Solution Construction to a Class of Riemann Problems of Multiphase Flow in Porous Media

Abstract

Fluid displacement in porous media can usually be formulated as a Riemann problem. Finding the solution to such a problem helps shed light on the dynamics of flow and consequently optimize operational parameters such as injected fluid composition. We developed an algorithm to find solutions to a class of Riemann problems of multiphase flow in porous media. In general, the solution to a Riemann problem in state space is a curve connecting the left and right states of the problem. The solution curves studied here are composed of classical wave curves. For a given Riemann problem, our procedure to find the solution consists of three steps: (1) guess the initial lengths of the solution’s constitutive wave curves; (2) construct each wave curve off the last state of its antecedent wave curve; and (3) iterate over the lengths of the constitutive wave curves, using an iterative solver, until the solution curve ends at the right state of the problem. We used benchmark cases from literature to verify the accuracy of the developed algorithm. Using the developed algorithm, we found solutions to some challenging cases where otherwise numerical simulators would be needed to find the type of the involved waves (i.e., rarefaction, shock or composite waves) and the coordinates of the middle states in the state space. Saturation profiles, oil cut and oil recovery for all the studied cases were computed. This information will assist us to: gain insight about the dynamics of flow, interpret core flooding measurements, assess the accuracy of developed models for foam physical properties, and verify the results of numerical simulators.

Appendices

Appendix A: Umbilic Point

The $2\times 2$ system of conservation equations ${\mathbf {u}}_t + {\mathbf {f(u)}}_x = \mathbf {0}$ can be represented in the form of the quasilinear system ${\mathbf {u}}_t + {\mathbf {A(u)}}{\mathbf {u}}_x = \mathbf {0}$, where ${\mathbf {A(u)}}$ is the Jacobian of the flux functions ${\mathbf {f(u)}}$. Let present ${\mathbf {A(u)}}$ in the following general form

$$\begin{aligned} {\mathbf {A(u)}} = \left[ \begin{array}{cc} a &{}\quad b \\ c &{}\quad d \\ \end{array} \right] . \end{aligned}$$

(A.1)

The eigenvalues of the system are

$$\begin{aligned} \lambda _{1,2} = \frac{(a+d)\pm \sqrt{(a-d)^2+4bc}}{2}. \end{aligned}$$

(A.2)

At point ${\mathbf {u}}$ in the state space depending on the sign of the discriminant of Eq. (A.2), the following cases may occur:

• 1. The discriminant is larger than zero and the system has two distinct eigenvalues. In this case, there are two linearly independent eigenvalues which serve as the characteristic directions at that point.

• 2. The discriminant is less than zero and the system has no real eigenvalues. There is no characteristic direction at that point.

• 3. The discriminant is zero and there are two real coincident eigenvalues. In this case, either, there is only one linearly independent eigenvector, or, there are infinite pairs of eigenvectors which are linearly independent. Below we investigate this case in more detail.

At a given point ${\mathbf {u}}$ in the state space, the discriminant of Eq. (A.2) may be zero under one of the following conditions:

• $bc<0$: In this case the matrix ${\mathbf {A(u)}}$ is defective and has only one linearly independent eigenvector, $\mathbf {r}=[1, (d-a)/2b]^T$.

• $b=0,c\ne 0, a=d$: In this case the matrix ${\mathbf {A(u)}}$ is defective and has only one linearly independent eigenvector, $\mathbf {r}=[0, 1]^T$.

• $b\ne 0,c=0, a=d$: In this case the matrix ${\mathbf {A(u)}}$ is defective and the only linearly independent eigenvector is $\mathbf {r}=[1, 0]^T$.

• $b=0, c=0, a=d$: In this case the matrix ${\mathbf {A(u)}}$ is diagonalizable and there are infinite pairs of linearly independent eigenvectors. The matrix ${\mathbf {A(u)}}$ at such a point is a multiple of the identity matrix.

An umbilic point is a point at which two characteristic speeds (i.e., eigenvalues) are coincident. In the above, we showed that at an umbilic point the general Jacobian matrix ${\mathbf {A(u)}}$ can be either defective or diagonalizable. When the matrix is diagonalizable at an umbilic point, the three conditions $b=0$, $c=0$, and $a=d$ have to be satisfied. The umbilic points that occur in $2\times 2$ systems of conservation laws—in particular, three-phase flow problems—has been studied extensively [see Schaeffer and Shearer (1987); Mederios (1992)]. For example, it is well-known [see Schaeffer and Shearer (1987); Holden (1990)] that when the relative mobility of each phase is a function of only its own saturation, there is only one umbilic point ${\mathbf {u}}^*$ in the state space at which:

$$\begin{aligned} \lambda _{rw}^{'}({\mathbf {u}}^*)=\lambda _{ro}^{'}({\mathbf {u}}^*)=\lambda _{rg}^{'}({\mathbf {u}}^*). \end{aligned}$$

(A.3)

The relative mobility of water, oil and gas phases are denoted via $\lambda _{rw}$, $\lambda _{ro}$, and $\lambda _{rg}$, respectively. Under the same assumption, condition (A.3) on the derivative of relative mobilities can be obtained from applying three conditions $b=0, c=0,{ \mathrm {and}} \,a=d$ to the Jacobian matrix of the three-phase flow system of conservation equations.

