Quantitatively Evaluating Formation Pressure Distribution After Hydraulic Fracturing in Tight Sand Oil

Abstract

Hydraulic fracturing with a horizontal well is the core technology for the efficient development of unconventional oil and gas resources such as tight oil. Quantitative characterization of formation pressure changes in tight oil reservoirs is of great significance for improving the development efficiency of tight oil reservoirs. In response to the difficulty of quantitatively characterizing the range, size, and release process of formation pressure control in the fractured wells of tight oil reservoirs, this work proposes a numerical simulation method to quantitatively evaluate reservoir and fluid elastic properties. Based on a simulation, the elastic energy control zone was divided into a fracture network control zone and a matrix control zone, which achieved the accurate calculation of different zones and elastic energies. The effects of fracturing parameters, formation and fluid elastic parameters, and well spacing on the elastic energy control range were analyzed, and elastic energy calculation charts were drawn under different permeability, half fracture length, and fluid elastic parameter conditions. Based on analysis of the elastic energy release process, the elastic recovery rate of this type of reservoir was predicted. These research results are of great significance for optimizing the parameters of unconventional oil and gas hydraulic fracturing and their development system.

1. Introduction

Global technically recoverable tight oil reserves exceed 400 billion barrels, demonstrating significant development prospects [1,2]. Substantial progress has been achieved in hydrocarbon-bearing basins such as the Ordos Basin and Junggar Basin in China [3,4]. Continental tight sandstone reservoirs in China exhibit strong heterogeneity (variation coefficient > 0.7), low formation pressure coefficients (<1.3), and high crude oil viscosity (>5 mPa·s) [5,6], resulting in natural elastic energy efficiency that is only 50–60% of North American shale reservoirs [7]. This reality compels a strategic shift toward synergistic pressure–elastic energy regulation driven by fracturing fluid retention. Retained fracturing fluid elevates pressure within fracture networks, creating localized high-pressure zones (fracturing fluid storage elevates fracture-network pressure to 1.2–1.5 times the original pressure [8]), confirming fracturing fluid’s dual role as both a stimulation agent and energy carrier. In the widely adopted depletion development mode using volume-fractured horizontal wells, elastic energy serves as the primary driving force. Early production stages are dominated by post-fracturing enhanced energy and reservoir-fluid elastic energy, contributing ~70% of production, while later stages transition to a matrix-dominated solution gas drive comprising reservoir-fluid elastic energy and dissolved gas energy [9]. The release of elastic energy directly reflects fluid migration and dynamic pressure changes within pore spaces, making quantitative characterization of post-fracturing formation pressure and elastic energy critically important for tight oil development.

Current quantitative characterization of post-fracturing formation pressure and elastic energy faces adaptability challenges in continental reservoir applications [10,11]. Widely used North American models such as RSM-DOE predict a 2.3-fold increase in fracture-network elastic energy (fracture half-length 200 m [12]), yet field measurements in Ordos tight sandstone reservoirs show 30–40% lower values than predictions, revealing pressure distribution prediction biases [13]. MIT’s FES coefficient quantifies fracture-network pressure elevation, but fails to couple stress-sensitivity effects—including >40% fracture conductivity attenuation and accelerated elastic energy decay [14]. Jia’s inversion model inadequately characterizes > six-month pressure propagation delays in matrix zones, constraining the dynamic prediction of elastic energy release boundaries [15]. Although domestic research has proposed innovative concepts like “fracturing energy-storage” (Changqing/Jilin Oilfields) and developed energy evaluation methods (e.g., Wang Yining’s carbonate reservoir model, Zheng Songqing’s quantitative evaluation of driving energy in fracture-cavity reservoirs) [16,17], dynamic process characterization requires enhancement: temporal evolution of pressure redistribution (e.g., imbibition time-delay effects) needs further quantification; coupling mechanisms for geomechanical dynamic responses (e.g., stress shadow-induced rock compressibility changes) require deepening; and the systematic establishment of threshold pressure gradient control (<0.005 MPa/m) over energy propagation boundaries remains incomplete [18].

