5 and Figure 6, respectively.

Example 2.

This example selects a model of two anomalous bodies in a homogeneous medium with a permeability of 4.25, and the anomalous bodies have the permeability of 6.18 and 7.55, see Figure 7. Figure 8, Figure 9, Figure 10 and Figure 11, respectively, show the identified permeability results of the homotopy method with constraints, with 5%, 10%, 15%, and 20% Gaussian noises.

To test the necessity of introduction of constraints and the wide convergence of the homotopy strategy, the homotopy method without constraints and basic iterative method with constraints are used for the above measured data, and the relative errors of all the identified permeability results by the three methods are listed in

Table 2. Relative errors of identified permeability $\mathbf{k}$ by three different methods. HM—homotopy method with constraints, HM—homotopy method without constraints, BIMC—basic iterative method with constraints.

Example Number	Noise Level	HMC	HM	BIMC
4.1	5%	6.22%	6.90%	×
	10%	6.38%	7.40%	×
	15%	7.13%	×	×
	20%	7.44%	×	×
4.2	5%	5.71%	6.47%	×
	10%	5.81%	7.39%	×
	15%	6.06%	×	×
	20%	7.19%	×	×

, where × means that there is no convergent result.

Table 2 shows that

(1)

The stability of the homotopy method with constraints is better than the homotopy method without constraints;

(2)

The region of convergence of the homotopy strategy is wider than the basic iterative method with constraints;

(3)

The homotopy method with constraints has wide convergence region and strong anti-noise ability.

5. Conclusions

To restrain the noise and improve identified model quality, the constraint condition is introduced to the inverse problem of the nonlinear convection-diffusion equation in the multiphase porous media flow. Then, Tikhonov regularization is applied to the constrained optimization problem obtained by the discretization to overcome the ill-posed property. We developed a homotopy method that is widely convergent. The results from the numerical experiments show that this method is feasible and effective. Compared with the homotopy method without constraints, our approach has better stability; compared with the basic iterative method with constraints, our approach has a wider convergence region. So far, there is no literature on the use of fractional operators for constraint inverse problems, which will be an interesting future work.