Motivation

Reservoir pressure dynamics away from wellbore and boundaries is sensitive to the two specific complex reservoir properties: transmissibility and pressure diffusivity .

In case the reservoir flow has been created by a well (vertical or horizontal) it will trend to form a radial flow away from boundaries and well itself.

In this case a pressure drop and well flowrate can be roughly related to each other by means of a simple analytical homogeneous reservoir flow model with wellbore and boundary effects neglected.

Since the well radius is neglected the well is modeled as a vertical 0-thickness line, sourcing the fluid from a reservoir, giving a model a specific name Line Source Solution.

Inputs & Outputs

Inputs		Outputs
	total sandface rate		reservoir pressure
	initial formation pressure
	transmissibility
	pressure diffusivity

	transmissibility			dynamic fluid viscosity
	pressure diffusivity			time
	absolute permeability			radial direction
	porosity			total compressibility

Physical Model

Radial fluid flow	Homogenous reservoir	Infinite boundary	Zero wellbore radius	Slightly compressible fluid flow	Constant rate production

Mathematical Model

Motion equation

Initial condition

Boundary conditions

\frac{\partial p}{\partial t} = \chi \, \left[  \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} \right]

p(t=0,r) = p_i

p(t, r=\infty) = p_i

\left[ r \frac{\partial p}{\partial r} \right]_{r=0} =  \frac{q_t}{2 \pi \sigma}

Computational Model

p(t,r) = p_i + \frac{q_t}{4 \pi \sigma} {\rm Ei} \left(-\frac{r^2}{4 \chi t} \right)

– exponential integral

Approximations

Infinite Acting Radial Flow (IARF)

p(t,r) \sim p_i + \frac{q_t}{4 \pi \sigma} \left[  
\gamma + \ln \left(\frac{r^2}{4 \chi t} \right) \right] 

= p_i - \frac{q_t}{4 \pi \sigma} \ln \left(\frac{2.24585 \, \chi  t}{r^2} \right)

Diagnostic Plots

GLOSSARY > Pressure Line Source Solution (LSS) @model > LSS.png

Pressure Drop

Log derivative

\delta p = p_i - p_{wf}(t) = \frac{q_t}{4 \pi \sigma} {\rm Ei} \left(-\frac{r^2}{4 \chi t} \right)

p' = t \frac{d (\delta p)}{dt} = \frac{q_t}{2\pi\sigma} \exp \left( - \frac{r^2}{4\chi t} \right)

\delta p   \sim  \ln t + {\rm const}, \ t \gg \frac{r^2}{4\chi}

p'  \sim \rm const, \ t \gg \frac{r^2}{4\chi}

Fig. 1. PTA Diagnostic Plot for LSS pressure response for the 0.1 md reservoir in a close line source vicinity (0.1 m), which is about a typical wellbore size.

One can easily see that with wellbore effects neglected even for a very low permeability reservoir the IARF regime is getting formed very early at 0.01 hr (36 s).