To address the current challenges of low accuracy in quantitative formation pressure characterization and difficulty in determining elastic energy influence ranges in tight oil fractured wells, this article proposes a quantitative characterization model for formation pressure changes based on percolation mechanics and geomechanics theories. It investigates the variation patterns of elastic energy in fractured wells, incorporating energy-controlled scopes under different fracture lengths and permeabilities along with energy-release stages. This approach enables quantitative calculation of single-well elastic energy, provides a novel theoretical methodology for formation pressure and elastic energy characterization, and offers a new analytical framework for energy replenishment in tight reservoirs. These results will establish a theoretical foundation for “fracturing energy-storage” development in continental tight reservoirs, advancing elastic recovery rates from the current 4–6% to North American levels (8–12%) [19].

2. Methodology

After hydraulic fracturing, a massive amount of fracturing fluid is trapped in the formation, which is the main factor causing the increase in formation pressure. In the primary stage, it is the main energy for crude oil displacement. During the depletion development process, as the formation pressure continues to decrease, the elastic energy of the formation and fluids is continuously released. The energy for crude oil displacement in the early and middle stages of depletion development mainly comes from the elastic energy of the reservoir and fluids. The local formation pressure is lower than the saturation pressure of crude oil, and natural gas dissolved in crude oil is separated from the crude oil. During the process of reducing formation pressure, more and more natural gas is separated from the crude oil, forming a continuous phase. At this stage, the displacement energy of crude oil mainly comes from the elastic properties of this part of natural gas. Therefore, the elastic energy after hydraulic fracturing of tight oil reservoirs includes increased elastic energy after fracturing, reservoir and fluid elastic energy, and dissolved gas elastic performance. During the depletion development and production process of hydraulic fracturing, the elastic energy release process is divided into three stages: increased elastic energy release after fracturing, reservoir and fluid elastic energy release, and dissolved gas elastic performance release [20].

2.1. Formation Pressure Distribution

The elastic energy derived from fracturing fluid primarily originates from formation pressure elevation within the fracture-network-dominated region. For the near-wellbore region subjected to hydraulic fracturing, formation pressure variation can be derived from the material balance equation, as expressed in Equation (1). By incorporating formation-specific elastic parameters, this relationship simplifies to Equation (2), which enables quantitative determination of the post-stimulation pressure elevation magnitude (∆p₁).

$$V_{por}+C_f{V}_b\Delta p_1=V_{os}{B}_o+(V_{ws}+V_{wy})B_w$$

(1)

$$V_{por}+C_f{V}_b\Delta p_1={\displaystyle \frac{V_{por}{S}_o{B}_o}{B_{oi}}}+({\displaystyle \frac{V_{por}{S}_w}{B_{wi}}}+V_{frac})B_w$$

(2)

where V_por is pore volume, m³; C_f is the rock compressibility coefficient, MPa⁻¹; V_b is the matrix volume, m³; Δp₁ is the pressure elevation magnitude induced by effective injected fluid, MPa; V_os is the in situ oil volume, m³; V_ws is the formation water volume, m³; V_wy is the effective injected fluid volume, m³; B_o is the current oil formation volume factor (post-fracturing), m³/m³; B_w is the current water formation volume factor (post-fracturing), m³/m³; B_oi is the initial oil formation volume factor (pre-fracturing), m³/m³; B_wi is the initial water formation volume factor (pre-fracturing), m³/m³; S_o is oil saturation, %; S_w is water saturation, %; and V_frac is the fracture fluid volume, m³.

Based on reservoir parameters from

Table 1. Well and reservoir parameters for horizontal wells in the L well block.

Parameter	Northern Region	Southern Region	Unit
Lateral Length	1500	1500	m
Well Spacing	1000	600	m
Reservoir Thickness	18.90	11.20	m
Initial Formation Pressure	16.50	18.62	MPa
Bubble-Point Pressure	9.56	10.72	MPa
Porosity	11.50	10.80	%
Matrix Permeability	0.24	0.16	mD
Initial Oil Saturation (So)	56.34	57.14	%
Initial Water Saturation (Sw)	43.56	32.86	%
Initial Oil FVF (Boi)	1.33	1.28	m3/m3
Initial Water FVF (Bwi)	1.09	1.08	m3/m3
Rock Compressibility (Cf)	0.58	0.54	10−6/MPa−1

for the L well, the fracturing fluid density used in the hydraulic fracturing was 216–480 m³ per stage, and corresponded to an average formation pressure elevation of 1.12–1.78 MPa, demonstrating significant energy replenishment through fluid retention.

