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 pseudo-steady state flow model.
Inputs & Outputs
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Physical Model
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Mathematical Model
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| r_{wf} < r \leq r_e |
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| \frac{\partial p}{\partial t} |
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0 Leftrightarrow,left[ \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} |
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=0p(t,\left[ \frac{\partial p}{\partial r} \ |
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rightarrow \infty ) = p_iEM415 | \left[ r\frac{\partial p(t, |
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)}{\partial r} \right]_{r |
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\rightarrow= r_w} = \frac{q_t}{2 \pi \sigma} |
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pr _i +(t, r_w) - S \cdot \left[ r \frac{ |
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q_t4 pi\sigma, \ln \frac{r_e}{| LaTeX Math Block |
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| p_{wf}_i +4\, \bigg[ - 2S + 1 + \ln \frac{r_e}{r_w} \bigg]
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Applications
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It is important to note that equations |
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– | LaTeX Math Block Reference |
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| do not constitute a complete CVP as it does not specify the initial condition. |
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| title | Line Source Solution |
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In case of infinite homogeneous reservoir, produced by a infinitely small vertical well with no skin and no wellbore storage the
function has an exact analytical formula, given by exponential integral | LaTeX Math Inline |
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| body | F(z) = {\rm Ei}_1 (z) |
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(see Line Source Solution (LSS) @model).| Expand |
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PTA – Pressure Transient Analysis The Productivity Index for single-phase low-compressibility fluid and low-compressibility rocks does not depend on formation pressure, bottom-hole pressure and the flow rate and can be expressed as: | LaTeX Math Block |
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J(t) = \frac{q_t}{p_i - p_{wf}(t)} =\frac{ 4 \pi \sigma }{ 2S - F \bigg( Pressure Drop| LaTeX Math Block |
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anchor | 1EWTY\delta pp_{wf}(t) \sim \ln\frac{{\rm w \,} q_t }{V_e \, \phi \, c_t} \, t + \frac{{\rm |
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const} Image Removed
| 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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w \,} q_t }{4\pi \sigma} \left[ 2 \ln \frac{r}{r_e} - \frac{r^2}{r_e^2} + 1 \right]
, \quad r_{wf} < r \leq r_e,
\quad {\rm w }= 1 - \frac{r_w^2}{ |
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Isobar equation for a constant-rate production:Q7VZX,r+4pi \sigma} \, F \bigg(
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| p_{wf}(t) = p_e(t) - \frac{ |
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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}}
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See Also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Radial fluid flow
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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Approximations
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| {\rm w} = 1 - \frac{r_w^2}{r_e^2} \approx 1 |
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, t + \frac{q_t}{4\pi \sigma} \bigg[ 2 \ln \frac{r}{r_e} - \frac{r^2}{r_e^2} + 1 \bigg]
, \quad r_{wf} < r \leq r_e |
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| p_{wf}(t) \approx p_e(t) - \frac{q_t}{2 \pi \sigma} \, \left[ \ln \frac{r_e}{r_w} + 0.5 + S \right] |
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| p_e(t) \approx p_i - \frac{q_t}{V_e \phi c_t}t |
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Applications
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Equation
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shows how the basic diffusion model parameters impact the 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 and plays important methodological role as they are used in many algorithms and express-methods of Pressure Testing.
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| title | Productivity Index Analysis |
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The Total Sandface 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 as: | LaTeX Math Block |
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| J_t = \frac{q_t}{p_e(t) - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + 0.5 +S} = {\rm const} |
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 as: | LaTeX Math Block |
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| J_t = \frac{q_t}{p_r(t)Ei} \bigg( - \frac{r^2}{4 \chi t} \bigg)Рассмотрим плоскопараллельный аксиально-симметричный однородный пласт постоянной толщины , с радиальной координатой в перпендикулярной к оси скважины плоскости, который вскрыт бесконечно тонкой скважиной в точке (где – радиальная координата в перпендикулярной к оси скважине плоскости) и начальным пластовым давлением .
Пусть в момент времени скважина запускается с дебитом (в пересчете на пластовые условия).Диффузия давления описывается решением уравнения однофазного радиального течения в бесконечном однородном пласте: | LaTeX Math Block |
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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: | LaTeX Math Block |
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| p(t = 0, r) = p_i |
и граничными условиями: | LaTeX Math Block |
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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 | LaTeX Math Inline |
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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 | LaTeX Math Inline |
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| body | {\rm Ei}(-x) \sim \ln (x) + \gamma \sim \ln (1.781 x) |
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, где – постоянная Эйлера. Режим радиального течения к линейному источнику примет вид:| LaTeX Math Block |
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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) |
Отсюда следует, что уже вскоре после запуска скважины динамическая депрессия на пласт начинает логарифмически расти во времени:| LaTeX Math Block |
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| | \delta p = p_i - p_{wf}(t) \sim { \rm const } + =\frac{q_t}{42 \pi \sigma} {\, \ln tа логарифмическая производная становится постоянной во времени: | LaTeX Math Block |
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| t \frac{d (\delta p)}{dt} \sim \frac{q_t}{4 \pi \sigma}В лог-лог координатах лог-производная депрессии будет горизонтальной, что является характерным для радиальной фильтрации в бесконечном пласте.r_e}{r_w} + 0.75 +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 ]
