Identifying Space-Dependent Coefficients and the Order of Fractionality in Fractional Advection–Diffusion Equation

Abstract

Tracer tests in natural porous media sometimes show abnormalities that suggest considering a fractional variant of the advection–diffusion equation supplemented by a time derivative of non-integer order. We are describing an inverse method for this equation: It finds the order of the fractional derivative and the coefficients that achieve minimum discrepancy between solution and tracer data. Using an adjoint equation divides the computational effort by an amount proportional to the number of freedom degrees, which becomes large when some coefficients depend on space. Method accuracy is checked on synthetical data, and applicability to actual tracer test is demonstrated.

Appendices

Appendix 1: A Discrete Version of the Fractional MIM

Standard approximations to fractional integrals and derivatives define the discrete problem (6) whose solutions approximate those of (1).

1.1 Approximating Temporal Integrals and Derivatives

In $p_2\partial _t+p_3\partial _tI_{0,+}^{1-\alpha }$ and in (2), the Riemann–Liouville integral $I_{0,+}^{1-\alpha }$ accounts for past history since time $t=0$. An approximation of order $O(\Delta t^{2})$ is (Diethelm et al. 2005)

$$\begin{aligned} I_{0,+}^{1-\alpha }y(k\Delta t)\approx & {} \sum _{j=0}^{k}{\mathcal I}^{j, k}y^{k-j}\quad \text {with} \quad {\mathcal I}^{0, k}= {\mathcal I}^0\equiv \frac{\Delta t^{1-\alpha }}{\varGamma (3-\alpha )},\nonumber \\ {\mathcal I}^{j, k}= & {} {\mathcal I}^{0}\left[ (j+1)^{2-\alpha }-2j^{2-\alpha }+(j-1)^{2-\alpha }\right] \quad \hbox {for}\,\,0<j<k, \nonumber \\ {\mathcal I}^{0,0}= & {} 0\quad \text {and }\quad {\mathcal I}^{k, k}={\mathcal I}^{0}\left[ (2-\alpha )k^{1-\alpha }-k^{2-\alpha }+(k-1)^{2-\alpha }\right] \quad for \,\, 0<k.\nonumber \\ \end{aligned}$$

(15)

Combining (15) with the standard backward finite difference approximation $\frac{y^{k}-y^{k-1}}{\Delta t}$ to the first-order derivative yields

$$\begin{aligned} \left( p_2\partial _t+p_3\partial _tI_{0,+}^{1-\alpha }\right) y(k\Delta t)= \frac{1}{\Delta t}\sum _{j=0}^kW_s(j,k)y((k-j)\Delta t)+O(\Delta t), \end{aligned}$$

(16)

with

$$\begin{aligned} W_s(0,k)= & {} p_{2}+p_{3}{\mathcal I}^0 \quad \text { and } \nonumber \\ W_s(j,k)= & {} p_{3} \left[ {\mathcal I}^{j, k}-{\mathcal I}^{j-1, k-1}\right] +p_{2}\delta _{j,1}\quad for \,\, 1\le j\le k. \end{aligned}$$

(17)

1.2 Approximating Spatial Derivatives and Boundary Conditions

At each time step $k>0$ and for each $s\in \{1,\ldots ,N_{Sp}\}$, standard central finite differences

$$\begin{aligned} \frac{p_1}{\Delta x^{2}}\left[ -u_{s+1}^{k}+2u_{s}^{k}-u_{s-1}^{k}\right] +\frac{V}{2\Delta x}\left[ u_{s+1}^{k}-u_{s-1}^{k}\right] , \end{aligned}$$

(18)

approximate $-\partial _{x}(\partial _{x}p_1u-Vu)|_{s\Delta x}^{k\Delta t}$ at order $O(\Delta x^2)$. Applying non-centered finite differences to the boundary conditions in (4) links $u_{0}^{k}$ and $u_{N_{Sp}+1}^{k}$ to immediately neighboring $u_{s}^{k}$, but is accurate at first order only:

$$\begin{aligned} {\Delta xV{+}p_1}u_{0}^{k}= {p_1u_{1}^{k}}+{C_0V}{\Delta xV{+}p_1}H(t_\mathrm{f}-k\Delta t){+}O(\Delta x),\quad u_{N_{Sp}{+}1}^{k}=u_{N_{Sp}}^{k}{+}O(\Delta x).\nonumber \\ \end{aligned}$$