2.2. Elastic Energy Increment Induced by Formation Pressure Elevation

The elastic energy derived from fracturing fluid primarily originates within the fracture-network-dominated region. Fluid production in this zone results from pore volume contraction of the rock–fluid system due to formation pressure depletion. Although studies have indicated that fracturing fluid imbibition contributes to recovery enhancement [18], these works exclusively attributed production to elastic energy mechanisms without considering imbibition effects. Meanwhile, saturation heterogeneity effects are minimized in the fracture-proximal zone due to enhanced fluid redistribution. Hence, the elastic energy calculation formula for this stage is expressed as follows.

$$E_1=100\times {\displaystyle \frac{A_{f}h\varphi C_f\Delta p_{1}N\mathrm{ET}}{B_{oi}}}$$

(3)

where E₁ is the elastic energy increment induced by formation pressure elevation, 10⁴ m³; $A_f$ is the area of the fracture-network-dominated region, km²; h is reservoir thickness, m; ϕ is porosity, %; and NET is the net-to-gross ratio (dimensionless).

2.3. Elastic Energy for Rock and Fluid in Reservoir

The elastic energy stored within reservoir rock and fluids is liberated through pressure depletion across both fracture-dominated compartments and matrix-dominated compartments, driving rock matrix compaction and pore fluid expansion, with spatial propagation efficiency governed by fundamentally divergent threshold pressure gradients demarcating these domains. Fracture-dominated compartments exhibit low threshold pressure gradients (0.001–0.01 MPa/m) due to fracture-enhanced connectivity, enabling efficient Darcy flow and rapid energy transmission, whereas matrix-dominated compartments suffer elevated threshold pressure gradients (0.012–0.053 MPa/m) imposed by nanoscale pore-throat capillarity, inducing non-Darcy behavior where energy propagation becomes thermodynamically ineffective below compartment-specific thresholds. This significant contrast in threshold pressure gradients (typically 5–50× difference, e.g., 0.005 MPa/m in fractures vs. 0.03 MPa/m in the matrix) stems from intrinsic reservoir heterogeneity and fracture network complexity, necessitating physics-based partitioning wherein regions with threshold pressure gradients < 0.01 MPa/m define fracture-dominated flow regimes and zones with threshold pressure gradients > 0.012 MPa/m characterize capillary-dominated matrix flow. This dichotomy is leveraged in numerical simulations resolving multiscale flow interactions by distinguishing proximal wellbore/fracture regions (where elevated fracture density suppresses threshold pressure gradients to ~0.005 MPa/m) from distal zones (where nanoscale capillarity elevates threshold pressure gradients to > 0.03 MPa/m), thereby adopting Mongalvy et al.’s [21] drainage volume partitioning into stimulated reservoir volume (equivalent to fracture-dominated compartments) and unstimulated reservoir volume (equivalent to matrix-dominated compartments) based on threshold pressure gradient discontinuity, as illustrated in Figure 1. The key geometric parameters include the following:

R_f: Fracture-controlled distance perpendicular to wellbore (equal to fracture half-length);

R_mv: Matrix-controlled distance perpendicular to wellbore;

R_mh: Matrix-controlled distance parallel to wellbore.

The fracture-network region exhibited significantly enhanced flow capacity due to fracturing, with R_f directly equivalent to the fracture half-length. Conversely, the matrix-dominated region retained poor flow capacity governed by non-Darcy flow dynamics characteristic of tight formations. Figure 2 and Figure 3 present pressure distributions and gradient contour maps for a representative fractured horizontal well numerically generated considering tight oil’s non-Darcy flow behavior.

3. Results and Discussion

3.1. Utilization Law of Elastic Energy Control Region

To investigate elastic energy propagation in volume-fractured horizontal wells within tight oil reservoirs, we employed reservoir engineering and seepage mechanics methodologies coupled with geological/fluid parameters from Table 1. This integrated approach elucidates energy-controlled domain dynamics. The flow field of a volume-fractured horizontal well was simulated through the superposition of finite-length strip sources. By convolving strip source functions via the Newman product method, we generated planar strip sources (Figure 4). Subsequent superposition of these sources accurately reconstructed the 2D flow field, rigorously incorporating tight oil’s threshold pressure gradient effects.

