Wavelet Transforms for the Simulation of Flow Processes in Porous Geologic Media

Abstract

Wavelets are localized small waves that exhibit the characteristic oscillatory behavior of waves with an amplitude that declines rapidly to zero. Their properties include orthogonality or bi-orthogonality, a natural multiresolution and, often, compact support. These properties can be used to repeatedly rescale a signal or a function, decomposing it to a desirable level, and obtaining and preserving trend and detail data at all scales that allow re-composition of the original signal. The overall goal of this work is to create a set of wavelet-based (WB) numerical methods using different wavelet bases for application to the solution of the PDEs of interest to petroleum engineering, namely the solution of the PDEs of fluid flow through porous and fractured media. A particular emphasis of the study is in processes associated with ultra-low permeability media such as shale oil and shale gas reservoirs. To address the problem, we developed WTFS (wavelet transform flow simulator), a new flow simulator written in MATLAB that couples wavelet transform with a standard finite-difference scheme. In the current state of development, the WB numerical solution is verified against analytical solutions of 1D problems for liquid flow through porous media and is validated through comparisons to numerical solutions for problems of 2D and 3D flow through porous media obtained from a conventional numerical simulator.

Article Highlights

• Derivation of a general approach that involves the use of different appropriate wavelet bases and the application of their multiresolution property to solve problems of single- and multi-phase flow through multidimensional porous media domains.

• Development of a compact MATLAB program that implements the WB solutions and which has been validated against analytical solutions and predictions from a conventional numerical reservoir simulator.

Appendix A

The equations governing 1D fluid flow through porous media and the associated initial and boundary conditions are as follows:

$$ \left\{ \begin{gathered} \frac{{\partial^{2} p_{{\text{D}}} }}{{\partial x_{{\text{D}}}^{2} }} = \frac{{\partial p_{{\text{D}}} }}{{\partial t_{D} }} \hfill \\ p_{{\text{D}}} \left( {x_{{\text{D}}} ,t_{{\text{D}}} = 0} \right) = 0 \hfill \\ \frac{{\partial p_{D} }}{{\partial x_{{\text{D}}} }}\left( {x_{{\text{D}}} = 0,t_{{\text{D}}} } \right) = - 1 \hfill \\ p_{{\text{D}}} \left( {x = 1,t_{{\text{D}}} } \right) = 0 \hfill \\ \end{gathered} \right. $$

(A1)

Discretizing the 1D dimensionless diffusivity equation using a standard backward finite-difference (FD) scheme yields:

$$ \frac{{\partial^{2} p_{{\text{D}}} }}{{\partial x_{{\text{D}}}^{2} }}\left( {x_{{\text{D}}} ,t_{{\text{D}}}^{s + 1} } \right) = \frac{{p_{{\text{D}}} \left( {x_{{\text{D}}} ,t_{{\text{D}}}^{s + 1} } \right) - p_{{\text{D}}} \left( {x_{{\text{D}}} ,t_{{\text{D}}}^{s} } \right)}}{\Delta t} $$

(A2)

Following the approach of Lepik (2005, 2007), at each time sub-interval $t_{{_{{\text{D}}} }}^{{}} \in \left[ {t_{{\text{D}}}^{s} ,t_{{\text{D}}}^{s + 1} } \right]$, $\frac{{\partial^{2} p_{{\text{D}}} }}{{\partial x_{{\text{D}}}^{2} }}\left( {x_{{\text{D}}} ,t_{{\text{D}}}^{s + 1} } \right)$ is approximated by a series of Haar wavelets, resulting in:

$$ \frac{{\partial^{2} p_{D} }}{{\partial x_{D}^{2} }}\left( {x_{D} ,t_{D}^{s + 1} } \right) = \sum\limits_{i = 1}^{2M} {a_{i} h_{i} \left( {x_{D} } \right)} $$

(A3)

Integrating Eq. (A.3) with respect to x_D from 0 to x_D yields:

$$ \frac{{\partial p_{D} }}{{\partial x_{D}^{{}} }}\left( {x_{D} ,t_{D}^{s + 1} } \right) = \frac{{\partial p_{D} }}{{\partial x_{D}^{{}} }}\left( {0,t_{D}^{s + 1} } \right) + \sum\limits_{i = 1}^{2M} {a_{i} I_{i,1} \left( {x_{D} } \right)} $$

(A4)

Next, integration of Eq. (A.4) from 0 to x_D results in:

$$ p_{D} \left( {x_{D} ,t_{D}^{s + 1} } \right) = p_{D} \left( {0,t_{D}^{s + 1} } \right) + \left( {x_{D} - 0} \right) \cdot \frac{{\partial p_{D} }}{{\partial x_{D}^{{}} }}\left( {0,t_{D}^{s + 1} } \right) + \sum\limits_{i = 1}^{2M} {a_{i} I_{i,2} \left( {x_{D} } \right)} $$

(A5)

Based on the initial and boundary conditions in Eq. (A.1), we can derive the general wavelet-based expression for pressure at a time $t_{D}^{s + 1}$ as:

$$ p_{D} \left( {x_{D} ,t_{D}^{s + 1} } \right) = \left( {1 - x_{D} } \right) + \left( {\sum\limits_{i = 1}^{2M} {a_{i} I_{i,2} \left( {x_{D} } \right)} - \sum\limits_{i = 1}^{2M} {a_{i} I_{i,2} \left( 1 \right)} } \right) $$

(A6)

Next, inserting Eqs. (A.3) and (A.6) into Eq. (A.2) and rearranging terms leads to:

$$ \sum\limits_{i = 1}^{2M} {a_{i} \left[ {h_{i} \left( {x_{D} } \right) \cdot \Delta t - I_{i,2} \left( {x_{D} } \right) + I_{i,2} \left( 1 \right)} \right]} = \left( {1 - x_{D} } \right) - p_{D} \left( {x_{D} ,t_{D}^{s} } \right) $$

(A7)

When $s = 0, \, t_{D}^{s} = t_{st} = 0$, $p_{D} \left( {x_{D} ,t_{D}^{s} } \right)$ is given by the initial condition $p_{D} \left( {x_{D} ,t_{D} = 0} \right) = 0$, and the boundary grid points are constrained by the known boundary conditions.