Comprehensive Solution for Transient Flow in Heterogeneous Porous Media

Abstract

Solutions of the hydraulic diffusivity equation are of utmost importance for many reservoir engineering problems. Despite all the efforts, there is still a need for the development of rigorous and comprehensive solutions for transient flow problems in heterogeneous oil reservoirs. This study demonstrates the use of an integral transform approach to obtain such a rigorous and comprehensive solution for the hydraulic diffusivity equation in heterogeneous porous domain. The reservoir heterogeneities can be approximated by any continuous differentiable function. The presented general solution and its derivation are valid for multi-dimensional problems in any orthogonal coordinate system. It has the advantage of rigorously solving the hydraulic diffusivity equation for transient, late-transient and steady-state (or pseudo-steady-state) flow regimes in a single formulation that allows the consideration of variable flowrates. In this work, applications of the general solution for one-dimensional problems in the Cartesian and radial coordinate systems are presented, showing comparisons of the results obtained with a finite difference numerical scheme. The solution presented can be used to analyze buildup, drawdown and interference test data, making it a useful tool for pressure transient analysis applied to reservoir engineering problems.

Appendices

Appendix 1

The eigenvalues, eigenfunctions and norms of the auxiliary eigenvalue problem used to solve the problem given by Eqs. (13a) through (13d) are:

$$\begin{aligned} \varOmega _{n\left( x \right) }= & {} \cos \left( {\lambda _n x} \right) \end{aligned}$$

(18)

$$\begin{aligned} \lambda _n= & {} \frac{2\pi \left( {n-1} \right) }{2L_x },\quad n=1,2,3\ldots \end{aligned}$$

(19)

$$\begin{aligned} N_{\varOmega _n }= & {} L_x /2 \end{aligned}$$

(20)

Appendix 2

The eigenvalues, eigenfunctions and norms of the auxiliary eigenvalue problem used to solve the problem given by Eqs. 16a–16d are:

$$\begin{aligned} \varOmega _{n\left( r \right) }= & {} J_0 \left( {\lambda _n r} \right) \end{aligned}$$

(21)

$$\begin{aligned} \lambda _n= & {} \hbox {non\,negative\,roots\,of}\left( {J_0 \left( {\lambda _n r_e } \right) =0} \right) \end{aligned}$$

(22)

$$\begin{aligned} N_{\varOmega _n }= & {} \frac{r_e^2 J_0^2 \left( {\lambda _n r_e } \right) }{2} \end{aligned}$$

(23)

Appendix 3: Integral Balance

The integral balance can be used as a tool to accelerate the convergence of the eigenfunction expansions, as presented by Scofano Neto et al. (1990) and Leiroz and Cotta (1990). This technique is used in order to attain a faster convergence of the spatial derivatives of pressure. The integral balance is based on the integration of the original partial differential equation (Eq. 4a) in the spatial domain followed by a manipulation of the boundary conditions. Applying the operator $\int _0^x \mathrm{d}x$ in Eq. 4a:

$$\begin{aligned} \frac{\partial }{\partial t}\left[ {\int _0^x \phi \left( x \right) c_\mathrm{t} p\left( {x,t} \right) \mathrm{d}x} \right]= & {} \frac{k\left( x \right) }{\mu }\left. {\frac{\partial p\left( {x,t} \right) }{\partial x}} \right| _{x=x} -\frac{k\left( x \right) }{\mu }\left. {\frac{\partial p\left( {x,t} \right) }{\partial x}} \right| _{x=0}\nonumber \\&+\int _0^x W_\mathrm{p} \left( {x,t} \right) \mathrm{d}x \end{aligned}$$

(24)

substituting a boundary condition with known value of $\partial p(x, t)/\partial x$ in Eq. 24, one can obtain an expression for $\partial p(x, t)/\partial x$ with excellent convergence behavior.

Figure 9 presents $\partial p(x, t)/\partial x$ for the problem presented in Case 2 as calculated by two ways: (a) deriving the solution given by Eq. 6 and using 300 terms in the eigenfunction expansion and (b) by applying the integral balance with 100 terms in the eigenfunction expansion.

Fig. 9

Pressure derivative throughout the reservoir calculated by: a deriving Eq. 6 and b integral balance

Table 8 Convergence behavior of spatial pressure derivative at $t=0.01$ days (Case 2) calculated using the spatial derivative of Eq. 6

Table 9 Convergence behavior of spatial pressure derivative at $t = 0.01$ days (Case 2) calculated using the integral balance

The convergence behavior of $\partial p/\partial x$ as calculated by deriving Eq. 6 and by using the integral balance is presented in Tables 8 and 9, respectively.

As can be seen in the analysis of Fig. 9 and Tables 8 and 9, the integral balance greatly enhances the convergence behavior of $\partial p/\partial x$ and can be used to attain converged solutions of the flow velocities during transient flow in heterogeneous porous media.