The actual pressure drawdown induced at location (x, y) at time t by the k-th fracture with historical flow rate Q_k is expressed as:

$$\begin{array}{l}\Delta p\left(x,y,x_k,y_k,t\right)=p_{\mathrm{e}}-p\left(x,y,x_k,y_k,t\right)=-G\sqrt{{r_{\mathrm{x},k}}^2+{r_{\mathrm{y},k}}^2}+{\displaystyle \frac{B}{2h{x}_{\mathrm{f}}{y}_{\mathrm{f}}\varphi c}}\\ \times {\displaystyle \sum _{i=1}^{\mathrm{n}}\left(Q_{k,i}-Q_{k,i-1}\right){\displaystyle {\int }_0^{t_{\mathrm{n}}-t_{i-1}}\left[\mathrm{erf}\frac{\frac{x_{\mathrm{f}}}{2}+\left(x-x_k\right)}{\sqrt{4\eta t}}+\mathrm{erf}\frac{\frac{x_{\mathrm{f}}}{2}-\left(x-x_k\right)}{\sqrt{4\eta t}}\right]\cdot \left[\mathrm{erf}\frac{\frac{y_{\mathrm{f}}}{2}+\left(y-y_k\right)}{\sqrt{4\eta t}}+\mathrm{erf}\frac{\frac{y_{\mathrm{f}}}{2}-\left(y-y_k\right)}{\sqrt{4\eta t}}\right]\mathrm{d}t}}\end{array}$$

(4)

where P_e is the initial formation pressure, MPa; G is the threshold pressure gradient, MPa/m; η is hydraulic diffusivity, m²/s; and t is time, s.

The crucial aspect of this model’s solution lay in determining the flow rate for each fracture at discrete time steps. The governing equation for the flow rate q_k(t) of the k-th fracture at time t was expressed as:

$$\Delta p\left(x,y,x_k,y_k,t\right)=A\left(x,y,x_k,y_k,t\right)Q_{k,\mathrm{n}}+B\left(x,y,x_k,y_k,t\right)$$

(5)

Given the pressure drawdown distribution Δp(x, y, x_k, y_k, t), the corresponding flow rate Q_k,n can be determined. As the pressure drawdown at the k-th fracture location Δp(x_k, y_k, t) resulted from interference effects among all N fractures, the matrix equation for solving flow rates under constant production pressure drop (Δp = p_e − p_wf) was formulated as:

$$\left[\begin{array}{ccc}A\left(x_1,y_1,x_1,y_1,t\right)& \dots & A\left(x_1,y_1,x_{\mathrm{N}},y_{\mathrm{N}},t\right)\\ ⋮& \ddots & ⋮\\ A\left(x_{\mathrm{N}},y_{\mathrm{N}},x_1,y_1,t\right)& \dots & A\left(x_{\mathrm{N}},y_{\mathrm{N}},x_{\mathrm{N}},y_{\mathrm{N}},t\right)\end{array}\right]\left[\begin{array}{c}{Q}_{1,\mathrm{n}}\\ ⋮\\ Q_{\mathrm{N},\mathrm{n}}\end{array}\right]+\left[\begin{array}{c}{\displaystyle \sum _{k=1}^{\mathrm{N}}B\left(x_1,y_1,x_k,y_k,t\right)}\\ ⋮\\ {\displaystyle \sum _{k=1}^{\mathrm{N}}B\left(x_{\mathrm{N}},y_{\mathrm{N}},x_k,y_k,t\right)}\end{array}\right]=\left[\begin{array}{c}\Delta p\\ ⋮\\ \Delta p\end{array}\right]$$

(6)

An iterative solution for Equation (6) yields the pressure distribution for a volume-fractured horizontal well in an infinite-acting reservoir. Applying the principle of pressure superposition enables the simulation of multi-well scenarios. Subsequent differentiation of the pressure field determines moving-boundary locations.

For a fractured horizontal well in an infinite-acting tight reservoir with a fixed fracture half-length of 100 m, the propagation range of the elastic energy moving boundary was calculated under varying matrix permeabilities and threshold pressure gradients. Figure 5 and Figure 6 demonstrate that in infinite formations, prolonged production led to continuous outward diffusion of the pressure front. The moving boundary expanded progressively both perpendicularly and parallel to the wellbore orientation. Notably, an initial rapid propagation phase transitioned to decelerated growth in later stages, ultimately approaching asymptotic stabilization. Enhanced propagation distances were observed under higher matrix permeability and lower threshold pressure gradient conditions.