Appendix B: Rarefaction Curve Parametrization

Rarefaction curves or composition paths are families of curves in the state space which satisfy the eigenvalue problem given in (3). For a given set of conservation equations ${\mathbf {A(u)}}$ and its eigenpairs $(\lambda ({\mathbf {u}}), \mathbf {r(u)})$ are known. Setting $\xi$ equal to $\lambda ({\mathbf {u}}(\xi ))$ and taking derivative of both sides with respect to $\xi$ results in

$$\begin{aligned} \nabla \lambda .{\mathbf {u}}_\xi = 1. \end{aligned}$$

(B.1)

As both ${\mathbf {u}}_\xi$ and $\mathbf {r(u)}$ are eigenvectors of the Jacobina matrix ${\mathbf {A(u)}}$, we will have

$$\begin{aligned} \frac{{\mathbf {u}}_\xi }{||{\mathbf {u}}_\xi ||} = \frac{\mathbf {r}}{||\mathbf {r}||}. \end{aligned}$$

(B.2)

Multiplying both sides of the equality given in (B.2) by $\nabla \lambda$ and using equality (B.1), we will have

$$\begin{aligned} \frac{||\mathbf {r}||}{||{\mathbf {u}}_\xi ||}=\nabla \lambda .\mathbf {r} \end{aligned}$$

(B.3)

From equalities (B.2) and (B.3), we obtain the following set of ODE’s for the rarefaction curves associated with eigenpair $(\lambda ({\mathbf {u}}), \mathbf {r(u)})$.

$$\begin{aligned} {\mathbf {u}}_\xi = \frac{\mathbf {r(u)}}{\nabla \lambda \mathbf {(u)}.\mathbf {r(u)}} \end{aligned}$$

(B.4)

Using the chain rule, we can parametrize equality (B.2) with respect to arc-length s and obtain the following system of ODE’s for the family of rarefaction curves associated with eigenvector $\mathbf {r(u)}$.

$$\begin{aligned} \frac{{\mathbf {u}}_s}{||{\mathbf {u}}_s||} = \frac{\mathbf {r}}{||\mathbf {r}||}. \end{aligned}$$

(B.5)

Note that ${\mathbf {u}}_s$ is the unit tangent vector to the rarefaction curve. Hence, we will have

$$\begin{aligned} {\mathbf {u}}_s = \frac{\mathbf {r}}{||\mathbf {r}||}. \end{aligned}$$

(B.6)

Appendix C: Coherence Condition

Here we obtain the coherence condition from eigenvalue problem (3). From the k-th row of system (3), we have

$$\begin{aligned} \frac{\partial f_k}{\partial u_1}\frac{{\mathrm {d}}u_1}{{\mathrm {d}}\xi } + \cdots + \frac{\partial f_k}{\partial u_N}\frac{{\mathrm {d}}u_N}{{\mathrm {d}}\xi } = \xi \frac{{\mathrm {d}}u_k}{{\mathrm {d}}\xi }. \end{aligned}$$

(C.1)

The left hand side of the above equation is equal to the total derivative of k-th flux function $f_k({\mathbf {u}})$ with respect to $\xi$. Hence, it follows

$$\begin{aligned} \frac{{\mathrm {d}}f_k}{{\mathrm {d}}\xi } = \xi \frac{{\mathrm {d}}u_k}{{\mathrm {d}}\xi }. \end{aligned}$$

(C.2)

Equation (C.2) can be further simplified to the following equation

$$\begin{aligned} \frac{{\mathrm {d}}f_k}{{\mathrm {d}}u_k} = \xi . \end{aligned}$$

(C.3)

The above equality is valid for any rows of the system (i.e., any k). Hence, it follows

$$\begin{aligned} \frac{{\mathrm {d}}f_i}{{\mathrm {d}}u_i} = \xi , \quad i=1,\ldots ,N. \end{aligned}$$

(C.4)

The system of equations given in (C.4) is known as coherence condition and its solution represents the family of curves associated with eigenvalue $\xi$.