Motivation
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In many practical cases the reservoir fluid flow created by well is getting aligned with a radial direction towards or away from well.
This type of reservoir fluid flow is called radial fluid flow and corresponding pressure diffusion models provide a diagnostic basis for pressure-rate base reservoir flow analysis.
The radial flow can be infinite acting or boundary dominated or transiting from one to another.
Although the actual reservoir fluid flow may not have an axial symmetry around the well-reservoir contact or around reservoir inhomogeneities (like boundary and faults and composite areas) but still in many practical cases the long-term correlation between the flowrate and bottom-hole pressure response can be approximated by a radial flow pressure modelthe Radial Flow Pressure Diffusion is evolving towards pressure stabilization and can be efficiently analyzed using the steady state flow model.
Inputs & Outputs
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| Expand |
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| ttotal compressibility, | LaTeX Math Inline |
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body | c_t = c_r + c |
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Physical Model
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Mathematical Model
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| Expand |
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52112
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| p(t, r ) = p(r) \Leftrightarrow \frac{\partial p}{\partial t} = 0 |
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\Leftrightarrow, \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} =0 |
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t, r \rightarrow \infty |
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| \left[ r\frac{\partial p( |
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t, )}{\partial r} \right]_{r \rightarrow r_w} = \frac{q_t}{2 \pi \sigma} |
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| p_{wf}= p(r_w ) - S \cdot r_w \, \frac{\partial p}{\partial r} \Bigg|_{r=r_w} |
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p-+ \frac{q_t}{2 \pi \sigma} \, \ln \frac{r}{r_e} = p(r_w) + \frac{q_t}{ |
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42 \pi \sigma} \, \ln \frac{r}{r_w} , \quad r_{wf} < r \leq r_e |
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| p_{wf} = p_i - \frac{q_t}{ |
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4-2S + \ln \frac{r_e}{r_w} \bigg] |
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Applications
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Equation Equations
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and | LaTeX Math Block Reference |
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show shows how the
basic diffusion model parameters impact the
pressure response while other diffusion parameters are encoded in relation between drawdown | LaTeX Math Inline |
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| body | \Delta p = p_i - p_{wf} |
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and total sandface flowrate function and play and plays important methodological role as they are used in many algorithms and express-methods of
Pressure Testing.
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| title | Line Source Solution |
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| F(z) = {\rm Ei}_1 (z) | (see Line Source Solution (LSS) @model).| Expand |
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| PTA – Pressure Transient Analysis
Pressure Drop| LaTeX Math Block |
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\delta p = p_i - p_{wf}(t) \sim \ln t + {\rm const} |
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| Log derivative| LaTeX Math Block |
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t \frac{d (\delta p)}{dt} \sim \rm const |
| Fig. 2. PTA Diagnostic plot for radial fluid flow |
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| title | Productivity Index Analysis |
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The Total Sandface Productivity Index for single-phase low-compressibility fluid and low-compressibility rocks does not depend on formation pressure, bottom-hole bottomhole pressure and the flow rate flowrate and can be expressed as: | LaTeX Math Block |
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| J(_t) = \frac{q_t}{p_i - p_{wf}(t)} =\frac{ 42 \pi \sigma }{ 2S - F \bigg( -\ln \frac{r_w^2e}{4 \chi t} \bigg) } |
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The Field-average Productivity Index for low-compressibility fluid and low-compressibility rocks does not depend on formation pressure, bottomhole pressure and the flowrate and can be expressed asIsobar equation for a constant-rate production: | LaTeX Math Block |
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| anchor | Q7VZXJ |
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| alignment | left |
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| p(t,r)J_t = p_i + \frac{q_t}{4 \pi \sigma} \, F \bigg(p_r(t) - \fracp_{r^2}{4 \chi twf}(t)} =\bigg) = {\rm const} \quad \rightarrow \quad \frac{r^2}{4 \chi t}= {\rm const} |
Since the pressure disturbance at moment was at well walls then the formula for constant-pressure front propagation becomes:| LaTeX Math Block |
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| r(t) = r_w + 2 \sqrt{\chi t} |
This leads to estimation of isobar velocity: | LaTeX Math Block |
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| u_p(t) = \sqrt{\frac{\chi}{t}}
frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + 0.5 +S} = {\rm const} |
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See Also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Radial fluid flow / Pressure diffusion / Pressure Diffusion @model / Radial Flow Pressure Diffusion @model
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing
[ Well & Reservoir Surveillance ] [ Pressure diffusion @model ][ Line Source Solution (LSS) @model ][ Linear Flow Pressure Diffusion @model ]
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but this only works for the middle-times and long-times as early times are influenced by wellbore storage and non-linear effects of skin.
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p(t,r) = p_i + \frac{q_t}{4 \pi \sigma} \, {\rm Ei} \bigg( - \frac{r^2}{4 \chi t} \bigg) |
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Диффузия давления описывается решением уравнения однофазного радиального течения в бесконечном однородном пласте:
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\frac{\partial p}{\partial t} = \chi \, \Delta p = \chi \, \frac{1}{r} \frac{\partial}{\partial r} \bigg( r \frac{\partial p}{\partial r} \bigg) |
с начальным условием:
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p(t = 0, r) = p_i |
и граничными условиями:
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p(t, r \rightarrow \infty ) = p_i |
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| anchor | Boundary_q |
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| alignment | left |
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r \frac{\partial p(t, x )}{\partial r} \bigg|_{r \rightarrow 0} = \frac{q_t}{2 \pi \sigma} |
где
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| body | \sigma = \frac{k \, h}{\mu} |
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– гидропроводность пласта, | LaTeX Math Inline |
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| body | \chi = \frac{k}{\mu} \, \frac{1}{\phi \, c_t} |
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– пьезопроводность пласта, – проницаемость пласта, – пористость пласта, – сжимаемость пласта, – сжимаемость порового коллектора, – сжимаемость насыщающего пласт флюида, – вязкость насыщающего пласт флюида.При анализе отклика давления на самой скважине (
) после включения на достаточно больших временах, удовлетворяющих условию:| LaTeX Math Block |
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t \gg \frac{r_w^2}{4 \chi}
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которые на практике наступают очень быстро, можно воспользоваться приближением
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| body | {\rm Ei}(-x) \sim \ln (x) + \gamma \sim \ln (1.781 x) |
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, где – постоянная Эйлера. ...
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p(t,r_w) = p_i + \frac{q_t}{4 \pi \sigma} \, \ln \bigg( 1.781 \, \frac{r_w^2}{4 \chi t} \bigg) |
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\delta p = p_i - p_{wf}(t) \sim { \rm const } + \frac{q_t}{4 \pi \sigma} \, \ln t |
а логарифмическая производная становится постоянной во времени:
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t \frac{d (\delta p)}{dt} \sim \frac{q_t}{4 \pi \sigma} |
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