(19)

Hence, Eqs. (16) and (18) yield approximations to

$$\begin{aligned} -\partial _{x}\left( \partial _{x}p_1u-Vu\right) |_{s\Delta x}^{k\Delta t}+\left[ \left( p_2\partial _t+p_3\partial _tI_{0,+}^{1-\alpha }\right) u\right] |_{s\Delta x}^{k\Delta t} \end{aligned}$$

at order $O(\Delta x)+O(\Delta t)$ for each $s\in \{1,\ldots ,N_{Sp}\}$ representing an interior point of $[0,\ell ]$ and each index $k>0$. Equating them to zero and eliminating $u_{0}^{k}$ and $u_{N_{Sp}+1}^{k}$ with the help of (19) yields a system of equations determining the $u_s^k$ that approximate the $u(s\Delta x, k\Delta t)$ at interior points of $[0,\ell ]$. It accounts for solute injection at column inlet. Remembering the initial condition, we set these equations in the compact form (6) reproduced below

$$\begin{aligned} A(\mathbf q ,\mathbf u )=\mathbf r (\mathbf q ), \quad \mathbf u \in {\mathcal D}( A) \end{aligned}$$

by defining linear mapping A and array $\mathbf r$. Each array $\mathbf u$ of ${\mathcal D}(A)\equiv \{{\mathbf {u}}\in X{/}{\mathbf {u}}^{0}=0\}$ satisfies the initial condition included in (4) and A maps it onto array $A({\mathbf {q}}, \mathbf{u })$ whose each rank k column $(A({\mathbf {q}}, \mathbf{u }))^{k}$ is

$$\begin{aligned} (A({\mathbf {q}}, \mathbf{u }))^{k}\equiv \mathbf{G }{} \mathbf u ^{k}+\sum _{j=1}^{k} \mathbf W (j,k) \mathbf u ^{k-j}, \end{aligned}$$

(20)

where $\mathbf u ^k=(u_{1}^{k},\ldots ,u_{N_{Sp}}^{k})^\dag$ represents the rank k column of $\mathbf u$. Matrices $\mathbf{G }$ and $\mathbf W (j,k)$ are defined at the end of the section, and array $\mathbf r$ recollects the $r_s^k$ defined by

$$\begin{aligned} r_{1}^{k}=\Delta t\frac{C_{0}V}{2\Delta x}\frac{2p_1+V\Delta x}{V\Delta x+p_1}H(k\Delta t)H(t_{f}-k\Delta t),\quad r_{s}^{k}=0\quad \text {for}\,\, s>1\quad \text {or}\,\, k=0. \end{aligned}$$

(21)

Each rank k column $(r_{1}^{k},\ldots ,r_{N_{Sp}}^{k})^\dag$ of $\mathbf r$ being noted $\mathbf r ^k$, (6) is equivalent to the system of equations

$$\begin{aligned} (A({\mathbf {q}}, \mathbf{u }))^{k}=\mathbf r ^{k}\quad \text { for}\quad 0\le k\le N_T. \end{aligned}$$

(22)

With $\mu =\frac{p_1\Delta t}{\Delta x^{2}}$ and $\nu =\frac{V\Delta t}{2\Delta x}$, the entries $g_{s,s^{\prime }}$ of $\mathbf G$ are

$$\begin{aligned} g_{s,s^{\prime }}= & {} 0\quad for \,\, |s-s^{\prime }|>1,\quad g_{s,s-1}=-\nu -\mu \quad for \,\, s>1,\nonumber \\ g_{s,s+1}= & {} \nu -\mu \quad for \,\, s<N_{Sp}, \quad g_{s,s}= p_{2}(s\Delta x)+p_{3}(s\Delta x) {\mathcal I}^0+\varOmega _s, \end{aligned}$$