Employing the principle of pressure superposition, this study simulated multiple fractured horizontal wells within tight reservoirs at well spacings of 400 m, 500 m, and 600 m. Under consistent reservoir conditions—a matrix permeability of 0.1 mD and a threshold pressure gradient of 0.01 MPa/m—with identical bottom hole flowing pressure, moving-boundary propagation dynamics were simulated. Figure 7 demonstrates that individual well drainage boundaries initially propagated outward with production time. Upon the onset of well interference, these boundaries underwent progressive retraction. The retraction rate diminished continuously over time until the boundaries stabilized into steady-state configurations. This transition from expansion to retraction and ultimate stabilization reflects the dynamic pressure equilibrium established between adjacent wells.

In this simulation, while maintaining reservoir base parameters (a matrix permeability of 0.1 mD and a threshold pressure gradient of 0.01 MPa/m) and consistent bottomhole flowing pressure, the temporal evolution characteristics of production under three well spacings (400 m, 500 m, and 600 m) were quantified based on the principle of pressure superposition. As shown in Figure 7, the drainage boundary of a single well exhibited three stages of dynamic behavior over production time: initial outward expansion (advancement of the pressure wave front), inward contraction after interference (superposition of pressure drawdown funnels from adjacent wells), and eventual stabilization (constant drainage radius). This boundary movement pattern directly controlled the production decline behavior, as shown in Figure 8. For the 400 m well spacing, strong interference (with a stable radius of 130 m) led to a geometric decline in production after interference, with production decreasing by 48.4% from its peak by the end of the second year, highlighting the risk of boundary lock-in of oil caused by the TPG effect under narrow well spacing. In contrast, for the 600 m well spacing, autonomous flow remained dominant (with a stable radius of 230 m), resulting in a gentle production decline, reflecting steady-state seepage characteristics under weak interference.

3.2. Elastic Energy Propagation Distance

For 1500 m lateral length volume-fractured horizontal wells in the L well block, predictive nomographs quantify elastic energy propagation distances across varying matrix permeabilities, fracture half-lengths, and stage counts. As shown in Figure 9, Figure 10 and Figure 11, these design charts capture the dependency of energy-controlled distances on reservoir and completion parameters, enabling rapid determination of both propagation distances and corresponding drainage areas per well. By integrating permeability, fracture geometry, and staging configurations, these tools facilitate scientifically grounded well spacing optimization to maximize reservoir drainage while mitigating interference risks.

4. Application

The effective elastic energy diagram of the L well block was calculated using the average parameters obtained from comprehensive geological and reservoir characterization studies. These parameters reflect the overall characteristics of the entire well area, which is divided into the southern section and the northern section. The specific parameters are shown in Table 1. An effective elastic energy map of the L well area was drawn based on data from the area and considering different reservoir thicknesses and single well control areas combined with the L well area’s spacing model (Figure 12 and Figure 13). The so-called effective elastic energy refers to the elastic energy controlled when the depleted development reached a certain formation pressure. When the formation pressure was low, the single well production capacity was low, there was no economic benefit, and the depleted development ended. In the research process of this study, the average formation pressure was 6 MPa, and the actual value could be determined based on the oilfield site. As shown in the figure, with an increase in reservoir thickness or the single well control area, the elastic energy controlled by a single well increased. The control area range of a single well at a distance of 1000 m in the northern area of the L well zone was 0.39 km²~0.64 km², and the thickness range of the reservoir was 8.90 m~18.60 m. According to the chart, the effective elastic energy range controlled by a single well was 3.65 × 10⁴ m³~8.42 × 10⁴ m³. The control area range of a single well at a distance of 600 m in the southern area of the L well zone was 0.39 km²~0.41 km², and the thickness range of the reservoir was 8.40 m~10.90 m. According to the chart, the effective elastic energy range controlled by a single well was 2.95 × 10⁴ m³~3.72 × 10⁴ m³.

