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Numerical Investigation of Memory-Based Diffusivity Equation: The Integro-Differential Equation

  • Research Article - Petroleum Engineering
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Abstract

The classical momentum balance equation discovered by Darcy in 1856 is expressed as the flux is proportional to the pressure gradient. However, the passage of the fluid through the porous matrix is very complex in general and hence may cause a local variation of the permeability. Thus, a one-dimensional model for an oil reservoir is introduced by considering the modification of conventional momentum balance equation. The modification is performed by introducing a derivative of fractional distributed orders as memory formalism. The fractional order is equivalent to a time-dependent diffusivity, and the distributed orders represent a variety of memory mechanisms to model the pressure response with a varied distribution of porosity and permeability. The time-domain and space-domain solutions are obtained by means of a numerical solution of the model equation. Results show that memory-based diffusivity equation has less pressure drops compared to Darcy model for a given distance and time. The differences in pressure drop between the two models become more significant when reservoir life becomes longer. The memory has an effect on the reservoir porosity and permeability which increases with time. If reservoir production continues, memory effect becomes more visible and contributes more in pressure response, which may be considered as a memory-driven mechanism. The proposed model is validated using Middle East filed data. The findings of this research establish the contribution of memory in reservoir fluid flow through porous media.

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Abbreviations

a :

Lithology factor, dimensionless

A yz :

Cross- sectional area of rock perpendicular to the flow of flowing fluid, \({{m}^{2}}\)

c f :

\({c_o + c_{w } =}\) total fluid compressibility of the system, 1/pa

c s :

Formation rock compressibility of the system, 1/pa

c t :

\({c_f + c_{s } =}\) total compressibility of the system, 1/pa

c w :

Formation water compressibility of the system, 1/pa

k :

Variable reservoir permeability, \({{m}^{2}}\)

k o :

Initial reservoir permeability, \({{m}^{2}}\)

L :

Distance between production well and outer boundary along x- direction, m

p :

Pressure of the system, \({{N/m}^{2}}\)

p o :

Initial or reference pressure of the system, N/m2

q   :

Volume production rate at time t, \({{m}^{3}/{s}}\)

q i :

Au = initial volume production rate, \({{m}^{3}/{s}}\)

t :

Time, s

x :

Flow direction for 1D for a particular point, x from wellbore, m

\({\alpha }\) :

Fractional order of differentiation, dimensionless

\({\rho }\) :

Density at pressure p, \({{kg/m}^{3}}\)

\({\rho _o }\) :

Density at a reference pressure p o , \({{kg/m}^{3}}\)

\({\phi }\) :

Porosity of fluid media at pressure p, \({{m}^{3}/{m}^{3}}\)

ϕ o :

Initial porosity of fluid media at reference pressure \({p_o, {m}^{3}/{m}^{3}}\)

\({\mu }\) :

Fluid dynamic viscosity, \({{pas}}\)

\({\mu _0 }\) :

Initial fluid viscosity, \({{pas}}\)

\({\eta }\) :

\({\frac{k}{\mu } \left( t \right)^{\alpha } =}\) ratio of the pseudopermeability of the medium with memory to fluid viscosity, \({{m}^{3} {s}^{1+\alpha }/{kg}}\)

\({\xi }\) :

A dummy variable for time, i.e., real part in the plane of the integral, s

\({{\Gamma }}\) :

Euler gamma function

\({\Delta {t}}\) :

Time step, s

\({{\Delta }x}\) :

Grid block size, m

\({{\Delta \xi }}\) :

Dummy time step, s

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Authors and Affiliations

  1. Oil and Gas Program, Department of Process Engineering, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, 240 Prince Philip Drive, St. John’s, NL, A1B 3X5, Canada

    M. Enamul Hossain

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Hossain, M.E. Numerical Investigation of Memory-Based Diffusivity Equation: The Integro-Differential Equation. Arab J Sci Eng 41, 2715–2729 (2016). https://doi.org/10.1007/s13369-016-2170-y

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