(23)

where all interior diagonal entries exhibit the same $\varOmega _s=2\mu$ (for $s\ne 1$ and $s\ne N_{Sp}$), while boundary conditions (19) result into $\varOmega _1=\mu (1+\frac{1}{2+\mu {/}\nu })$ and $\varOmega _{N_{Sp}}=\mu +\nu$ at both ends. We see that matrix $\mathbf G$ depends on $\mathbf q$, $\Delta x$ and $\Delta t$. The $\mathbf W (j,k)$ are $N_{Sp}\times N_{Sp}$ diagonal matrices of entries $(\mathbf W (j, k))_{s,s}=W_s(j,k)$. In (20), each $\mathbf W (j,k)$ operates on columns of rank smaller than k stored in $\mathbf u$.

Using (20) will help us seeing that problem (6) has exactly one solution in ${\mathcal D}(A)$ provided these parameters make matrix $\mathbf G$ invertible.

1.3 Well-Posedness of (6)

If $\mathbf G$ is invertible, we easily determine the single solution ${\mathbf {u}}_{\mathbf {q}}$ of (22) in ${\mathcal D}(A)$ by setting $\mathbf{u }^{0}=0$ and recursively solving Eq. (22), increasing k from 1 to $N_T$. Gershgorin circle theorem (Gerschgorin 1931) gives us a sufficient condition for matrix $\mathbf G$ invertibility in the form of

$$\begin{aligned} \left| g_{s,s}\right| >\left| g_{s-1,s}\right| (1-\delta _{s,1})+\left| g_{s+1,s}\right| \left( 1-\delta _{s,N_{Sp}}\right) , \end{aligned}$$

(24)

$\delta _{i,j}$ being Kronecker index. This condition is satisfied when $\mathbf{q }$ belongs to any closed convex set ${Q}_\varepsilon = {R}_+\times [\frac{V\Delta t}{\Delta x}+\varepsilon ,+\infty [^{n+1}\times {R}_+^{n+1}\times [0,1]$ with $\varepsilon >0$ arbitrarily small.

By implicit function theorem (Jittorntrum 1978), the mapping $\mathbf q \mapsto \mathbf u _{{\mathbf {q}}}$ is of ${\mathcal C}^\infty$ class in $Q_\varepsilon$, matrices $\mathbf G$ and $\mathbf W (j,k)$ having elements of class ${\mathcal C}^\infty$ in ${Q}_\varepsilon$ with respect to the components of $\mathbf{q }$.

1.4 Numerical Scheme Accuracy

We validate the above scheme and fix $\Delta x$ and $\Delta t$ by considering a ${\mathcal R}\ne 0$ so that the continuous problem (1 and 4) with $C_0=0$ has an exact solution which we compare with the still noted $u_s^k$ issues of the above scheme. We set

$$\begin{aligned} \varphi _0(x)=e^{\frac{Vx}{2p_1}}\left( \sin {\sigma x}+\frac{2p_1\sigma }{V}\cos {\sigma x}\right) \end{aligned}$$

and take $\sigma$ defined by

$$\begin{aligned} \frac{2p_1}{V}\sigma ^{2}\sin {(\sigma L)}-2\sigma \cos {(\sigma L)}-\frac{V}{2p_1}\sin {(\sigma L)}=0{:} \end{aligned}$$

$u(x,t)\equiv t\varphi _0(x)$ solves (1 and 4) provided we set

$$\begin{aligned} {\mathcal R}(x,t)=\varphi _0(x)\left[ p_2(x)+p_3(x)\frac{t^{1-\alpha }}{\varGamma (2-\alpha )}+Mt\right] ,\quad M=p_1\sigma ^{2}+\frac{V^{2}}{4p_1}. \end{aligned}$$

Steps $\Delta x$ and $\Delta t$ must satisfy $p_{2}(s\Delta x)> \frac{V\Delta t}{\Delta x}$ to ensure $\mathbf q \in Q_\varepsilon$. For instance, with $L=1$, $p_1=0.05$, $V=1$, $\alpha =0.8$, $p_2(x)=0.5+0.3\sin {(4\pi x})$ and $p_3(x)=0.5-0.4\sin {(4\pi x)}$, taking $N_{Sp}=360$ and $\Delta t=\Delta x/10$ results into relative errors $|\frac{u_s^k-u(s\Delta x,k\Delta t)}{u(s\Delta x,k\Delta t)} |$ smaller than $2\times 10^{-4}$, for $t<1$.