The northern section of the L well block began production in September 2021, and as of January 2025, it has been in production for 40 months, yielding a cumulative liquid production of 15.1 × 10⁴ m³ and a cumulative oil production of 12.9 × 10⁴ m³. Utilizing the quantitative characterization model for elastic energy, the initial effective elastic energy of the region (calculated up to an average reservoir pressure of 6 MPa) was determined to be 41.8 × 10⁴ m³. As shown in Figure 14, following the fracturing fluid-induced pressure increase in June 2022, the elastic energy increment (from the fracture-controlled zone) was fully released, with all elastic energy release originating from the fracture-controlled zone, accounting for 21.1%. From June 2022 to March 2023, the elastic energy release was primarily from the fracture-controlled zone. By March 2023, the elastic energy release from the fracture-controlled zone accounted for 55.7%, while that from the matrix-controlled zone accounted for 4.4%. From March 2023 to January 2025, the matrix-controlled zone experienced significant elastic energy release. By January 2025, the elastic energy release from the fracture-controlled zone accounted for 77.4%, and that from the matrix-controlled zone accounted for 34.8%. The cumulative effective elastic energy released was 25.2 × 10⁴ m³, representing 60.2% of the total effective elastic energy. Subsequent measures should be taken to replenish energy and enhance its utilization efficiency to improve recovery rates. The quantitative characterization model for elastic energy was validated through the decline in reservoir pressure, with calculation errors less than 15%.

5. Conclusions

(1)

Retained fracturing fluid is the primary factor inducing formation pressure increase and subsequent elastic energy enhancement in tight sand reservoirs. Quantitative analysis of the L well block confirmed that effective fluid retention (216–480 m³ per stage) elevates local pressure by 1.12–1.78 MPa within fracture networks. This pressure redistribution facilitates partitioning the elastic energy domain into fracture-network-controlled and matrix-controlled zones—a critical framework for evaluating post-fracturing energy distribution dynamics.

(2)

In tight sandstone reservoirs, elastic energy release during production—driven by rock compression and fluid expansion in response to dynamic pressure changes—exhibits compartmentalized behavior between fracture-network-dominated domains (characterized by Darcy flow) and matrix-dominated domains (governed by non-Darcy flow, where energy propagation ceases below critical pressure gradient thresholds, thereby delineating energy control boundaries). Within infinite reservoirs, the elastic energy control boundary undergoes rapid initial expansion followed by asymptotic stabilization. Enhanced permeability (>0.5 mD) and reduced threshold pressure gradients (<0.01 MPa/m) significantly extend the propagation range. Under multi-well configurations (well spacing: 400–500 m), dynamic pressure redistribution initiates boundary expansion until interference occurs, subsequently inducing progressive contraction that stabilizes upon equilibration of fracture-network and matrix pressure gradients. These fundamentally interrelated mechanisms—permeability-dominated diffusion efficiency, threshold gradient-constrained expansion decay, and well spacing-modulated dynamic redistribution—collectively govern elastic energy utilization boundaries.

(3)

Field-validated nomographs provide rapid operational guidance for well spacing and fracture design by quantifying elastic energy propagation distances across key parameters (permeability: 0.1–1 mD; fracture half-length: 50–200 m; stage count: 6–15), maximizing drainage while mitigating interference. Application in the L block demonstrated thickness-controlled recoverable energy (3.65–8.42 × 10⁴ m³ at 6 MPa abandonment pressure), confirming the model’s efficacy for continental tight oil development. Furthermore, 40-month field validation revealed compartmentalized energy liberation dynamics: fracture networks rapidly released 77.4% of their stored energy within 40 months, while matrix zones contributed 34.8% of matrix-zone stored energy through sustained flow, collectively liberating 60.2% (25.2 × 10⁴ m³) of the 41.8 × 10⁴ m³ total energy with <15% prediction error. This confirms a critical 18–30 month window for energy replenishment to recover the residual matrix potential.

Critical limitations restrict universal deployment: excluding fluid heterogeneity may overestimate energy in stratified reservoirs; neglecting vertical heterogeneity (e.g., interbeds, permeability contrasts) systematically overestimates elastic energy propagation, distorts matrix supply timing, and risks underestimating dead oil zones in reservoirs with significant vertical heterogeneity; and neglecting solution gas expansion reduces accuracy in high-GOR wells and late-production stages dominated by gas drive.