This gives us an idea of which grid we can use. A posteriori checks are applied, especially when parameter identification returns large Péclet numbers. In this case, the centered finite difference approximation used in “Approximating Spatial Derivatives and Boundary Conditions” section for the advection term $V\partial _xu$ may be flawed by numerical diffusion. Hence, we compare with a modified version of the discrete direct problem (6) in which $\frac{V}{6\Delta x}(3u_s^n+u_{s-2}^n-6u_{s-1}^n+2u_{s+1}^n)$ approximates $V\partial _xu$. This is Eq. (4.9) of Cada and Torrilhon (2009) which corresponds to a flux limiter given by the first line of (3.8) in this reference, in view of the small $\partial ^2_{x^2}u$ which we observe. Other a posteriori checks are comparisons with random walks whose probability density function is $p_2u$, as in Ouloin et al. (2013).

Appendix 2: Discrete Adjoint State for Discrete Direct Problem (6)

The linear operator A is described by (20), which implies

$$\begin{aligned} \left\langle A(\mathbf q ,\mathbf u )\cdot \mathbf w \right\rangle _{X}=\sum _{k=1}^{N_{T}} \left\langle \mathbf G {} \mathbf u ^{k}\cdot \mathbf w ^{k}\right\rangle _{ {R}^{N_{Sp}}}+ \sum _{k=1}^{N_{T}}\sum _{j=1}^{k} \left\langle \mathbf W (j,k) \mathbf u ^{k-j}\cdot \mathbf w ^{k}\right\rangle _{ {R}^{N_{Sp}}}, \end{aligned}$$

(25)

for each $(\mathbf u ,\mathbf w )$ in ${\mathcal D}(A)\times X$. Matrices $\mathbf G$ and $\mathbf W (j,k)$ are defined in “Approximating Spatial Derivatives and Boundary Conditions” section, and each $\langle \mathbf G {} \mathbf u ^{k}\cdot \mathbf w ^{k}\rangle _{ {R}^{N_{Sp}}}$ is equal to $\langle \mathbf u ^{k}\cdot \mathbf G ^{\dagger } \mathbf w ^{k}\rangle _{ {R}^{N_{Sp}}}$, superscript ${\dagger }$ denoting transpose. Re-arranging the double sum in the right-hand side of (25) proves that the adjoint $A^{*}$ of A is the operator that transforms each array $\mathbf w$ of X into array $A^{*}(\mathbf q ,\mathbf w )$ whose each column of rank k is

$$\begin{aligned} \left( A^{*}(\mathbf q ,\mathbf w )\right) ^k=\mathbf G ^{\dagger }{} \mathbf w ^{k}+ \sum _{j=1}^{N_{T}-k}{} \mathbf W (j,k+j) \mathbf w ^{k+j}\quad {\text{ for }}\,\,0\le k\le N_T. \end{aligned}$$

(26)

This implies that Eq. (13) is equivalent to the set of all equations

$$\begin{aligned} \mathbf G ^{\dagger }\varvec{\psi }^{k}+\sum _{j=1}^{N_{T}-k}{} \mathbf W (j, k+j)\varvec{\psi }^{k+j}=-\sum _{{\mathcal M}^{(d)}}\left( \frac{\partial f_{{\bar{s}}}^{{\bar{k}}}}{\partial {\mathbf {u}}} \right) ^{k} (\mathbf q ,\mathbf u _{{\mathbf {q}}}) \end{aligned}$$

(27)

obtained for $0\le k\le N_T$. When $\mathbf G$ is invertible it is the same for $\mathbf G ^{\dagger }$; hence, the system of all these equations has exactly one solution in because the final simulation time T is such that $\max \{{\bar{k}} / ({\bar{k}},{\bar{s}})\in {\mathcal M}^{(d)}\}<N_{T}$ which implies $(\frac{\partial f_{{\bar{s}}}^{{\bar{k}}}}{\partial \mathbf u } )^{N_{T}}(\mathbf q , \mathbf u )=0$, hence $\varvec{\psi }^{N_T}=0$ because $\mathbf G$ is invertible. Then, for each $k<N_T$ Eq. (27) is of the form of

$$\begin{aligned} \mathbf G ^{\dagger }\varvec{\psi }^{k}={\mathcal F}\left( \theta ,\mathbf q ,\mathbf u _{{\mathbf {q}}},\mathbf C ,\varvec{\psi }^{k+1},\ldots ,\varvec{\psi }^{N_T}\right) \end{aligned}$$

and has exactly one solution in ${R}^{N_{Sp}}$ entirely determined by $\varvec{\psi }^{k+1},\ldots ,\varvec{\psi }^{N_T}$, $\mathbf q$, $\theta$ and $\mathbf C$. All these $\varvec{\psi }^{k}$ form the unique solution of Eq. (13) in ${\mathcal D}(A^*)$, which we call $\varvec{\psi }_\mathbf{q }$.

Appendix 3: The Derivatives of E w.r.t. the Parameters

The adjoint problem (13) depends on the differential of f w.r.t. $\mathbf u$, and determining the gradient of E w.r.t. $\mathbf q$ also needs the derivatives of f, A, and $\mathbf r$ w.r.t. the entries $q_h$ of $\mathbf q$. Though standard algebra returns these derivatives, two points are worth being mentioned.

First, for $j=2,3$, the $\partial E{/}\partial p_{j}^{(i)}$ are obtained by applying chain rule to (14) and (5).

Then, the derivative w.r.t. $\alpha$ involves $\frac{\partial {\mathcal I}^0}{\partial \alpha }$, $\frac{\partial ({\mathcal I}^{j-1,k-1}-{\mathcal I}^{j-1,k-1})}{\partial \alpha }$ and $\frac{\partial f_{{\bar{s}}}^{{\bar{k}}}}{\partial \alpha }$ for which we need the digamma function $\frac{\varGamma ^{\prime }}{\varGamma }$ (Abramowitz and Stegun 1972)

$$\begin{aligned} \frac{\varGamma ^{\prime }}{\varGamma }(z+1)=\gamma \sum _{k=1}^{\infty }\frac{z}{k(z+k)}, \end{aligned}$$

(28)

where $\gamma$ is Euler constant. We obtain

$$\begin{aligned} \frac{ \partial {\mathcal I}^{0}}{\partial \alpha }={\mathcal I}^{0} \left[ -\ln (\Delta t)+\frac{\varGamma ^{\prime }}{\varGamma }(3-\alpha ) \right] , \end{aligned}$$

and

$$\begin{aligned} \frac{\partial {\mathcal I}^{j, k}}{\partial \alpha }= & {} {\mathcal I}^{0} \left[ (j+1)^{2-\alpha }-2j^{2-\alpha }+(j-1)^{2-\alpha } \right] \left( -\ln (\Delta t)+ \frac{\varGamma ^{\prime }}{\varGamma }(3-\alpha ) \right) \\&- \left[ \ln (j+1)(j+1)^{2-\alpha }-2\ln (j)j^{2-\alpha }+ \ln (j-1)(j-1)^{2-\alpha } \right] \quad \text{ for }\,\, 1<j<k,\\ \frac{\partial {\mathcal I}^{k, k}}{\partial \alpha }= & {} {\mathcal I}^{0} \left[ (2-\alpha )k^{1-\alpha }-k^{2-\alpha }+(k-1)^{2-\alpha } \right] \left( -\ln (\Delta t)+ \frac{\varGamma ^{\prime }}{\varGamma }(3-\alpha ) \right) \\&- \left[ k^{1-\alpha }+(2-\alpha )\ln (k)k^{1-\alpha }-\ln (k)k^{2-\alpha }+ \ln (k-1)(k-1)^{2-\alpha } \right] . \end{aligned}$$